{"text":"\\section{Introduction}\n\n\nDuring the last two years a race of industrial and research organizations has been opened to develop a ready-to-implement engineering solution for quantum computing (QC). It resulted in the QC market closely resembling the ascent ages of classical computing industry.\nNamely, there were many underdeveloped computing architectures which being incompatible with each other required significant efforts in porting software and algorithmic solutions between them. Given a broadly supported opinion that in the near term we are unlikely to become witnesses to flexible large-scale quantum architectures, there is a critical need to develop  portable, architecture-agnostic hybrid quantum-classical frameworks that will allow solving large-scale computational problems on small-scale quantum architectures. \n\n\nThere are multiple emerging quantum computation paradigms. The performance comparison of these paradigms is an important research topic. In this paper, we present for the first time a performance comparison of two leading quantum computation paradigms - D-Wave  quantum annealing and gate-based universal quantum computation. Both approaches have great potential for achieving quantum speedup for a number of important  problems~\\cite{king2018observation,romero2018strategies,ambainis2018quantum,dunjko2018computational}.\n\n\n\nThe first approach, quantum annealing (QA), is based on adiabatic quantum computation (AQC)~\\cite{mcgeoch2014adiabatic}. QA solves computational problems by using a guided quantum evolution~\\cite{yang2017optimizing}. \nThe evolution starts with an initial Hamiltonian with an easy-to-prepare ground state and ends up in the ground state of the problem Hamiltonian. QA is based on the adiabatic theorem that guarantees that if the Hamiltonian is evolved slowly then transitions to excited states are suppressed during the adiabatic evolution~\\cite{yang2017optimizing}.\nThe D-Wave quantum annealer uses superconducting flux qubits \\cite{amin2004,dwave2018} and has been shown to solve optimization problems on graphs \\cite{ushijima2017graph}, machine learning \\cite{omalley2017}, traffic flow optimization \\cite{neukart2017}, and simulation problems \\cite{harris2018}.\nQuantum and hybrid quantum-classical approaches have been employed.\n\nThe second approach is often referred to as the gate-based or universal QC. This mode of QC was theoretically demonstrated to have a great potential for exponential speedups over best known classical algorithms~\\cite{nielsen2002quantum}. In the near term, the capability of the quantum devices is limited by the number of qubits, low fidelity of gates, and lack of error correction. These limitations constrain us to using low-depth quantum circuits (i.e., quantum circuits with few gates) on a small number of qubits. Within the constraints of near-term intermediate-scale quantum (NISQ) technology~\\cite{preskill2018quantum}, a number of hybrid quantum-classical algorithms were developed and experimentally demonstrated to solve small problems. One of the most promising of such algorithms is Quantum Approximate Optimization Algorithm (QAOA)~\\cite{farhi2014quantum,farhi2016quantum}. QAOA is inspired by adiabatic quantum computation. Similarly to AQC and QA, the evolution path starts with an easy-to-prepare Hamiltonian in the ground state and evolves to the final Hamiltonian that encodes the solution of the problem remaining in the ground state. However, unlike QA in QAOA the evolution is performed by applying a series of parametrized gates called ansatz~\\cite{mcclean2016theory} which is parametrized by a set of variational parameters. This is accomplished by a hybrid approach that combines quantum evolution and classical variational optimization for optimal QAOA parameters~\\cite{yang2017optimizing} with the goal of finding the evolution path that prepares the ground state of the problem Hamiltonian. \n\n\\section{Methodology}\n\nThis work addresses three main challenges. First, we show how to use quantum computing to solve the community detection problem, a well known NP-hard problem. Second, we present an approach to solving realistic large problems using the NISQ hardware with a limited number of noisy qubits.  Third, we demonstrate a method that is portable across two leading quantum computation paradigms and can be easily extended to future hardware.\n\nThe community detection problem (or modularity graph clustering) has a variety of applications ranging from biology to social network analysis~\\cite{palla2005uncovering,su2010glay,bardella2016hierarchical,nicolini2017community}. \nIts complexity \\cite{brandes2006maximizing} and practical importance justify an attempt to solve it using QC. The goal of the community detection is to split nodes of a graph $G=(V,E)$ into communities by maximizing its modularity~\\cite{2006PNAS..103.8577N}:\n\n\\begin{equation}\nH = \\frac{1}{4|E|}\\Sigma_{ij}(A_{ij} - \\frac{k_ik_j}{2|E|})s_is_j =  \\frac{1}{4|E|}\\Sigma_{ij}B_{ij}s_is_j,\n\\label{eq:mod}\n\\end{equation}\n\n\\noindent where $s_i\\in\\{-1,+1\\}$ are variables indicating node $i$th community assignment, $k_i$ is a degree $i\\in V$, and $A$ is the adjacency matrix of $G$.\nIn this paper, we will focus on clustering the graph into two communities. There are several approaches to generalize the problem for cases when the number of communities is greater than 2.\n\nThe clustering of large networks is currently impossible with existing quantum computers because of the small number of available qubits.\nThis limitation applies both to quantum annealing~\\cite{ushijima2017graph} and universal quantum computing~\\cite{otterbach2017unsupervised}. To tackle large problems using available quantum hardware, we use a hybrid quantum-classical local-search approach. Our approach is inspired by existing numerous local-search heuristics (see~\\cite{rotta2011multilevel} for a review). Our algorithm finds a solution to the global community detection problem by selecting subproblems small enough to fit on the target quantum computer, solving them using a quantum algorithm and iterating until the solution to the global problem is found. The outline is presented in Algorithm~\\ref{alg:outline}.\n\n\\begin{algorithm}\n \\caption{Community Detection}\\label{alg:outline}\n\\begin{algorithmic}\n\\Procedure{Community detection}{Graph $G$}\n\\State solution = initial\\_guess($G$)\n \\While{not converged}\n  \t\\State $X$ = populate\\_subset($G$)\n  \t\\State \/\/ \\textit{using QAOA or D-Wave QA}\n  \t\\State candidate = solve\\_subproblem($G$, $X$)\n  \t\\If{$\\mbox{candidate} > \\mbox{solution}$}\n  \t\\State solution = candidate\n    \\EndIf\n \\EndWhile\n \\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nIn particular, we start with a random community assignment. At each step we select a subproblem (subset of vertices $X\\subset V$) by taking the vertices with highest potential gain if moving them from one community to another. The gain for each vertex can be computed efficiently~\\cite{2006PNAS..103.8577N}. Then we fix the community assignment of all $i\\not\\in X$, encode them into the problem as boundary conditions (denoted by $\\tilde{s}_j$, a typical technique in many heuristics \\cite{leyffer2013fast,hager2018multilevel}) and maximize\n\n\\begin{equation}\n\\label{eq:subproblem}\n\\arraycolsep=1.4p\n\\begin{array}{r c l}\nQ_{s} & = & \\sum_{i>j | i,j\\in X}2B_{ij}s_is_j +  \\sum_{i\\in X}\\sum_{j\\not\\in X}2B_{ij}s_i\\tilde{s}_j \\\\\n & = & \\sum_{i>j| i,j\\in X}2B_{ij}s_is_j + \\sum_{i\\in X}C_{i}s_i. \n\\end{array}\n\\end{equation}\n\nThe subproblems are solved using QC. To satisfy the constraints of available hardware, we fix the subproblem size to some small number (in our experiments, it was 25). \n\n\n\\section{Implementation details and Results}\n\nWe implement our local search algorithm in Python using the graph methods provided by NetworkX~\\cite{hagberg2008}. The novelty of our approach is that it allows to use D-Wave QA, QAOA and classical Gurobi \\cite{optimization2014inc} solvers interchangeably simply by passing different flags, enabling rapid prototyping and direct comparison of different methods as the hardware and its capabilities evolve. Additionally, Gurobi was used as a global optimization solver for the sake of quality comparison. To our knowledge this is the first attempt to directly compare universal quantum computing and quantum annealing. Our framework is also easily extendable, making it possible for researchers to add new backends as they become available. We plan to release the framework as an open-source project.\n\nOur results are presented in Figure \\ref{fig:results}. In these experiments, we used the Intel-QS~\\cite{smelyanskiy2016qhipster} simulator for QAOA (at the time our group did not have access to a universal quantum computer of sufficient size). We use six real-world networks from the KONECT dataset~\\cite{kunegis2013konect} with up to 400 nodes as our benchmark. For each network, we ran 30 experiments with different random seeds. The same set of seeds is used between three backend solvers, making the results directly comparable. The subproblem size is fixed at 25 (i.e., 25 qubits are used). Our results demonstrate that the quantum local search approach with both quantum methods is capable of achieving results comparable to state-of-the-art, with a potential to outperform as hardware evolves.\n\n\n \n\n\n\n\n \\vspace{-0.78cm}\n \n\\begin{figure}[htb]\n \\begin{tikzpicture} \n  \\node (img)  {\\includegraphics[width=0.8\\linewidth]{.\/fig\/mod_iter.pdf}};\n  \\node (temp) [left=of img]{};\n  \\node[below=of img, node distance=0cm, yshift=1.1cm,font=\\color{black}] {Network Name};\n \\node[below=of temp, node distance=2cm, anchor=center,yshift=-1.0cm,,xshift=1.2cm, rotate=90,font=\\color{black}] {Num. Solver Calls};\n  \\node[above=of temp, node distance=2cm, xshift=1.2cm,yshift=0.8cm, rotate=90, anchor=center,font=\\color{black}] {Modularity};\n \\end{tikzpicture}\n  \\vspace{-0.4cm}\n \\caption{Box-plots comparing modularity scores (greater is better) and number of solver calls (less is better) respectively for the three different solvers. For the graph {\\tt oz}, Gurobi and D-Wave returned a modularity score greater than the Global Solver (best known value)}\n \\label{fig:results}\n\\end{figure}\n \\vspace{-0.6cm}\n\\section{Discussion}\n\n\n\n\nIn the near term, quantum hardware will be in a constant state of change. Many different NISQ-era hardware solutions will appear and some will be abandoned. In the midst of such evolutionary times, we want to be able to continue research in quantum algorithms and head towards solving real-world problems. To accomplish this, we need portable, architecture-agnostic hybrid quantum-classical frameworks that will allow solving large-scale computational problems on small-scale quantum architectures. Moreover, these frameworks need to be robust and future-proof. In this work, we have presented a prototype of such a framework for solving the problem of community detection in networks on two distinctively different architectures: D-Wave quantum annealer and universal quantum computer. We suggest extending this approach for solving other types of problems in science.\n\nThe constant change of hardware and overall immaturity of the existing technology leads to many risks in QC. In spite of major effort, it has not been experimentally demonstrated yet an ability to achieve speedups over state-of-the-art classical supercomputers and there are valid concerns about scalability of existing implementations~\\cite{brugger2018quantum}. Advances in material design and engineering will allow the community to overcome those hurdles. We expect QC to eventually become a part of the HPC ecosystem with an initial role as an accelerator providing a new layer of parallelism. Our approach will provide for co-design exploration towards the best QC accelerator choice for an application mix.\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\appendices\n\n\\section*{Acknowledgment}\nThis research used the resources of the Argonne Leadership Computing Facility, which is a U.S. Department of Energy (DOE) Office of Science User Facility supported under Contract DE-AC02-06CH11357. We gratefully acknowledge the computing resources provided and operated by the Joint Laboratory for System Evaluation (JLSE) at Argonne National Laboratory.  The authors would also like to acknowledge the NNSA's Advanced Simulation and Computing (ASC) program at Los Alamos National Laboratory (LANL) for use of their Ising D-Wave 2X quantum computing resource and D-Wave Systems Inc. for use of their 2000Q resource. The LANL research contribution has been funded by LANL Laboratory Directed Research and Development (LDRD).  LANL is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. DOE under Contract DE-AC52-06NA25396. Clemson University is acknowledged for generous allotment of compute time on Palmetto cluster.\n\n\n\n\\bibliographystyle{.\/IEEEtranS}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction and Motivation}\n\n\n This paper is motivated by work of Szlam, Maggioni, Coifman \\& Bremer \\cite{szlam2} and an observation made explicit by\nSzlam \\cite{szlam}: taking iterated spectral cuts induced by the nodal set of the first non-constant eigenfunction for the Neumann $p$-Laplacian seems to converge to rectangles in shape. \nIt is already observed in \\cite{szlam} that starting with an isosceles right triangle will lead to a spectral cut along the symmetry axis and produce two smaller isoceles right triangles: no convergence to rectangles takes place. However, this is unstable under small perturbations of the initial domain. \n\\begin{center}\n\\begin{figure}[ht!]\n\\includegraphics[width=0.85\\textwidth]{Fig1}\n\\caption{Iterated spectral cuts of the standard graph Laplacian seem to lead to rectangles in a generic, non-convex, non-smooth domain.}\n\\end{figure}\n\\end{center}\n\\vspace{-20pt}\n\nWe believe this to be a fascinating question in itself but an affirmative answer would also be useful in guaranteeing that iterative spectral partitioning is an effective method to partition domains and, ultimately, graphs and data (see Irion \\& Saito \\cite{saito} where this phenomenon is exploited).  More importantly, a better understanding of this problem will shed light on the more general data analysis algorithms based on the $p$-Laplacian. More precisely, let $\\Omega \\subset \\mathbb{R}^2$ be an open, bounded, and connected domain. We now propose an iterative subdivision of $\\Omega$ as follows.\nFor any $1 \\leq p < \\infty$, the ground state of the $p-$Laplacian can be written as\n\\begin{equation} \\label{definition p Lap ground state}\n\\lambda_{1,p}(\\Omega) = \\inf_{\\int_{\\Omega}{f dx} = 0} \\frac{\\int_{\\Omega}{|\\nabla f|^p dx}}{ \\int_{\\Omega}{|f|^p dx}}.\\end{equation}\nIt is known that the function $f$ minimizing this functional exists and we can use it to iteratively define\n\\begin{equation}\n\\label{speccut}\n \\Omega_{n+1} = \\left\\{ x \\in \\Omega_n: f(x) \\geq 0\\right\\}\\,,\n \\end{equation}\nwhere $n=0,1,2,\\ldots$ and $\\Omega_0:=\\Omega$. \nThe function $f$ is only defined up to sign, so restricting to the part of the domain where it is positive is without loss of generality. We raise the following conjecture (and refer to the subsequent paragraphs for clarification and obvious obstacles).\n\\begin{quote} \\textbf{Main Conjecture.} If $\\Omega_0$ is not the isosceles right triangle (having angles 45-90-45), then the sequence of sets $(\\Omega_n)_{n=1}^{\\infty}$ converges to the set of rectangles with eccentricity\nbounded by 2 in the Gromov-Hausdorff distance.\n\\end{quote}\nIt is clear that one cannot expect convergence to a fixed rectangle: in general, the spectral cut of an $a \\times b$ rectangle with $a > b$ will be given by two $(a\/2) \\times b$ rectangles and, as long as $a \\leq 2b$,\nthe next cut would then yield two $(a\/2) \\times (b\/2)$ rectangles. This motivates a refined question.\n\n\\begin{quote} \\textbf{Question.} Do $(\\Omega_{2n})_{n=1}^{\\infty}$ and $(\\Omega_{2n+1})_{n=1}^{\\infty}$ converge in shape to a fixed rectangle?\n\\end{quote}\n\nThere is one obvious obstruction: if $\\Omega_{0}$ is not already a rectangle, then while performing iterated subdivisions, there is always a sequence of choices for the sign of the\neigenfunction that ensures that part of the boundary of $\\Omega_{n}$ coincides with part of the boundary of $\\Omega$ which could possibly be quite ill-behaved (say, fractal).\nHowever, for a sequence of choices of signs that leads to domains bounded away from $\\partial \\Omega_0$ ('deep' inside the domain) this should not be the case.\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=.4\\textwidth]{Fig2-1}\\hspace{60pt}\n\\includegraphics[width=.4\\textwidth]{Fig2-2}\n\\caption{Left: the spectral cut of a quadrilateral determined by four corners, $(0, 0)$, $(\\pi\/25, 3\/5)$, $(1,0)$, and $(1, 3\/5-\\exp(1)\/100)$ provided by the graph 1-Laplacian. Right: the spectral cut of a shape close to a quadrilateral provided by the graph 1-Laplacian.}\\label{Fig:One}\n\\end{figure}\n\n\nWhile this particular question seems to be novel, the problem of trying to understand the geometry of nodal cuts induced by the $p-$Laplacian or general nonlinear operators has\nbeen studied for a long time. We refer to \\cite{plap,plap2,plap3,plap4,plap5,plap6,plap7,plap8,plap9,plap0,plap10,plap11} and references therein. We especially\nemphasize the works of Gajewski \\& G\\\"artner \\cite{plap7, plap8} who study the behavior of the cut as $p \\rightarrow 1$ as a means of finding effective ways of separating the\ndomain into two roughly equally sized domains, as well as the work of Parini \\cite{plap11} studying the limit of the cut as $p \\rightarrow 1$ under Dirichlet boundary conditions.\nMany of these results are posed for Dirichlet conditions where the effective functional as $p \\rightarrow 1$ is given by\n$$ \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)}} \\quad \\mbox{while the Neumann case induces} \\quad \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)\\mathcal{H}^{n}(\\Omega \\setminus E)}}.$$\nWhen the domain has Neumann boundary conditions, the quantity above is referred to as the {\\it Ratio Cut}.\nNew phenomena arise as a consequence. One interesting problem, that may also be of interest in the Dirichlet case, is the stability of the nodal set of the $p-$Laplacian\nas a function of $p$. \n\nIn the result presented here, we will focus on the $1$-Laplacian, where we can establish some preliminary results that suggest stability of rectangular domains as attractors for the proposed spectral dynamical system on domains.  In particular, from henceforward, we refer to the {\\it $1$-spectral cut iteration} of a domain as the operation defined in \\eqref{speccut} with $p=1$ and the {\\it $1$-spectral cut} as the set $C_{n} := \\left\\{ x \\in \\Omega_n: f(x) = 0\\right\\}$. To simplify the terminology, we use {\\it spectral cut} and {\\it $1$-spectral cut} interchangeably. \n\n\\textbf{Numerics.} For the reader's convenience, we briefly recall here the numerical implementation of the p-Laplacian and its related spectral cut \\cite{Buhler_Hein:2009,Hein_Buhler:2010}.\nTake a point cloud $\\mathcal{X}:=\\{x_i\\}_{i=1}^N\\subset (\\mathcal M,d)$, a metric space $\\mathcal M$ with the metric $d$. Construct an affinity matrix ${\\bf W}\\in \\mathbb{R}^{N\\times N}$, where $\\bf W_{ij}$ is the affinity between $x_i$ and $x_j$. It is associated with an undirected affinity graph with $N$ vertices, where the affinity between $x_i$ and $x_j$, $w_{ij}$, is ${\\bf W}_{ij}$ for $i,j=1,\\ldots,N$. The {\\em graph $p$-Laplacian} is defined by\n\\begin{equation}\n\\Delta_p f(i)=\\sum_{j=1}^N w_{ij}\\phi_p(f(i)-f(j)),\n\\end{equation}\nwhere $f\\in\\mathbb{R}^N$ is a function defined on the vertices and $\\phi_p(x)=\\texttt{sign}(x)|x|^{p-1}$ for $x\\in \\mathbb{R}$. When $p=2$, this gives the bilinear form defined by the standard graph Laplacian ${\\bf D} - {\\bf W}$, where ${\\bf D}=\\texttt{diag}({\\bf W}{\\bf 1})$ and ${\\bf 1}$ is a $N$-dim vector with all entries $1$. Clearly, in general the graph p-Laplacian is nonlinear. We have\n\\begin{equation}\n\\langle f,  \\Delta_p f \\rangle = \\frac12 \\sum_{i,j=1}^N w_{ij} |f_i - f_j|^p\n\\end{equation}\nfor $1 \\leq p < \\infty$.  See for instance \\cite{Buhler_Hein:2009} for a discussion of the relationship between the variational formulation and the discrete operator formulation.\n\nA real number $\\lambda$ is called an  eigenvalue for the graph $p$-Laplacian if there exists a non-zero vector $f\\in\\mathbb{R}^N$ so that \\cite[Definition 3.1]{Buhler_Hein:2009}\n\\begin{equation}\n(\\Delta_p f)_i=\\lambda \\phi_p(f_i)\\,,\n\\end{equation}\nwhere $i=1,\\ldots,N$. We call $f$ the $p$-eigenfunction of of the graph $p$-Laplacian associated with the eigenvalue $\\lambda$. We know that ${\\bf 1}$ is the trivial eigenvector with eigenvalue $0$, and we have \\cite[Lemma 3.2]{Buhler_Hein:2009}\n\\begin{equation}\n{\\bf 1}^T\\phi_p(f)=0\n\\end{equation} \nif the eigenvector $f$ is non-trivial.\nIt is shown in \\cite{Buhler_Hein:2009} that a non-zero function $f$ is an eigenvector if and only if it is a critical point (local minima) of the functional \n\\begin{equation}\nF_p(f)=\\frac{\\langle f,  \\Delta_p f \\rangle}{\\|f\\|_{\\ell^p}^p}\\,.\n\\end{equation}\nNote that $F_p(f)$ is the discretization of the functional shown in \\eqref{definition p Lap ground state}.\nIn this work, our main interest is the second $p$-eigenvector of the graph $p$-Laplacian for the spectral clustering purpose. In \\cite{Buhler_Hein:2009}, the second eigenvalue is shown to be the global minimum of the functional\n\\begin{equation}\nF^{(2)}_p(f):=\\frac{\\langle f,  \\Delta_p f \\rangle}{\\texttt{var}_p(f)},\n\\end{equation}\nwhere \n\\begin{equation}\n\\texttt{var}_p(f)=\\min_{c\\in\\mathbb{R}}\\sum_{i=1}^N |f_i-c|^p\\,,\n\\end{equation} \nand the corresponding eigenvector is then given by \n\\begin{equation}\nf_p^{(2)}=f^*-c^*{\\bf 1}, \n\\end{equation}\nwhere $f^*$ is any global minimizer of $F^{(2)}_p$ and $c^*=\\arg\\min_{c\\in \\mathbb{R}} \\sum_{i=1}^N |f^*_i-c|^p$. \nLike the usual spectral clustering, once we have $f_p^{(2)}$, we cluster the point cloud by the signs of its entries.  \nFor $p=1$, \nthe consistency of spectral cut with the graph $1$-Laplacian was studied in \\cite{trillos2016consistency}.\nNumerically, we apply the nonlinear inverse power method proposed in \\cite{Hein_Buhler:2010} to evaluate the iterative bi-partition of a given 2-dimensional domain.\nIn Figure \\ref{Fig:One}, we present some numerically computed Ratio Cuts for nearly rectangular domains. \n\n\n\\textbf{Outline.} The paper proceeds as follows.  In Section \\ref{sec:results}, we highlight the two main theorems we can prove on stability of near rectangular domains.  We also present some open problems to be considered naturally as generalizations of these theorems.   In Section \\ref{sec:pf1}, we give the full proof that the spectral cut algorithm converges to a rectangle with bounded aspect ratio if the initial domain is near a rectangle in Gromov-Hausdorff sense as will be carefully laid out below.  Section \\ref{sec:pf3} provides the details for the proof that quadrilaterals near the rectangle of aspect ratio $2$ in terms of small angle deviations from $90$ degrees will converge under the spectral cut algorithm to rectangles with bounded aspect ratio.   In the appendix we gather some long calculations that are useful in analyzing the Ratio Cut in a neighborhood of a quadrilateral.\n\n\n\n\n\\section*{Acknowledgements} \nWe thankfully acknowledge the generous support of NSF CAREER Grant DMS-1352353  (J.L. Marzuola \\& W. Hamilton).  We thank Stefan Steinerberger for starting this project with us and for many helpful conversations along the way.  \n\n\n\\section{results} \n\\label{sec:results}\nWe prove two results that, while not establishing the main conjecture, do seem to suggest a mechanism by which this procedure happens.\n$$ \\mbox{certain shapes} \\underbrace{\\implies}_{\\mbox{Theorem 1}} \\mbox{nearly straight cuts} \\implies \\mbox{curved quadrilaterals} \\underbrace{\\implies}_{\\mbox{Theorem 2}} \\mbox{rectangles.}$$\nThe missing steps are as follows: (1) we do not know whether a generic $\\Omega_0 \\subset \\mathbb{R}^2$ will ever produce domains $\\Omega_n$ for which Theorem 1 becomes applicable, for example, when $\\Omega_0$ has a fractal boundary; \nand (2) we do not know whether the dynamical system on the space of rectangles ever produces a quadrilateral sufficiently close to the set of rectangles for Theorem 2 to become applicable.\n\n\n\\subsection{Rectangular stability.} The first of the two main results states that the spectral cut of domains that 'roughly' look like rectangles are being cut\nin the middle. More formally, let us carefully describe the domains we will consider.\n\\begin{assumption}\n\\label{rectangle_assumptions}\nWe will work with domains that are small perturbations of a rectangle in the following sense:\n\\begin{itemize} \n\\item The domain $Q$ is a perturbed\nrectangle that is close in the Gromov-Hausdorff distance to a reference rectangle $R$: \n$$d_{GH}(Q, R) \\leq \\varepsilon \\left( \\mbox{length of the shorter side of}~R\\right)\\,,$$\nwhere $\\varepsilon>0$ is sufficiently small.\n\\item\nIn a roughly $10\\sqrt{\\varepsilon}-$neighborhood of the two intersection points of the $1$-spectral cut of $R$ with the boundary, the boundary of $Q$ \ncan be written as graphs of functions of the associated boundary segments of $R$ (see Fig. \\ref{fig:cut}). Moreover, each of these functions can be well approximated by a parabola with bounded, small curvature and small Lipschitz constant.\n\\end{itemize}\n\\end{assumption}\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale = 2]\n\\draw [] (0,0) -- (1.61,0) -- (1.61, 1) -- (0, 1) -- (0,0);\n\\draw [dashed] (0.8,-0.2) -- (0.8, 1.2);\n\\draw [ultra thick] (0,0) to[out=10,in=170] (1, 0);\n\\draw [ultra thick] (1,0) to[out=340,in=200] (1.61, 0);\n\\draw [ultra thick] (1.61,0) to[out=80,in=280] (1.61, 1);\n\\draw [ultra thick] (1.61,1) to[out=170,in=10] (0, 1);\n\\draw [ultra thick] (0,1) to[out=260,in=100] (0, 0);\n\\draw [ultra thick, dashed] (0.8, -0.2) to[out=85,in=277] (0.81, 1.2);\n\\end{tikzpicture}\n\\caption{A perturbed rectangle $Q$ close to a reference rectangle $R$: the longer part of the boundary of $Q$ can be written as the graph of a Lipschitz function. The goal is to show that the spectral cut of $Q$ is close to that of $R$.}\n\\label{fig:cut}\n\\end{figure}\n\\end{center}\n\n\\begin{theorem}\n\\label{thm1} \nFor a domain satisfying Assumption \\ref{rectangle_assumptions}, the spectral cut occurs in a $\\sqrt{\\varepsilon}-$neighborhood of the midpoint of the long axis.  Moreover, there is a function $c(\\varepsilon)$ tending to\n0 as $\\varepsilon \\rightarrow 0$ such that, if said part of the boundary can be written as a Lipschitz function with Lipschitz constant $L \\leq \\sqrt{2} - c(\\varepsilon)$ in the $\\sqrt{\\varepsilon}-$neighborhood of the spectral cut of $R$, then the spectral cut of $Q$\nis a circular arc near a straight line.\n\\end{theorem}\n\nNumerical examples suggest that this result starts becoming applicable rather quickly: already a small number of cuts seems to suffice to produce roughly\nrectangular shapes. As soon as Theorem 1 becomes applicable the spectral cuts will start being closer and closer to straight lines, which immediately implies that many shapes will end\nup approaching quadrilaterals after several additional steps. The next result shows that as soon as we are dealing with quadrilaterals close to a rectangle, the procedure\nis smoothing and produces quadrilaterals closer to the rectangle; this has to be understood in the usual sense of 'cuts' away from the boundary: a quadrilateral\nhaving a $91^{\\circ}$ degree angle will always pass this angle on to one of its descendants.  \nWe note a certain similarity of Theorem \\ref{thm1} to the result of Grieser-Jerison \\cite{GrieserJerison} for the standard Laplacian on rectangular domains of high aspect ratio with one side described by a curve.  See also the recent work \\cite{BCM} where general curvilinear quadrilaterals were considered.  \n\n\\subsection{Near Curvilinear Quadrilaterals} We settle our conjecture in the case of $\\Omega$ being near a quadrilateral that is close to a rectangle: we prove convergence to a rectangular shape in the Gromov-Hausdorff distance 'away from the boundary' (in the sense\ndiscussed above: rectangles incorporating parts of the initial boundary or boundaries created by very early initial spectral cuts need never be regular).\nWe introduce a qualitative way of measuring distance to quadrilaterals as follows: given a curvilinear quadrilateral $Q$ with angles $\\alpha, \\beta, \\gamma, \\delta$ and sides described by the curves $y = \\gamma_1 (x)$, $y = \\gamma_2(x)$, $x = \\gamma_3(y)$ and $x  = \\gamma_4 (y)$, we\ndefine the functional\n\\begin{equation}\\label{Definition I(Q)}\nI(Q) =  \\left| \\alpha- \\frac{\\pi}{2} \\right| + \\left| \\beta- \\frac{\\pi}{2} \\right| + \\left| \\gamma- \\frac{\\pi}{2} \\right| + \\left| \\delta- \\frac{\\pi}{2} \\right| + \\max_{1 \\leq j \\leq 4} \\sup_s (|\\gamma_j' |(s)   + |\\gamma_j'' |(s)+ |\\gamma_j''' |(s)) .\n\\end{equation}\nWe particularly want our domains to be well approximated by parabolic curves on the sides intersecting the ratio cut, and will refer to such domains as {\\it approximately parabolic curvilinear quadrilaterals}. \n\n\\begin{remark}\nThis is actually quite a bit more restrictive than one needs, though is certainly sufficient; otherwise, the theorem becomes harder to frame as one must introduce a large number of cases for behaviors of each boundary component of the domain.  However, in the calculations below, we will work with domains that are perturbations of a base rectangle with a reasonable aspect ratio, and whose top and bottom boundary components can be well-approximated by parabolas in a neighborhood near the axis of symmetry of the base rectangle.  As a result, we can dramatically change regularity requirements for the left and right curves in our model calculations as long as the perturbations are nearly symmetric in area and bounded by a sufficiently small constant.  \n\\end{remark}\n\n\\begin{theorem} \n\\label{thm2}\nFor an approximately parabolic curvilinear quadrilateral of aspect ratio near $1:2$, there exists $\\varepsilon_1 > 0$ such that if $I(Q) \\leq \\varepsilon_1$, then the $1$-spectral cut induced\nby the $L^1-$Laplacian is a circular arc with opening angle bounded above by $\\varepsilon_1\/8$.  \n\\end{theorem}\nThe proof also shows that the constant $1\/8$ cannot be further improved.\nThis result implies that near-quadrilateral regions have $1$-spectral cuts that are closer to a quadrilateral in Gromov-Hausdorff distance. This implies exponentially fast convergence to rectangles in shape. The way the result is obtained actually allows for a fairly precise understanding of what happens (in particular, it can be used to show that there is a choice of signs such that $I(Q_n)$ grows substantially along the subsequence). We refer to the proof for details. \n\n\n\\subsection{Open problems} These results naturally suggest several open problems; we only name a few that seem \nparticularly natural.\n\n\\begin{enumerate}\n\n\\item Prove the main conjecture for $p=1$; that is, show that a generic $\\Omega_0 \\subset \\mathbb{R}^2$ will produce domains $\\Omega_n$ for which Theorem 1 can be applied, and show that the dynamical system on the space of rectangles produces a quadrilateral sufficiently close to the set of rectangles for Theorem 2 to become applicable. Is it possible to transfer some of the arguments to the range $p \\in (1,1+\\varepsilon_0)$?  What happens\nas $p \\rightarrow \\infty$? \n\\item What happens in higher dimensions, or even to domains with curvature, like manifolds? One would still expect the cut to have a smoothing effect but the types of geometric obstruction that\none could encounter in the process may be more complicated. Is the generic limit given by a rectangular box? If not, is there a finite set of shapes\nthat can arise in the limit?\n\\item It seems natural to expect that similar results should hold true under Dirichlet conditions. This would require the study of the variational problem \n$$ \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)}}.$$\nCan the results be extended to this case?\n\\item Experimentally, we observe that the nodal line is (generically) fairly stable under small perturbations of the value of $p$. Can any result in this\ndirection be made precise? Does it help to assume  the domain $\\Omega$ to be convex?\n\\end{enumerate}\n\n\n\n\n\\section{proof of theorem 1}\n\\label{sec:pf1}\n\n\\subsection{Preliminaries.} The crucial ingredient in our approach is that the $p$-Laplacian degenerates as $p \\rightarrow 1^+$. The minimum of the associated energy functional characterizing the ground state is not assumed by any continuous\nfunction: reinterpreting the functional in terms of total variation, the extremal function is constant on two sets in the domain that are separated by\nan $(n-1)-$dimensional hypersurface. More precisely, we have the following consequence of the coarea formula (see \\cite{stein} for further implications\nof this result).\n\n\n\\begin{thm}[Cianchi, \\cite{Cianchi}] Let $\\Omega \\subset \\mathbb{R}^{n}$ be open, bounded and connected. Then we have the sharp Poincar\\'e-type inequality for any sufficiently smooth functions $u$:\n$$\\left\\|u-\\frac{1}{|\\Omega|}\\int_{\\Omega}{u(z)dz}\\right\\|_{L^{1}(\\Omega)} \\leq\n \\left(\\sup_{\\Gamma \\subset \\Omega}{\\frac{2}{\\mathcal{H}^{n-1}(\\Gamma)}\\frac{|S||\\Omega \\setminus S|}{|\\Omega|}}\\right)\\left\\|\\nabla u\\right\\|_{L^{1}(\\Omega)},$$\nwhere $\\Gamma \\subset \\Omega$ ranges over all surfaces which divide $\\Omega$ into two connected open subsets $S$ and $\\Omega \\setminus \\overline S$. \n\\end{thm}\n\nThis means that the nodal line $\\Gamma$ defining the $1$-spectral cut is a hypersurface partitioning $\\Omega$ into two sets $S, \\Omega \\setminus \\overline S$\n$$ \\mbox{so as to minimize the quantity} \\qquad \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|}.$$\nIt is relatively easy to check that this value is assumed if we relax the $L^1$-norm of the gradient and re-interpret it as total variation. Let us assume that $f(x) = a \\chi_{S} + b \\chi_{\\Omega \\setminus S}$.\nWe want $f$ to have mean value 0, which leads to\n$$ a |S| + b |\\Omega \\setminus S| = 0 \\quad \\mbox{and thus} \\quad b = -a\\frac{|S|}{|\\Omega \\setminus S|} \\quad \\mbox{implying} \\quad \\|f\\|_{L^1} = 2a |S|,$$\nwhere we take $a>0$ without loss of generality.\nMoreover, the ``formal'' contribution to the total variation interpretation of the gradient is given by\n$$ \\|\\nabla f \\|_{L^1} = |\\Gamma| (a - b) =|\\Gamma|  \\left(a + a\\frac{|S|}{|\\Omega \\setminus S|} \\right).$$\nThis implies\n$$ \\frac{  \\|\\nabla f \\|_{L^1}  }{ \\|f\\|_{L^1}} =  \\frac12 |\\Gamma|  \\left( \\frac{1}{|S|} + \\frac{1}{|\\Omega\\setminus S|}\\right) =  \\frac12 |\\Gamma| \\frac{|\\Omega|}{|S| |\\Omega \\setminus S|}.$$\nTo be rigorous, convolution with a smooth mollifier shows that one can get arbitrarily close to the optimal constant with smooth functions. We will now show that, for a rectangle, the optimal spectral cut splits the domain via a straight line cut intersecting the longest sides at right angles.  The crucial ingredient here is that the argument does not appeal to symmetry,\nis stable under perturbations and easily implies the same results for domains that are merely close to rectangles in the Gromov-Hausdorff distance.\n\n\\begin{lemma} \\label{Lemma: rectangle result}\nLet $R = [0,a] \\times [0,b] \\subset \\mathbb{R}^2$  with $a > b$. The $1$-spectral cut is exactly at $\\left\\{a\/2\\right\\} \\times [0,b]$.\n\\end{lemma}\n\\begin{proof}\nThe cut in the middle of the longer side yields\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} = \\frac{b (ab)}{(ab)^2\/4} = \\frac{4}{a}.$$\nIt is clear that other spectral cuts touching the longer sides have $ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{4}{a},$ since $\\mathcal{H}^1(\\Gamma)\\geq b$ and $|S| |\\Omega \\setminus S|\\leq (ab)^2\/4$. \n\\begin{center}\n\\begin{figure}[ht!] \n\\begin{tikzpicture}[scale = 2]\n\\draw [ultra thick] (0,0) -- (1.61,0) -- (1.61, 1) -- (0, 1) -- (0,0);\n\\draw [dashed] (0.8,0) -- (0.8, 1);\n\\draw [thick] (0.7,0) to[out=19,in=140] (1.61, 0.6);\n\\node at (1.5,0.2) {$S$};\n\\node at (0.3,0.6) {$\\Omega \\setminus S$};\n\\end{tikzpicture}\n\\caption{The geometric construction in the proof of Lemma 1. }\n\\label{fig4}\n\\end{figure}\n\\end{center}\n\nIf $\\Gamma$ touches two opposite sides, then\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|}  \\geq b \\frac{ |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{b (ab)}{(ab)^2} \\min_{0 \\leq x \\leq 1}{\\frac{1}{x(1-x)}} \\geq \\frac{4}{a}.$$\nEquality can only arise if the cut has length $b$ (forcing it to be a line) and $x=1\/2$ (forcing a split into two domains of the same area). This characterizes\nexactly the cut in the middle.\n\n\nIt remains to deal with the case where the spectral cut touches two adjacent sides; this case is illustrated in Figure \\ref{fig4}. Let us assume that the enclosed domain is denoted by $S$, $|S| = x |\\Omega| = x ab$ for some $0 < x < 1$, and $\\Gamma = \\partial S \\cap \\mbox{int}~R$ is the part of the boundary curve strictly inside the rectangle.\nThe isoperimetric inequality implies that\n$$ \\mathcal{H}^1(\\Gamma) \\geq  \\sqrt{\\pi}\\sqrt{x a b},$$\nand thus\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{ \\sqrt{\\pi x a b} ab }{a^2 b^2 x (1-x)} = \\frac{\\sqrt{\\pi}}{\\sqrt{x}(1-x)} \\frac{1}{\\sqrt{a}\\sqrt{b}}.$$\nAn explicit computation shows that for all $0 < x < 1$\n$$ \\frac{1}{\\sqrt{x}(1-x)}  \\geq \\frac{3\\sqrt{3}}{2}$$\nand therefore, using $b \\geq a,$\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{3\\sqrt{3 \\pi}}{2} \\frac{1}{a} \\geq \\frac{4.6}{a}.$$\nThis is always a constant fraction worse than the cut along the longest axes, concluding the argument.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk1}\nIt is easily seen that all aspects of the argument are stable under (even moderately large) perturbations of the domain in the sense of Gromov-Hausdorff distance.  To be specific, take a domain $Q$ satisfying Assumption \\ref{rectangle_assumptions}, and denote the length of the shorter side of the associated rectangle $R$ by $h$. By Assumption \\ref{rectangle_assumptions}, the Gromov-Hausdorff distance between $Q$\nand $R$ is $d_{GH}(Q, R) \\leq \\varepsilon h$. There is clearly a straight line that is parallel to the shorter side of $R$ and bisects the area: the\nbound on the Gromov-Hausdorff distance implies that this straight line has length at most $h + 2h\\varepsilon$. Therefore,\n$$   \\inf_{\\Gamma}{\\frac{\\mathcal{H}^1(\\Gamma)}{|S| |\\Omega \\setminus S|}} \\leq 4h + 8h \\varepsilon.$$\nArguments as in Lemma 1 show that it has to cut the longer side roughly in the middle. \n\\end{remark}\n\n\n\nThe second ingredient is a straightforward consequence of the isoperimetric inequality that we need to make quantitative.\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale = 6]\n\\draw [ultra thick] (0,0) -- (1,0);\n\\node at (-0.02, -0.04) {$a$};\n\\node at (1.02, -0.04) {$b$};\n\\node at (0.5, 0.05) {$\\Omega$};\n\\draw [ultra thick] (0,0) to[out=20,in=180] (0.4, 0.08);\n\\draw [ultra thick] (0.4,0.08) to[out=20,in=160] (1, 0);\n\\end{tikzpicture}\n\\caption{A curve slightly longer than the straight line.}\n\\label{fig:areagain}\n\\end{figure}\n\\end{center}\n\n\n\n\n\\begin{lemma} Consider a simply connected domain $\\Omega$ as shown in Figure \\ref{fig:areagain} that is comprised of a straight line segment between $a$ and $b$ and some arbitrary curved line segment enclosing a domain $\\Omega$. For every $\\varepsilon_0 >0$\nthere exists an $\\varepsilon_1 > 0$ such that if\n\\begin{equation}\n\\label{assump1}\n|\\partial \\Omega|  \\leq  \\left(1+\\varepsilon_1\\right) 2\\|a - b\\|,\n\\end{equation}\nthen\n$$ |\\Omega| \\leq  \\frac{1 + \\varepsilon_0}{\\sqrt{6}} \\| a - b\\|^{3\/2} \\sqrt{|\\partial \\Omega| - 2 \\|a-b\\|}.$$\n\\end{lemma}\n\\begin{proof} The isoperimetric inequality immediately implies that the optimal curve defining the Ratio Cut for $Q$ satisfying Assumption \\ref{rectangle_assumptions} must take the form of a circular arc; otherwise, we could fix $a,b$ at the boundary of a circle\nand use the novel shape to create a set in the plane that contains more area than the disk whose boundary has the same size. \n\n It remains to compute the constants for the circular arc.\nChoose a coordinate system with $a,b$ at $(0,0),(\\|a-b\\|,0)$ respectively. The opening angle $\\alpha$ of the sector of the disk with radius $r>0$ that produces such an arc is given by\n$$ \\alpha = 2 \\arcsin{\\left( \\frac{\\|a-b\\|}{2r} \\right)},$$\nwhere $0 < \\alpha < \\frac{\\pi}{2}$ is sufficiently small by assumption \\eqref{assump1}.\nAs a consequence, the length of the circular arc is given by\n$$ r \\alpha =  2r \\arcsin{\\left( \\frac{\\|a-b\\|}{2r}\\right)} \\geq \\|a-b\\| + \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2},$$\nand so\n\\begin{equation}\n\\label{pomegabd}\n |\\partial \\Omega| \\geq 2\\|a-b\\| +   \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2}.\n \\end{equation}\nIn particular, by assumption \\eqref{assump1}, we have that \n$$  \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2} \\leq  2\\varepsilon_1\\|a-b\\| \\qquad \\mbox{and thus} \\qquad \\frac{\\|a-b\\|^2}{r^2}  \\leq 48 \\varepsilon_1.$$\n\nThe enclosed area captured by the line segment and the circular arc is \n$$\\frac{ r^2 \\pi \\alpha}{2\\pi} - r^2 \\cos{\\left(\\frac{\\alpha}{2}\\right)} \\sin{\\left(\\frac{\\alpha}{2}\\right)}.$$\nWe have\n$$ \\frac{ r^2 \\pi \\alpha}{2\\pi} =  r^2 \\arcsin{\\left( \\frac{\\|a-b\\|}{2r} \\right)} = r \\frac{\\|a-b\\|}{2} + \\frac{\\|a-b\\|^3}{48 r}  + \\mbox{higher order terms}$$\nas well as\n$$ \\cos{\\left(\\frac{\\alpha}{2}\\right)} = \\sqrt{1 - \\frac{\\|a-b\\|^2}{4r^2}} \\quad \\mbox{and} \\quad \\sin{\\left(\\frac{\\alpha}{2}\\right)} =  \\frac{\\|a-b\\|}{2r}.$$\nThe higher order terms in the expansion are an infinite series in $\\|a-b\\|\/r$ with exponents decaying fast enough. For $\\varepsilon_1$ sufficiently\nsmall depending on $\\varepsilon_0$, we can bound\n$$ \\mbox{higher order terms}  \\leq \\varepsilon_0  \\frac{\\|a-b\\|^3}{48 r}  .$$\nAltogether this implies that\n$$\\mbox{area} \\leq (1 + \\varepsilon_0) \\frac{\\|a-b\\|^3}{12r}$$\nand plugging in the bound on $1\/r$ from \\eqref{pomegabd} gives\n$$\\mbox{area} \\leq  \\frac{1 + \\varepsilon_0}{\\sqrt{6}} \\| a - b\\|^{3\/2} \\sqrt{|\\partial \\Omega| - 2 \\|a-b\\|}.$$\n\\end{proof}\n\n\n\n\n\n\\subsection{Proof of Theorem 1}\n\n\\begin{proof} \nWe assume without loss of generality $|Q| = 1$. For this proof we minimize, over all possible set partitions $Q = S \\cup (Q \\setminus S)$, the quotient\n$$ \\inf_{\\Gamma} \\frac{\\mathcal{H}^1(\\Gamma) }{|S| | Q \\setminus S|},$$\nwhere $\\Gamma=\\overline S\\cap \\overline{Q \\setminus S}$.\n\nLemma \\ref{Lemma: rectangle result} and Remark \\ref{rmk1} show that the cut must intersect opposite sides; if the aspect ratio is large enough, then these opposite sides will be the longest sides. \n\n\n\n\n\nWe now show that the distance to the axis of symmetry of $R$ is of order $\\lesssim \\sqrt{\\varepsilon}$.\nSince the Gromov-Hausdorff distance between the domain and the rectangle is $\\varepsilon$, by the same calculation in Remark \\ref{rmk1} we see that any cut has to have length at least\n$ h (1- 2\\varepsilon)$, where $h$ is the length of the shorter side of the associated rectangle $R$. Note that the shortest line between two points may not make the optimal Ratio Cut, though, as that may result in a less favorable area splitting. Hence, one possibility is that we get a gain from having a slightly longer curve to encompass a more favorable area, but then by the isoperimetric inequality, it can be seen that circular arcs are favorable to any other configuration in terms of length to area trade-offs.  A key example of this is the trapezoidal domain with angled top and bottom boundary curves. Therefore the optimal spectral cut $Q = A \\cup B$ satisfies\n$$  \\frac{h(1-2\\varepsilon)}{|A| |B|} = \\frac{h(1-2\\varepsilon)}{|A| (1-|A|)} \\leq  \\frac{\\mathcal{H}^1(\\Gamma)}{|S| |Q \\setminus S|} \\leq 4h(1 + 2 \\varepsilon)$$\nfor any $\\Gamma$.\nThus, for $\\varepsilon \\leq 0.1$\n\\begin{equation}\n\\frac{1}{|A| (1-|A|)} \\leq 4\\frac{1 + 2\\varepsilon}{1 - 2 \\varepsilon} \\leq 4  + 20\\varepsilon.\\label{bound of A(1-A)}\n\\end{equation}\nWhen combined with the elementary inequality\n$$ \\frac{1}{x(1-x)}  \\geq   4 + 16\\left(x-\\frac{1}{2}\\right)^2,$$\nwe have that the optimal spectral cut yields two sets satisfying\n$$ \\frac{1}{2} - \\sqrt{\\frac{5\\varepsilon}{4}} \\leq |A| \\quad \\mbox{and}\\quad |B| \\leq \\frac{1}{2} + \\sqrt{\\frac{5\\varepsilon}{4}}.$$\nThe Lipschitz bound then ensures that the spectral cut's intersection with $\\partial Q$ occurs in a $\\sqrt{\\epsilon}$ small neighborhood of $Q$'s axis of symmetry. \n\nSince the functional itself only contains the length $\\mathcal{H}^1(\\Gamma),$ as well as a term depending on the partition of the areas $|S||\\Omega\\setminus S|$, we can conclude that the optimal $\\Gamma$ is a circular arc; otherwise, we can minimize the length of the curve while keeping the bounded area fixed.  Indeed, this boils down to the question of minimizing the arc-length of a curve of fixed area,\n\\[\n\\text{arg min}_{y = \\gamma (x)} \\left\\{  \\int_a^b \\sqrt{1 + (\\gamma'(x) )^2} dx \\ \\bigg| \\ \\int_a^b |\\gamma| dx \\,\\,\\mbox{ fixed.}\\right\\},\n\\] \nwhich is minimized by a circular arc via the isoperimetric inequality.\nWe already know from above that $\\mathcal{H}^1(\\Gamma) \\leq (1+2\\varepsilon) h$, so Lemma 2 implies that the area captured by the curved\narc satisfies\n$$ \\mbox{captured area} \\leq  \\frac{1 + c_{\\varepsilon}}{\\sqrt{6}} h^{3\/2} \\sqrt{\\mathcal{H}^1(\\Gamma) - h} \\,\\leq  \\frac{1 + c_{\\varepsilon}}{\\sqrt{3}} h^2 \\sqrt{\\varepsilon} ,$$\nwhere $c_{\\varepsilon} > 0$ depends on $\\varepsilon$, but does tend to 0 as $\\varepsilon$ tends to $0$. \n\nIn summary, the above conclusions show that the Ratio Cut $\\Gamma$ must be a circular arc in a $\\sqrt{\\epsilon}$ neighborhood of the axis of symmetry of the reference rectangle from Assumption $1$. \n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm2} for Curvilinear Quadrilaterals with Parabolic Top and Bottom Curves}\n\\label{sec:pf3}\n\nTo prove Theorem $2$, we will first analyze how the Ratio Cut depends on parameters determining the domain $Q$ and circular arc cut $\\Gamma$ for a true quadrilateral. Then, we will demonstrate for a simple example of a curvilinear trapezoid with one parabolic edge and three flat edges that the curvature has a smaller order impact on the Ratio Cut than the trapezoidal feature. Lastly, we will prove the theorem for even more general domains with parabolic top and bottom bounding curves. These domains will be shown to be generic in section 5, at least to leading order in how the Ratio Cut depends upon the structure of the curves that make up the longest sides of our approximately parabolic curvilinear rectangles. \n\nConsider the quadrilateral $Q$ determined by the vertices\n$$ (0,0), (x_1, a), (1,0) \\quad \\mbox{and} \\quad (1+x_2, a+x_3),$$\nwhere we assume that $ 0 < a < 1$ (implying that we have normalized the quadrilateral and put the longer side on the $x-$axis). The quantities $x_1, x_2, x_3$\nare perturbation parameters and assumed to be small, in the sense that $|x_i| \\ll \\min\\left\\{a,1- a\\right\\}$. The condition $|x_i| \\ll a$ is clear; we want the perturbation to be small with respect to length and height of the rectangle. The other condition is slightly more subtle: if it is not satisfied, then the quadrilateral might be close to a square and the location of the spectral cut will depend nonlinearly on the perturbation parameters. The cut is still going to be a circular arc nearly bisecting the domain, but a slight perturbation of $x_1, x_2, x_3$ may send $\\Gamma$  from being nearly vertical to nearly horizontal (see Fig. \\ref{fig:quad}).\nThe same arguments used to prove Theorem 1 show that the spectral cut will be a circular arc connecting two points ${\\bf q} = (q,0)$ and ${\\bf p} = (p,y(p))$.\n\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,0) -- (1,0) -- (1.1, 0.8) -- (-0.1, 0.7) -- (0,0);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\node at (0.08, 0.08) {$\\alpha$};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (0.95, 0.05) {$\\beta$};\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1.1,0.8) circle (0.015cm);\n\\node at (1.2, 0.8+0.08) {($1+x_2$,$a+x_3$)};\n\\filldraw (-0.1,0.7) circle (0.015cm);\n\\node at (-0.24, 0.8-0.08) {($x_1$,$a$)};\n\\filldraw (0.5,0) circle (0.015cm);\n\\node at (0.5, -0.08) {${\\bf q}:=(q,0)$}; \n\\filldraw (0.45,0.75) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thick,dashed] (0.45, 0.75) -- (0.5,0);\n\\draw [thin,dashed] (0.5,0) to[out=45,in=330] (0.45, 0.75);\n\\node at (0.42, 0.35) {$\\Gamma$};\n\\node at (0.69, 0.35) {$\\tilde{\\Gamma}$};\n\\node at (1, 0.7) {$\\gamma$};\n\\node at (-0.03, 0.63) {$\\delta$};\n\\node at (0.43, 0.05) {$\\eta$};\n\\node at (0.55, 0.1) {$\\nu$};\n\\node at (0.4, 0.67) {$\\phi$};\n\\node at (0.51, 0.65) {$\\mu$};\n\\end{tikzpicture}\n\\caption{A general quadrilateral $Q$ after rescaling.}\n\\label{fig:quad}\n\\end{figure}\n\\end{center}\n\n\n\\subsection{Circular Arc to Triangle in terms of the sector angle}\n\n\nLet us assume the cut, $\\tilde E$, is a circular arc from $\\bf p$ to $\\bf q$, where $\\bf p$ and $\\bf q$ appear nearly opposite each other on the longest sides.   We will take $|\\tilde \\Gamma| = r \\theta$ for some radius $r>0$ and sector angle $\\theta\\geq0$, to be determined, and observe then the area contained between the line $\\Gamma$ connecting $\\bf p$ and $\\bf q$ and the circular arc $\\tilde \\Gamma$, which we will call $\\tilde \\Omega$, satisfies\n\\[\n| \\tilde \\Omega | = \\frac{r^2 \\theta}{2} - \\frac{\\| {\\bf p}- {\\bf q}\\|^2}{4 \\tan (\\theta\/2)}.\n\\]\nWe also have that\n\\[\nr = \\frac{\\| {\\bf p}- {\\bf q} \\|}{2 \\sin (\\theta\/2)},\n\\]\nfrom which we conclude, after Taylor expanding in $\\theta$, that\n\\[\n| \\tilde \\Omega | = \\|{\\bf p}- {\\bf q}\\|^2 \\left( \\frac{\\theta^2}{48} + O( \\theta^4) \\right).\n\\]\nAlso, Taylor expanding once more, we have\n\\[\n|\\tilde \\Gamma| = r \\theta = \\| {\\bf p}- {\\bf q}\\| \\left( 1 + \\frac{\\theta^2}{24} + O( \\theta^4) \\right).\n\\]\n\nWith these identities in hand, we can proceed to explore how the ratio cut depends upon the curves defining the longest aspect ratio sides of our curvilinear quadrilateral.  To illustrate how to compute the dependence of the cut locally on the parametrization of a curve, we will first work with a toy model that is flat on $3$ sides and has a smooth quadratic curve on one of the long sides.  \n\n\\subsection{The parabolic trapezoid}\n\\label{partrap}\n\nWe take a domain that is a parabolic trapezoid bounded by a straight line from $(0,0)$ to $(1,0)$, a straight line from $(0,0)$ to $(0,1\/2)$, the straight line from $(1,0)$ to $(1, 1\/2+a)$ and the curve $y(x) = \\epsilon x^2 + (a - \\epsilon) x + 1\/2$; note that, for convenience, we have chosen our aspect ratio to be approximately $2$.  Our goal is to track how the cut changes as a function of $\\epsilon,a$ close to $0$.   The Ratio Cut will intersect the bottom and top boundary curves at the points $(q,0)$ and $(p,y(p))$ respectively, with the cut itself a circular arc.  Splitting the domain by a circular arc, we can decompose one of the cut domains into two curvilinear quadrilaterals, see Figure \\ref{fig:quad2} for an illustration.\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,1\/2)--(0,0) -- (1,0) -- (1, 1\/2+0.1) \n\\draw [ultra thick] (0,0.5) to[out=25,in=160] (1, 0.6);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1,1\/2+0.1) circle (0.015cm);\n\\node at (1.08, 1\/2+0.1+0.08) {($1$,$0.6$)};\n\\filldraw (0,0.5) circle (0.015cm);\n\\node at (-0.1, 0.5+0.08) {($0$,$0.5$)};\n\\filldraw (0.5,0) circle (0.015cm);\n\\node at (0.5, -0.08) {${\\bf q}:=(q,0)$}; \n\\filldraw (0.52,0.67) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thin,dashed] (0.5,0) to[out=75,in=285] (0.52, 0.67);\n\\node at (0.45, 0.35) {$\\Gamma$};\n\\end{tikzpicture}\n\\caption{A parabolic trapezoid $Q$ as a simplified model.}\n\\label{fig:quad2}\n\\end{figure}\n\\end{center}\n\nIt can be easily seen that to compute the Ratio Cut, we can split the domains into components given by the region $[0,p]$, a triangle with base $[p,q]$, and the cap of a circular arc that is either added or subtracted depending upon orientation ($p<q$ or $q<p$ and $\\theta > 0$ or $\\theta < 0$ respectively). Decomposing the domain into these components, we compute\n\\begin{align*}\n&A_{total}  = \\int_0^1 y(x) dx = \\frac{\\epsilon}{3} + \\frac{a-\\epsilon}{2} + \\frac12, \\ \\ \\text{(Total Area)} \\\\\n&A_1 (p,q,\\theta)  = \\int_0^p y(x) dx + \\frac12 y(p) (q-p)+\\frac12 R^2 (p,q,\\theta) \\theta \\\\\n& \\hspace{4cm} - \\frac14 \\left[   (q-p)^2 + y(p)^2 \\right] \\cot \\left( \\frac{\\theta}{2} \\right), \\ \\ \\text{(Left Area)} \\\\\n& A_2 (p,q,\\theta)  = A_{total} - A_1 (p,q,\\theta), \\ \\ \\text{(Right Area)} \n\\end{align*}\nwhere $R(p,q,\\theta)$ is the radius of the circle, given by\n\\begin{align}\nR(p,q,\\theta)  = \\frac{\\sqrt{  (q-p)^2 + y(p)^2 }}{ 2 \\sin \\left( \\frac{\\theta}{2} \\right)}\\,.\n\\end{align}\nThe above quantities give the ratio cut, denoted by $\\text{RC} (p,q,\\theta)$:\n\\begin{align*}\n \\text{RC} (p,q,\\theta) = \\frac{ R(p,q,\\theta) \\theta}{ A_1 (p,q,\\theta) A_2 (p,q,\\theta)}.\n\\end{align*}\nIt follows via direct calculation that $ \\text{RC} (p,q,\\theta)$ is a smooth function of $(p,q,\\theta)$ in a neighborhood of $(1\/2,1\/2,0)$. Indeed, since $\\frac12 R^2 (p,q,\\theta) \\theta - \\frac14 \\left[   (q-p)^2 + y(p)^2 \\right] \\cot \\left( \\frac{\\theta}{2} \\right)\\to 0$ and $R(p,q,\\theta) \\theta\\to \\sqrt{  (q-p)^2 + y(p)^2 }$ when $\\theta\\to 0$, $\\text{RC} (p,q,\\theta)$ is smooth in a neighborhood of $(1\/2,1\/2,0)$. Hence we can explore behaviors nearby using the Implicit Function Theorem.   \n\nAs an illustrative calculation, let us simply take the full quadratic approximation in $\\theta$, $p-\\frac12$, $q-\\frac12$, $a$ and $\\epsilon$ to the Ratio Cut. A direct calculation gives:\n\\begin{small}\n\\begin{align*}\n& \\text{RC} (p,q,\\theta)  {\\color{red}=} \\left(  8 + 24 \\left(q - \\frac12 \\right)^2 - 16 (q-\\frac12) (p-\\frac12) + 24 (p - \\frac12)^2 \\right) \\\\\n& - a \\left(8 + 8\\left(q - \\frac12 \\right) - 8 \\left(p - \\frac12 \\right) - 56\\left(q - \\frac12 \\right) ^2 - 56 \\left(p - \\frac12 \\right)^2 + 80 \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)    \\right) \\\\\n& + \\epsilon \\left(  \\frac43 + \\frac{52}{3} \\left(q - \\frac12 \\right)^2 + \\frac{100}{3} \\left(p - \\frac12 \\right)^2    - 40 \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)  \\right) + \\frac43 \\theta \\left(   \\left(q - \\frac12 \\right) + \\left(p - \\frac12 \\right) \\right) \\\\\n& + a^2  \\left( 10 + 16 \\left(q - \\frac12 \\right) - 16 \\left(p - \\frac12 \\right)  + 120 \\left(q - \\frac12 \\right)^2 + 104 \\left(p - \\frac12 \\right)^2  -192 \\left(q - \\frac12 \\right) \\left(p - \\frac12 \\right) \\right)  \\\\\n& + \\epsilon^2 \\left(  \\frac{122}{9}  \\left(q - \\frac12 \\right)^2 + \\frac{218}{9} \\left(p - \\frac12 \\right)^2  - \\frac{284}{9} \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)   \\right)  +  \\frac{7}{18} \\theta^2   \\\\\n& -a \\epsilon \\left(  \\frac83 + \\frac{8}{3} \\left(q - \\frac12 \\right) - 8 \\left(p - \\frac12 \\right)   + 72 \\left(q - \\frac12 \\right)^2 + 104 \\left(p - \\frac12 \\right)^2 - \\frac{464}{3} \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)   \\right)  \\\\\n&  -  a \\theta \\left(  \\frac23 + 8 \\left(q - \\frac12 \\right)^2 - \\frac{32}{3}  \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)  \\right)  - \\frac89 \\epsilon \\theta \\left( \\left(q - \\frac12 \\right)  + \\left(p - \\frac12 \\right)   \\right)  \n\\end{align*}\\end{small} \nup to a higher order error, when $a$ and $\\epsilon$ are sufficiently small.\nWhile this may not look so useful, we get a great deal of information by looking at the system when $a = \\epsilon = 0$.  In such a case, the equations for a critical point in $p,q,\\theta$ become\n\\begin{align*}\n48 \\left(q - \\frac12 \\right)  - 16 \\left(p - \\frac12 \\right) + \\frac43 \\theta & = 0 \\\\\n-16 \\left(q - \\frac12 \\right)  + 48 \\left(p - \\frac12 \\right) + \\frac43 \\theta & = 0 \\\\\n\\frac43 \\left(q - \\frac12 \\right)  + \\frac43 \\left(p  - \\frac12 \\right) + \\frac79 \\theta & = 0.\n\\end{align*}\nThus, the Jacobian matrix is\n\\begin{equation*}\nJ_{a=0,\\epsilon=0} = \\left[  \\begin{array}{rrr}\n48 & -16 & \\frac43 \\\\\n-16 & 48 & \\frac43 \\\\\n\\frac43 & \\frac43 & \\frac79\n\\end{array}\n\\right],\n\\end{equation*}\nwhich is non-singular.  As a result, we observe that if $a = \\epsilon = 0$, the optimal solutions is $p = q = \\frac12$, $\\theta = 0$.  We know this from symmetry arguments, but now we also have set ourselves up for an application of the Implicit Function Theorem in order approximate the RC for near rectangular domains.  \n\nWe next observe what happens if $a \\neq 0$, $\\epsilon = 0$. This gives the modified system\n\\begin{equation*}\nJ_{a,\\epsilon=0} \\left[ \\begin{array}{c}\n\\left(q - \\frac12 \\right)  \\\\\n\\left(p - \\frac12 \\right) \\\\\n\\theta \n\\end{array} \\right] = a \\left[ \\begin{array}{r}\n8 \\\\\n-8 \\\\\n\\frac23\n\\end{array} \\right]  + \\text{Quadratic Error in $a, q-\\frac12, p-\\frac12$},\n\\end{equation*}\nwhere\n\\begin{equation*}\nJ_{a,\\epsilon=0} = \\left[  \\begin{array}{ccc}\n48 -112 a & -16 + 80 a& \\frac43 \\\\\n-16 +80 a & 48 -112 a & \\frac43 \\\\\n\\frac43 & \\frac43 & \\frac79\n\\end{array}\n\\right],\n\\end{equation*}\nHence, $\\vec v$ satisfying $J_{a,\\epsilon=0} \\vec{ v }= \\vec{0}$ is given by \n\\begin{equation*}\n\\vec{v} = a J_{a=0,\\epsilon=0}^{-1} \\left[ \\begin{array}{c}\n8 \\\\ -8 \\\\ \\frac23\n\\end{array} \\right] + O(a^2) = a \\left[  \\begin{array}{c}\n\\frac{1}{12} \\\\ -\\frac16 \\\\ 1 \\end{array}  \\right] + O(a^2).\n\\end{equation*}\nNote that the distance between $p$ and $q$ is then $\\frac{a}{4} < a$, with the maximal amplitude of the bulge from the circular arc of size \n\\[  \\frac{a}{8}\\sqrt{y(p)^2+(p-q)^2} < \\frac{a}{8}. \\]  \nSince the terms that are linear in $\\epsilon$ are quadratic or higher in $q-1\/2,p-1\/2$, a similar analysis including $\\epsilon$ shows that the curvature of the parabolic curve is actually a lower order deformation for the Ratio Cut than the trapezoidal deflection.  Importantly, this argument demonstrates that the trapezoidal deflection is decreasing in the new cut domain.  Though the new cut domain will be closer to aspect ratio $1$, the overall deflections are still decreasing on subsequent domains.\n\n\nUsing the Implicit Function Theorem, we are able to compute comparable results for the full Ratio Cut.  Indeed, to turn this into a rigorous argument, we need first observe that the minimum Ratio Cut at $a=\\epsilon = 0$ is uniquely $p=q=\\frac12$, $\\theta=0$, which is easily seen by looking at the Hessian.  Then, we look at $ \\text{RC} (p,q,\\theta; a, \\epsilon)$ as a map from $\\mathbb{R}^5 \\to \\mathbb{R}^3$, and use the Implicit Function Theorem to construct the desired local map from $(p,q,\\theta;\\,a,\\epsilon)$ in an open set around $(a,\\epsilon) = (0,0)$. \n\n\\subsection{The Ratio Cut with sides given by parabolic approximations}\n\\label{arcs}\n\n\nNow, we proceed to handle a more general family of domains.  Given our assumption on the smoothness of the curves, we can assume that the top and bottom curves are approximated by quadratic curves to high accuracy near points of intersection with the ratio cut.  Specifically, let us consider an arbitrary domain $Q$ that can be approximated (in the Gromov-Hausdorff sense) by a parabolic trapezoid $Q_0$ with vertices $(0,0),(0,\\frac{1}{2}+a_1),(1,\\frac{1}{2}+a_2),$ and $(1,0)$. Note, the inclusion of $a_1$ and $a_2$ here will allow us to vary the aspect ratio. We fix the width of $Q_0$ to $1$, however, as can always be done by a scaling of the domain.  The top paraboloid of $Q_0$ is parametrized as \n$$\ny_{T}(x) = \\epsilon_{t}x^2 + (a_2 - a_1 - \\epsilon_{t})x + a_1 + \\frac{1}{2},\n$$ \nand the bottom paraboloid is parametrized as \n$$\ny_{B}(x) = \\epsilon_{b}x^2 -\\epsilon_b x.\n$$ \nWe assume that $Q$ and $Q_0$ differ by two sufficiently small and bounded ``black-box'' regions on the left and right.\nThe areas of these two small regions will be denoted $A_{\\texttt{WL}}$ and $A_{\\texttt{WR}}$. \n\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,1\/2)--(0,0) ;\n\n\\draw [red,ultra thick] (0,0) to[out=5,in=195] (0, 0.5);\n\\node[red] at (-0.15, 0.25) {$A_{\\texttt{WL}}$};\n\n\\draw [ultra thick] (1,0) -- (1, 1\/2+0.1) ;\n\n\\draw [red,ultra thick] (1,0) to[out=105,in=335] (1, 0.6);\n\\node[red] at (1.15, 0.25) {$A_{\\texttt{WR}}$};\n\n\\draw [ultra thick] (0,0.5) to[out=25,in=160] (1, 0.6);\n\\draw [ultra thick] (0,0) to[out=-10,in=190] (1, 0);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1,1\/2+0.1) circle (0.015cm);\n\\node at (1.08, 1\/2+0.1+0.08) {$(1,0.6)$};\n\\filldraw (0,0.5) circle (0.015cm);\n\\node at (-0.1, 0.5+0.08) {($0$,$0.5$)};\n\\filldraw (0.5,-0.05) circle (0.015cm);\n\\node at (0.5, -0.15) {${\\bf q}:=(q,y_B(q))$}; \n\\filldraw (0.52,0.67) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thin,dashed] (0.5,-0.05) to[out=65,in=295] (0.52, 0.67);\n\\draw [dotted] (0.5,-0.05) -- (0.52, 0.67) ;\n\\node at (0.65, 0.48) {$\\Gamma$};\n\\node at (0.55, 0.35) {$\\Omega$};\n\\end{tikzpicture}\n\\caption{A more general domain $Q$ approximated by a parabolic trapezoid.}\n\\label{fig:quad3}\n\\end{figure}\n\\end{center}\n\nWe start by preparing quantities associated to $Q_0$. A circular arc $\\Gamma$ passing through the points $(p,y_{T}(p))$ and $(q, y_B(q))$, with angle $\\theta$, cuts the parabolic trapezoid into a left and right domain, denoted by $S$ and $Q_0\\backslash S$.\n\tTo compute an equivalent analytic expression for the ratio cut, we use Stoke's theorem\n\tto compute the left area $A_L = |S|$ and total $A_T = |Q_0|$, which we use to compute the right area $A_R = |Q_0\\backslash S| = |Q_0|-|S|$. In particular, \n\t$$\n\tA_T = |Q_0| = \\int_{Q_0} dA = \\frac{1}{2} \\int_{\\partial Q_0} xdy - ydx,\n\t$$\n\twhere we integrate along the left vertical boundary $\\{(0,t): 0\\leq t\\leq \\frac{1}{2}+a_1\\}$, along the top parabolic curve $\\{(x,y_T(x)): 0\\leq x\\leq 1 \\},$ along the right vertical boundary $\\{(1,t): \\frac{1}{2}+a_2\\leq t\\leq 0 \\}$ (with the indicated orientation), and finally along the bottom parabolic curve $\\{(x,y_B(x)): 1\\leq x\\leq 0 \\}$ (with the indicated orientation). \n\tWe can compute the (indefinite) integrals for the two parabolic pieces fairly easily:\n\t\\begin{align*}\n\t\tdy_T \t&= (2\\epsilon_t x + a_2-a_1-\\epsilon_t)dx,\\\\\n\t\tdy_B \t&= (2\\epsilon_b x -\\epsilon_b)dx,\n\t\\end{align*}\n\tand so\n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int xdy_T - y_T dx \t&= \\frac{1}{2} \\int x(2\\epsilon_t x + a_2-a_1-\\epsilon_t)dx - (\\epsilon_{t}x^2 + (a_2 - a_1 - \\epsilon_{t})x + a_1 + \\frac{1}{2})dx\\\\\n\t\t\t&= \\frac{1}{2}\\int \\epsilon_t x^2-(a_1+\\frac{1}{2}) dx\n\t\t\t= \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x;\\\\\n\t\t\\frac{1}{2} \\int xdy_B - y_B dx \t&= \\frac{1}{2}\\int x(2\\epsilon_b x -\\epsilon_b)dx - (\\epsilon_{b}x^2 + -\\epsilon_b x)dx\\\\\n\t\t\t&= \\frac{1}{2} \\int \\epsilon_b x^2 dx\n\t\t\t= \\frac{\\epsilon_b}{6}x^3.\n\t\\end{align*}\nThe $1$-forms for the vertical components are \n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int 0dt - t d(0) &= \\frac{1}{2} \\int 0 = 0,\\\\\n\t\t\\frac{1}{2} \\int 1 dt - t d(1) \t&= \\frac{1}{2} \\int dt - 0 = \\frac{t}{2}.\n\t\\end{align*}\n\tPutting it all together, we have\n\t\\begin{align*}\n\t\tA_T &= 0 + \\left[ \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x \\right]_{x=0}^{x=\\frac{1}{2}+a_1} + \\left[\\frac{t}{2} \\right]_{t=\\frac{1}{2}+a_2}^{0} + \\left[ \\frac{\\epsilon_b}{6}x^3 \\right]_{x=1}^{x=0}\\\\\n\t\t\t&= \\frac{\\epsilon_t a_1^3}{6} + \\frac{\\epsilon_t a_1^2}{4} + \\frac{\\epsilon_t a_1}{8} + \\frac{\\epsilon_t}{48} -\\frac{\\epsilon_b}{6} - \\frac{a_2}{2} - \\frac{a_1^2}{2} - \\frac{a_1}{2} - \\frac{3}{8}.\n\t\\end{align*}\n\t\n\tFor $A_L = |S|$, we use Stokes' theorem again but split $S$ into two parts divided by the straight line connecting $(p,y_t(p))$ and $(q,y_b(q))$; these parts are a curvilinear quadrilateral and circular cap. The area of the curvilinear quadrilateral is computed by Stokes' theorem, while the area of the circular cap is determined by an elementary formula. Adding these two quantities gives us $A_L$.\n\t%\n\t\n\t\n\tThe line connecting $(p,y_T(p))$ to $(q,y_B(q))$ is parametrized as $\\{ (1-t)(p,y_T(p)) + t(q,y_B(q)): 0\\leq t \\leq 1 \\},$ which componentwise becomes\n\t$$ \n\t( (1-t)p + t q, (1-t)y_T(p) + ty_B(q)), \n\t$$\n\tand so the integral along this straight line becomes\n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int xdy - ydx \t=\\,& \\frac{1}{2} \\int ((1-t)p + t q)d((1-t)y_T(p) + ty_B(q))\\\\ &\\quad - ((1-t)y_T(p) + ty_B(q))d((1-t)p + t q)\\\\\n\t\t\t=\\,& \\frac{1}{2} \\int ((1-t)p + t q)(-y_T(p) dt + y_B(q) dt)\\\\\n\t\t\t\t&\\quad - ((1-t)y_T(p) + ty_B(q)) ( -p dt + q dt)\\\\\n\t\t\t=\\,& t q (p (\\epsilon_b q-\\epsilon_b-\\epsilon_t p+\\epsilon_t)-a_2 (p+1))\\,,\n\t\\end{align*}\n\twhere we denote the last quantity by $SL(t)$.\n\t\n\tGiven two points $p$ and $q$, and an angle $\\theta$, the radius of the circle containing points $p$ and $q$ separated by angle $\\theta$ is \n\t$$\n\tR = \\frac{\\|p-q\\|_2}{2\\sin(\\frac{\\theta}{2})}.\n\t$$ \n\tThus, the area of the circular segment is\t{\\allowdisplaybreaks\n\t$$\n\t|\\Omega|= \\frac{R^2}{2}(\\theta - \\sin(\\theta)) = \\frac{(p - q)^2 + (y_T(p) - y_B(q))^2}{2}(\\theta - \\sin(\\theta)),\n\t$$ \n\tand so\n\t\\begin{align*}\n\t\tA_L =\\,& 0 + \\left[ \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x \\right]_{x=0}^{x=p} + [SL(t)]_{t=0}^{t=1} + \\left[ \\frac{\\epsilon_b}{6}x^3 \\right]_{x=q}^{x=0} + |\\Omega|\\\\\n\t\t=\\,& \\frac{1}{6} \\left( -\\epsilon_b q^3 - \\frac{3}{2}(1+2a_1)p + \\epsilon_t p^3 + 6q (-a_2(1+p) + p( - \\epsilon_b + \\epsilon_t + \\epsilon_t q - \\epsilon_t p))\\right.\\\\\n\t\t& \\left.+ 3\\left( (q-p)^2 + \\left( a_2  + \\epsilon_b q - \\epsilon_b q^2 + a_2p - \\epsilon_t p + \\epsilon_t p^2\\right)^2\\right) (\\theta - \\sin(\\theta))\\right).\n\t\\end{align*}\n\tWe can use this to get the area of the right portion:\n\t\\begin{align*}\n\t\tA_R =& A_T - A_L \\\\\n\t\t\t=& \\frac{1}{48} \\bigg(-18-24 a_1-24 a_1^2-24 a_2-8 \\epsB+\\epsT+6 a_1 \\epsT+12 a_1^2 \\epsT\\\\\n\t\t\t&+8 a_1^3 \\epsT+8 \\epsB \\tBott^3+12 (1+2 a_1) \\tTop-8 \\epsT \\tTop^3\\\\\n\t\t\t&-48 \\tBott \\big(-a_2 (1+\\tTop)+\\tTop (\\epsT+\\epsB (-1+\\tBott)-\\epsT \\tTop)\\big)\\\\\n\t\t\t&-24 \\big((\\tBott-\\tTop)^2+(a_2+\\epsB \\tBott-\\epsB \\tBott^2+a_2 \\tTop-\\epsT \\tTop+\\epsT \\tTop^2)^2\\big) (\\theta-\\sin(\\theta))\\bigg).\n\t\\end{align*}\n\t}\n\tFinally, the length of the ratio cut takes the form $$|\\Gamma| = R\\theta = \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2 \\frac{\\theta}{2}}{\\sin(\\frac{\\theta}{2})} = \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2}{\\sinc(\\frac{\\theta}{2})}.$$\n\n\tWith the above pieces from $Q_0$, the ratio cut of $Q$\n\n\tcan be expressed in terms of variables $p, q,$ and $\\theta$, together with parameters $a_1,a_2, \\epsT, \\epsB, A_{\\texttt{WL}}, A_{\\texttt{WR}}$ that determine the domain.  Specifically, by letting $\\sigma = (a_1,a_2, \\epsT, \\epsB, A_{\\texttt{WL}}, A_{\\texttt{WR}})$ be the parameter vector, we can express the ratio cut as \n\t\\begin{align*}\n\t\tRC(q, p, \\theta; \\sigma) \t&= \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2}{\\sinc(\\frac{\\theta}{2})(A_L + A_{\\texttt{WL}})(A_R+A_{\\texttt{WR}})}.\n\t\\end{align*}\n\t%\n\tThe series expansion of $RC(q, p, \\theta;\\sigma)$ near $\\left( \\frac{1}{2}, \\frac{1}{2}, 0;   \\vec{0} \\right)$ is\n\t\\begin{align*}\n\t\tRC \t=& 8 + 24 \\tBCent^2 - 16 \\tBCent \\tTCent + \\frac{4}{3} \\tBCent \\Th\\\\\n\t\t\t&+ 24 \\tTCent^2 + \\frac{4}{3} \\tTCent \\Th + \\frac{7}{18}\\Th^2\\\\\n\t\t\t&+ a_1 p_{a_1}(q, p, \\theta) + a_2 p_{a_2}(q, p, \\theta) + a_3 p_{a_3}(q, p, \\theta)\\\\\n\t\t\t&+ \\epsilon_t p_{\\epsilon_t}(q, p, \\theta) + \\epsilon_b p_{\\epsilon_b}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}} p_{A_{\\texttt{WL}}}(q, p, \\theta) + A_{\\texttt{WR}} p_{A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_1^{\\,2} p_{a_1 a_1}(q, p, \\theta) + a_1a_2 p_{a_1 a_2}(q, p, \\theta) + a_1a_3 p_{a_1 a_3}(q, p, \\theta)\\\\\n\t\t\t&+ a_2^{\\,2} p_{a_2a_2}(q, p, \\theta) + a_2a_3 p_{a_2a_3}(q, p, \\theta) + a_3^{\\, 2} p_{a_3a_3}(p, q, \\theta)\\\\\n\t\t\t&+ a_1A_{\\texttt{WL}} p_{a_1A_{\\texttt{WL}}}(q, p, \\theta) + a_1A_{\\texttt{WR}} p_{a_1A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_1 \\epsT p_{a_1 \\epsT}(q, p, \\theta) + a_1 \\epsB p_{a_1 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ a_2 A_{\\texttt{WL}} p_{a_2A_{\\texttt{WL}}}(q, p, \\theta) + a_2A_{\\texttt{WR}} p_{a_2 A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_2 \\epsT p_{a_2 \\epsT}(q, p, \\theta) + a_2 \\epsB p_{a_2 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ a_3 A_{\\texttt{WL}} p_{a_3A_{\\texttt{WL}}}(q, p, \\theta) + a_3A_{\\texttt{WR}} p_{a_3 A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_3 \\epsT p_{a_3 \\epsT}(q, p, \\theta) + a_3 \\epsB p_{a_3 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}}A_{\\texttt{WL}} p_{A_{\\texttt{WL}} A_{\\texttt{WL}}}(q, p, \\theta) + A_{\\texttt{WL}} A_{\\texttt{WR}} p_{A_{\\texttt{WL}} A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WR}} A_{\\texttt{WR}} p_{A_{\\texttt{WR}} A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}} \\epsT p_{A_{\\texttt{WL}} \\epsT}(q, p, \\theta) + A_{\\texttt{WL}} \\epsB p_{A_{\\texttt{WL}} \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WR}} \\epsT p_{A_{\\texttt{WR}} \\epsT}(q, p, \\theta) + A_{\\texttt{WR}} \\epsB p_{A_{\\texttt{WR}} \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ \\epsT \\epsT p_{\\epsT \\epsT}(q, p, \\theta) +  \\epsT \\epsB p_{\\epsT \\epsB}(q, p, \\theta) +  \\epsB \\epsB p_{\\epsB \\epsB}(q, p, \\theta)\\,,\n\t\\end{align*}\nup to a higher order error, where the polynomial $p_{\\sigma^{\\alpha}}$ is the partial derivative $\\left. \\frac{\\partial^{\\alpha} RC}{\\partial \\sigma^{\\alpha}} \\right|_{\\sigma = 0}$ for a multi-index $\\alpha$.  Detailed fomulae for these terms are given in the Appendix.\t\n\t\n\t\n\tNext, we compute the linearization of $RC$ near the point $(q,p,\\theta) = (\\frac{1}{2}, \\frac{1}{2}, 0)$:\n\t\\begin{align*}\n\t\t\\left.\\frac{\\partial RC}{\\partial q}\\right|_{\\sigma = 0} &= 48\\big (q - \\frac{1}{2}\\big) - 16 \\big(p - \\frac{1}{2}\\big) + \\frac{4}{3}\\theta,\\\\\n\t\t\\left.\\frac{\\partial RC}{\\partial p}\\right|_{\\sigma = 0} &= -16 \\big(q - \\frac{1}{2}\\big) +48 \\big(p - \\frac{1}{2}\\big) + \\frac{4}{3}\\theta,\\\\\n\t\t\\left.\\frac{\\partial RC}{\\partial \\theta}\\right|_{\\sigma = 0} &= \\frac{4}{3} \\big(q - \\frac{1}{2}\\big) + \\frac{4}{3} \\big(p - \\frac{1}{2}\\big) + \\frac{7}{9}\\theta.\n\t\\end{align*}\n\t%\n\tThe Jacobian of $RC$ at $\\sigma = 0$ is thus\n\t$$\n\tJ = \\begin{pmatrix} 48 & -16 & \\frac{4}{3}\\\\ -16 & 48 & \\frac{4}{3} \\\\ \\frac{4}{3} & \\frac{4}{3}  & \\frac{7}{9}  \\end{pmatrix}.\n\t$$\n\t%\nHere, and below, we have kept the $0$-coefficient terms solely as a place keeper to demonstrate that we have actually computed the coefficients of all the terms in our expansion, as well as to make it easier to verify the formulae for the interested reader.\n\tNext, we explore the other pieces of the linearization of $RC$, namely all first-order terms in the variables, together with the parameters.\n\t{\\allowdisplaybreaks\n\tExplicitly, we have\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial q} \t&= \\left( 48 - 112 a_1 - 112 a_2 + 112 a_3 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{200}{3} \\epsilon_b + \\frac{104}{3} \\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(-16 + 80 a_1 + 80a_2 - 80a_3 + 40 \\epsilon_b - 40\\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad  + \\Big(8a_1 -8a_2 - 8a_3 + 32 A_{\\texttt{WL}} - 32 A_{\\texttt{WR}} - 32 a_1 a_1 + 0 a_1a_2 + 32 a_1a_3\\\\\n\t\t&\\quad\\quad +32 a_2a_2 + 0 a_2a_3 - 32 a_3a_3 - 128a_1 A_{\\texttt{WL}} + 0 a_1 A_{\\texttt{WR}} + \\frac{8}{3} a_1 \\epsilon_t - \\frac{8}{3} a_1\\epsilon_b\\\\\n\t\t&\\quad \\quad  + 0a_2A_{\\texttt{WL}} + 128 a_2 A_{\\texttt{WR}} - \\frac{8}{3} a_2 \\epsilon_t + \\frac{8}{3} a_2 \\epsilon_b\\\\\n\t\t&\\quad \\quad  + 64 a_3 A_{\\texttt{WL}} - 64 a_3 A_{\\texttt{WR}} - 8 a_3 \\epsilon_t + 8 a_3 \\epsilon_b\\\\\n\t\t&\\quad \\quad  - 512 A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + 512 A_{\\texttt{WR}}A_{\\texttt{WR}} \\\\\n\t\t&\\quad \\quad  + \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_t - \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_b - \\frac{32}{3} A_{\\texttt{WR}} \\epsilon_t + \\frac{32}{3} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\Big)\\\\\n\t\t&\\quad + \\text{ cubic terms},\n\t\\end{align*}\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial p} \t&= \\left(-16 + 80 a_1 + 80a_2 - 80a_3 + 40 \\epsilon_b - 40\\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad + \\left(48 - 112 a_1 - 112 a_2 +112 a_3 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{104}{3} \\epsilon_b + \\frac{200}{3} \\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad  + \\left(-8a_1 + 8a_2 + 8a_3 + 32A_{\\texttt{WL}} - 32 A_{\\texttt{WR}} +32 a_1 a_1 + 0 a_1a_2 - 32 a_1a_3\\right.\\\\\n\t\t&\\quad\\quad \\left.-32 a_2a_2 + 0 a_2a_3 + 32 a_3a_3 - 64 a_1 A_{\\texttt{WL}} + 64 a_1 A_{\\texttt{WR}} - 8 a_1 \\epsilon_t + 8 a_1\\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad  \\left. - 64 a_2A_{\\texttt{WL}} + 64 a_2 A_{\\texttt{WR}} + 8 a_2 \\epsilon_t - 8 a_2 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. + 0 a_3 A_{\\texttt{WL}} - 128 a_3 A_{\\texttt{WR}} + \\frac{8}{3} a_3 \\epsilon_t - \\frac{8}{3} a_3 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - 512 A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + 512 A_{\\texttt{WR}}A_{\\texttt{WR}} \\right.\\\\\n\t\t&\\quad \\quad \\left. + \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_t - \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_b + \\frac{32}{3} A_{\\texttt{WR}} \\epsilon_t + \\frac{32}{3} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\right)\\\\\n\t\t&\\quad + \\text{ cubic terms},\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial \\theta} \t&= \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{7}{9} - \\frac{5}{9} a_1 - \\frac{5}{9}a_2 + \\frac{5}{9} a_3 - \\frac{32}{9}A_{\\texttt{WL}} - \\frac{32}{9}A_{\\texttt{WR}} + \\frac{1}{54}\\epsilon_b - \\frac{1}{54} \\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad + \\left(\\frac{2}{3}a_1 - \\frac{2}{3}a_2 + \\frac{2}{3} a_3 + \\frac{8}{3}A_{\\texttt{WL}} - \\frac{8}{3}A_{\\texttt{WR}} - \\frac{4}{3} a_1 a_1 + 0 a_1a_2 + 0 a_1a_3\\right.\\\\\n\t\t&\\quad\\quad \\left.+ \\frac{4}{3} a_2a_2 - \\frac{4}{3} a_2a_3 + \\frac{4}{3} a_3a_3 - 8 a_1 A_{\\texttt{WL}} - \\frac{8}{3} a_1 A_{\\texttt{WR}} - \\frac{1}{9} a_1 \\epsilon_t + \\frac{1}{9} a_1\\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad  \\left.+ \\frac{8}{3} a_2A_{\\texttt{WL}} + 8 a_2 A_{\\texttt{WR}} + \\frac{1}{9} a_2 \\epsilon_t - \\frac{1}{9} a_2 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{8}{3} a_3 A_{\\texttt{WL}} - 8 a_3 A_{\\texttt{WR}} - \\frac{1}{9} a_3 \\epsilon_t + \\frac{1}{9} a_3 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{128}{3} A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + \\frac{128}{3} A_{\\texttt{WR}}A_{\\texttt{WR}} \\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{4}{9} A_{\\texttt{WL}} \\epsilon_t + \\frac{4}{9} A_{\\texttt{WL}} \\epsilon_b + \\frac{4}{9} A_{\\texttt{WR}} \\epsilon_t - \\frac{4}{9} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\right)\\\\\n\t\t&\\quad + \\text{ cubic terms}.\n\t\\end{align*}\n\t%\n\tWith these expansions, we can express the linearization as $J+ J_{\\sigma}$, where the terms $J_{\\sigma,ii}$ come from the partials computed above. Explicitly,\n\t\\begin{align*}\n\t\tJ_{\\sigma,11} \t&= - 112 a_1 - 112 a_2 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{200}{3} \\epsilon_b + \\frac{104}{3} \\epsilon_t,\\\\\n\t\tJ_{\\sigma,12}\t&= 80 a_1 + 80a_2 + 40 \\epsilon_b - 40\\epsilon_t,\\\\\n\t\tJ_{\\sigma,13} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,21} \t&= 80 a_1 + 80a_2 + 40 \\epsilon_b - 40\\epsilon_t,\\\\\n\t\tJ_{\\sigma,22} \t&= - 112 a_1 - 112 a_2 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{104}{3} \\epsilon_b + \\frac{200}{3} \\epsilon_t,\\\\\n\t\tJ_{\\sigma,23} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,31} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,32} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,33} \t&= - \\frac{5}{9} a_1 - \\frac{5}{9}a_2 - \\frac{32}{9}A_{\\texttt{WL}} - \\frac{32}{9}A_{\\texttt{WR}} + \\frac{1}{54}\\epsilon_b - \\frac{1}{54} \\epsilon_t.\n\t\\end{align*}\n\t\n\t\n\tWith these pieces we can express the criterion for being a critical point of the Ratio Cut, incorporating linear terms in the parameters, as\n\t\\begin{align*}\n\t\t(J+J_{\\sigma}) \\begin{pmatrix} q - \\frac{1}{2}\\\\ p - \\frac{1}{2}\\\\ \\theta \\end{pmatrix} &= L(\\sigma)\\\\\n\t\t&:= a_1 \\begin{pmatrix} 8 \\\\ -8 \\\\ \\frac{2}{3}\\end{pmatrix} + a_2 \\begin{pmatrix}-8\\\\ 8\\\\ -\\frac{2}{3}\\end{pmatrix} + a_3 \\pmat{-8 \\\\ 8 \\\\ \\frac{2}{3}} + A_{\\texttt{WL}} \\begin{pmatrix}32 \\\\ 32 \\\\ \\frac{8}{3} \\end{pmatrix} + A_{\\texttt{WR}} \\begin{pmatrix}-32\\\\ -32 \\\\ -\\frac{8}{3}\\end{pmatrix}\\\\\n\t\t&\\quad  + a_1a_1 \\pmat{-32 \\\\ 32 \\\\ - \\frac{4}{3}} + a_1a_2 \\pmat{0 \\\\ 0 \\\\ 0} + a_1a_3 \\pmat{32 \\\\ -32 \\\\ 0} \\\\\n\t\t&\\quad  + a_2a_2 \\pmat{32 \\\\ -32 \\\\ \\frac{4}{3}} + a_2a_3 \\pmat{0 \\\\ 0 \\\\ -\\frac{4}{3}} + a_3a_3 \\pmat{ - 32 \\\\ 32 \\\\ \\frac{4}{3} } \\\\\n\t\t&\\quad \t+ a_1A_{\\texttt{WL}} \\pmat{ -128 \\\\ -64 \\\\ -8 } + a_1 A_{\\texttt{WR}} \\pmat{ 0 \\\\ 64 \\\\ - \\frac{8}{3} } + a_1 \\epsilon_t \\pmat{ \\frac{8}{3} \\\\ -8 \\\\ - \\frac{1}{9} } + a_1 \\epsilon_b \\pmat{ - \\frac{8}{3} \\\\ 8 \\\\ \\frac{1}{9} } \\\\\n\t\t&\\quad \t+ a_2 A_{\\texttt{WL}} \\pmat{ 0 \\\\ -64 \\\\ \\frac{8}{3} } + a_2 A_{\\texttt{WR}} \\pmat{ 128 \\\\ 64 \\\\ 8 } + a_2 \\epsilon_t \\pmat{ - \\frac{8}{3} \\\\ 8 \\\\ \\frac{1}{9} } + a_2 \\epsilon_b \\pmat{ \\frac{8}{3} \\\\ - 8 \\\\ -\\frac{1}{9} } \\\\\n\t\t&\\quad  + a_3 A_{\\texttt{WL}} \\pmat{ 64 \\\\ 0 \\\\ -\\frac{8}{3} } + a_3 A_{\\texttt{WR}} \\pmat{ -64 \\\\ - 128 \\\\ -8 } + a_3 \\epsilon_t \\pmat{ -8 \\\\ \\frac{8}{3} \\\\ -\\frac{1}{9} } + a_3 \\epsilon_b \\pmat{ 8 \\\\ -\\frac{8}{3} \\\\ \\frac{1}{9} } \\\\\n\t\t&\\quad  + A_{\\texttt{WL}} A_{\\texttt{WL}} \\pmat{ -512 \\\\ -512 \\\\ -\\frac{128}{3}  } + A_{\\texttt{WL}} A_{\\texttt{WR}} \\pmat{ 0 \\\\ 0 \\\\ 0 } + A_{\\texttt{WR}} A_{\\texttt{WR}} \\pmat{ 512 \\\\ 512 \\\\ \\frac{128}{3}  } \\\\\n\t\t&\\quad  + A_{\\texttt{WL}} \\epsilon_t \\pmat{\\frac{32}{3} \\\\ \\frac{32}{3} \\\\ - \\frac{4}{9}  } + A_{\\texttt{WL}} \\epsilon_b \\pmat{ - \\frac{32}{3} \\\\ - \\frac{32}{3} \\\\ \\frac{4}{9} } \\\\\n\t\t&\\quad  + A_{\\texttt{WR}} \\epsilon_t \\pmat{ - \\frac{32}{3} \\\\ \\frac{32}{3} \\\\ \\frac{4}{9} } + A_{\\texttt{WR}} \\epsilon_b \\pmat{ \\frac{32}{3} \\\\ \\frac{32}{3} \\\\ -\\frac{4}{9}  } \\\\\n\t\t&\\quad  + \\epsilon_t \\epsilon_t \\pmat{0\\\\0\\\\0   } + \\epsilon_t \\epsilon_b \\pmat{0\\\\ 0\\\\ 0  } + \\epsilon_b \\epsilon_b \\pmat{0\\\\ 0 \\\\ 0  }\\\\\n\t\t&\\quad + \\text{ cubic terms in the parameters}.\n\t\\end{align*}\n\t}\n\t\\subsection*{Using the Implicit Function Theorem}\n\t\n\tSuppose $v$ is a critical triple of values for the Ratio Cut, i.e. $(J+J_{\\sigma})v = 0$. Then near $\\sigma = 0$, we can solve for $v$ in terms of the parameters in $\\sigma$:\n\t\\begin{align*}\n\t\tv \t&= J^{-1} L(\\sigma)\\\\\n\t\t\t&= \\pmat{\\frac{1}{12}(a_1 - a_2 - 2a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ -\\frac{1}{12} (2a_1 - 2a_2 - a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ a_1 - a_2 + a_3} + \\text{quadratic terms in the parameters}.\n\t\\end{align*}\n\t%\n\tWe will not write out the full $J^{-1}L(\\sigma)$ here, though it is used for the approximations later and its form becomes clear in the representations below.\n\t%\n\t\n\t\n\tTo summarize, we have shown in this section that, near the critical point $(p,q,\\theta) = \\left( \\frac{1}{2}, \\frac{1}{2}, 0 \\right)$ of the ratio cut, and for $\\sigma=(a_1, a_2, \\epsilon_b, \\epsilon_t, A_{\\texttt{WL}}, A_{\\texttt{WR}})$ near $0$, we have \n\t\\begin{align*}\n\t\t\\begin{pmatrix} q - \\frac{1}{2} \\\\ p - \\frac{1}{2} \\\\ \\theta \\end{pmatrix} = \\pmat{\\frac{1}{12}(a_1 - a_2 - 2a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ -\\frac{1}{12} (2a_1 - 2a_2 - a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ a_1 - a_2 + a_3}+ \\text{quadratic terms in the parameters}.\n\t\\end{align*}\n\t\n\t\t\n\t\\section{Are our domains generic?}\n\t\n\tSince Ratio Cuts are circular, it seems more natural to prove Theorem 2 for circular, instead of parabolic, boundary curves. Using circular arcs, however, proved to be intractable for our computational methods, whereas parabolic arcs could be used. This section shows that such approximations only incur third order errors, and thus Theorem 2 is still applicable for circular boundary curves. This first lemma shows this approximation is valid near the intersection of the cut and the original domain's boundary. \n\t\n\t\\begin{lemma}\n\tProvided $I(Q)$ defined in \\eqref{Definition I(Q)} is sufficiently small, we may approximate the top and bottom curves by parabolas in a neighborhood of the Ratio Cut.\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\tThe small $I(Q)$ makes sure the domain is approximately trapezoidal with a, for example, $1:2$, aspect ratio, and that the top and curves are $C^3$ with small oscillations on a uniform spatial scale.  Hence, Taylor expanding the top and bottom curves out to $2$nd order on this scale near $p=1\/2, q = 1\/2$, we can create a parabolic approximation that is accurate up to $O( |p-\\frac12|^3, |q-\\frac12|^3)$ in this region. Potentially modifying the left and right wings in order to ensure the wings of domain have the appropriate error outside this uniform neighborhood, we see that these curvilinear trapezoids will have ratio cuts that are indeed approximated up to lower order terms in the same fashion as our exact parabolic trapezoid calculations.   \n\t\\end{proof}\n\t\t\n\tIn practice, we are given a domain $Q$ whose top and bottom boundary curves are more general to start, and upon iteration of our domains we are most interested when sides are given by circular arcs. To handle smooth enough domains of this form in full generality, we can construct a parabolic trapezoid by: rescaling the domain to fit our aspect ratio; locally using a parabolic approximation for the top and bottom curves; and finally cutting off left and right portions of $\\Omega$, which are treated as black-box regions. The Ratio Cut function sees these regions as general wing areas, denoted $A_{\\texttt{WL}}$ and $A_{\\texttt{WR}}$ for the left- and right-wing areas respectively.  The measure of distance from the rectangle, $I(Q)$, in which we measure the deflection of our domains, allows us to approximate our quadrilateral by a parabolic curve with errors that are higher order in the parameter space. \n\t\t\n\t\\subsection{Can a circular arc be parabolically approximated?}\n\t\n\tOptimal cuts for generic parabolic trapezoids are circular. Because of this, we would expect the top and bottom boundary curves of the domains in the next iteration we have been considering to be circular, but not parabolic. Here we show that parabolic arcs approximate circular arcs well, and that the ratio cut for circular boundary curves differs from the ratio cut for parabolic boundary curves at third order and higher. Thus, working with parabolic curves does not incur any significant error in the ratio cut series expansion.\n\t\n\t\\begin{lemma}\n\t   Up to higher order error terms in the curve parameters, we may approximate the top and bottom circular arc curves as paraboloids encompassing the correct area.\n\t   \\end{lemma}\n\t   \n\t   The remainder of this section will be devoted to the proof of this Lemma.\n\t\n\t\\subsubsection{Approximating Circular Arcs} \n\tFirst let us consider the most important example for our iterated domain conjecture, in which the top and bottom boundaries are circular arcs.  In what follows, the top and bottom parabolic curves will be, respectively,\n\t\\begin{align*}\n\t\ty_T^P(x) \t&= \\epsilon_t x^2 + (a_1 - a_2 - \\epsilon_t)x+ \\left( a_1 + \\frac{1}{2} \\right),\\\\\n\t\ty_B^P(x) \t&= \\epsilon_b x^2 + (a_3 - \\epsilon_b)x.\n\t\\end{align*}\n\t\n\tThe parameters $a_1, a_2, a_3$ denote horizontal perturbations from the top-left, top-right, and bottom-right vertices of a rectangle, so our parabolic trapezoid has vertices $(0,0)$, $\\left(0, \\frac{1}{2}+a_1 \\right)$, $\\left(1, \\frac{1}{2}+a_2 \\right),$ and $\\left(1, a_3 \\right)$. The terms $\\epsilon_t$ and $\\epsilon_b$ are curvature parameters that specify the shape of the two boundary curves.\n\t\n\tThe circular arcs are given by the formulas\n\t\\begin{align*}\n\t\ty_T^C(x) &= c_{y,t} + \\sqrt{r_t^2 - (x-c_{x,t})^2}, \\\\\n\t\ty_B^C(x) \t&= c_{y,b} - \\sqrt{r_b^2 - (x-c_{x,b})^2},\n\t\\end{align*}\n\twhere $r_i$ and $(c_{x,i},c_{y,i})$ are the radius and center of the corresponding top ($i=T$) or bottom ($i=B$) circle:\n\t\n\t\\begin{align*}\n\t\tr_t &= \\frac{\\sqrt{1 + (a_1-a_2)^2}}{2 \\sin(\\frac{\\theta}{2})},\\\\\n\t\tr_b &= \\frac{\\sqrt{1+a_3^2}}{2 \\sin(\\frac{\\theta}{2})},\n\t\\end{align*}\n\tand the center point coordinates are found from the systems of equations (for $i=T$ and $i=B$)\n\t\\begin{align*}\n\t\t&\\begin{cases}\n\t\t(c_{x,t})^2 + (\\frac{1}{2} + a_1 - c_{y,t})^2 \t= r_t^2,\\\\\n\t\t(1-c_{x,t})^2 + (\\frac{1}{2}+a_2 - c_{y,t})^2 \t= r_t^2,  \n\t\t\\end{cases}\\\\\n\t\t\\text{and } \n\t\t&\\begin{cases}\n\t\t(c_{x,b})^2 + (c_{y,b})^2 \t= r_b^2,\\\\\n\t\t(1-c_{x,b})^2 + (a_3 - c_{y,b})^2 \t= r_b^2.\n\t\t\\end{cases}\n\t\\end{align*}\n\t\n\tAs an explicit example, expressing $y_B^C$ in terms of the parameters gives\n\t\\begin{align*}\n\n\t\ty_B^C(x) \t&= \\frac{a_3}{2} + \\cot\\left( \\frac{\\theta_b}{2} \\right) - \\left( \\frac{1}{1+a_3^2} \\left( a_3^4 - (1-2x)^2 - 4a_3^2 (x-1) x \\right. \\right. \\\\\n\t\t&\\quad \\left.\\left. + 2a_3 (1+a_3^2) (1-2x) \\cot\\left( \\frac{\\theta_b}{2} \\right) + (1+a_3^2) \\csc^2\\left( \\frac{\\theta_b}{2} \\right)  \\right) \\right)^{\\frac{1}{2}}.\n\t\\end{align*}\n\t\n\tThe next lemma shows how to choose $\\epsilon_t$ and $\\epsilon_b$ so that the parabolic trapezoid's area approximates the circular trapezoid's area up to third order. These curvature terms are chosen so that the quadrilateral's parabolic caps have, up to third order terms, the same area as corresponding circular arcs.\n\t\n\t\\begin{lemma}\n\tLet $l_T(x) = (1-t)\\left(\\frac{1}{2}+a_1 \\right) + t \\left(\\frac{1}{2}+a_2 \\right)$, and write \n\t\\begin{equation}\n\tS_T^C = \\{ (x,y) \\colon l_T(x) \\leq y \\leq y_T^C(x) \\}, \\,\\, S_T^P(x) = \\{ (x,y) \\colon l_T(x) \\leq y \\leq y_T^P(x) \\}\n\t\\end{equation} \n\tfor the regions bounded below by the straight line $\\{\\{ t, l_T(x) \\} \\colon t \\in [0,1] \\}$ and above by the curves $\\{\\{t, y_T^i(x) \\} \\colon x\\in[0,1], i = C \\text{ or } P \\}$ respectively. Similarly, let $l_B(x) = t a_3,$ and write\n\t \\begin{equation}\n\tS_B^C = \\{ (x,y) \\colon l_B(x) \\leq y \\leq y_B^C(x) \\},\\,\\, S_B^P(x) = \\{ (x,y) \\colon l_B(x) \\leq y \\leq y_B^P(x) \\}\n\t\\end{equation} \n\tfor the regions bounded above by the straight line $\\{\\{ t, l_B(x) \\} \\colon t \\in [0,1] \\}$ and below by the curves $\\{\\{t, y_B^i(x) \\} \\colon x\\in[0,1], i = C \\text{ or } P\\}$ respectively.\n\t\n\t\tFor $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$, we have\n\t\t\\begin{align*}\n\t\t\t|S_T^C| -|S_T^P| \t&= O(\\theta_t^3)\n\t\t\t\\end{align*}\n\t\t\tand\n\t\t\t\\begin{align*}\n\t\t\t\t\t\t|S_B^C| - |S_B^P| \t&= O(\\theta_b^3) .\n\t\t\\end{align*}\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\t\tBoth $|S_T^C|$ and $|S_B^C|$ can be found using basic trigonometry: the area of a circular segment with radius $r$ and angle $\\theta$ is $\\frac{1}{2} r^2 (\\theta - \\sin(\\theta))$. In our case,\n\t\t\\begin{align*}\n\t\t\t|S_T^C| \t&= \\frac{1}{2} \\left(\\frac{\\sqrt{1+(a_1-a_2)^2}}{2\\sin(\\frac{\\theta_t}{2})} \\right)^2 (\\theta_t - \\sin(\\theta_t) )\\\\\n\t\t\t\t&= \\frac{1+(a_1-a_2)^2}{12}\\theta_t + \\frac{1+(a_1-a_2)^2}{360}\\theta_t^3 + O(\\theta_t^4),\\\\\n\t\t\t|S_B^C| \t&= \\frac{1}{2} \\left(\\frac{\\sqrt{1+a_3^2}}{2\\sin(\\frac{\\theta_b}{2})} \\right)^2(\\theta_b - \\sin(\\theta_b))\\\\\n\t\t\t\t&= \\frac{1+a_3^2}{12} \\theta_b + \\frac{1+a_3^2}{360} \\theta_b^3 + O(\\theta_b^4).\\\\\n\t\t\\end{align*}\n\t\t\n\t\tFor $|S_T^P|$ and $|S_B^P|$, we integrate:\n\t\t\\begin{align*}\n\t\t\t|S_T^P| \t&= \\int_{0}^1 y_T^P(x) - l_T(x)dx\\\\\n\t\t\t\t&= \\int_0^1 \\epsilon_t x^2 + (a_2 - a_1 - \\epsilon_t)x + \\left( \\frac{1}{2} + a_1 \\right) - (1-x)\\left( \\frac{1}{2}+a_1 \\right) - x\\left(\\frac{1}{2}+a_2 \\right) dx\\\\\n\t\t\t\t&= -\\frac{\\epsilon_t}{6}\n\t\t\t\t\\end{align*}\n\t\tand \n\t\t\\begin{align*}\n\t\t |S_B^C| \t&= \\int_0^1 l_B(x) - y_B^C(x) dx\\\\\n\t\t\t&= \\int_0^1 a_3 x -\\epsilon_b x^2 - (a_3 - \\epsilon_b)x dx\n\t\t\t= \\frac{\\epsilon_b}{6}.\n\t\t\\end{align*}\n\t\tWe want to choose constants $C_t$ and $C_b$ so that setting $\\epsilon_t = C_t \\theta_t$ and $\\epsilon_b = C_b \\theta_b$ gives us $|S_T^C|-|S_T^P| = O(\\theta_t^3)$ and $|S_B^C| - |S_B^P| = O(\\theta_b^3)$. Indeed, letting $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$ gives us this result.\n\t\\end{proof}\n\t\n\tSince the parabolic and circular trapezoids only differ in the type of caps on each, making these choices for $\\epsilon_t$ and $\\epsilon_b$ ensures the area of the circular and parabolic trapezoids are very well approximated. In addition, the boundary curves agree up to third order, and so the left areas and ratio cut lengths also agree to at least third order (in the parameters).\n\t\\begin{cor} \n\t\tSetting $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$, we get the following approximations:\n\t\t\\begin{align*}\n\t\ty_T^C(x) - y_T^P(x) &= \\frac{1}{48}(1+(a_1-a_2)^2) \\theta_t^2 \\left( O(a_1) + O(a_2) + O(\\theta_t)  \\right),\\\\\n\t\ty_B^C(x) - y_B^P(x) &= \\frac{1}{48} (1+a_3^2) \\theta_b^2 (O(a_3) + O(\\theta_b)).\n\t\t\\end{align*}\n\t\t\n\t\tMoreover, we have\n\t\t\\begin{align*}\n\t\tA_T^C \t&= A_T^P + O(\\theta_t^3) + O(\\theta_b^3),\\\\\n\t\t\\text{and } A_L^C \t&= A_L^P + O(\\theta_t^2)(O(a_1) + O(a_2)) + O( \\theta_b^2)O(a_3),\n\t\t\\end{align*}\n\t\twhere $A_T^i$, $A_L^i$ is the total area and left area respectively of the parabolic ($i=P$) or circular ($i=C$) trapezoid, and\n\t\t\\begin{align*}\n\t\t\t\\|(p,y_T^C(p)) - (q,y_B^C(q)) \\| \t&= \\|(p,y_T^P(p)) - (q,y_B^P(q))\\| + O(\\theta_t^3)+O(\\theta_b^3).\n\t\t\\end{align*}\n\t\\end{cor}\n\t\n\tThe corollary justifies our use of only quadratic terms in the Ratio Cut Taylor expansion, since the parabolic and circular Ratio Cuts (i.e. Ratio Cut functions of parabolic or circular trapezoids) agree up to third order in the parameters of the domains, given the proper curvature term substitutions.\n\t\n\t\\begin{proof}\n\t\tThe parabolic and circular boundary curve approximations come by Taylor expansions.\n\t\t%\n\t\tFor the areas, recall that the parabolic and circular trapezoids differ only in their ``caps''. Thus, assuming $a_1<a_2$ and $a_3<0$, the total area of the circular trapezoid is\n\t\t\\begin{align*}\n\t\t\tA_T^C \t&= \\left(\\frac{1}{2} + a_1\\right) + \\frac{1}{2} \\left(a_2-a_1 \\right) + \\frac{1}{2}\\left(-a_3 \\right) + \\area(S_T^C) + \\area(S_B^C)\\\\\n\t\t\t\t&= \\left(\\frac{1}{2} + a_1\\right) + \\frac{1}{2} \\left(a_2-a_1 \\right) + \\frac{1}{2}\\left(-a_3 \\right) + \\area(S_T^P) + \\area(S_B^P) + O(\\theta_t^3) + O(\\theta_b^3)\\\\\n\t\t\t\t&= A_T^P + O(\\theta_t^3) + O(\\theta_b^3).\n\t\t\\end{align*}\n\t\t%\n\t\tWrite \n\t\t\\begin{equation}\n\t\tL^C := \\|(p,y_T^C(p)) - (q,y_B^C(q))\\|,\\,\\,L^P := \\|(p,y_T^P(p)) - (q,y_B^P(q))\\|; \n\t\t\\end{equation}\n\t\tthese quantities are the lengths of the line segments connecting the intersection points of intersection between the ratio cut and boundary of the region. Expanding $L^C$ into a Taylor series in $\\theta_b$ and $\\theta_t$ gives \n\t\t\\begin{align*}\n\t\t\tL^C = L^P + O(\\theta_b^3) + O(\\theta_t^3).\n\t\t\\end{align*}\n\t\t\n\t\tFinally, write \n\t\t\\begin{equation}\n\t\t\\bar{y}_j^P(x) := y_j^P(x) + r_j(x), \n\t\t\\end{equation}\n\t\twhere $r_j$ is quadratic in the parameters, for $j = T,B$. Let $\\gamma_1$ be the left boundary segment of the (circular or parabolic) trapezoid, $\\gamma_2$ the bottom boundary segment, $\\gamma_3$ the right boundary segment, and finally $\\gamma_4$ the top boundary segment, such that $\\sum_i \\gamma_i$ is oriented counter-clockwise.\n\t\t\n\t\tIn general, if a function $y=f(x)$ is expressed as a sum $f(x) = g(x) + h(x)$, then in utilizing Stokes' theorem we get\n\t\t\\begin{align*}\n\t\t\t\\frac{1}{2} \\int x dy - ydx \t&= \\frac{1}{2} \\int x d(g(x) + h(x)) - (g(x)+h(x)) dx\\\\\n\t\t\t\t&= \\frac{1}{2} \\int x dg + xdh - g dx - h dx\\\\\n\t\t\t\t&= \\frac{1}{2} \\int xdg - g dx + \\frac{1}{2} \\int x dh - h dx.\n\t\t\\end{align*}\n\t\tIn short, the line integrals involved in Stokes' theorem are linear in the input $y$.\n\t\t\n\t\tSince we are approximating the circular arcs as parabolic arcs plus correction terms, we only need to account for the pieces of the area line integral that incorporate the correction pieces. The left boundary curve is the same for both the parabolic and circular trapezoids, so splitting up the line integrals into parabolic pieces plus correction terms gives \n\t\t\\begin{align*}\n\t\t\tA_L^C &= \\sum_i \\frac{1}{2} \\int_{\\gamma_i} xdy - ydx + A_S^C\\\\\n\t\t\t&=\\, A_L^P + \\frac{1}{2} \\int_{0}^{q} x d (r_b(x)) - r_b(x) dx\\\\\n\t\t\t&\\quad + \\frac{1}{2} \\int_{q}^{p} l_{right,x}(x) d(l_{right,y}(x)) - l_{right,y}(x) d(l_{right,x}(x))\\\\\n\t\t\t&\\quad + \\frac{1}{2} \\int x d(r_t(x)) - r_t(x) dx + O(\\theta_b^3)+O(\\theta_t^3),  \n\t\t\\end{align*}\n\t\twhere the $A_S^i$'s are the the circular segment areas bounded by the straight line connecting the two intersection points and the ratio cut, and \n\t\t\\begin{align*}\n\t\t(l_{right,x}(x),l_{right,y}(x))& \\\\\n\t\t:= \\frac{p - x}{p - q}(p, y_T^C(p))& + \\frac{x - q}{p-q}(q, y_B^C(q)) - \\left( \\frac{p - x}{p - q}(p, y_T^P(p)) + \\frac{x - q}{p-q}(q, y_B^P(q)) \\right)\n\t\t\\end{align*} \n\t\tis the error between the line segments connecting the parabolic arc intersection points and the circular arc intersection points. \n\t\t\n\t\tSince the functions $r_i(x), l_{right,x}(x), l_{right,y}(x)$ are at least third order in the parameters, the line integrals involving these terms also agree up to third order. Finally, as indicated in the computation, the circular segment area $A_S^C$ is cubic in the parameters because the lengths of the line segments agree up to cubic terms. Thus, $A_L^C = A_L^P + O(\\theta_b^2) O(a_3)+O(\\theta_t^2)( O(a_1) + O(a_2))$. \n\t\\end{proof}\n\t\n\tPutting the above results together, we conclude that the ratio cuts corresponding to circular and parabolic boundary curves differ only in cubic terms of the parameters. Putting these pieces together, we see the same is true for the full ratio cut.\n\t\n\t\\begin{lemma}\n\t\tLet $\\Gamma^C, A_T^C,$ and $A_L^C$ be the ratio cut, total area, and left-area respectively of a domain constructed from circular boundary arcs. Then we can construct a new domain with parabolic arcs such that\n\t\t\\begin{align*}\n\t\t\tRC^C - RC^P \t&= \\frac{|\\Gamma^C|}{(A_T^C-A_L^C)A_L^C} - \\frac{|\\Gamma^P|}{(A_T^P-A_L^P)A_L^P} = O(|r|^3),\n\t\t\\end{align*}\n\t\twhere $\\Gamma^P, A_{T}^P,$ and $A_{L}^P$ are the ratio cut, total area, and left-area corresponding to a domain with parabolic boundary arcs.\n\t\\end{lemma}\n\t\n\t\\begin{proof} \n\t\tBy the previous lemma (and its corollary), we can write\n\t\t\\begin{align*}\n\t\t\tA_T^C \t&= A_T^P + O(\\theta_t^3)+O(\\theta_b^3),\\\\\n\t\t\tA_L^C \t&= A_L^P + O(\\theta_t^2)(O(a_1)+O(a_2)) + O(\\theta_b^2)O(a_3),\\\\\n\t\t\tl^C \t&= l^P + O(\\theta_b^3) + O(\\theta_t^3),\n\t\t\\end{align*}\n\t\tand so\n\t\t\\begin{tiny}\n\t\t\\begin{align*}\n\t\t\tRC^C \t&= \\frac{|\\Gamma^C|}{(A_T^C - A_L^C)A_L^C}\\\\\n\t\t\t\t\t&= \\frac{ \\frac{l^P + O(\\theta_b^3) + O(\\theta_t^3)}{\\sinc(\\frac{\\theta}{2})}}{(A_T^P + O(\\theta_t^3)+O(\\theta_b^3) - A_L^P - O(\\theta_t^2)(O(a_1)+O(a_2)) - O(\\theta_b^2)O(a_3) )(A_L^P + O(\\theta_t^2)(O(a_1)+O(a_2)) + O(\\theta_b^2)O(a_3))}.\n\t\t\\end{align*}\n\t\t\\end{tiny}\n\t\tTaylor expanding the difference between $RC^P$ and $RC^C$ in terms of the parabolic approximations gives\n\t\t\\begin{align*}\n\t\t\tRC^P - RC^C \t&= O(a_1\\theta_t^2) + O(a_2\\theta_t^2) + O(\\theta_t^3) + O(a_3\\theta_b^2) + O(\\theta_b^3).\n \t\t\\end{align*}\n \t\tSince the parabolic approximation incurs only third (and higher) order errors, so does the full ratio cut function as seen in the Taylor series expansion.\n\t\\end{proof}\n\t\n\tThis lemma lets us safely use the parabolic approximation when investigating the dependency of the ratio cut on the parameters.\n\n\n\n\n\n\t\t\n\t\t\n\t\t\\section{Discussion and Conclusion}\n\n\t\\subsection{Pictures comparing optimizers with approximations}\n\t\n\tFirst, let us demonstrate how well our approximation does to compute the Ratio Cut when the curves are parabolic arcs.  The set of plots in Figures \\ref{compplot1}-\\ref{compplot8} compare optimal values of $RC$ with the linear approximations in $p, q,$ and $\\theta$.\n\t%\n\tEach plot has the optimal ratio cut value (solid blue) for a family of parameters, indicated next to the $x$-axis. The circle parameters for the optimal cuts were computed numerically using Mathematica's {\\texttt{FindMinimum}} command. The dot-dashed red curves correspond to the linear approximations for optimal $q, p,$ and $\\theta$. The figure below each plot corresponds to an extremal configuration used to compare optimal and approximate ratio cut values. \n\t%\n\tAll parameters $a_1, a_2, a_3, \\epsilon_b, \\epsilon_t, A_{\\texttt{WL}}$, and $A_{\\texttt{WR}}$ are zero unless otherwise stated.\n\t\n\t\n\t\t\\subsection{Potential Applications}\n\nGiven domains that are \"nearly rectangular\" in the appropriate sense, we have shown here that spectral cuts determined by the $1$-Laplacian in geometric domains consist of area minimizing curves that reduce the distance to being rectangular through a series of qualitative geometric results and a careful asymptotic expansion in a parametrization of the cut near a point of (near) symmetry in the domain.  Given the isoperimetric nature of the cuts, we propose that $1$-Laplacian partitioning could be useful in data analysis, however we acknowledge that the numerical tools for computing the Ratio Cut would likely need to be improved computationally to make large data sets tractable.  Indeed, the isoperimetric nature of the Ratio Cut makes it clear to imagine how partitioning could occur, but also shows that approximating a continuum domain by a graph accurately requires great care due to the subtlety with which graph distances can fluctuate in point clouds.\n\nConsistency of graph $1$-Laplacian spectral cuts was studied in \\cite{trillos2016consistency}.  See \\cite{Buhler_Hein:2009,Hein_Buhler:2010,BHcode} for current computational methods for computing $1$-Laplacian states using proximal-dual type numerical schemes.   Similar ideas have been proposed in \\cite{bresson2013adaptive}.  See also \\cite{nossek2018flows} for similar schemes originating from a diffusion-type equation, with the intent of forcing the equation to converge to a nonlinear eigenfunction.  \n\t\t\n\t\n\t\n\n\t\\begin{figure}\n\t\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_positive_plot-eps-converted-to.pdf}\n\t\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_positive_errors-eps-converted-to.pdf}  \\\\\n\t\t\t\\includegraphics[width=0.2\\textwidth]{final_pictures\/a1_positive_figure-eps-converted-to.pdf}\n\t\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = 0.1$.}\n\t\t\t\t\\label{compplot1}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_negative_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_negative_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_negative_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1$ values between $-0.1$ and $0$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -0.1$.}\t\t\\label{compplot2}\n\t\t\\end{figure}\n\t\t\t\n\t\\begin{figure}\t\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/AWL_positive_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/AWL_positive_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/AWL_positive_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $A_{WL}$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $A_{WL} = 0.1$.}\n\t\t\t\t\\label{compplot3}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/epsT_negative_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/epsT_negative_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/epsT_negative_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $\\epsilon_t$ values between $-0.5$ and $0$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $\\epsilon_t = -0.5$.}\n\t\t\t\t\\label{compplot4}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_together_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_together_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_a3_together_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=a_3$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = a_3 = 0.1$.}\n\t\t\t\t\\label{compplot5}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_opposite_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_opposite_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_a3_opposite_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=-a_3$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -a_3 = 0.1$.}\n\t\t\t\t\\label{compplot6}\n\t\\end{figure}\n\t\n\t\\begin{figure}\t\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsT_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsT_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_epsT_figure-eps-converted-to.pdf}\n\t\t\\caption{ (Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=-\\epsilon_t\/5$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -\\epsilon_t\/5 = 0.1$.}\n\t\t\t\t\\label{compplot7}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsB_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsB_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_epsB_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=\\epsilon_b\/5$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = \\epsilon_b\/5 = 0.1$.}\n\t\t\t\t\\label{compplot8}\n\t\\end{figure}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nGiven a sequence $w=w_1w_2\\dots w_n$ of members of a totally ordered\nset, we say $j$ is a \\emph{strongly outstanding element} of $w$ if\nwhenever $i < j$ we have $w_i < w_j$. In this case we call $w_j$ a\n\\emph{strongly outstanding value}.  We say $j$ is a \\emph{weakly\noutstanding element} if $w_i \\leq w_j$ whenever $i < j$, and call\n$w_j$ a \\emph{weakly outstanding value}. In this paper we will explore\nthe contexts in which $w$ is a permutation, a multiset permutation, or a\nword over some finite alphabet. A famous theorem of R\\'{e}nyi\n\\cite{ren} (see also \\cite{lc}) states that the number of\npermutations of $[k]$ with $r$ strongly outstanding elements is\nequal to the number of such permutations with $r$ cycles, the latter\nbeing given by $\\left[\\begin{array}{ll}k\\\\r\\end{array}\\right]$, the\nunsigned Stirling number of the first kind.\n\nIn this paper we investigate additional properties of the\noutstanding elements and values of permutations, and extend them to\nmultiset permutations and words on $[k]=\\{1,2,\\dots,k\\}$. An\ninteresting sidelight to our results is that we obtain a proof of\nGauss's celebrated $_2F_1$ evaluation by comparing two forms of one\nof our generating functions, in section \\ref{sec:str} below.\n\n\\section{Summary of results}\n\\subsection{Multiset permutations}\nFor permutations\\footnote{Note that for permutations, all outstanding elements and values are\nstrongly outstanding.} we have the following results.\n\\begin{theorem}\n\\label{th:sms} Let $M =\\{1^{a_1}, 2^{a_2}, \\dots, k^{a_k}\\}$ be a\nmultiset with $N=a_1+a_2+\\dots+a_k$, and let $f(M,r)$ denote the\nnumber of permutations of $M$ that contain exactly $r$ strongly\noutstanding elements. Then\n\\begin{equation}\n\\label{gfso}\nF_M(x)\\, =\\kern -12.7pt\\raise 6pt\\hbox{{\\tiny\\textrm{def}}}\\,\\,\\sum_r f(M,r)x^r=\\frac{(N-1)!a_k x}{a_1!a_2!\\dots\na_k!}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right).\n\\end{equation}\n\\end{theorem}\n\\begin{cor}\nThe generating function for the probability that\na randomly selected permutation of $M$ has exactly $r$ strongly\noutstanding elements is\n\\begin{equation}\n\\label{pgfso}\nP_M(x)=\\frac{a_kx}{N}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right),\n\\end{equation}\nand the average number of strongly outstanding elements among\npermutations of $M$ is\n\\begin{equation}\n\\label{eq:pmpe}\nP_M'(1)=\\sum_{i=1}^{k}\\frac{a_i}{a_i+a_{i+1}+\\dots+a_k}.\n\\end{equation}\n\\end{cor}\n\n\\begin{theorem}\n\\label{th:woep}\n  The generating function for the number of permutations of $M$\nthat contain exactly $t$ weakly outstanding elements is given by\n\\begin{equation}\n\\label{gfwo}\nG_M(x)=\\phi_{N,a_1}(x)\\phi_{N-a_1,a_2}(x)\\dots\n\\phi_{N-a_1-\\dots-a_{k-2},a_{k-1}}(x)x^{a_k}\n\\end{equation}\nwhere\n\\[\\phi_{N,a}(x)=\\sum_{m=0}^a{N-m-1\\choose a-m}x^m.\\]\n\\end{theorem}\n\\begin{cor}\nThe average number of weakly outstanding elements among permutations\nof $M$ is\n\\begin{equation}\n\\label{eq:avgweak}\n\\sum_{i=1}^k\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}.\n\\end{equation}\n\\end{cor}\n\\begin{cor}\nLet $A=\\max_i\\{a_i\\}$.  The amount by which the average number of\nweakly outstanding elements exceeds the average number of strongly\noutstanding elements is $\\le\\frac{\\pi^2}{6}A(A-1)$.\n\\end{cor}\n\\subsection{Words}\nThe next theorem involves Stirling numbers of the second kind,\ndenoted by ${n \\brace m}$. This is defined as the number of ways to\npartition a set of $n$ elements into $m$ nonempty subsets.\n\\begin{theorem}\n\\label{th:wordsstrong}\n  The number of $n$-letter words over an alphabet of $k$\nletters which have exactly $r$ strongly outstanding elements is\ngiven by\n\\begin{equation}\n\\label{wordso}\nf(n,k,r) = \\sum_m{k\\choose m}{m\\brack r}{n\\brace m},\n\\end{equation}\nand the average number of strongly outstanding elements among such\nwords is\n\\[H_k=1+\\frac12+\\frac13+\\dots+\\frac1k+o(1)\\quad(n\\to\\infty).\\]\n\\end{theorem}\n\\begin{theorem}\n\\label{th:wordsweak}\nThe generating function for the number $g(n,k,t)$ of $n$-letter words\nover an alphabet of $k$ letters which have exactly $t$ weakly\noutstanding elements is given by\n\\begin{equation}\n\\label{wordwo} G_k(n,x)\\, =\\kern -12.7pt\\raise 6pt\\hbox{{\\tiny\\textrm{def}}}\\,\\, \\sum_tg(n,k,t)x^t=\n\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose t}{x+t-1\\choose k-1},\n\\end{equation}\nand the average number of weakly\noutstanding elements among these words is\n\\[\\frac{n}{k}+H_{k-1}+O\\left(\\left(\\frac{k-1}{k}\\right)^n\\right).\\]\n\\end{theorem}\n\n\nNext we introduce the notion of a \\emph{template} for words on\n$[k]$. A permutation of 5 or more letters \\emph{matches the\ntemplate} `$YN*YY$', for example, if 1, 4, and 5 are outstanding\nelements, 2 is not an outstanding element, and 3 is unconstrained.\nFor example the permutation 2145763 matches this template. In this\ncase we think of the letters of the template $Y$, $N$,\n* as representing yes, no, and unconstrained, respectively.  We\ngeneralize the $Y$ and $N$ constraints to $S$, $W$, $\\overline{S}$,\nand $\\overline{O}$; where $S$ indicates a strongly outstanding\nelement, $W$ a weakly outstanding element, $\\overline{S}$ indicates\nthe absence of a strongly outstanding, and $\\overline{O}$ the\nabsence of an outstanding element.  We provide an algorithm for\nproducing the generating function for the number of words that match\na given template.  When the word is a permutation, we have:\n\\begin{theorem}\n\\label{th:tmp1}\nLet $\\tau$ be a given template of length at most $n$, and let\n$\\tau_j$ denote the letter that appears in position $j$, counting\nfrom the left, of the template $\\tau$.  Since every element of a\npermutation is either strongly outstanding or not outstanding, the\nletters of $\\tau$ are chosen from $\\{Y, N, *\\}$. The probability\nthat a permutation of at least $n$ letters matches the template\n$\\tau$ is\n\\begin{equation}\n\\label{notm} \\prod_{j:\\tau_j={\\mathrm{ `N\\kern-1pt\\raisebox{-.4pt}{\\scalebox{.7}{\\mbox{'}}}}}}\\left(1-{1\\over\n{j}}\\right)\\prod_{j:\\tau_j={\\mathrm{ `Y\\kern-1pt\\raisebox{-.4pt}{\\scalebox{.7}{\\mbox{'}}}}}}{1\\over {j}}\n.\\end{equation}\n\\end{theorem}\nThe corresponding result for words is:\\\\\n\\begin{theorem}\n\\label{th:tmp2}\n Let $\\tau$ be a word on $\\{S, W, *, \\overline{S},\n\\overline{O}\\}$.  Suppose $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$ is\nthe ordinary generating function for $f(k,\\tau)$, the number of\nwords over the alphabet $[k]$ that match the template $\\tau$.\nConsider the adjunction of one new symbol, $A \\in \\{S, W, *,\n\\overline{S}, \\overline{O}\\}$, at the right end of $\\tau$.  The\ngenerating function, $F(k,\\tau A,x)$ can be obtained from\n$F(k,\\tau,x)$ by applying an operator $\\Omega_A$, i.e.,\n\\[F(k,\\tau A,x)=\\Omega_AF(k,\\tau,x),\\]\nwhere\n\\begin{eqnarray}\n\\Omega_SF(k,\\tau, x) & = & xF(k,\\tau, x)\/(1-x),\\label{omegas}\\\\\n\\Omega_WF(k,\\tau, x) & = & F(k,\\tau, x)\/(1-x),\\label{omegaw}\\\\\n\\Omega_*F(k,\\tau, x) & = & x\\frac{d}{dx}F(k,\\tau, x),\\label{omegastar}\\\\\n\\Omega_{\\overline{S}}F(k,\\tau, x) & = &\n\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)F(k,\\tau, x),\\ \\text{and}\\label{omegaovers}\\\\\n\\Omega_{\\overline{O}}F(k,\\tau, x) & = &\n\\left(x\\frac{d}{dx}-\\frac{1}{1-x}\\right)F(k,\\tau,\nx)\\label{omegaovero}.\n\\end{eqnarray}\n\\end{theorem}\n\\section{Notation}\nIn the following sections we will count permutations, multiset\npermutations, and words that contain a given number of strongly or\nweakly outstanding elements.  Let $f(k,r)$ denote the number of\npermutations of $[k]=\\{1,2,\\dots,k\\}$ that contain exactly $r$\nstrongly outstanding elements.  For a multiset\n$M=\\{1^{a_1},2^{a_2},\\dots,k^{a_k}\\}$, let $f(M,r)$ denote the\nnumber of permutations of $M$ that contain exactly $r$ strongly\noutstanding elements. Finally let $f(n,k,r)$ denote the number of\n$n$ letter words on the alphabet $[k]$ that contain exactly $r$\nstrongly outstanding elements.  When counting weakly outstanding\nelements, we use $g$ in place of $f$ and $t$ in place of $r$.\n\nUsing the above notation, we define the following generating\nfunctions for strongly outstanding elements: $F_k(x) = \\sum_r\nf(k,r)x^r$, $F_M(x) = \\sum_r f(M,r)x^r$, and $F_k(n,x) = \\sum_r\nf(n,k,r)x^r$. We define analogous generating functions for weakly\noutstanding elements and use $G$ in place of $F$.\n\n\\section{Strongly outstanding elements of multiset permutations}\nIn this section we establish Theorem \\ref{th:sms} and its corollaries.\n\nGiven a\nmultiset $M=\\{1^{a_1},2^{a_2},\\dots ,k^{a_k}\\}$, let $N=\\sum_ja_j$.\nWe construct the permutations of $M$ that have exactly $r$ strongly\noutstanding elements as follows. We have $N$ slots into which we\nwill put the $N$ elements of $M$ to make these permutations.\n\nTake the $a_1$ 1's that are available and place them in some\n$a_1$-subset of the $N$ slots that are available. There are two\ncases now. If the set of slots that we chose for the 1's did not\ninclude the first (leftmost) slot, then we can fill in the remaining\nslots with any permutation of the multiset $M\/1^{a_1}$ that has\nexactly $r$ strongly outstanding elements. On the other hand, if we\ndid place a 1 into the first slot, then after placing all of the\n1's, the remaining slots can be filled in with any permutation of\nthe multiset $M\/1^{a_1}$ that has exactly $r-1$ strongly outstanding\nelements.\n\nRecall that $f(M,r)$ denotes the number of permutations of the\nmultiset $M$ that have exactly $r$ strongly outstanding elements.\nThe argument in the preceding paragraph shows that\n\\[f(M,r)=\\left({N\\choose a_1}-{N-1\\choose a_1-1}\\right)f(M\/1^{a_1},r)+{N-1\\choose a_1-1}f(M\/1^{a_1},r-1).\\]\nWhen we define\n\\[F_M(x)=\\sum_rf(M,r)x^r,\\]\nwe have the recurrence\n\\begin{eqnarray*}\nF_M(x)&=&\\left(\\left({N\\choose a_1}-{N-1\\choose\na_1-1}\\right)+{N-1\\choose a_1-1}x\\right)F_{M\/1^{a_1}}(x)\\\\\n&=&\\left({N-1\\choose a_1}+{N-1\\choose a_1-1}x\\right)F_{M\/1^{a_1}}(x)\n\\end{eqnarray*}\nThis shows that the generating polynomial resolves into linear\nfactors over the integers. Indeed we get the explicit form\n\\begin{eqnarray*}\nF_M(x)&=& \\left({N-1\\choose a_1}+{N-1\\choose\na_1-1}x\\right)\\left({N-a_1-1\\choose a_2}+{N-a_1-1\\choose\na_2-1}x\\right)\\dots\\\\\n&=&\\prod_{i=1}^k\\left({N-a_1-a_2-\\dots -a_{i-1}-1\\choose\na_i}+{N-a_1-a_2-\\dots -a_{i-1}-1\\choose a_i-1}x\\right)\\\\\n&=&\\frac{(N-1)!a_k x}{a_1!a_2!\\dots\na_k!}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right)\n\\end{eqnarray*}\nThis gives us Theorem \\ref{th:sms}. Its corollaries follow by obvious calculations.\n\nSince the generating polynomial has real zeros only, the\nprobabilities $\\{p_r(M)\\}$ are unimodal and log concave. By\nDarroch's Theorem \\cite{dar}, the value of $r$ for which $p_r(M)$ is\nmaximum differs from $P_M'(1)$ of (\\ref{eq:pmpe}) above by at most 1.\n\n\\section{Weakly outstanding elements of multiset permutations}\nNext we prove Theorem \\ref{th:woep}.\n\nConsider $g(M,t)$, the number of permutations of\n$M=\\{1^{a_1},2^{a_2},\\dots, k^{a_k}\\}$ that have exactly $t$ weakly\noutstanding elements, and $G_M(x)=\\sum_tg(M,t)x^t$. To find\n$g(M,t)$, suppose the permutation begins with a block of exactly\n$m\\ge 0$ 1's. Since the value that follows the last 1 in the block\nis not available for a 1, there remain $N-m-1$ slots into which the\nremaining 1's can be put, in ${N-m-1\\choose a_1-m}$ ways. Once all\nof the 1's have been placed, if the remaining permutation of the\nmultiset $M\/1^{a_1}$ has exactly $t-m$ weakly outstanding elements\nthen the whole thing will have $t$ weakly outstanding elements.\nHence we have\n\\[g(M,t)=\\sum_{m\\ge 0}{N-m-1\\choose a_1-m}g(M\/1^{a_1},t-m),\\]\nif $M\/1^{a_1}$ is nonempty, whereas if $M=1^{a_1}$ only, then\n$g(M,t)=\\delta_{t,a_1}$. If we multiply by $x^t$ and sum on $t$ we\nfind that\n\\[G_M(x)=\\left\\{\\sum_{m\\ge 0}{N-m-1\\choose a_1-m}x^m\\right\\}G_{M\/1^{a_1}}(x),\\]\nexcept that if $M=1^{a_1}$ only, then $G_M(x)=x^{a_1}$. So if we\nwrite\n\\[\\phi_{N,a}(x)=\\sum_{m=0}^a{N-m-1\\choose a-m}x^m,\\]\nthen we have\n\\begin{equation}\n\\label{eq:gm}\n G_M(x)=\\phi_{N,a_1}(x)\\phi_{N-a_1,a_2}(x)\\dots\n\\phi_{N-a_1-\\dots-a_{k-2},a_{k-1}}(x)x^{a_k}.\n\\end{equation}\nThis gives (\\ref{gfwo}).\n\nThe two sums $\\phi_{N,a}(1)={n\\choose a}$ and\n$\\phi_{N,a}'(1)={n\\choose a}\/(n-a+1)$ are elementary, and they imply\nthat $\\phi_{N,a}'(1)\/\\phi_{N,a}(1)=1\/(n-a+1)$. Then logarithmic\ndifferentiation of (\\ref{eq:gm}) and evaluation at $x=1$ shows that\n the average number of weakly outstanding elements in permutations of\n$M$ is\n\\begin{equation}\n\\label{eq:wb}\n\\sum_{i=1}^k\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}.\n\\end{equation}\nLet $A=\\max_i\\{a_i\\}$. If we compare (\\ref{eq:wb}) and (\\ref{eq:pmpe})\nwe find that the amount by which the average number of weakly\noutstanding elements exceeds the average number of strongly\noutstanding elements is\n\\begin{eqnarray*}\n\\sum_{i=1}^k&&\\kern-25pt\\left(\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}\n-\\frac{a_i}{a_i+a_{i+1}+\\dots+a_k}\\right)\\\\\n\\qquad&=&\\sum_{i=1}^k\\frac{a_i(a_i-1)}{(a_{i+1}+a_{i+2}+\\dots+a_k+1)(a_i+a_{i+1}+\\dots+a_k)}\\\\\n\\qquad&\\le& A(A-1)\\sum_{i=1}^k\\frac{1}{(k-i+1)^2}\\le \\frac{\\pi^2}{6}A(A-1).\n\\end{eqnarray*}\nThis estimate is best possible when $A=1$, i.e., when every element\noccurs just once.\n\n\\section{Strongly outstanding elements of words}\n\\label{sec:str}\n Next we investigate $f(n,k,r)$, the number of\n$n$-letter words over an alphabet of $k$ letters that have exactly\n$r$ strongly outstanding elements.  In so doing we will prove\nTheorem \\ref{th:wordsstrong}.\n\nNote first that the number of strongly outstanding elements of such\na word depends only on the permutation of the distinct letters\nappearing in the word that is achieved by the \\textit{first}\nappearances of each of those letters, because a value $j$ can be\nstrongly outstanding in a word $w$ only if it is the first (i.e.,\nleftmost) occurrence of $j$ in $w$.\n\nHence associated with each $n$-letter word $w$ over an alphabet of\n$k$ letters which has exactly $r$ strongly outstanding elements\nthere is a triple $(S,{\\cal P},\\sigma)$ consisting of\n\\begin{enumerate}\n\\item a subset $S\\subseteq [k]$, which is the set of all of the distinct\n letters that actually appear in $w$, and\n\\item a partition ${\\cal P}$ of the set $[n]$ into $m=|S|$ classes,\nnamely the $i$th class of ${\\cal P}$ consists\n of the set of positions in the word $w$ that contain the $i$th letter\n  of $S$, and\n\\item a permutation $\\sigma\\in S_m$, $m=|S|$, which is the sequence of \\textit{first} appearances in $w$ of each of the $k$ letters that occur in $w$. $\\sigma$ will have $r$ strongly outstanding elements.\n\\end{enumerate}\n\nConversely, if we are given such a triple $(S,{\\cal P},\\sigma)$, we\nuniquely construct an $n$-letter word $w$ over $[k]$ with exactly\n$r$ strongly outstanding elements as follows.\n\nFirst arrange the classes of the partition ${\\cal P}$ in ascending\norder of their smallest elements. Then permute the set $S$ according\nto the permutation $\\sigma$, yielding a list $\\tilde{S}$. In all of\nthe positions of $w$ that are described by the first class of the\npartition ${\\cal P}$ (i.e., the class in which the letter `1' lives)\nwe put the first letter of $\\tilde{S}$, etc., to obtain the required\nword $w$.\n\nThus the number of words that we are counting is equal to the number\nof these triples, viz.\n\\begin{equation}\n\\label{eq:fnkr}\n f(n,k,r)=\\sum_m{k\\choose m}{m\\brack r}{n\\brace m}.\n\\end{equation}\nIt is noteworthy that three flavors of ``Pascal-triangle-like''\nnumbers occur in this formula.\n\nLet $\\rho(n,k)$ denote the average number of strongly outstanding\nelements among the $n$-letter words that can be formed from an\nalphabet of $k$ letters. To find $\\rho(n,k)$ for large $n$, we have\nfirst that \\begin{eqnarray*} \\sum_rf(n,k,r)&=&\\sum_m{k\\choose\nm}{n\\brace m}\\sum_r{m\\brack r}= \\sum_m{k\\choose m}{n\\brace\nm}m!\\\\\n&\\sim& \\sum_m{k\\choose m}{n\\brace m}m!\\sim \\sum_m{k\\choose m}n^m\\sim\nn^k,\n\\end{eqnarray*}\nfor large $n$, where we have used the facts that ${n\\brace m}\\sim\nn^m\/m!$ and $\\sum_r{m\\brack r}=m!$. Similarly,\n\\begin{eqnarray*}\\sum_rrf(n,k,r)&=&\\sum_m{k\\choose m}{n\\brace m}\\sum_rr{m\\brack\nr}= \\sum_m{k\\choose m}{n\\brace m}m!H_m\\\\\n&\\sim& \\sum_m{k\\choose m}{n\\brace m}m!H_m\\sim \\sum_m{k\\choose\nm}n^mH_m\\sim n^kH_k,\\end{eqnarray*}\n where we have used the additional fact that\n$\\sum_rr{m\\brack r}=m!H_m$. If we divide these last two equations we\nfind that\n\\[\\lim_{n\\to\\infty}\\rho(n,k)=H_k=1+\\frac12+\\frac13+\\dots+\\frac1k.\\]\nThe proof of Theorem \\ref{th:wordsstrong} is complete. $\\Box$\n\nFrom eq. (\\ref{eq:fnkr}) we can use the standard generating\nfunctions for the two kinds of Stirling numbers to show that\n\\[\\sum_{r,n}f(n,k,r)y^rt^{n-r}=\\prod_{j=1}^k\\left(1+\\frac{y}{1-jt}\\right).\\]\nBut from (\\ref{eq:fnkr}) we also have\n\\begin{eqnarray*}\n\\sum_{r,n}f(n,k,r)y^rt^{n-r}&=&\\sum_{\\ell,r}{k\\choose \\ell}{\\ell\\brack r}\\left(\\frac{y}{t}\\right)^r\\sum_n{n\\brace \\ell}t^n\\\\\n&=&\\sum_{\\ell,r}{k\\choose \\ell}{\\ell\\brack r}\\left(\\frac{y}{t}\\right)^r\\frac{t^{\\ell}}{(1-t)(1-2t)\\dots (1-\\ell t)}\\\\\n&=&\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=0}^{\\ell-1}\\left(\\frac{y}{t}+j\\right)\\frac{t^{\\ell}}{(1-t)(1-2t)\\dots (1-\\ell t)}\\\\\n&=&\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=1}^{\\ell}\\frac{y+(j-1)t}{1-jt}.\n\\end{eqnarray*}\n\nComparison of these two evaluations shows that we have found and\nproved the following identity:\n\\begin{equation}\n\\label{eq:ident}\n\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=1}^{\\ell}\\frac{y+(j-1)t}{1-jt}=\\prod_{j=1}^k\\left(1+\\frac{y}{1-jt}\\right).\n\\end{equation}\nBut Gauss had done it earlier, since it is the evaluation of his\nwell known\n\\[ _2F_1\\left[ \\begin{array}{cc}\n    -k & y\/t \\\\\n    1-1\/t&\n     \\end{array}\\bigg|\\,1 \\right].\\]\n\n\\section{A calculus of templates}\n\\subsection{Templates and permutations}\nIn this section we prove Theorems \\ref{th:tmp1} and \\ref{th:tmp2}.\n  We begin by establishing\nequation (\\ref{notm}), which appeared first in \\cite{wil}.\n\nThis result is a generalization of a theorem of R. V. Kadison\n\\cite{rvk}, who discovered the case where the template is `NN$\\cdots\n$NY', and proved it by the sieve method. To make this paper\nself-contained, we include a proof of (\\ref{notm}).\n\nOur proof is by induction on $n$. Suppose it has been proved that,\nfor all templates $\\tau$ of at most $n-1$ letters, the number of\npermutations of $n-1$ letters that match $\\tau$ is correctly given\nby (\\ref{notm}), and let $\\tau$ be some template of $\\le n$ letters.\nIf in fact the length of $\\tau$ is $<n$ then the formula\n(\\ref{notm}) gives the same result as it did when applied to\n$(n-1)$-permutations, which is the correct probability.\n\nIn the case where the length of $\\tau$ is $n$ and the rightmost\nletter of $\\tau$ is `Y', every matching permutation $\\sigma$ must\nhave $\\sigma (n)=n$. Hence the number of matching permutations is\n$(n-1)!p_{n-1}(\\tau')$, where $\\tau'$ consists of the first $n-1$\nletters of $\\tau$, which is equal to $n!p_n(\\tau)$, proving the\nresult in this case.\n\nIn the last case, where the length of $\\tau$ is $n$ and the\nrightmost letter of $\\tau$ is `N', the probability of a permutation\nmatch must be $p_{n-1}(\\tau')(1-1\/n)$, since this case and the\npreceding one are exhaustive of the possibilities and the preceding\none had a probability of $p_{n-1}(\\tau')\/n$. But this agrees with\nthe formula (\\ref{notm}) for this case, completing the proof of the\ntheorem.\n\n\\subsection{Templates on words}\nNext we include results for words that are analogous to those we\nfound for permutations.  A preview of the kind of results that we\nwill get is the following. Suppose $F(k,\\tau,x)=\\sum_{k\\geq 1}\nf(k,\\tau)x^k,$ where $f(k,\\tau)$ denotes the number of words over\nthe alphabet $[k]$ that match the template $\\tau$. Then consider the\nadjunction of one new symbol, let's call it $A$, at the right end of\n$\\tau$. Then we will show that the new generating function,\n$F(k,\\tau A,x)$ can be obtained from $F(k,\\tau,x)$ by applying a\ncertain operator $\\Omega_A$. That is,\n\\[F(k,\\tau A,x)=\\Omega_AF(k,\\tau,x).\\]\nThe operator $\\Omega_A$ will depend only on the letter $A$ that is\nbeing adjoined to the template $\\tau$.\n\nHence, to find the generating function for a complete template\n$\\tau$, we begin with $F(k,\\emptyset,x)=1\/(1-x)$, and we read the\ntemplate $\\tau$ from left to right. Corresponding to each letter in\n$\\tau$ we apply the appropriate operator $\\Omega$. When we have\nfinished scanning the entire template the result will be the desired\ngenerating function for $\\tau$.\n\nThe letters that we will allow in a template\n$\\tau=\\tau_1\\tau_2\\dots\\tau_l$ are\n$\\{S,W,O,*,\\overline{S},\\overline{O}\\}$. Their meanings are that if\n$w=w_1\\dots w_l$ is a word of length $l$ over the alphabet $[k]$\nthen for $w$ to match the template it must be that $w_i$ is\n\\begin{enumerate}\n\\item a strongly outstanding value whenever $\\tau_i=S$, or\n\\item a weakly outstanding value whenever $\\tau_i=W$, or\n\\item unrestricted whenever $\\tau_i=*$, or\n\\item not a strongly outstanding value whenever $\\tau_i=\\overline{S}$, or\n\\item neither a weakly nor a strongly outstanding value whenever $\\tau_i=\\overline{O}$.\n\\end{enumerate}\nLet's consider what happens to the count of matching words when we\nadjoin one of these letters to a template whose counting function is\nknown. Let $f(k,\\tau)$ denote the number of words of length $l=$\nlength$(\\tau)$, on the alphabet $[k]$ that match the template\n$\\tau$.\n\\begin{enumerate}\n\\item If $w$ is one of the words counted by $f(k,\\tau S)$, and if we delete its last letter, we obtain one of the words that is counted by $f(i,\\tau)$ for some $1\\le i<k$, and consequently\n\\[f(k,\\tau S)=\\sum_{i=1}^{k-1}f(i,\\tau).\\]\nIf $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$, then we have\n\\[F(k,\\tau S,x)=\\frac{x}{1-x}F(k,\\tau ,x).\\]\nThus we have found the operator $\\Omega_S$, and it is defined by\n$\\Omega_SF(x)=xF(x)\/(1-x)$.  This is equation (\\ref{omegas}).\n\\item Similarly, if $w$ is one of the words counted by $f(k,\\tau W)$, and if we delete its last letter, we obtain one of the words that is counted by $f(i,\\tau)$ for some $1\\le i\\le k$, and consequently\n\\[f(k,\\tau W)=\\sum_{i=1}^{k}f(i,\\tau).\\]\nIf $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$, then we have\n\\[F(k,\\tau W,x)=\\frac{F(k,\\tau ,x)}{1-x}.\\]\nThus we have found the operator $\\Omega_W$, and it is defined by\n$\\Omega_WF(x)=F(x)\/(1-x)$.  This is (\\ref{omegaw}).\n\\end{enumerate}\nSince the argument in each case is easy and similar to the above we\nwill simply list the remaining three operators, equations\n(\\ref{omegastar}), (\\ref{omegaovers}), and (\\ref{omegaovero}), as\nfollows:\n\\begin{eqnarray*}\n\\Omega_*F(x)&=&x\\frac{d}{dx}F(x)\\\\\n\\Omega_{\\overline{S}}F(x)&=&\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)F(x)\\\\\n\\Omega_{\\overline{O}}F(x)&=&\\left(x\\frac{d}{dx}-\\frac{1}{1-x}\\right)F(x)\n\\end{eqnarray*}\n\nThe successive applications of these operators can be started with\nthe generating function for the empty template,\n\\[F(\\emptyset,x)=\\frac{1}{1-x}.\\]\n\nThus to find the generating function for some given template $\\tau$,\nwe begin with the function $1\/(1-x)$, and then we read one letter at\na time from $\\tau$, from left to right, and apply the appropriate\none of the five operators that are defined above.\n\nAs an example, how many words of length 3 over the alphabet $[k]$\nmatch the template $\\tau=S*\\overline{S}$? This is the coefficient of\n$x^k$ in\n\\begin{eqnarray*}\nF(S*\\overline{S},x)&=&\\Omega_{\\overline{S}}\\Omega_*\\Omega_S\\frac{1}{1-x}\\\\\n&=&\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)\\left(x\\frac{d}{dx}\\right)\n\\left(\\frac{x}{1-x}\\right)\\frac{1}{1-x}\\\\\n&=&\\frac{x+3x^2}{(1-x)^4}\n\\end{eqnarray*}\nThe required number of words over a $k$ letter alphabet that match\nthe template $\\tau$ is the coefficient of $x^k$ in the above. Since\nthese generating functions will always be of the form\n$P(t)\/(1-x)^r$, with $P$ a polynomial, we note for ready reference\nthat\n\\[[x^k]\\left\\{\\frac{\\sum_ja_jx^j}{(1-x)^r}\\right\\}=\\sum_ja_j{r+k-j-1\\choose\nr- 1}.\\] In the example above we have $r=4$, $a_1=1$, $a_2=3$, so\n\\[f(k,S*\\overline{S})={k+2\\choose 3}+3{k+1\\choose 3}.\\]\n\n\n\\section{Weakly outstanding elements of words}\nFinally we prove Theorem \\ref{th:wordsweak}.\n\n Recall that $g(n,k,t)$ is the number\nof $n$-letter words over the alphabet $[k]$ which have exactly $t$\nweakly outstanding elements. Consider just those words $w$ that\ncontain exactly $m$ 1's, $m<n$. If $w$ begins with a block of\nexactly $l$ 1's, $0\\le l\\le m$ then by deleting all $m$ of the 1's\nin $w$ we find that the remaining word is one with $n-m$ letters\nover an alphabet of $k-1$ letters and it has exactly $t-l$ weakly\noutstanding elements. Thus we have the recurrence\n\\[\ng(n,k,t)=\\sum_{m=0}^{n-1}{n-l-1\\choose\nm-l}g(n-m,k-1,t-l)+\\delta_{t,n}.\n\\]\nWhen we set $G_k(n,x)=\\sum_tg(n,k,t)x^t$, we find that\n\\begin{equation}\n\\label{eq:f2rec} G_k(n,x)=\\sum_{m=0}^{n-1}\\sum_{l=0}^k{n-l-1\\choose\nm-l}x^lG_{k-1}(n-m,x)+x^n.\\qquad(n,k\\ge 1;\\, G_0(n,x)=0)\n\\end{equation}\nTo discuss this recurrence, let $\\Delta=\\Delta_x$ be the usual\nforward difference operator on $x$, i.e., $\\Delta_x\nf(x)=f(x+1)-f(x)$. Then from the recurrence above we discover that\n\\[G_1(n,x)=x^n;\\ G_2(n,x)= \\Delta_x x^n(x-1);\\ G_3(n,x)=\\Delta_x^2x^n{x-1\\choose 2}.\\]\nThis leads to the conjecture that\n\\begin{eqnarray*}\nG_k(n,x)&=&\\Delta_x^{k-1}\\left\\{x^n{x-1\\choose k-1}\\right\\}\\\\\n&=&\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose t}{x+t-1\\choose\nk-1}.\n\\end{eqnarray*}\nTo prove this it would suffice to show that the function $G_k(n,x)$\nabove satisfies the recurrence (\\ref{eq:f2rec}). If we substitute\nthe conjectured form of $G_k$ into the right side of\n(\\ref{eq:f2rec}) we find that the sums over $l$ and $m$ can easily\nbe done, and the identity to be proved now reads as\n\\begin{eqnarray*}\n&&\\sum_{t=0}^{k-2}(-1)^{k-t}{k-2\\choose t}{x+t-1\\choose k-2}\\frac{(x+t)}{t+1}((x+t+1)^n-x^n)+x^n\\\\\n&&\\qquad=\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose\nt}{x+t-1\\choose k-1}.\n\\end{eqnarray*}\nIf we replace the dummy index of summation $t$ by $t-1$ on the left\nside, the identity to be proved becomes\n\\begin{eqnarray*}\n&&\\sum_{t=1}^{k-1}(-1)^{k-t-1}{k-2\\choose t-1}{x+t-2\\choose k-2}\\frac{(x+t-1)}{t}((x+t)^n-x^n)+x^n\\\\\n&&\\qquad=\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose\nt}{x+t-1\\choose k-1}.\n\\end{eqnarray*}\nIt is now trivial to check that for $1\\le t\\le k-1$, the coefficient\nof $(x+t)^n$ on the left side is equal to that coefficient on the\nright. If we cancel those terms and divide out a factor $x^n$ from\nwhat remains, the identity to be proved becomes\n\\[-\\sum_{t=1}^{k-1}(-1)^{k-t-1}{k-2\\choose t-1}{x+t-2\\choose k-2}\\frac{(x+t-1)}{t}+1=(-1)^{k-1}{x-1\\choose k-1}.\\]\nThe sum that appears above is a special case of Gauss's original\n$_2F_1[a,b;c|1]$ evaluation, and the proof of (\\ref{wordwo}) is\ncomplete.  A straightforward calculation now shows that the average\nnumber of weakly outstanding elements among $n$-letter words over\nthe alphabet $[k]$ is\n\\[=\\frac{n}{k}+H_{k-1}+O\\left(\\left(\\frac{k-1}{k}\\right)^n\\right).\\]\n\n\\section{Related results}\nIn the literature, the outstanding elements and values of sequences\ngo by various names: \\emph{\\'{e}l\\'{e}ments salients} (R\\'{e}nyi),\n\\emph{left-to-right maxima}, \\emph{records}, and others; and appear\nin several different contexts and results, for example:\n\nIt is well known that the probability of obtaining at least $r$\nstrongly outstanding elements in a sequence $X_1,X_2,\\dots ,X_n$ of\n$n$ independent, identically distributed, continuous random\nvariables approaches 1 as $n\\rightarrow\\infty$.  (Glick's survey\n\\cite{gl}, for example, contains this result and those that follow\nin this paragraph and the next.) Let $Y_i = 1$ if $i$ is a strongly\noutstanding element of such a sequence, and 0 otherwise. The\nexpected value is $E[Y_i]=1\/i$ and the variance is $V[Y_i]=1\/i -\n1\/i^2$. The number of strongly outstanding elements in a sequence of\ncontinuous iid random variables is therefore $R_i = \\sum_i Y_i$ with\nexpectation $E[R_i]=\\sum_{i=1}^n 1\/i$ and variance\n$V[R_i]=\\sum_{i=1}^n 1\/i - \\sum_{i=1}^n 1\/i^2$. Note $\\sum_{i=1}^n\n1\/i - \\ln(n) \\rightarrow$ Euler's constant $=.5772\\dots$, and\n$\\sum_{i=1}^n 1\/i^2 \\rightarrow \\pi^2\/6 = 1.6449\\dots$ for\n$n\\rightarrow \\infty.$\n\nLet $N_r$ denote the $r^{\\text{th}}$ outstanding element in a\nsequence of $n$ iid continuous random variables.  Then $N_1 = 1$,\n$E[N_r] = \\infty$ for $r\\geq 2$, and $E[N_{r+1} - N_r] = \\infty$ as\nwell.  The probability $P[N_2 = i_2, N_3 = i_3, \\dots , N_r = i_r] =\n1\/(i_2 -1)(i_3-1)\\dots (i_r-1)i_r$ for $1 < i_2 < i_3 < \\dots <i_r$.\nThe probability that an iid sequence of $n$ continuous random\nvariables has exactly $r$ strongly outstanding elements is\n$\\left[\\begin{array}{ll}k\\\\r\\end{array}\\right]\/(r-1)!$ for large\n$n$.\n\nChern and Hwang \\cite{ch} consider the number $f_{n,k}$ of $k$\nconsecutive records (strongly outstanding elements) in a sequence of\n$n$ iid continuous random variables.  They improve upon known\nresults for the limiting distribution of $f_{n,2}$.  In particular\nthey show $f_{n,k}$ is asymptotically Poisson for $k=1,2$, and this\nis not the case for $k\\geq 3$.  They give the probability generating\nfunction for $f_{n,2}$, and observe that the distribution of\n$f_{n,2}$ is identical to that for the number of fixed points $j$ in\na random permutation of $[n]$ for $1 \\leq j < n$.   They give a\nrecurrence for the probability generating function for $f_{n,k}$,\nand compute the mean and variance for this number.\n\nAre there similar results for discrete distributions?  Prodinger\n\\cite{prod} considers left-to-right maxima in both the strict\n(strongly outstanding) and loose (weakly outstanding) senses for\ngeometric random variables. He finds the generating function for the\nprobability that a sequence of $n$ independent geometric random\nvariables (each with probability $pq^{v-1}$ of taking the positive\ninteger value $v$, where $q = p-1$) has $k$ strict left-to-right\nmaxima. This probability is the coefficient of $z^ny^k$ in\n$$\nF(z,y) = \\prod_{k \\geq 1}\n\\left(1+\\frac{yzpq^{k-1}}{1-z(1-q^k)}\\right).\n$$\nFor loose left-to-right maxima the analogous generating function is\n$$\n\\prod_{k\\geq 0}\\frac{1-z(1-q^k)}{1-z+zq^k(1-py)}.\n$$\nIn this paper, Prodinger also finds the asymptotic expansions for\nboth the expected numbers of strict and loose left-to-right maxima,\nand the variances for these numbers.  He does all of the above for\nuniform random variables as well.\n\nKnopfmacher and Prodinger \\cite{kp} consider the \\emph{value} and\n\\emph{position} of the $r^{\\text{th}}$ left-to-right maximum for $n$\ngeometric random variables. The \\emph{position} is the\n$r^{\\text{th}}$ strongly outstanding element, and the \\emph{value}\nis that taken by the random variable in that position (again the\nvalue $v$ is taken with probability $pq^{v-1}$, where $q=1-p$). For\nthe $r^{\\text{th}}$ strong left-to-right maximum, the asymptotic\nformulas for value and position are $\\frac{r}{p}$ and\n$\\frac{1}{(r-1)!}\\left(\\frac{p}{q}\\log_{\\frac{1}{q}}n\\right)^{r-1}$,\nrespectively. For the \\emph{r}th weak left-to-right maximum, the\nvalue and position are asymptotically $\\frac{rq}{p}$ and\n$\\frac{1}{(r-1)!}\\left(p\\log_{\\frac{1}{q}}n\\right)^{r-1}$.  These\nresults are obtained by first computing the relevant generating\nfunctions.\n\nA number of additional properties of outstanding elements of permutations\n are in Wilf \\cite{wil}.\n\nKey \\cite{key} describes the asymptotic behavior of the number of\nrecords (strongly outstanding elements) and weak records (weakly\noutstanding elements) that occur in an iid sequence of integer\nvalued random variables.\n\nWhen the sequence $w_1, w_2, \\dots, w_n$ under consideration is a\nrandom permutation on $n$ letters, then the expected number of\nstrongly outstanding elments is the \\emph{n}th harmonic number $H_n\n= 1 + \\frac{1}{2} + \\frac{1}{2} + \\dots + \\frac{1}{n}$. The variance\nis $H_n = H_n^{(2)}$, where\n$H_n^{(2)}=1+\\frac{1}{4}+\\dots+\\frac{1}{n^2}$ denotes the \\emph{n}th\nharmonic number of the second order.\n\nBanderier, Mehlhorn, and Brier \\cite{bmb} show that the average\nnumber of left-to-right maxima (strongly outstanding elements) in a\n\\emph{partial permutation} is $\\log(pn) + \\gamma +\n2\\frac{1-p}{p}+\\left(\\frac{1}{2}+\\frac{2(1-p)}{p^2}\\right)\\frac{1}{n}+O\\left(\\frac{1}{n^2}\\right)$\nwhere $\\gamma = .5772\\dots$ is Euler's constant.  To obtain a\n\\emph{partial permutation}, we begin with the sequence $1,2,\\dots\n,k$ and select each element with probability $p$; then take one of\nthe $(pn)!$ permutations of $[pn]$ uniformily at random and let it\nact on the selected elements, while the nonselected elements stay in\nplace.\n\nFoata and Han \\cite{fh} investigate the (right-to-left) \\emph{lower\nrecords} of \\emph{signed permutations}.  A \\emph{signed permutation}\nis a word $w=w_1w_2\\dots w_n$ for which the letters $w_i$ are\npositive or negative integers, and $|w_1||w_2|\\dots |w_n|$ is a\npermutation of $\\{1,2,\\dots,n\\}$.  A \\emph{lower record} of such a\nword is a letter $w_i$ such that $w_i < w_j$ for all $j$ with $i+1\n\\leq j \\leq n$. The authors consider the \\emph{signed subword}\nobtained by reading the lower records of $w$ from left to right; and\nderive generating functions for signed permutations in terms of\nsigned subwords, numbers of positive and negative letters in signed\nsubwords, and other statistics.\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Results}\n   \n   \n   \n   \n\n\\section{Introduction}\nThe objective of the study is to propose a system-level framework with quantitative measures to assess the resilience of road networks to inform transportation planning and project development processes in practice. Road networks are the backbone of society that enables the movement of people and goods safely and efficiently. A reliable road network does not only allow improved access by the workforce to business, but also result in long-term cost saving and increase of safety and viability of residential neighborhoods \\citep{weilant_incorporating_2019}. Disruptions of road networks would reduce economic productivity, harm local commercial activities and community well-being, and restrict people's mobility and access to critical facilities \\citep{weilant_incorporating_2019,frizzelle_importance_2009,jenelius_importance_2006}. The US National Response Plan (NRP) urges each state to have a plan for responding to \"\u2026 reduce the vulnerability to all natural and manmade hazards; and minimize the damage and assist in the recovery from any type of incident that occurs\"  \\citep{us_department_of_homeland_security_national_2004}. The need for incorporating resilience assessments in transportation planning and project development processes is increasingly recognized by transportation organizations in the United States and worldwide. For example, the US Federal Highway Administration (FHWA) has established mandates and guidelines for transportation agencies regarding the incorporation of resilience into transportation systems through their asset management planning efforts \\citep{flannery_resilience_2018}.\n\nWhile the need for system-level resilience assessment has been recognized, the measures for quantifying system-level resilience to inform transportation planning processes are rather limited in practice \\citep{esmalian_operationalizing_2022}. Existing literature has shown that the evaluation of road network resilience is multi-dimensional, including vulnerability assessment, criticality assessment, and interdependency and cascading failure assessment. \\citet{sharma_resilience_2018} suggested measuring resilience using vulnerability, severity of consequences, and time to recover from the disruptions. Considering the post-disaster network access to critical facilities, studies \\citep{dong_robust_2019, dong_integrated_2020} proposed the concept of robust components to measure communities' risk disparity in terms of losing access to hospitals in both earthquake and flooding scenarios using a network topology analysis approach. \\citet{gao_universal_2016} used a single metric to measure the resilience of multi-dimensional complex networks. The author found that a resilient system is (1) densely connected in the system, (2) highly heterogeneous in the connections, and (3) relatively symmetrical. Using a percolation approach, \\citet{dong_measuring_2020} measured the resilience of different cities' and states' road networks in the face of random failure using the giant component metrics. This topological metric enables the ranking of road networks based on the derived resilience score. \\citet{weilant_incorporating_2019} summarized the resilience measurement process into an AREA approach, which represents Absorptive capacity, Restorative capacity, Equitable access, and Adaptive capacity. \\citet{bruneau_framework_2003} proposed that a resilience framework should include measurement of robustness, rapidity, resourcefulness, and redundancy, integrated with technical, organizational, social, and economic dimensions \\citep{davis_establishing_2018, flannery_resilience_2018, norris_community_2008, reggiani_network_2013}. \\citet{da_silva_city_2014} suggested a resilience system should be equipped with qualities such as robustness (can anticipate potential failures and make provisions), redundancy (has spare capacity to accommodate changes), and integratedness (promotes consistency by pursuing integration and alignment between systems). This study explored various quantitative metrics and identified those in this multi-dimensional and system-level assessment framework based on connectivity of road segments in road networks; vulnerability of road segments to extreme events, such as flooding and hurricanes; disrupted access to critical facilities; and cascading impact of critical facility networks. Quantitative metrics that can consider all of these dimensions in resilience assessment are essential for a system-level understanding of the resilience of road networks and also inform transportation planning and project development.\n\nWhile the scholarly literature is rich in terms of methods and measures for assessment of vulnerability and resilience in transportation infrastructure, little of the existing work has been adopted in practice. The literature has put efforts into developing assessment measures to understand the resilience and vulnerability of road networks \\citep{hsieh_highway_2020,singh_vulnerability_2018,dave_extreme_2021,martin_assessing_2021}. Several conceptual frameworks have been proposed in the literature to promote consideration of resilience into infrastructure assessment, including road networks \\citep{jenelius_road_2015,morelli_measuring_2021,papilloud_vulnerability_2021}. However, in the United States, only about half of the state departments of transportation (DOTs) and metropolitan planning organizations (MPOs) started incorporating resilience considerations in their plans, and only some of them explicitly listed resilience as a goal \\citep{dix_integrating_2018}. This limitation is mainly due to two reasons: (1) transportation planners and decision-makers do not know what method and measures to use among several proposed in the literature; (2) the methods and measures for vulnerability and resilience assessment should be obtainable using the existing data available to most transportation agencies. \n\nTo address this gap, this study identified and implemented a framework incorporating quantitative metrics to classify the criticality of road segments for understanding the system-level vulnerability and criticality of the road network of the state of Texas to inform the planning and project development processes. As shown in Figure \\ref{fig:fig1}, the proposed framework includes four quantitative metrics that quantify vulnerability and criticality for each segment of the state road network. The first metric for connectivity of road segments adopts percolation analysis to quantify the criticality of road segments under the impact of external perturbations on road networks. The second metric for vulnerability to extreme events examines the proximity of road segments to the disaster impact areas. In this study, we identified areas exposed for flood hazards using 100- and 500-year floodplains to quantify the probability of flooding on road segments. The third and fourth metrics are based on interdependency analysis and proximity to critical facilities and infrastructure networks to quantify the criticality of road segments for accessing critical facilities. Finally, we combined the four metrics into integrated metrics to obtain the overall criticality of the road segments in road networks. In addition, this study presents a case study by applying the proposed framework in Texas road networks to demonstrate the effectiveness of implementing this framework in a practical scenario.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.6\\linewidth]{fig\/Fig_1.png}\n    \\caption{Four metrics for understanding the resilience of road networks. The quantitative and systematic evaluation metrics are informed by the existing literature.}\n\t\\label{fig:fig1}\n\\end{figure}\n\nThe remainder of the paper is structured as follows: Section \\ref{sec:Research Approach} outlines the research approach of the four quantitative metrics used for the resilience assessment of road networks. Sections \\ref{sec:Connectivity of Road Segments} to Section \\ref{sec:Cascading Impact of Critical Infrastructure Networks} provide detailed information and discussion of each quantitative metric. Section \\ref{sec:Integrated Criticality Metric} demonstrates details of integrated metrics that consider all individual metrics in different aspects. Section \\ref{sec:Case Study for Texas Road Network} presents a case study in Texas road networks by implementing the proposed framework. Finally, Section \\ref{sec:Concluding Remarks} presents conclusions gleaned from the study.\n\n\\section{Research Approach}\n\\label{sec:Research Approach}\nThis study implemented quantitative and computational research methods for the resilience assessment of road networks. As shown in Figure \\ref{fig:fig2}, the research approach first identified quantitative metrics for the resilience assessment of road networks. With those metrics, we then incorporated additional information, such as evacuation routes, to understand the criticality of road networks. Lastly, all of the results and information were integrated into a geographic information system-based map for practical usage. In this section, we briefly outline how the quantitative metrics are calculated for road networks; further specific discussions can be found in the sections devoted to these metrics.\n\nFirst, we evaluate the criticality of individual road segments for the connectivity of the road network, a characteristic that is particularly important in disasters. For example, residents rely on connected roads to evacuate, and first responders need to access impacted areas to save lives and protect properties. During disasters, fragmented networks may isolate areas, block transportation resources, and create traffic congestion. In addition, disruptions or floodwaters on road networks may spread over time. Measures considering static road network failures only may fail to capture the interconnections of road segments and consequent impacts of flood propagation. Hence, in this task, we adopted percolation analysis to simulate how road failures spread from one road segment to another and the extent to which failures of road networks influence their connectivity. In the case study, we collected road network data from the Texas Department of Transportation (TxDOT) roadway inventory On-System dataset updated on August 8, 2021, to perform these analyses and compute this criticality metric for road segments. This dataset includes details about the geometries of the road segments, road types, speed limits, and names of the roads. Also, the data includes both highways and streets in urban and rural areas in Texas.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_2.png}\n    \\caption{Research approach for conducting the vulnerability and resilience assessment of state road infrastructure networks.}\n\t\\label{fig:fig2}\n\\end{figure}\n\nThe second metric is the criticality of road segments to disaster impacts, which, in this study, are caused by flooding. The area of the case study, the state of Texas, especially South Texas, is prone to extreme events, such as flooding and hurricanes. Determining the criticality of road segments during flooding events is important for planning and prioritization in response to disruptive impacts of flooding. Hence, we collected 100- and 500-year floodplain data, which is the most recent nationwide extract of the National Flood Hazard Layer (NFHL) for Texas. The NFHL dataset was obtained by the Federal Emergency Management Agency (FEMA) for examining road exposure to flooding. The data includes flood map panel boundaries, flood hazard zone boundaries, and other information related to flood control zones and areas. Using this data, we identified the areas of flood zones and calculated the proximity of road networks to the nearest flood zones. Based on the proximity, we classified the distance into five categories representing the level of criticality of the road segments to flood.\n\nThird, we analyzed the criticality of road segments based on impacts of road disruption on access to (1) critical facilities such as hospitals and fire stations and (2) critical infrastructure networks such as gas lines and oil pipelines. One essential role of the road networks is to transport resources and provide public services. Accessing critical facilities is thus an important component of road network resilience assessment. We implemented two metrics to evaluate the criticality of road segments based on the interdependencies, measured based on distance buffers, between road segments and facilities. In particular, one metric is measured using the proximity of road segments to criticality facilities, while another metric is measured based on the closeness to critical infrastructure networks. We collected facility location data from Homeland Infrastructure Foundation Level Data (HIFLD) in the ArcGIS data repository, as well as energy sector data and evacuation route data from TxDOT, which include information of facility types and corresponding locations. We then extracted the location information for each of the critical facilities, such as electrical substations; power plants; hospitals; national shelter facilities; seaports; petroleum, oil and lubricant (POL) terminals; fire stations; and police stations, as well as for each of the critical infrastructure networks, such as railroad network, transmission lines, gas lines, hydrocarbon gas lines, and oil pipelines, which are essential for resiliency. Disruption in access to these facilities and infrastructure networks can exacerbate the impacts faced by communities during and in the aftermath of disasters. \n\nFinally, it is important to evaluate the overall criticality by considering all dimensions of the resilience of road networks. To this end, we devised two integration approaches to show the overall criticality of the road segments. The first method is to rank overall criticality based on the evenly weighted sum of all individual criticality metrics of each road segment. The second approach is based on the root squared sum ranking, by which the overall criticality score is calculated by taking the square root of the sum of squares of individual metrics. In the case study, we computed the integrated criticality metrics for all road segments in Texas. The results for all metrics \u2014 including four individual and two integrated metrics \u2014 can be mapped onto GIS-based maps for visualization and to provide opportunities for further incorporation with the other resilience dimensions or critical considerations. Detailed information for each metric, including individual and integrated metrics, is discussed in the following sections.\n\n\\section{Connectivity of Road Segments}\n\\label{sec:Connectivity of Road Segments}\n\\subsection{Metric Definition and Calculation}\nThe connectivity of a road segment is determined by the percolation analysis performed on the road network. The percolation analysis is an analytical technique that determines to what extent a road segment contributes to the network's overall connectivity. The procedure for percolation analysis is to remove nodes and links based on specific criteria and to quantify changes in network structure after a disturbance. When more critical road segments are perturbed, they are more likely to cause significant disconnection in the network.\n\nWhile natural hazards, such as flooding, perturb road networks, some components within the networks may fail and other parts of the networks could be disconnected and isolated. The percolation analysis mimics this process by sequentially removing nodes in topological networks. Standard criteria of node attributes rank the criticality of components within a network: degree, weighted degree, link betweenness, and size of the giant component. Figure \\ref{fig:fig3} shows an example of removing failure links and recording the size of giant components to understand the impact of perturbations on road networks. To obtain the worst scenario, we applied the node degree link removal strategy for simulation. That is, to identify failure links we removed nodes in topological networks in order from those with high degrees (node having a greater number of connections) to low degrees (node having fewer connections). In the real-world situation, it is unlikely that all the nodes with higher degrees would fail at the early stage simultaneously. During each removal stage in this analysis, essential characteristics were recorded, including the proportion of isolated links and critical transitions. Isolated links are those that become disconnected from any other part of a network; and critical transitions occur where network connectivity significantly drops (proportion of isolated links suddenly increase). Figure \\ref{fig:fig4} depicts the steps for computing this metric. In this study, the road segments are grouped into five criticality levels. The first 20\\% of isolated road segments are assigned level 1, and the last 20\\%, level 5.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.85\\linewidth]{fig\/Fig_3.png}\n    \\caption{Example of percolation analysis by removing failure links and recording the size of the giant component.}\n\t\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_4.png}\n    \\caption{Steps for evaluating the criticality of road segments based on network connectivity.}\n\t\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nIn the case study, we utilized roadway inventory data updated on August 8, 2021, as the road network for computing this metric. The original road network data contains multiple lanes and complicated intersections. To perform network analysis, we simplified the original road network data, including merging multiple lanes on the same road segment into a single central line if they are close enough to each other. The simplified road network is then transformed into a topological network where all intersections and road segments are represented as nodes and links.\n\n\\section{Vulnerability to Flooding}\n\\label{sec:Vulnerability to Flooding}\nThis metric promotes understanding of the criticality of road segments to disaster impacts: flooding, in the case study. According to the literature, flooding is the most prominent hazard of concern for the Texas DOT and Texas MPOs. Therefore, we included a metric focusing on the exposure to flood hazards determined by the proximity of road segments to floodplains. This metric can be adopted to account for other disaster impacts to evaluate the criticality of road segments in road networks.\n\n\\subsection{Metric Definition and Calculation}\nThis metric determines the criticality level of road segments based on their proximity to floodplains. ArcGIS was used to perform geoprocessing tasks, including: (1) distinct road segments whose midpoints are located in the 100-year floodplain and the 500-year floodplain, and (2) calculating the distances from the midpoint of the rest of the road segments to the boundary of the 500-year floodplain. As shown in Figure \\ref{fig:fig5}, this metric determines the road segments belonging to one of the five criticality levels. The road segments with their midpoint located in the 100-year floodplain are assigned to level 1. The road segments with their midpoint located between 100- and 500-year floodplain are assigned to level 2. Level 3 covers the road segments whose midpoints are 0 to 200 meters away from the 500-year floodplain. Level 4 covers the road segments whose midpoints are 200 to 400 meters away from the 500-year floodplain. Level 5 covers the road segments more than 400 meters away from the 500-year floodplain. Figure \\ref{fig:fig5} demonstrates an example of assigning criticality levels to road segments in central Houston. Figure \\ref{fig:fig6} depicts the steps for computing this metric.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.75\\linewidth]{fig\/Fig_5.png}\n    \\caption{Definition of the metric of vulnerability to extreme events and an example of criticality levels of road segments in central Houston.}\n\t\\label{fig:fig5}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_6.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to flooding areas.}\n\t\\label{fig:fig6}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nThe floodplain data used in this case study was obtained from the NFHL provided by FEMA on Data.gov. However, floodplain data is available for only some districts, including major urban areas in Texas, and missing in some northwestern counties in Texas (Figure \\ref{fig:fig7}). FEMA flood zones are geographic areas that FEMA has defined according to varying levels of flood risk. Each zone reflects the severity or type of flooding in the area based on the definition of flood zones, which distinguish the 100-year floodplain, 500-year floodplain, and others. Each of the 100-year and 500-year floodplains consists of several subcategories. The 100-year floodplain means there is a 1\\% annual flooding probability and is identified as high risk, while the 500-year floodplain refers to a 0.2\\% annual flooding probability and moderate risk. Regions located outside the 500-year floodplain will be identified as minimal risk or unknown risk.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.7\\linewidth]{fig\/Fig_7.png}\n    \\caption{Floodplain data availability in Texas. The major urban areas in Texas, such as Houston, Dallas\/Fort Worth, and Austin, have available floodplain data, whereas floodplain data is missing in some of the northwestern counties in Texas.}\n\t\\label{fig:fig7}\n\\end{figure}\n\n\\section{Proximity to Essential Facilities}\n\\label{sec:Proximity to Essential Facilities}\nThe previous two metrics evaluate the criticality of road segments based on network connectivity and vulnerability to flooding. Essential facilities, however, also play an important role during and in the aftermath of disasters; thus, road segments that are close to those essential facilities become critical because of the need to transport resources and to provide public services. This section evaluates the criticality of road segments based on the capability to provide access to critical facilities.\n\n\\subsection{Metric Definition and Calculation}\nTo evaluate the criticality of road segments in road networks, we implemented the steps shown in Figure \\ref{fig:fig8}. The idea is to evaluate the number of critical facilities to which a road segment provides direct access. We took the simplified road network as described in the previous section, calculated a 2,000-meter buffer around it, and identified the number of critical facilities that lie in this buffer. The purpose of this step is (1) to arrive at an estimated number of critical facilities distributed close to road networks and (2) to shortlist facilities to be used for computing the criticality metric. A buffer value of 2,000 meters was selected because most of the critical facilities lie within this buffer range. Moreover, we assumed that if a facility is within a 2,000-meter buffer range of a road network, it can be considered spatially co-located (co-location interdependency). This buffer value was selected based on our case study area; however, the buffer value may differ based on the density of the critical facilities in an area and the density of road infrastructure. After extracting the information of all the critical facilities that lie close to road networks, we computed the criticality metric by counting the number of facilities that lie in the nearest proximity (2,000-meter buffer range) of each road segment to evaluate the importance of each road segment. These road segments are then assigned a criticality score from level 1 to level 5, where level 1 is the most critical level for road segments and level 5 is the least critical level. In this study, if a road segment had more than ten critical facilities in its nearest proximity, it was assigned a score of level 1; if it had five to ten facilities, it was assigned to level 2; if it had two to five facilities, it was assigned to level 3; if it had only one facility, it was assigned to level 4; and if the road segment did not have any facility in its nearest proximity, it was assigned a score of level 5. The criticality levels of 1 and 2 mean that a road segment is critical and needs to be maintained for ensuring capable access to critical infrastructure in the vicinity, whereas levels 4 and 5 imply that there are none or less significant numbers of critical infrastructure in the nearest proximity to the road segment.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_8.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to critical facilities.}\n\t\\label{fig:fig8}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nIn the case study, we collected location-based data of eight types of critical facilities, which are electrical substations, power plants, hospitals, national shelter facilities, seaports, fire stations, police stations, as well as petroleum, oil and lubricant (POL) terminals, from HIFLD and ArcGIS open data portal. The data consist of the coordinates of the respective critical facilities. It is important to maintain functionality of each of these facilities, including electricity, safety, health, and disaster relief. Access to critical facilities, especially during disruptions caused by disasters or other factors, is paramount for resiliency. Figure \\ref{fig:fig9} shows the spatial distribution of the critical facilities in the state of Texas. Aggregating all the facilities together, we collected location information for around 15,000 critical facilities.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_9.png}\n    \\caption{Spatial distribution of the eight critical facilities in the State of Texas.}\n\t\\label{fig:fig9}\n\\end{figure}\n\n\\section{Cascading Impact of Critical Infrastructure Networks}\n\\label{sec:Cascading Impact of Critical Infrastructure Networks}\nThe criticality metric proposed in the previous section, proximity to essential facilities, evaluates the criticality of road segments based on the locations of critical facilities that can be demonstrated as points on maps. Some critical infrastructures, however, such as the electrical power grid, railroads, and oil pipelines, are not conducive to be captured by the metric of proximity to essential facilities because they are spatially distributed as networks instead of single location attributes. Failure of a link within an infrastructure network can lead to severe cascading impacts on the entire infrastructure network \\citep{fekete_critical_2019,lee_community-scale_2021,serre_assessing_2018}. For this reason, we calculated the criticality of road segments according to their closeness to critical infrastructure networks to understand the criticality related to cascading impacts. Figure \\ref{fig:fig10}a shows the approach used in the previous metric where the criticality score is based on proximity to essential facilities, whereas Figure \\ref{fig:fig10}b demonstrates how we calculated the criticality score based on cascading impact of critical infrastructure networks.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.65\\linewidth]{fig\/Fig_10.png}\n    \\caption{Schematic of assessing criticality of road segments based on (a)critical facilities and (b) critical facility networks.}\n\t\\label{fig:fig10}\n\\end{figure}\n\n\\subsection{Metric Definition and Calculation}\nTo compute the criticality scores based on cascading impacts of critical infrastructure networks, we adopted an approach similar that described in Section \\ref{sec:Proximity to Essential Facilities} in which we used the nodes of infrastructure networks as a proxy to determine the interdependency of infrastructure networks with road networks. Figure \\ref{fig:fig11} shows the analysis steps used to compute this metric. Based on the criticality measure (i.e., counts of critical facility nodes), we assigned a criticality score from level 1 to level 5, where level 1 is the most critical level for road segments and level 5 is the least critical level. In this study, if a road segment had more than ten critical infrastructure network nodes in its nearest proximity, it was assigned a score of level 1; if it had five to ten infrastructure network nodes, it was assigned to level 2; if it had two to five infrastructure network nodes, it was assigned to level 3; if it had only one infrastructure network node, it was assigned to level 4; and if the road segment did not have any infrastructure network node in its nearest proximity, it was assigned a score of level 5.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_11.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to infrastructure networks.}\n\t\\label{fig:fig11}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nWe collected critical infrastructure network data for five types of critical infrastructure networks, which are railroad networks, transmission lines, gas lines, hydrocarbon gas lines (HGL), and oil pipelines. These datasets were acquired from sources such as HIFLD and ArcGIS open data portal. We simplified infrastructure networks into a node-and-link network model using two steps: (1) we created a buffer between 100 and 500 meters for each infrastructure network and merged overlapping features. This was done to ensure that multiple parallel lines in the network could be merged. (2) Centerlines for the buffered networks were computed, merging some parallel lines into one and simplifying some intersections where multiple nodes are combined into one. Figure \\ref{fig:fig12} uses Harris County for illustration. Figure \\ref{fig:fig12} (row 1) shows the approximation of converting infrastructure networks into nodes and edges. Figure \\ref{fig:fig12} (row 2) shows superimposed simplified infrastructure networks with the road network in Harris County.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_12.png}\n    \\caption{Example of the facility networks and road network in Harris County. Row 1: Infrastructure networks that were simplified to node-edge networks. Row 2: Mapping of simplified infrastructure networks with road networks.}\n\t\\label{fig:fig12}\n\\end{figure}\n\n\\section{Integrated Criticality Metric}\n\\label{sec:Integrated Criticality Metric}\nAfter calculating the four criticality metrics we developed, the integrated criticality metrics that were calculated to help decision-makers understand the overall criticality of every road segment within the road network. While every criticality metric provides unique insight, a single integrated metric that comprises all information from different dimensions can be informative as well. In this section, we introduce two approaches to calculate metrics for the overall criticality scores and present the results of the integrated metrics. In addition, we compare the results of the two integrated criticality metrics and discuss the findings based on the comparison.\n\n\\subsection{Metric Development}\nTo understand the overall criticality of each road segments in networks, we calculated two integrated criticality metrics, which are based on evenly weighted sum (EWS) ranking and root squared Sum (RSS) ranking. The equations for EWS (Equation (\\ref{eq:eq1})) and RSS (Equation (\\ref{eq:eq2})) ranking calculations are: \n\n\\begin{equation}\n\tEWS_{RS}= \\sum {Level_{RS,CA}}\n\t\\label{eq:eq1}\n\\end{equation}\n\n\\begin{equation}\n\tRSS_{RS}= \\sqrt{\\sum {{Level_{RS,CA}}^2}}\n\t\\label{eq:eq2}\n\\end{equation}\n\nwhere, $EWS_{RS}$ and $RSS_{RS}$ are the ranking values for each road segment $RS$ and $Level_{RS,CA}$ is the criticality levels of each road segment and all criticality aspects $CA$ examined. For instance, if a road segment has a criticality level of 3 for connectivity, level 2 for vulnerability to flooding, level 4 for proximity to essential facilities, and level 4 for cascading impact, the EWS ranking value of this road segment will be 13, whereas the RSS ranking value will be about 6.71. After ranking values for all road segments, these values are reclassified into five levels. We assigned an integrated criticality level to each road segment. The classification method divides the range of all ranking values (i.e., maximum subtracts minimum of ranking values of all road segments) into five equal intervals, representing five classes. The level 1 category, comprising the most critical segments in the network, is the highest 20\\% of the ranking values. The class of level 5 category comprising the least critical segments in the network, is the lowest 20\\% of the ranking values.\n\n\\section{Case Study for Texas Road Network}\n\\label{sec:Case Study for Texas Road Network}\nThis section presents the results for the four individual and two integrated metrics used to assess the criticality of road segments in the state of Texas. \n\n\\subsection{Connectivity of Road Segments}\nThis metric reveals which road segments are critical in terms of connecting different parts of the network. The more critical a road segment is, the higher impact it will cause on overall connectivity when it is damaged; thus, this metric is a key indicator for informing road infrastructure planning. Figure \\ref{fig:fig13} shows the remaining links after removing 20\\% of nodes with the highest degrees at each stage. Figure \\ref{fig:fig14} shows the percentage of links being isolated during the removal process. During the link removal process that simulates the failure of road segments, critical transitions (sudden increases in the percentage of links becoming isolated) are not observed, which indicates that no particular segment has substantial impacts to the network when failing. From observing the visual results in Texas, most of the road segments that belong to higher criticality levels (levels 1 and 2) are located in the outskirts of major urban areas or in coastal areas. These road segments may be part of the busy commuting routes or the weakly connected parts of the network. The criticality level for this metric depends solely on the topology of the road network. Figure \\ref{fig:figa1} shows the criticality score of the road segments in Texas based on the connectivity metric. Most of the highly critical links, links with levels 1 and 2 criticalities, appear on the outskirts of major cities and in the coastal regions where fewer alternative substitute routes exist in case a link is removed.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_13.png}\n    \\caption{Remaining links after removing 20\\% of nodes with the highest degrees and their corresponding failure links at each stage in Texas.}\n\t\\label{fig:fig13}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.5\\linewidth]{fig\/Fig_14.png}\n    \\caption{Percentage of links being isolated during the removal process. There is no abrupt change in the percentage of isolated links with the percentage change in nodes removed.}\n\t\\label{fig:fig14}\n\\end{figure}\n\n\\subsection{Flooding Vulnerability}\nAs aforementioned, flooding is the most prominent hazard in the state of Texas. This metric shows road segments that are more vulnerable to flooding-related hazards. Even though we made our best efforts to have floodplain data for all counties in Texas, floodplain data in some counties are not available, leaving the criticality of some road segments unclassified. In the case study, the road segments located in the 100- and 500-year floodplain, which are identified as high- and moderate-risk road segments by the definition of FEMA, are assigned to levels 1 and 2, and which form only about 20\\% of the total number of road segments. The rest of the road segments are unlikely to be affected by flooding and are relatively less vulnerable. In the Appendix \\ref{sec:Visualization Maps for the Case Study Area}, Figure \\ref{fig:figa2} shows the criticality of road segments based on this metric. The road segments that are identified as levels 1 and 2 may need additional attention to enhance their resiliency and to plan for alternative routes while affected by the hazards. \n\n\\subsection{Proximity to Essential Facilities}\nThis criticality metric captures the importance of road segments that can provide access to critical and essential facilities. We find that the critical road segments identified by this metric are more concentrated in the centers of the urban areas. In particular, the critical road segments are found in the high-density metropolitan areas of Houston, San Antonio, Austin, and Dallas\/Fort Worth, which are the four major urban areas in Texas. Since the concentration of critical facilities is relative denser in urban and populated areas with respect to the road density, there are more critical road segments in terms of proximity to essential facilities present in urban areas. Figure \\ref{fig:figa3} illustrates important road segments (levels 1 and 2) in Texas. The road segments having higher criticality in terms of proximity to essential facilities are located mainly in the major urban areas in Texas. Additionally, most level 3 road segments that have moderate criticality are located at major road junctions. On the other hand, rural areas consist mostly of low criticality road segments (levels 4 and 5) due to a lower concentration of critical facilities.\n\n\\subsection{Cascading Impact on Critical Infrastructure Networks}\nThe metric of cascading impact on critical infrastructure networks, evaluates the criticality of infrastructure networks based on their colocation interdependency with the road network. That is, the more infrastructure network nodes are in close proximity to the road segment, the higher the criticality of cascading impact on critical infrastructure networks is for a road segment. Here, unlike the metric of proximity to critical facilities, which demonstrates that the critical road segments are more concentrated in the urban areas, we found that most of the critical road segments in terms of cascading impact on critical infrastructure networks are near the coastline and western Texas instead of concentrated around urban areas. Moderately critical road segments are mostly present in urban areas. Figure \\ref{fig:figa4} shows the critical road segments in red and orange (levels 1 and 2) based on their interdependency to critical infrastructure networks. \n\n\\subsection{Integrated Criticality Metrics}\nThe results of implementing the two integrated criticality metrics in Texas, as an example, are demonstrated in Figures \\ref{fig:figa5} and \\ref{fig:figa6}. More than 75\\% of the road segments are classified as levels 4 and 5 for both integrated criticality metrics. As shown in Figure \\ref{fig:fig15}, the EWS method tends to have more levels 1, 2, and 3 criticality road segments, which is more conservative than the RSS method. By implementing the integrated criticality metrics, the comparison of criticality levels can be conducted at any scale and location in Texas (e.g., metropolitan area and county level) to understand the overall criticality of each road segment. Figure \\ref{fig:fig16} demonstrates an example in Harris County that the level 1 and 2 road segments are generally located close to intersections of road segments. In addition, the road segments at the northwestern corner of Harris County, where the city of Houston is located, have relatively minor criticality compared to the road segments in the other area of Harris County.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.6\\linewidth]{fig\/Fig_15.png}\n    \\caption{Distribution of different criticality levels of the RSS and EWS methods for the state of Texas. }\n\t\\label{fig:fig15}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.9\\linewidth]{fig\/Fig_16.png}\n    \\caption{Results of EWS ranking (left) and RSS ranking (right) in Harris County.}\n\t\\label{fig:fig16}\n\\end{figure}\n\nBased on the integrated criticality metrics, we can specify the overall criticality of different road segments and prioritize resilience investments accordingly. Thus, it is essential to know the most common criticality concern among the four metrics if a road segment is classified as level 1 (i.e., the most critical level) for the overall criticality level. Figure \\ref{fig:fig17} shows the distribution of the four individual metrics for the EWS and RSS integrated criticality metrics in level 1. More than 70\\% of the road segments in the Texas road networks assigned an EWS integrated criticality of level 1 are critical (i.e., level 1 criticality) in terms of flood exposure and network connectivity. Similarly, more than 80\\% of the road segments with RSS integrated criticality of level 1 are also critical in flood exposure and network connectivity. On the other hand, the criticality in terms of cascading impacts appears to be relatively less influential for the road segments with the EWS and RSS integrated criticality metrics of level 1, of which the criticality related to cascading impacts are mostly at level 2. With these results, agencies and organizations can prioritize maintenance plans proactively and ensure the required functionality and serviceability of road networks to mitigate the impact of disasters and recover from impact rapidly.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.8\\linewidth]{fig\/Fig_17.png}\n    \\caption{Distribution of the influence of individual metrics for the road segments with EWS and RSS integrated criticality of level 1.}\n\t\\label{fig:fig17}\n\\end{figure}\n\n\\section{Concluding Remarks}\n\\label{sec:Concluding Remarks}\nThe objective of the study is to propose a framework with quantitative measures incorporating resilience assessment into transportation planning and project development processes in practice. While the literature discusses several methods and measures for resilience assessment of transportation infrastructure, just a small fraction of the existing work has been adopted in practice to inform transportation planning and project development. This limitation is mainly due to two reasons: (1) transportation planners and decision makers do not know which to use among several methods and measures proposed in the literature; (2) the methods and measures for vulnerability and resilience assessment should be obtainable using the available data. To address this gap, in this study, we proposed and tested a framework for system-level assessment of transportation infrastructure networks based on four measures: (1) loss of connectivity of road segments in road networks, (2) vulnerability of road segments to extreme events, such as flooding and hurricanes, (3) disrupted access to critical facilities, and (4) cascading impacts of critical infrastructure networks. In addition, two integrated metrics were proposed to combine all metrics into one measure that can show the overall performance of road segments in terms of resilience. To demonstrate the use of the framework, we implemented individual and integrated metrics in the state of Texas. We collected datasets related to the topology of the Texas road network, hazard characteristics, and the location of critical facilities (e.g., power substations, oil\/gas lines, and hospitals) from open sources and calculated the values of each metric for all road network links in Texas. The results showed that the framework can effectively provide useful insights with quantitative measures. In addition, the visualization of the results can make the interpretation of resilience assessment on road networks more straightforward and comprehensive. Also, the case study demonstrates the ability of the framework to be conducted at any scale and location to reveal the resilience and criticality of each road segment on networks.\n\nThe framework proposed in this paper can help transportation agencies incorporate resilience considerations into planning and project development proactively and to understand the resilience performance of the current road networks effectively. For example, road segments with high criticality and vulnerability levels could be prioritized for adaptation and hazard mitigation retrofit. Roads with high criticality and vulnerability to flooding impacts could be effectively identified and appropriate projects (such as drainage improvement or road elevation increase) could be implemented to reduce the vulnerability of those road segments. The demonstration of the results of this study to planners and decision makers in Texas has confirmed that the produced measures are deemed useful by practitioners. In fact, at the time of writing this manuscript, a state-wide transportation resilience planning effort is underway, and the results of this study are a major component in informing the planning process. Since the data used in this study is available to other states and countries, the framework presented in this study can be adopted by other transportation agencies across the globe for regional transportation resilience assessments. \n\n\\section{Data availability}\nAll datasets used in this study are publicly available. The road network data can be acquired from the TxDOT data repository. Flood impact data is extracted from the National Flood Plain Layer on the FEMA website. Data on critical facilities and critical infrastructure networks can be accessed from publicly available Homeland Infrastructure Foundation Level Data (HIFLD) and ArcGIS open data portals.\n\n\\section{Code availability}\nThe code that supports the findings of this study is available from the corresponding author upon request.\n\n\\section{Acknowledgements}\nThis research is part of the study supported by the Texas Department of Transportation (TxDOT). The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official view or policies of TxDOT. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the organizations or the individuals listed here.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nBorn from the gravitational core collapse of massive stars in supernova explosions, neutron stars are the densest known stars in the universe (see, e.g., \nRef.~\\cite{haensel2007}). Most of them have been discovered as isolated radio pulsars. Neutron stars can also belong to binary systems, in which the \nneutron star accretes matter from a stellar companion. These systems are observed as x-ray pulsars. Neutron stars may be endowed with very strong magnetic \nfields. Soft $\\gamma$-ray repeaters \nand anomalous x-ray pulsars are thought to be such so-called ``magnetars'' (see, e.g., Ref.~\\cite{woods2006} for a review). Their surface magnetic field, as inferred \nfrom spin-down and spectroscopic studies, are of the order of $10^{14}-10^{15} $~G \\cite{olausenkaspi2014, tiengo2013, hongjun2014}. Numerical simulations \nsuggest that their internal magnetic field could be even stronger, reaching $10^{18}$~G~\\cite{pili2014,chatterjee2015}. \n\nApart from a thin atmospheric plasma layer of light elements (mainly hydrogen and helium) possibly surrounding a Coulomb liquid of electrons and ions, a \nneutron star is thought to contain at least three distinct regions. The outermost region consists of a solid crust made of a crystal lattice of fully ionized \natoms arranged on a body-centered cubic lattice (see, e.g., Ref.~\\cite{chamelhaensel2008} for a review). With increasing density, nuclei become progressively \nmore neutron rich due to electron captures until neutrons start to drip out of nuclei at some threshold density $\\rho_{\\rm drip}$, thus marking the boundary \nbetween the outer crust and the inner crust. In unmagnetized or weakly magnetized nonaccreting neutron stars (with magnetic fields $\\ll 10^{14}$~G), the neutron-drip \ntransition is found to occur at density $\\rho_{\\rm drip} \\approx 4 \\times 10^{11}$~g~cm$^{-3}$ (see, e.g., Refs.~\\cite{bps1971, haensel2007, roca2008, pearson2011, \nwolf2013, kreim2013, chamel2015c}). On the other hand, this density can be shifted in magnetars, due to the presence of strong magnetic fields (see, e.g., \nRefs.~\\cite{sek1977, lai91, nandi2011, chamel2012, nandi2013, vishal2014, chamel2015b,bas2015}), and in accreting neutron stars, due to the accretion from a companion \n\\cite{hz1990, hz2003, chamel2015a}.\nThe crust dissolves into an homogeneous liquid mixture at about half the density prevailing in heavy atomic nuclei. This transition is expected to be realized \nthrough different nuclear structures, such as cylinders, rods, and plates (also called ``nuclear pasta'', see Ref.~\\cite{haensel2007,chamelhaensel2008} and references therein). Further \ninside, at very high densities in the neutron-star core, additional degrees of freedom, such as hyperons, meson condensates, and\/or deconfined quarks, could be \npresent (see, e.g., Ref.~\\cite{haensel2007, page2006, weber2007}). \n\n\nFrom the nuclear physics point of view, the neutron-star crust is also a unique ``laboratory'' to probe the properties of infinite homogeneous asymmetric nuclear matter, in particular \nthe symmetry energy, at subsaturation densities (see, e.g. Ref.~\\cite{steiner2015}). The symmetry energy is generally defined as \n\\begin{equation}\n\\label{eq:e-inm}\nS_1(n) = \\frac{1}{2}\\frac{\\partial^2 (\\mathcal{E}\/n)}{\\partial \\eta^2}\\biggr\\vert_{\\eta=0} \\ ,\n\\end{equation}\nwhere $\\mathcal{E}(n,\\eta)$ is the energy density of infinite homogeneous nuclear matter with proton (neutron) density $n_p$ ($n_n$), baryon density \n$n=n_n + n_p$, and charge asymmetry $\\eta = (n_n-n_p)\/n$. Alternatively, the symmetry energy often refers to the difference between the energy of \npure neutron matter and that of symmetric matter:\n\\begin{equation}\n\\label{eq:s2}\nS_2(n) =\\frac{\\mathcal{E}(n,\\eta=1) - \\mathcal{E}(n,\\eta=0)}{n} \\ .\n\\end{equation} \nBecause $\\mathcal{E}(n,\\eta)$ generally contains terms of order $\\eta^4$ and higher, the two definitions $S_1(n)$ and $S_2(n)$ of the symmetry energy \ndo not exactly coincide (see, e.g.  the discussion in Sec.~III of Ref.~\\cite{goriely2010}). From now on, we adopt the first definition. \nThe symmetry energy of infinite homogeneous nuclear matter $S_1(n)$ can be expanded around the saturation density $n_0$ (whose value is $\\approx 0.16$~fm$^{-3}$) as\n\\begin{equation}\n\\label{eq:s1}\nS_1(n) \\approx J + \\frac{1}{3} L \\left( \\frac{n-n_0}{n_0} \\right) + \\frac{1}{18} K_{\\rm sym} \\left( \\frac{n-n_0}{n_0} \\right)^2\\ .\n\\end{equation}\nWhereas the value of the symmetry energy at saturation, $J$, is fairly well constrained by nuclear physics experiments to lie around 30~MeV, the values of the \nslope of the symmetry energy, $L$, and of higher order coefficients like $K_{\\rm sym}$ at saturation,\n are still very uncertain and poorly constrained (see, e.g., the discussion in \nRefs.~\\cite{tsang2012,lattimerlim2013}). The symmetry energy has been shown to affect the composition of neutron-star crusts and the crust-core transition~(see, e.g., \nRefs.~\\cite{horowitz2001,oyamatsu2007, roca2008, vidana2009,ducoin2011,grill2012,newton2013,sulaksono2014,provid2014,bao2014, grill2014,seif2014,provid2014}). \nComparatively few studies have been devoted to the role of the symmetry energy on the boundary between the outer and inner crusts~\\cite{bao2014} (some discussions can \nalso be found, e.g., in Refs.~\\cite{roca2008,provid2014,grill2014}). Moreover, all these studies focused on nonaccreting and unmagnetized neutron stars. \n\n\nIn this paper, we study the role of the symmetry energy on the onset of neutron drip in both nonaccreting and accreting neutron stars, allowing for the \npresence of a strong magnetic field as in magnetars. To this end, we have made use of the recent set of Brussels-Montreal microscopic nuclear mass \nmodels~\\cite{goriely2013}. Our model of neutron-star crust is presented in Sec.~\\ref{sec:model}~: after discussing our assumptions in Sec.~\\ref{sec:assumptions}, \nthe nuclear mass models are briefly described in Sec.~\\ref{sec:hfb-models}. The determination of the neutron-drip transition is discussed in Sec.~\\ref{sec:neutron-drip}, \nand numerical results are presented in Sec.~\\ref{sec:results}. \n\n\n\\section{Model of neutron-star crust}\n\\label{sec:model}\n\n\n\\subsection{Main assumptions}\n\\label{sec:assumptions}\n\nIn the region of the crust that we consider here, the pressure is high enough that atoms are fully ionized~\\cite{chamelhaensel2008}. We assume that the temperature $T$ is \nlower than the crystallization temperature $T_m$ so that nuclei are arranged in a regular crystal lattice. Considering that the crystalline structures are \nmade of only one type of ion $^{A} _{Z} X$, with mass number $A$ and atomic number $Z$, the crystallization temperature $T_m$ is given by \n(see, e.g., Ref.~\\cite{haensel2007}):\n\\begin{equation}\nT_m = \\frac{e^2}{a_e k_{\\rm B} \\Gamma_m} Z^{5\/3} \\ ,\n\\end{equation}\nwhere $e$ is the elementary electric charge, $a_e=(3\/(4\\pi n_e))^{1\/3}$ the electron-sphere radius expressed in terms of the electron number density $n_e$, \n$k_\\text{B}$ is the Boltzmann's constant, and $\\Gamma_m\\simeq 175$ is the Coulomb coupling parameter at melting. Typically, $T_m$ is much lower than the \nelectron Fermi temperature defined by\n\\begin{equation}\n T_{\\rm F}=\\frac{\\mu_e-m_e c^2}{k_{\\rm B}} \\ ,\n\\end{equation}\nwhere $m_e$ is the electron mass, $c$ is the speed of light, and $\\mu_e$ is the electron Fermi energy. \nTherefore, electrons are highly degenerate and from now on we will set $T=0$. To a very good approximation, electrons can be treated as an ideal relativistic \nFermi gas. Expressions for the electron energy density $\\mathcal{E}_e$ and electron pressure $P_e$ can be found in Ref.~\\cite{haensel2007}. \nThe main corrections arise from electron-ion interactions (see, e.g., Ref.~\\cite{pearson2011} and references therein for a discussion of other corrections). \nNeglecting the finite size of ions and the quantum zero-point motion of ions off their equilibrium position, the lattice contribution to the energy density \nis given by \n(see e.g. Chap. 2 in Ref.~\\cite{haensel2007})\n\\begin{equation}\n\\label{eq:EL}\n\\mathcal{E}_L  = C e^2 n_e^{4\/3} Z^{2\/3} \\ ,\n\\end{equation}\nwhere $C$ is a crystal structure constant, whose value for a body-centered cubic lattice is $C=-1.444231$~\\cite{baiko2001}. This expression still remains valid in the \npresence of a strong magnetic field, as a consequence of the Bohr-van Leeuwen theorem~\\cite{vanvleck1932}. \nThe matter pressure $P$ can be expressed as\n\\begin{equation}\n\\label{eq:ptot}\nP = P_e + P_L \\ ,\n\\end{equation}\nwhere $P_e$ is the pressure of a uniform electron gas, and the lattice pressure $P_L$ is given by \n\\begin{equation}\n\\label{eq:PL}\n P_L=\\frac{\\mathcal{E}_L}{3} \\ .\n\\end{equation}\nThe only microscopic inputs for the description of the outer crust are nuclear masses $M^\\prime(A,Z)$. They can be obtained from the corresponding \natomic mass $M(A,Z)$ after subtracting out the binding energy of the atomic electrons (see Eq.~(A4) of Ref.~\\cite{lpt03}):\n\\begin{equation}\n\\label{3}\nM^\\prime(A,Z)c^2 = M(A,Z)c^2 + 1.44381\\times 10^{-5}\\,Z^{2.39} + 1.55468\\times 10^{-12}\\,Z^{5.35} \\ ,\n\\end{equation}\nwhere both masses are expressed in units of MeV$\/c^2$. Nuclear masses may be changed in the presence of a strong magnetic field. On the other hand, \nfor the magnetic field strength we shall consider, namely $B < 10^{17} $~G, those changes are very small \\cite{pena2011} and will thus be ignored. \nFor the masses that have not yet been measured, we have made use of the microscopic mass tables computed by the Brussels-Montreal group  (for a recent review of these models, see, e.g., Ref.~\\cite{chamel2015c}). \n\n\\subsection{Microscopic nuclear mass models}\n\\label{sec:hfb-models}\n\nThe family of Brussels-Montreal nuclear mass models that we adopt here~\\cite{goriely2013} are based on the nuclear energy density functional (EDF) \ntheory using a generalized form of Skyrme zero-range effective interactions~\\cite{chamel2009}, supplemented with a microscopic contact pairing interaction~\\cite{chamel2010}. \nFor all these models, the masses of nuclei were obtained by adding to the \nHartree-Fock-Bogoliubov (HFB) energy a phenomenological Wigner term and a correction term for the rotational and vibrational spurious collective energy \n(see, e.g. Ref.~\\cite{goriely2010} for a discussion about the accuracy of this latter correction). The EDFs BSk22, BSk23, BSk24, BSk25, and BSk26 \nunderlying the nuclear mass models HFB-22, HFB-23, HFB-24, HFB-25, and HFB-26 were primarily fitted to the 2353 measured masses of nuclei with $N$ and $Z \\geq 8$ from \nthe 2012 Atomic Mass Evaluation \\cite{audi2012}, with a root-mean-square (rms) deviation of $0.63$~MeV, $0.57$~MeV, $0.55$~MeV, $0.54$~MeV, and $0.56$~MeV \nrespectively. At the same time, these EDFs were constrained to reproduce the equation of state (EoS) of homogeneous neutron matter, as obtained by many-body \ncalculations using realistic two- and three-nucleon interactions. Moreover, the incompressibility $K_v$ of infinite homogeneous symmetric nuclear matter at saturation was \nrequired to fall in the range $240\\pm10$~MeV~\\cite{colo2004}, and the isoscalar effective mass was fixed to the realistic value $M_s^*=0.8$. In addition, the \nEoS of symmetric nuclear matter obtained from these EDFs were found to be compatible with the empirical constraints inferred from the analysis of heavy-ion \ncollision experiments~\\cite{dan02,lynch09}. For all these reasons, we believe that these EDFs can be reliably applied to the description of neutron-star crusts. \n\nIn generating these five EDFs, different values of the symmetry energy coefficient $J$ \nwere imposed, thus making them suitable for a systematic study of \nthe role of the symmetry energy on the outer crust of a neutron star. In particular, BSk22, BSk23, BSk24 and BSk25 were constrained to the symmetry energy \ncoefficients $J = 32$, $31$, $30$ and $29$~MeV, respectively and were all fitted to the realistic neutron-matter EoS labeled ``V18\" in Ref.~\\cite{ls2008}, \nwhile BSk26 was fitted to the EoS labeled ``A18 + $\\delta\\,v$ + UIX$^*$\"in Ref.~\\cite{apr1998} under the constraint $J = 30$~MeV. The values of the symmetry energy coefficient $J$, as well as the higher-order coefficients \n$L$ and $K_{\\rm sym}$ for the different EDFs are indicated in Table \\ref{tab:esymcoef}. As already discussed in Ref.~\\cite{pearson2014}, $L$ is strongly \ncorrelated with $J$: increasing $J$ leads to higher values of $L$. The difference between the values of $L$ obtained with BSk24 and BSk26 (for which $J=30$ MeV) \narises from the constraining neutron-matter EoS. Indeed, as shown in Eqs.~(\\ref{eq:e-inm}) and (\\ref{eq:s1}) the softer the underlying neutron-matter EoS, \nthe lower $L$. On the other hand, the correlation between $L$ and $K_{\\rm sym}$ is less clear. As shown in Fig.~\\ref{fig:JvsL}, the values of $J$ and $L$ obtained \nwith the Brussels-Montreal EDFs are consistent with constraints coming from the combined analysis of various experiments \\cite{tsang2012, lattimerlim2013} \n(see also the discussion in Sec.~IIIC in Ref.~\\cite{goriely2013}): the constraint deduced in Ref.~\\cite{tsang2009} from heavy-ion collisions (HIC), the constraint derived \nin Ref.~\\cite{chen2010} from measurements of the neutron-skin thickness in tin isotopes, and finally the constraint obtained from the analysis of the giant dipole \nresonance (GDR) \\cite{trippa2008}. \n\nThe density dependence of the symmetry energy $S_1(n)$, Eq.~(\\ref{eq:s1}), for the Brussels-Montreal EDFs BSk22 to BSk25 is shown in Fig.~\\ref{fig:esym-vs-n}. The most notable feature, due to the mass fit, \nis the crossing of all curves at densities around $0.11$~fm$^{-3}$. Indeed, the binding energy of finite nuclei is mainly sensitive to the symmetry energy \nat such densities rather than to the symmetry energy at saturation density~\\cite{horowitz2001, zhangchen2013}. \nFor these functionals, which were fitted to the same \nneutron-matter EoS, we observe a clear correlation between the density dependence of the symmetry energy and its value at saturation with a change of hierarchy \nat the crossing point: the higher $J$, the higher (lower) the symmetry energy above (below) this point. From these considerations, the ``effective'' value of \nthe symmetry energy averaged over the volume of a nucleus is thus expected to increase with decreasing $J$, as found in previous studies~\\cite{goriely2005,gaidarov2012}. \n\n\nComparing various constraints from both nuclear physics and astrophysics~\\cite{pearson2014, fantina2015}, BSk24 (BSk22) was found to be the best (worst) in the series of Brussels-Montreal EDFs BSk22-BSk26. \nIn the following, we shall thus take BSk24 as the reference EDF. We will not consider BSk26 since it was fitted to a different neutron-matter EoS from the other EDFs. \nWe have made use of the mass tables from the BRUSLIB database~\\cite{bruslib}.\n \n\n\\begin{table}\n\\centering\n\\caption{Symmetry energy coefficient $J$ and higher-order symmetry energy coefficients of infinite \nhomogeneous nuclear matter at saturation for the Brussels-Montreal energy density functionals~\\cite{goriely2013}.}\\smallskip\n\\label{tab:esymcoef}\n\\begin{tabular}{|c|ccccc|}\n\\hline\n & BSk22 & BSk23 & BSk24 & BSk25 & BSk26 \\\\\n\\hline\n$J$ [MeV] & 32.0 & 31.0 & 30.0 & 29.0 & 30.0 \\\\\n$L$ [MeV] & 68.5 & 57.8 & 46.4 & 36.9 & 37.5  \\\\\n$K_{\\rm sym}$ [MeV] & 13.0 & -11.3 & -37.6 & -28.5 & -135.6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.35]{JvsL.eps}\n\\end{center}\n\\caption{(Color online) Experimental constraints on the symmetry energy parameters, taken from Ref.\\cite{lattimerlim2013}; the dashed line represents the constraint \nobtained from fitting experimental nuclear masses using the Brussels-Montreal Hartree-Fock-Bogoliubov models (including unpublished ones) with an root-mean-square deviation below \n$0.84$~MeV; star symbols correspond to the series of models from Ref.~\\cite{goriely2013} that we consider in this work. See text for details.}\n\\label{fig:JvsL}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.35]{esym1_bsk22-25.eps}\n\\end{center}\n\\caption{(Color online) Symmetry energy of infinite homogeneous nuclear matter $S_1(n)$ versus density, \nfor the Brussels-Montreal energy density functionals BSk22 to BSk25~\\cite{goriely2013}.}\n\\label{fig:esym-vs-n}\n\\end{figure*}\n\n\n\n\\section{Neutron-drip transition}\n\\label{sec:neutron-drip}\n\nNeutron stars are formed from the catastrophic gravitational core collapse of massive stars during supernova explosions. Under such extreme conditions, it is generally assumed that \nall kinds of nuclear and electroweak processes occur and that, as the neutron star cools down, matter remains in full thermodynamic equilibrium. Eventually, \nthe neutron star becomes cold and fully ``catalyzed'' \\cite{haensel2007} unless matter is accreted from a companion star. In the latter case, the accretion \nmay heat the neutron star and change the composition of its crust. These two different astrophysical scenarios will thus be treated separately. \n\n\n\\subsection{Nonaccreting neutron stars}\n\nThe usual procedure~\\cite{bps1971} to determine the equilibrium nucleus in any layer of the outer crust of a nonaccreting neutron star at pressure $P$ is to minimize the Gibbs \nfree energy per nucleon defined by \n\\begin{equation}\n\\label{eq:gibbs-general}\ng = \\frac{\\mathcal{E}+P}{n} \\ ,\n\\end{equation}\nwhere $\\mathcal{E}$ denotes the mean energy density of the crustal matter, and $n$ denotes the mean baryon number density. As shown in Ref.~\\cite{chamel2015d}, $g$ \nremains the suitable thermodynamic potential in the presence of a strong magnetic field. Equation~(\\ref{eq:gibbs-general}) can be equivalently expressed \nas~\\cite{bps1971,lai91} \n\\begin{equation}\n\\label{eq:gibbs}\ng=\\frac{M'(A,Z)c^2}{A}+\\frac{Z}{A}\\left(\\mu_e -m_e c^2+\\frac{4}{3} C e^2 n_e^{1\/3} Z^{2\/3}\\right)\\, .\n\\end{equation}\nIgnoring neutron band structure effects~\\cite{chamel2006,chamel2007}, \nthe onset of neutron drip is determined by the condition $g=m_n c^2$, where $m_n$ is the neutron mass~\\cite{bps1971} (see also the discussion in Ref.~\\cite{chamel2015a}). \nThis latter condition can be approximately expressed as \\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:n-drip-mue}\n\\mu_e(n_e^{\\rm drip}) + \\frac{4}{3}C e^2 (n_e^{{\\rm drip}})^{1\/3} Z^{2\/3} =  \\mu_e^{{\\rm drip}}\\, ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:muedrip}\n\\mu_e^{{\\rm drip}}(A,Z)\\equiv \\frac{-M'(A,Z)c^2+A m_n c^2}{Z} +m_e c^2 \\, .\n\\end{equation}\nNote that this condition remains the same in the presence or in the absence of a magnetic field. \nHowever, since the relation between the electron density and \nelectron Fermi energy does depend on the magnetic field, the neutron-drip density and pressure do depend on the magnetic field. \n\nFor weakly magnetized neutron stars, $B\\lesssim B_\\star$ where $B_\\star \\equiv B\/B_{\\rm crit}$ is the magnetic field expressed in units of the critical \nmagnetic field \n\\begin{equation}\nB_{\\rm crit} = \\left(\\frac{m_ec^2}{\\alpha \\lambda_e^3}\\right)^{1\/2} \\simeq 4.4 \\times 10^{13} {\\rm G} \\ , \n\\end{equation}\n$\\lambda_e=\\hbar\/(m_e c)$ being the electron Compton wavelength and $\\alpha=e^2\/(\\hbar c)$ the fine structure constant, \nthe baryon density and pressure at neutron drip are approximately given by \\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:ndrip-cat-approx}\nn_{\\rm drip}(A,Z) \\approx \\frac{A}{Z} \\frac{ \\mu_e^{\\rm drip}(A,Z)^3}{3\\pi^2 (\\hbar c)^3}\n\\biggl[1+\\frac{4 C \\alpha}{(81\\pi^2)^{1\/3}} Z^{2\/3} \\biggr]^{-3}\\, .\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-cat-approx}\nP_{\\rm drip}(A,Z) \\approx \\frac{\\mu_e^{\\rm drip}(A,Z)^4}{12 \\pi^2 (\\hbar c)^3}\\biggl[1+\\frac{4C \\alpha Z^{2\/3}}{(81\\pi^2)^{1\/3}} \\biggr]^{-3} \\, .\n\\end{equation}\nIn the presence of a strongly quantizing magnetic field such that \n$n_e<n_{e\\text{B}}$ and $T<T_\\text{B}$ with \n\\begin{equation}\\label{eq:neB}\nn_{e\\text{B}}= \\frac{B_\\star^{3\/2}}{\\sqrt{2} \\pi^2 \\lambda_e^3} \\, ,\n\\end{equation}\n\\begin{equation}\\label{eq:TB}\n T_\\text{B}=\\frac{m_e c^2}{k_\\text{B}} B_\\star\\, , \n\\end{equation}\nthe neutron-drip density and pressure are approximately given by \\cite{chamel2015b}\n\\begin{equation}\n\\label{eq:ndrip-cat-approx-b}\nn_{\\rm drip}(A,Z) \\approx \\frac{A}{Z} \\frac{ \\mu_e^{\\rm drip}(A,Z)}{m_e c^2} \\frac{B_\\star}{2\\pi^2 \\lambda_e^3}\n\\biggl[1-\\frac{4}{3} C \\alpha Z^{2\/3} \\left(\\frac{B_\\star}{2 \\pi^2}\\right)^{1\/3} \\left(\\frac{m_e c^2}{\\mu_e^{\\rm drip}}\\right)^{2\/3} \\biggr] \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-cat-approx-b}\nP_{\\rm drip}(A,Z) \\approx \\frac{B_\\star \\mu_e^{\\rm drip}(A,Z)^2}{4 \\pi^2 \\lambda_e^3 m_e c^2}\\biggl[1 - \\frac{1}{3} C \\alpha Z^{2\/3} \\left(\\frac{4 B_\\star}{\\pi^2}\\right)^{1\/3} \\left(\\frac{m_e c^2}{\\mu_e^{\\rm drip}}\\right)^{2\/3} \\biggr] \\, .\n\\end{equation}\nUsing Eqs.~(\\ref{eq:neB}) and (\\ref{eq:ndrip-cat-approx-b}), the condition $n_e<n_{e\\rm B}$ can be approximately expressed to first order in $\\alpha$  as \n\\begin{equation}\\label{eq:Bdrip}\nB_\\star > B_\\star^{\\rm drip}\\equiv \\frac{1}{2}\\left(\\frac{ \\mu_e^{\\rm drip}(A,Z)}{m_e c^2}\\right)^2 \n\\biggl[1-\\frac{8 C \\alpha Z^{2\/3}}{3(2 \\pi^2)^{1\/3}} \\biggr] \\, .\n\\end{equation}\n\n\n\\subsection{Accreting neutron stars}\n\nIn accreting neutron stars, the magnetic field is typically negligibly small ($B\\ll B_\\star$) and will thus be ignored.  \nFor an accretion rate $\\dot{M}=10^{-9}~{\\rm M_\\odot}$~yr$^{-1}$ the original outer crust is replaced by accreted matter in $10^4$~yr.\nFor low-mass binary systems, the accretion stage can last for $10^9$~yr.\nAt densities above $\\sim 10^8$~g~cm$^{-3}$, matter is highly degenerate and relatively cold  ($T\\lesssim 5\\times 10^8$~K) so that thermonuclear processes are strongly suppressed, since their rates are many orders of magnitude lower than the compression rate due to accretion \\cite{haensel2007}. The only relevant \nprocesses are electron captures and neutron-emission processes, whereby the nucleus $^A_ZX$ is transformed into a nucleus $^{A-\\Delta N}_{Z-1}Y$ with \nproton number $Z-1$ and mass number $A-\\Delta N$ by capturing an electron with the emission of $\\Delta N$ neutrons $n$ and an electron neutrino $\\nu_e$:\n\\begin{equation}\n\\label{eq:e-capture+n-emission}\n^A_ZX+ e^- \\rightarrow ^{A-\\Delta N}_{Z-1}Y+\\Delta N n+ \\nu_e\\, .\n\\end{equation}\nIn this case, the condition for the onset of neutron drip becomes~\\cite{chamel2015a}\n\\begin{eqnarray}\n\\label{eq:e-capture+n-emission-gibbs-approx}\n\\mu_e + C e^2 n_e^{1\/3}\\biggl[Z^{5\/3}-(Z-1)^{5\/3} + \\frac{1}{3} Z^{2\/3}\\biggr] = \\mu_e^{\\rm drip-acc} \\, ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{eq:muebetan}\n\\mu_e^{\\rm drip-acc}(A,Z)\\equiv M'(A-\\Delta N,Z-1)c^2-M'(A,Z)c^2 +m_n c^2 \\Delta N + m_e c^2 \\, .\n\\end{equation}\nThe neutron-drip density and pressure are approximately given by~\\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:ndrip-acc}\nn_{\\rm drip-acc}(A,Z) \\approx \\frac{A}{Z} \\frac{\\mu_e^{\\rm drip-acc}(A,Z)^3}{3\\pi^2 (\\hbar c)^3} \n \\biggl[1+\\frac{C \\alpha}{(3\\pi^2)^{1\/3}}\\left(Z^{5\/3}-(Z-1)^{5\/3}+\\frac{Z^{2\/3}}{3}\\right)\\biggr]^{-3}\\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-acc}\nP_{\\rm drip-acc}(A,Z) \\approx \\frac{\\mu_e^{\\rm drip-acc}(A,Z)^4}{12 \\pi^2 (\\hbar c)^3}\\biggl[1+\\frac{4C \\alpha Z^{2\/3}}{(81\\pi^2)^{1\/3}} \\biggr]\n \\biggl[1+\\frac{C \\alpha}{(3\\pi^2)^{1\/3}}\\left(Z^{5\/3}-(Z-1)^{5\/3}+\\frac{Z^{2\/3}}{3}\\right)\\biggr]^{-4}\\, .\n\\end{equation}\nAs discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus can be determined as follows. Given the mass number $A$ and the initial atomic number \n$Z_0$ of the ashes of x-ray bursts, the atomic number $Z$ at the neutron-drip point is the highest number of protons lying below $Z_0$ for which the $\\Delta N$-neutron \nseparation energy defined as\n\\begin{equation}\n\\label{eq:sn-dn}\nS_{\\Delta N n}(A,Z-1) \\equiv M(A-\\Delta N,Z-1)c^2 - M(A,Z-1)c^2 + \\Delta N m_n c^2\n\\end{equation}\nis negative. \n\n\n\n\\section{Numerical results}\n\\label{sec:results}\n\n\\subsection{Nonaccreting neutron stars}\n\\label{sec:results-nonaccreting}\n\nWe have calculated the properties of neutron-star crusts at the neutron-drip point by minimizing the Gibbs free energy per nucleon~(\\ref{eq:gibbs}), \nboth in the absence and in the presence of a strong magnetic field. In the latter case, we have set $B_\\star=500$, $1000$, $1500$ and $2000$ corresponding \nto magnetic fields in the range $2.2 \\times 10^{16}$~G to $8.8 \\times 10^{16}$~G. Since nuclear masses at this depth of the outer crust are not experimentally \nknown, the predictions for the dripping nucleus are model dependent. The neutron-drip properties are summarized in Table~\\ref{tab:drip-cat} in \nunmagnetized neutron stars, and in Tables~\\ref{tab:drip-cat-mag-500}-\\ref{tab:drip-cat-mag-2000} in strongly magnetized neutron stars (magnetars). \n\nFigure~\\ref{fig:cat-ndrip-B} shows that for any given value of the magnetic field strength, the neutron-drip density increases almost linearly with the slope \nof the symmetry energy $L$ (or equivalently with $J$ since the two coefficients are strongly correlated, as previously discussed in Sec.~\\ref{sec:hfb-models}). \nOn the other hand, the behavior of the neutron-drip density with respect to the magnetic field strength exhibits typical quantum oscillation whereas the \nneutron-drip pressure increases monotonically, as recently discussed in Ref.~\\cite{chamel2015b}. The errors of the analytical formulas~(\\ref{eq:ndrip-cat-approx})-(\\ref{eq:Pdrip-cat-approx}) amount to $0.1\\%$ at most, as compared to the numerical solution of Eq.~(\\ref{eq:n-drip-mue}). The proton fraction $Z\/A$ \nat the neutron-drip point is also found to be strongly correlated with the symmetry energy. As shown in the right panel of Fig.~\\ref{fig:cat-AZ-B}, $Z\/A$ decreases \nalmost linearly with increasing $L$ (or $J$). Similar behaviors of $Z\/A$ and $n_{\\rm drip}$ with $L$ have been recently obtained in Ref.~\\cite{bao2014}, and \ncan be inferred from the discussions in Refs.~\\cite{roca2008,provid2014,grill2014}. Nevertheless, in all cases they considered the limiting case $B_\\star=0$. \nIn Ref.~\\cite{bao2014}, the authors studied the role of the symmetry energy on the properties of neutron-star crusts around \nthe neutron-drip threshold using two sets of relativistic mean field (RMF) models based on the TM1 and IUFSU parametrizations respectively. They generated \nseries of models so as to achieve different values of $L$ keeping the symmetry energy at $n=0.11$~fm$^{-3}$ fixed. In our case, the fixed value of the symmetry \nenergy at $n\\approx 0.11$~fm$^{-3}$ results from the mass fit without any further constraint. Although the variations of $Z\/A$ and $n_{\\rm drip}$ they found \nare nonlinear over this range of values of $L$, the variations become almost linear on the narrower range we consider (from about 37~MeV to about 69~MeV). \nAlthough it has been found that a soft symmetry energy favors neutron drip in isolated nuclei~\\cite{todd2003}, this result does not necessarily imply the observed \ncorrelation between $n_{\\rm drip}$ and $L$. Indeed, as recently discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus in the crust is actually stable \nagainst neutron emission, but unstable against electron captures followed by neutron emission. Actually, as will be discussed in Sec.~\\ref{sec:results-accreting}, \naccreting neutron star crusts exhibit different correlations between $n_{\\rm drip}$ and $L$. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_cat_22-25_B_ndrip.eps}\n\\end{center}\n\\caption{(Color online) Neutron-drip density in nonaccreting neutron-star crusts as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation\nand for different magnetic field\nstrengths, as obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models~\\cite{goriely2013}.}\n\\label{fig:cat-ndrip-B}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_cat_22-25_AZ_B.eps}\n\\end{center}\n\\caption{(Color online) (a) Mass number $A$ and (b) proton fraction $Z\/A$ at the neutron-drip transition for nonaccreting neutron-star crusts as a function of the slope $L$ of the \nsymmetry energy of infinite homogeneous nuclear matter at saturation, \nas obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models~\\cite{goriely2013}. Squares (circles) correspond to $B_\\star=0, 500$ and $1500$ \n($B_\\star =1000$ and $2000$).}\n\\label{fig:cat-AZ-B}\n\\end{figure*}\n\n\nThe role of the symmetry energy on the properties of the crust at the neutron-drip transition can be understood as follows. Neglecting electron-ion interactions, \nthe neutron-drip condition~(\\ref{eq:n-drip-mue}) reduces to\n\\begin{equation}\n\\label{eq:mue-approx}\n\\mu_e\\approx \\mu_e^{\\rm drip} = m_e c^2 + \\frac{A}{Z}\\left(m_n c^2- \\frac{M^\\prime(A,Z)c^2}{A}\\right)\\, .\n\\end{equation}\nFor the sake of simplicity, let us consider a two-parameter mass formula: \n\\begin{equation}\n\\label{eq:2par-ldm}\nM^\\prime(A,Z)c^2=A \\left[a_{\\rm eff}+J_{\\rm eff} \\left(1-2 \\frac{Z}{A} \\right)^2+m_u c^2 \\right] + Z m_e c^2\\, ,\n\\end{equation}\nwhere $m_u$ is the unified mass unit, $a_{\\rm eff}<0$ is the contribution from charge-symmetric matter, while the deviations introduced by \nthe charge asymmetry are embedded in the coefficient $J_{\\rm eff}>0$. Note that due to nuclear surface effects the values of these coefficients \ndo not need to be the same as their corresponding values in infinite homogeneous nuclear matter at saturation. In particular, as already discussed \nin Sec.~\\ref{sec:hfb-models}, the ``effective'' symmetry energy coefficient $J_{\\rm eff}$ is expected to be smaller than $J$, and to decrease \nwith increasing $J$ or $L$. Minimizing $g$ using Eq.~(\\ref{eq:mue-approx}) and the mass formula~(\\ref{eq:2par-ldm}), the equilibrium proton fraction \nat the neutron-drip transition is approximately given by~\\cite{chamel2012} \n\\begin{equation}\n\\label{eq:zovera-cat}\n\\frac{Z}{A}\\approx \\frac{1}{2}\\sqrt{1+\\frac{a_{\\rm eff}}{J_{\\rm eff}}}\\, .\n\\end{equation}\nThis shows that $Z\/A$ decreases with increasing $L$ (decreasing $J_{\\rm eff}$). Note that in this analysis we have not made any assumption \nregarding the magnetic field. In other words, the correlation between $Z\/A$ and $L$ (or $J$) is thus expected to be independent of the magnetic \nfield strength (at least at the level of accuracy of the simple mass formula considered here), in agreement with the results plotted in the right panel of \nFig.~\\ref{fig:cat-AZ-B}. It follows from Eq.~(\\ref{eq:2par-ldm}) that decreasing $J_{\\rm eff}$ increases $M^\\prime(A,Z)$ (the energy cost associated \nwith charge asymmetry is reduced, therefore nuclei are more bound). Using Eq.~(\\ref{eq:mue-approx}), we find that $\\mu_e^{\\rm drip}$ increases with \n$L$. Since $n_{\\rm drip}$ and $P_{\\rm drip}$ increase with $\\mu_e^{\\rm drip}$, as shown in Eqs.~(\\ref{eq:ndrip-cat-approx}) and (\\ref{eq:Pdrip-cat-approx})\nin the absence of magnetic field, and in Eqs.~(\\ref{eq:ndrip-cat-approx-b}) and (\\ref{eq:Pdrip-cat-approx-b}) in the presence of a strongly quantizing \nmagnetic field, we can thus conclude that the neutron-drip transition is shifted to higher density and pressure with increasing the symmetry energy, as \nshown in Fig.~\\ref{fig:cat-ndrip-B}. \n\n\nThe equilibrium nucleus at the neutron-drip transition is less sensitive to the symmetry energy, as previously noticed in Ref.~\\cite{bao2014} in the absence \nof magnetic fields (see their Fig.~9). This can be understood as follows. The equilibrium with respect to weak interaction processes requires \n\\begin{equation}\n\\label{eq:betaeq}\n\\mu_p+\\mu_e=\\mu_n\\, ,\n\\end{equation}\nwhere $\\mu_p$ ($\\mu_n$) is the proton (neutron) chemical potential. Substituting the neutron-drip value of the neutron chemical potential $\\mu_n=m_n c^2$ in\nEq.~(\\ref{eq:betaeq}) and using Eq.~(\\ref{eq:mue-approx}), we obtain\n\\begin{equation}\n\\label{eq:mup}\n\\mu_p-m_p c^2 = Q_{n,\\beta}+\\frac{A}{Z}\\left(\\frac{M^\\prime(A,Z)c^2}{A}-m_n c^2 \\right)\\, ,\n\\end{equation}\nwhere $m_p$ is the proton mass and $Q_{n,\\beta} = 0.782$~MeV is the $\\beta$-decay energy of the neutron. The quantity on the left-hand side of Eq.~(\\ref{eq:mup}) \nis approximately equal to the opposite of the one-proton separation energy. This shows that the equilibrium nucleus is uniquely determined by nuclear masses only, \nand is sensitive to the details of the nuclear structure. As a consequence, the predicted nucleus depends on the nuclear mass model employed (see, e.g., \nRefs.~\\cite{roca2008,pearson2011,wolf2013,kreim2013,chamel2015c}). \nTo better illustrate this point, we have plotted in Fig.~\\ref{fig:mass_diff} the differences in \nthe mass predictions between HFB-22, HFB-25, and HFB-24 mass models for two isotopic chains, corresponding to the proton number at the neutron-drip \npoint (see also Table~\\ref{tab:drip-cat}). As shown in Fig.~\\ref{fig:mass_diff}, the HFB-22 model deviates more significantly from the ``reference'' \nmodel HFB-24 than HFB-25, thus explaining the quantitative differences in the dripping nucleus. The variations of $Z$ and $A$ with $L$ we find \nappear to be more irregular than those shown in Fig.~9 of Ref.~\\cite{bao2014}. This stems from the fact that in Ref.~\\cite{bao2014} nuclear masses were \ncalculated using the semi-classical Thomas-Fermi approximation, which does not take into account pairing and shell effects contrary to the fully quantum mechanical \nmass models~\\cite{goriely2013} employed here. \n\n\nThe presence of a strong magnetic field can change the composition at the neutron-drip point, as shown in the left panel of Fig.~\\ref{fig:cat-AZ-B} \n(see also Tables~\\ref{tab:drip-cat-mag-500}-\\ref{tab:drip-cat-mag-2000}). \nHowever, such behavior is only observed for the nuclear mass model HFB-22. In particular, the equilibrium nucleus is $^{122}$Kr for $B_\\star=0, 500$ and $1500$, \nwhile for $B_\\star=1000$ and $2000$ it is $^{128}$Sr. These results can be understood as follows. As discussed in Sec.~\\ref{sec:neutron-drip}, the equilibrium \nnucleus at the neutron-drip pressure $P_{\\rm drip}$ must be such as to minimize the Gibbs free energy per nucleon, therefore we must have \n\\begin{equation}\\label{eq:stability}\ng(A,Z,P_{\\rm drip}) < g(A^\\prime,Z^\\prime,P_{\\rm drip})\\, ,\n\\end{equation}\nfor any values of $A^\\prime\\neq A$ and $Z^\\prime\\neq Z$. This condition can be approximately expressed as~\\cite{chamel2015b}\n\\begin{equation}\\label{eq:dripping-nucleus}\n\\frac{M^\\prime(A^\\prime,Z^\\prime)c^2}{Z^\\prime} - \\frac{M^\\prime(A,Z)c^2}{Z} > \\left(\\frac{A^\\prime}{Z^\\prime}-\\frac{A}{Z}\\right) m_n c^2 + C e^2 n_e^{1\/3}\\left( Z^{2\/3}-Z^{\\prime\\, 2\/3}\\right)\\, ,\n\\end{equation}\nwhere the electron density $n_e$ has to be determined from Eq.~(\\ref{eq:n-drip-mue}). Equation~(\\ref{eq:dripping-nucleus}) can be equivalently written \nas $n_e < n_e^0$, where \n\\begin{equation}\\label{eq:ne0}\nn_e^0\\equiv \\biggl[\\frac{M^\\prime(A^\\prime,Z^\\prime)c^2}{Z^\\prime} - \\frac{M^\\prime(A,Z)c^2}{Z} - \\left(\\frac{A^\\prime}{Z^\\prime}-\\frac{A}{Z}\\right) m_n c^2\\biggr]^3\n\\biggl[C e^2 \\left( Z^{2\/3}-Z^{\\prime\\, 2\/3}\\right)\\biggr]^{-3}\\, .\n\\end{equation}\nThe HFB-22 nuclear mass model predicts very similar values for the  \nthreshold electron Fermi energy $\\mu_e^{\\rm drip}$ for nuclei $^{128}$Sr and $^{122}$Kr: $24.970$ and $25.006$~MeV respectively. Substituting the theoretical \nvalues of the masses of $^{128}$Sr and $^{122}$Kr in Eq.~(\\ref{eq:ne0}) with $Z=36$, $A=122$, $Z^\\prime=38$, $A^\\prime=128$, we obtain $n_e^0\\approx 8.54\\times 10^{-5}$~fm$^{-3}$. \nDue to Landau quantization of electron motion, $n_e$ varies non-monotonically with $B_\\star$. As a consequence, the lattice term in Eq.~(\\ref{eq:dripping-nucleus}) \ncan thus become comparable to the other terms depending on $B_\\star$ to the effect that the condition (\\ref{eq:stability}) may be violated (i.e. $n_e\\geq n_e^0$), as \nshown in Fig.~\\ref{fig:drip-nuc-hfb22}. \nTransitions between $^{128}$Sr and $^{122}$Kr are found to occur at magnetic field strengths $B_\\star\\approx 861, 1239$ and $1883$. As shown in the right panel of \nFig.~\\ref{fig:cat-AZ-B}, the proton fraction $Z\/A$ is barely affected by these changes of composition. In other words, the correlation between $Z\/A$ and the symmetry \nenergy is almost independent of the magnetic field strength, as previously discussed. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{mass_differ_cat.eps}\n\\end{center}\n\\caption{(Color online) Difference in mass predictions for two pairs of Brussels-Montreal nuclear mass models along the two isotopic chains $Z=36$ and $Z=38$ relevant at neutron drip.}\n\\label{fig:mass_diff}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_nuc_hfb22.eps}\n\\end{center}\n\\caption{Electron density $n_e$ (solid line) at the neutron-drip transition in nonaccreting neutron-star crusts as a \nfunction of the magnetic field strength, using the Brussels-Montreal nuclear mass model HFB-22 and assuming that the neutron-drip nucleus \nis $^{122}$Kr, as in the absence of magnetic field. The horizontal dotted line represents $n_e^0$, as given by Eq.~(\\ref{eq:ne0}). \nSee text for details. }\n\\label{fig:drip-nuc-hfb22}\n\\end{figure*}\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of nonaccreting and unmagnetized neutron stars, as predicted by the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models: mass and atomic numbers of the dripping nucleus, baryon density, and corresponding pressure. }\\smallskip\n\\label{tab:drip-cat}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.71 & 4.99  \\\\\nHFB-23 & 126 & 38 & 2.63 & 4.93  \\\\\nHFB-24 & 124 & 38 & 2.56 & 4.87  \\\\\nHFB-25 & 122 & 38 & 2.51 & 4.83 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of nonaccreting magnetized neutron stars with $B_\\star=500$, as predicted by the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models: mass and atomic numbers of the dripping nucleus, baryon density, and corresponding pressure.}\\smallskip\n\\label{tab:drip-cat-mag-500}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.74 & 5.52 \\\\\nHFB-23 & 126 & 38 & 2.66 & 5.45 \\\\\nHFB-24 & 124 & 38 & 2.61 & 5.39 \\\\\nHFB-25 & 122 & 38 & 2.55 & 5.35 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=1000$.}\\smallskip\n\\label{tab:drip-cat-mag-1000}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 128 & 38 & 3.06 & 6.70 \\\\\nHFB-23 & 126 & 38 & 2.98 & 6.63 \\\\\nHFB-24 & 124 & 38 & 2.91 & 6.56 \\\\\nHFB-25 & 122 & 38 & 2.85 & 6.52 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=1500$.}\\smallskip\n\\label{tab:drip-cat-mag-1500}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.30 & 8.66 \\\\\nHFB-23 & 126 & 38 & 2.24 & 8.60 \\\\\nHFB-24 & 124 & 38 & 2.20 & 8.56 \\\\\nHFB-25 & 122 & 38 & 2.16 & 8.52 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=2000$.}\\smallskip\n\\label{tab:drip-cat-mag-2000}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-3}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 128 & 38 & 3.06 & 11.6 \\\\\nHFB-23 & 126 & 38 & 3.00 & 11.6 \\\\\nHFB-24 & 124 & 38 & 2.95 & 11.5 \\\\\nHFB-25 & 122 & 38 & 2.89 & 11.4 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Accreting neutron stars}\n\\label{sec:results-accreting}\n\nFor accreting neutron-star crusts, we have studied the neutron-drip transition as explained in Sec.~\\ref{sec:neutron-drip} (see also Ref.~\\cite{chamel2015a}). \nWe have considered different initial compositions: the ashes produced by an $rp$-process during an x-ray burst \\cite{schatz2001}, and the ashes produced by \nsteady state hydrogen and helium burning \\cite{schatz2003} as expected to occur during superbursts \\cite{gupta2007}. After determining the dripping nucleus, we \nhave calculated the neutron-drip density and pressure by solving numerically Eq.~(\\ref{eq:e-capture+n-emission-gibbs-approx}) considering all possible \nneutron-emission processes. Results are summarized in Tables~\\ref{tab:drip-acc-22}-\\ref{tab:drip-acc-25} for different nuclear mass models. \nFor comparison with previous works~\\cite{hz1990, hz2003}, we have also considered ashes of x-ray bursts consisting of pure $^{56}$Fe. Results are indicated in \nTable~\\ref{tab:drip-acc-a56}. \n\nAs already pointed out in Ref.~\\cite{chamel2015a}, for a given nuclear mass model the neutron-drip transition in accreting neutron stars can occur at \neither lower or at higher densities and pressures than in nonaccreting neutron stars. Depending on the mass model adopted, the neutron-drip density thus \nranges from $1.60 \\times 10^{-4}$~fm$^{-3}$ to $3.90 \\times 10^{-4}$~fm$^{-3}$, \nand the corresponding pressure from $2.77 \\times 10^{-4}$~MeV~fm$^{-3}$ to $7.77 \\times 10^{-4}$~MeV~fm$^{-3}$. The numerical results \nobtained solving Eq.~(\\ref{eq:e-capture+n-emission-gibbs-approx}) are reproduced by the analytical formulas~(\\ref{eq:ndrip-acc}) and \n(\\ref{eq:Pdrip-acc}) with an error of $0.1\\%$ at most. \n\nThe change of the neutron-drip density with the slope $L$ of the symmetry energy is found to be very different from that obtained in \nnonaccreting neutron-star crusts. As shown in Fig.~\\ref{fig:acc-ndrip}, for some ashes like $^{104}$Cd, no obvious correlation is observed while for \nother ashes like $^{66}$Ni, $n_{\\rm drip-acc}$ appears to be \\emph{anticorrelated} with $L$: $n_{\\rm drip-acc}$ decreases with increasing $L$. \nThis behavior can be understood as follows. Ignoring electron-ion interactions, the neutron-drip condition~(\\ref{eq:e-capture+n-emission-gibbs-approx}) \nreduces to\n\\begin{equation}\n\\mu_e\\approx \\mu_e^{\\rm drip-acc} = M^\\prime(A-\\Delta N,Z-1)c^2-M^\\prime(A,Z)c^2 +m_n c^2 \\Delta N + m_e c^2\\, , \n\\end{equation}\nwhich can be more conveniently written as \n\\begin{equation}\\label{eq:muedrip-acc-s}\n\\mu_e^{\\rm drip-acc} = S_{\\Delta N n}(A,Z-1) + \\mu_e^\\beta(A,Z)\\, ,\n\\end{equation}\nusing Eq.~(\\ref{eq:sn-dn}) and introducing the threshold electron Fermi energy for the onset of electron captures (see, e.g. Ref.~\\cite{chamel2015d} for a recent \ndiscussion)\n\\begin{equation}\n\\mu_e^\\beta(A,Z)=M^\\prime(A,Z-1)c^2 -M^\\prime(A,Z)c^2 + m_e c^2\\, .\n\\end{equation}\nThe mass difference $\\Delta M^\\prime=M'(A,Z-1) -M'(A,Z)$, which represents the change of mass associated with the substitution of a proton by a neutron, \nis expected to be mainly determined by symmetry energy effects. On the other hand, the $\\Delta N$-neutron separation energy $S_{\\Delta N n}(A,Z-1)$ \nis likely to be more dependent on the details of the nuclear structure than on the symmetry energy. As shown in Tables~\\ref{tab:drip-acc-22}-\\ref{tab:drip-acc-a56}, \n$|S_{\\Delta N n}(A,Z-1)| \\ll \\Delta M^\\prime c^2$ therefore $\\mu_e^{\\rm drip-acc}\\approx \\mu_e^\\beta$. On the other hand, as discussed in Sec.~\\ref{sec:neutron-drip}, \nthe composition of accreting neutron-star crusts at the neutron-drip transition is directly determined by the condition $S_{\\Delta N n}(A,Z-1)<0$. Provided \nthe dependence on the symmetry energy of $S_{\\Delta N n}(A,Z-1)$ is weak enough, the dripping nucleus will thus be independent of $L$. In this case, the \nvariations of $S_{\\Delta N n}(A,Z-1)$ with $L$ are typically much smaller than the variations of $\\Delta M^\\prime c^2$, as shown in Figs.~\\ref{fig:deltamprime} and \n\\ref{fig:Sdeltan}. Using the simple mass formula~(\\ref{eq:2par-ldm}), we find\n\\begin{equation}\n\\mu_e^\\beta(A,Z)=\\Delta M^\\prime c^2 +m_e c^2 \\approx 4J_{\\rm eff}\\left(1+\\frac{1-2Z}{A}\\right)\\, .\n\\end{equation}\nThis means that with increasing $J$ or $L$ (decreasing $J_{\\rm eff}$), $\\Delta M^\\prime$ and $\\mu_e^\\beta(A,Z)$ both \\emph{decrease}. It thus follows from \nEqs.~(\\ref{eq:ndrip-acc}), (\\ref{eq:Pdrip-acc}), and Eqs.~(\\ref{eq:muedrip-acc-s}), that $\\mu_e^{\\rm drip-acc}$, $n_{\\rm drip-acc}$ and $P_{\\rm drip-acc}$ \nalso decrease with $L$, as shown in the upper panel of Fig.~\\ref{fig:acc-ndrip}. \nIn the peculiar case of $A=105$, the rather low value of $\\Delta M^\\prime c^2$ predicted by HFB-25 is compensated by a comparatively high value $S_{\\Delta N n}(A,Z-1)$, \nas can be seen in Figs.~\\ref{fig:deltamprime} and \\ref{fig:Sdeltan}. As a result, $n_{\\rm drip-acc}$ is still found to decrease with increasing $L$ despite the nonmonotonic \nvariation of $\\Delta M^\\prime$, as shown in Fig.~\\ref{fig:acc-ndrip}. \nFor some ashes, the variations of $S_{\\Delta N n}(A,Z-1)$ are comparable to those of $\\Delta M^\\prime c^2$, and large enough to even change the composition. This  \nleads to nonmonotonic variations of the neutron-drip density and pressure, as illustrated in the lower panel of Fig.~\\ref{fig:acc-ndrip}. In these cases, \neffects other than the symmetry energy play a role. The anticorrelation between $n_{\\rm drip-acc}$ (or $P_{\\rm drip-acc}$) and $L$ thus relies to \na large extent on the importance of nuclear structure effects far from the stability valley. \n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of accreting neutron stars, as predicted by the HFB-22 Brussels-Montreal nuclear mass model: mass and atomic numbers of the dripping nucleus, number of emitted neutrons, baryon density $n_{\\rm drip-acc}$  ($10^{-4}$ fm$^{-3}$),  and corresponding pressure $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$), $S_{\\Delta N n}(A,Z-1)$ (MeV), $\\Delta M^\\prime$ (MeV$\/c^2$), and $\\mu_e^{\\rm drip-acc}$ (MeV).\nThe mass numbers $A$ are listed from top to bottom considering that the ashes are produced by ordinary\nx-ray bursts (upper panel) or superbursts (lower panel). See text for details.}\\smallskip\n\\label{tab:drip-acc-22}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   & $ Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.71 &  5.31 & -0.79 & 25.14 & 24.86 \\\\\n 105  &   33 &  1 &   1.90 &  3.40 & -1.00 & 22.70 & 22.21 \\\\\n  68  &   22 &  1 &   2.31 &   4.64 & -0.28 & 24.14 & 24.37 \\\\\n  64  &   18 &  5 &   3.90 &   7.77 & -1.85 & 29.23 & 27.89 \\\\\n  72  &   22 &  1 &   2.89 &   5.78 & -0.31 & 25.55 & 25.75 \\\\\n  76  &   24 &  1 &   2.86 &   5.95 & -0.17 & 25.52 & 25.86 \\\\\n  98  &   32 &  1 &   1.93 &   3.66 & -0.20 & 22.34 & 22.65 \\\\\n 103  &   33 &  1 &   1.60 &  2.77 & -0.02 & 20.62 & 21.11 \\\\\n 106  &   32 &  1 &   2.92 &  5.71 & -0.69 & 25.50 & 25.32 \\\\\n  \\hline\n  66  &   22 &  1 &   1.99 &   3.95 & -0.19 & 23.09 & 23.41 \\\\\n  64  &   18 &  5 &   3.90 &   7.77 & -1.85 & 29.23 & 27.89 \\\\\n  60  &   20 &  1 &   1.83 &   3.55 & -1.67 & 24.02 & 22.86 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-23 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-23}\n\\begin{tabular}{cccccccc}\n\\hline\n   $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.83 &  5.62 & -1.02 & 25.73 & 25.22 \\\\\n 105  &   33 &  1 &   1.97 &  3.57 & -1.33 & 23.30 & 22.48 \\\\\n  68  &   22 &  1 &   2.35 &   4.73 & -0.56 & 24.54 & 24.49 \\\\\n  64  &   20 &  1 &   3.27 &   7.04 & -0.13 & 26.75 & 27.13 \\\\\n  72  &   22 &  1 &   3.06 &   6.24 & -0.11 & 25.84 & 26.24\\\\\n  76  &   24 &  1 &   2.94 &   6.18 & -0.38 & 25.98 & 26.11 \\\\\n  98  &   32 &  1 &   1.98 &   3.77 & -0.50 & 22.82 & 22.83 \\\\\n 103  &   31 &  1 &   2.45 &  4.50 & -0.91 & 24.29 & 23.89 \\\\\n 106  &   34 &  1 &   2.11 &  4.02 & -0.009 & 22.63 & 23.13 \\\\\n  \\hline\n  66  &   22 &  1 &   2.07 &   4.16 & -0.23 & 23.43 & 23.71 \\\\\n  64  &   20 &  1 &   3.27 &   7.04 & -0.13 & 26.75 & 27.13 \\\\\n  60  &   20 &  1 &   1.93 &   3.79 & -1.77 & 24.50 & 23.24 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-24 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-24}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.87 &  5.73 & -1.49 & 26.32 & 25.34 \\\\\n 105  &   33 &  1 &   2.10 &  3.89 & -1.11 & 23.57 & 22.97 \\\\\n  68  &   22 &  1 &   2.45 &   5.00 & -0.75 & 25.07 & 24.83 \\\\\n  64  &   22 &  1 &   1.66 &   3.22 & -0.07 & 21.81 & 22.25 \\\\\n  72  &   22 &  1 &   3.08 &   6.28 & -0.29 & 26.07 & 26.29 \\\\\n  76  &   24 &  1 &   3.10 &   6.61 & -0.46 & 26.50 & 26.55 \\\\\n  98  &   32 &  1 &   2.04 &   3.94 & -0.38 & 22.95 & 23.08 \\\\\n 103  &   31 &  3 &   2.49 &  4.59 & -0.89 & 24.39 & 24.01 \\\\\n 106  &   34 &  1 &   2.17 &  4.16 & -1.10 & 23.92 & 23.33 \\\\\n  \\hline\n  66  &   22 &  1 &   2.09 &   4.22 & -0.29 & 23.58 & 23.80 \\\\\n  64  &   22 &  1 &   1.66 &   3.22 & -0.07 & 21.81 & 22.25 \\\\\n  60  &   20 &  1 &   2.03 &   4.05 & -1.66 & 24.78 & 23.63 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-25 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-25}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   34 &  1 &  2.02 &  3.87 & -0.009 & 22.42 & 22.92 \\\\\n 105  &   33 &  1 &  2.18 &  4.07 & -0.57 & 23.30 & 23.24 \\\\\n  68  &   22 &  1 &   2.59 &  5.40 & -0.75 & 25.55 & 25.31 \\\\\n  64  &   22 &  1 &   1.79 &  3.59 & -0.18 & 22.52 & 22.85 \\\\\n  72  &   22 &  1 &   3.27 &  6.82 & -0.55 & 26.87 & 26.83 \\\\\n  76  &   24 &  1 &   3.14 &  6.73 & -0.58 & 26.74 & 26.67 \\\\\n  98  &   32 &  1 &   2.11 &  4.12 & -0.64 & 23.47 & 23.34 \\\\\n 103  &   33 &  1 &  1.78 &  3.18 & -0.76 & 22.10 & 21.85 \\\\\n 106  &   34 &  1 &  2.25 &  4.36 & -0.05 & 23.15 & 23.61 \\\\\n  \\hline\n  66  &   22 &  1 &   2.20 &   4.52 & -0.22 & 23.92 & 24.21 \\\\\n  64  &   22 &  1 &   1.79 &   3.59 & -0.18 & 22.52 & 22.85 \\\\\n  60  &   20 &  1 &   2.08 &   4.21 & -1.94 & 25.28 & 23.85 \\\\\n \\end{tabular}\n \\end{table}\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of accreting neutron stars, as predicted by different Brussels-Montreal nuclear mass models for $^{56}$Fe ashes: atomic number $Z$ of the dripping nucleus, number of emitted neutrons, density and corresponding pressure, $S_{\\Delta N n}(A,Z-1)$, $\\Delta M^\\prime$ (MeV$\/c^2$), and $\\mu_e^{\\rm drip-acc}$ (MeV). See text for details.}\\smallskip\n\\label{tab:drip-acc-a56}\n\\begin{tabular}{ccccc}\n\\hline\n                                              & HFB-22 & HFB-23  & HFB-24 &  HFB-25    \\\\\n\\hline\n$Z$                                           &   18   &  18     &  18    &  18   \\\\\n$\\Delta N$                                    &    1   &   1     &   3    &  1     \\\\\n$n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$) & 2.49   &  2.58   &  2.73  &  2.84   \\\\\n$P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$)  & 5.10   &  5.34   &  5.77  &  6.07   \\\\\n$S_{\\Delta N n}$ (MeV) & -1.17 & -1.32 & -3.27 & -1.61 \\\\\n$\\Delta M^\\prime$ (MeV$\/c^2$) & 25.76 & 26.20 & 28.64 & 27.32 \\\\\n$\\mu_e^{\\rm drip-acc}$ (MeV) & 25.10 & 25.39 & 25.88 & 26.22  \\\\\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_ndrip_new.eps}\n\\end{center}\n\\caption{(Color online) Neutron-drip density as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass \nmodels, for accreting neutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:acc-ndrip}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_deltam.eps}\n\\end{center}\n\\caption{(Color online) Mass difference $\\Delta M^\\prime$ (in units of MeV$\/c^2$) as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the \nHFB-22 to HFB-25 Brussels-Montreal nuclear mass models, for accreting \nneutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:deltamprime}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_Sdeltan.eps}\n\\end{center}\n\\caption{(Color online) $\\Delta N$-neutron separation energy $S_{\\Delta N n}$ (in units of MeV) as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the \nHFB-22 to HFB-25 Brussels-Montreal nuclear mass models, for accreting \nneutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:Sdeltan}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have studied the role of the symmetry energy on the neutron-drip transition in both accreting and nonaccreting neutron-star crusts. We have also allowed for  \nthe presence of a strong magnetic field, as in magnetars. The masses of nuclei encountered in this region of the neutron-star crust are experimentally unknown. \nFor this reason, we have employed a recent family of microscopic nuclear mass models, from HFB-22 to HFB-25, developed by the Brussels-Montreal collaboration~\\cite{goriely2013}. These models provide equally good fits to the 2353 measured masses of nuclei with $N$ and $Z \\geq 8$ from the 2012 Atomic Mass \nEvaluation~\\cite{audi2012}, with a root-mean-square deviation of about $0.6$~MeV. \nOn the other hand, these models lead to different predictions for the behavior of the symmetry energy in infinite homogeneous\nnuclear matter.\nIn particular, these functionals \nwere constrained so as to yield different \nvalues of the symmetry energy at saturation, from $J=29$~MeV to $J=32$~MeV, the slope of the symmetry energy ranging from $L=37$~MeV to $L=69$~MeV. \n\nFor nonaccreting weakly magnetized neutron stars, the neutron-drip density $n_{\\rm drip}$ is found to increase almost linearly with $L$ (or equivalently with $J$) \nwhile the proton fraction $Z\/A$ decreases, in agreement with previous studies~\\cite{bao2014} (see also Refs.~\\cite{roca2008,provid2014,grill2014}). \nIn the presence of a strong magnetic field, the dripping nucleus\nhence also $Z\/A$ is unchanged, as found in Refs.~\\cite{chamel2012,chamel2015b}, for all models but HFB-22. In this case, the dripping nucleus alternates between \n$^{122}$Kr and $^{128}$Sr depending on the magnetic field strength. This peculiar behavior arises from Landau quantization of electron motion and from the fact \nthat the threshold electron Fermi energy $\\mu_e^{\\rm drip}$ are almost equal. Their proton fraction are also very similar so that all in all the linear \ncorrelation between $Z\/A$ and $L$ is hardly affected by the magnetic field. The neutron-drip density $n_{\\rm drip}$ exhibits typical quantum oscillations as \na function of the magnetic field strength, as recently discussed in Ref.~\\cite{chamel2015b}. Still, $n_{\\rm drip}$ remains linearly correlated with $L$. \nAlthough a soft symmetry energy favors neutron drip in isolated nuclei~\\cite{todd2003}, this result does not necessarily imply the observed correlation \nbetween $n_{\\rm drip}$ and $L$. Indeed, as recently discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus in the crust is actually stable \nagainst neutron emission, but unstable against electron captures followed by neutron emission. In fact, such a correlation is not found in accreting neutron-star crusts. \nDepending on the initial composition of the ashes from x-ray bursts and superbursts, $n_{\\rm drip}$ \n\\emph{decreases} almost linearly with increasing $L$ while the dripping nucleus remains the same. In other cases, the symmetry energy does not seem \nto play any role. \n\nWe have qualitatively explained these different behaviors using a simple mass formula, and making use of the analytical expressions for the neutron-drip density and \npressure obtained in Refs.~\\cite{chamel2015a,chamel2015b}. In particular, we have shown that the anticorrelation between $n_{\\rm drip}$ and $L$ in accreting \nneutron stars depends to a large extent to the relative importance of nuclear structure effects (like shell effects and pairing) and symmetry energy effects on the \nneutron separation energy. More precisely, the anticorrelation is broken whenever the differences between the neutron separation energies predicted by the different \nmass models are large enough to change the dripping nucleus. \n\nIn any case, the composition of the deepest layers of the outer crust of a neutron star is very sensitive to the details of the nuclear structure far from the stability valley. In nonaccreting neutron-star crusts, the neutron-drip transition is mainly governed by the values of the masses of very neutron-rich strontium and krypton isotopes. \nAlthough the masses of these nuclei have not yet been measured, the composition of nonaccreting neutron-star crusts has been recently constrained by experiment \nto deeper layers~\\cite{wolf2013}. The nuclei thought to be present in accreting neutron star crusts span a much larger region of the nuclear chart, depending on \nthe ashes from x-ray bursts and superbursts. In these neutron stars, the neutron-drip transition is not directly determined by nuclear masses but rather by some \ncombinations of masses, namely the (multiple) neutron separation energies and the isobaric two-point mass differences. \n\nThe onset of neutron emission by nuclei marks the transition to the inner region of the neutron-star crust, where neutron-proton clusters coexist with a neutron liquid. \nIn turn, this neutron liquid, which becomes superfluid at low enough temperatures, is expected to play a role in various observed astrophysical phenomena  (see, e.g., \nRef.~\\cite{chamelhaensel2008} for a review) like sudden spin-ups and spin-downs (so-called ``glitches'' and ``antiglitches'', respectively)~\\cite{dib2008, gug2014, \narchibald2013, sasmaz2014, duncan2013, kantor2014}, quasiperiodic oscillations detected in the giant flares from soft $\\gamma$-ray repeaters~\\cite{passamonti2014}, \ncooling of strongly magnetized neutron stars~\\cite{aguilera2009}, deep crustal heating (most of the heat being released near the neutron-drip transition~\\cite{haensel2008}), \nand the thermal relaxation of quasipersistent soft x-ray transients~\\cite{shternin2007,brown2009,page2013}. By shifting the neutron-drip transition to higher or lower densities, the symmetry energy may thus leave its imprint on these astrophysical phenomena.  \n\n\n\n\\begin{acknowledgments}\nThis work has been mainly supported by Fonds de la Recherche Scientifique - FNRS (Belgium), and by the bilateral project between Fonds de la Recherche Scientifique - FNRS (Belgium), Wallonie-Bruxelles-International (Belgium) and the Bulgarian Academy of Sciences. \nThis work has been also partially supported by the Bulgarian National Science Fund under Contract No. DFNI-T02\/19t and the Cooperation in Science and Technology (COST Action) MP1304 ``NewCompStar''. The authors would like to thank J.~M. Pearson for very fruitful discussions.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}