diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfkor" "b/data_all_eng_slimpj/shuffled/split2/finalzzfkor" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfkor" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn the present paper we prove quantitative symmetry results for overdetermined problems involving the fractional Laplacian in unbounded exterior sets or bounded annular sets. These problems originate from the study of capacity of a set and relative capacity which, in the classical setting, are given by\n\\begin{equation*}\n {\\rm cap}(\\Omega) := \\inf \\left\\{ \\frac12 \\int_{\\mathbb{R}^n}|\\nabla v|^2 dx\\ :\\ v\\in C^{\\infty}_c(\\mathbb{R}^n),\\,\\,v{|_\\Omega} \\geq 1 \\right\\}\\,,\n\\end{equation*}\nand\n\\begin{equation*}\n{\\rm cap}(\\Omega;D) := \\inf \\left\\{ \\frac{1}{2} \\int_{D} |\\nabla v|^2 dx:\\ v \\in C_c^\\infty (\\Omega),\\ v_{|_D} \\geq 1\\right\\},\n\\end{equation*}\nrespectively; here $D$ and $\\Omega$ are bounded open sets, with $\\overline D \\subset \\Omega \\subset \\mathbb{R}^n$, $n \\geq 3$, and $\\nabla v$ is the gradient of the function $v$.\n\nInstead of the classical notion of capacity, in this paper we consider the capacity in a fractional setting. For a parameter $s \\in (0,1)$, the $\\textit{fractional capacity of order s}$ (or $\\textit{s-capacity}$) of the set $\\Omega$ is defined as follows:\n\\begin{equation} \\label{caps_def}\n {\\rm cap}_s(\\Omega) := \\inf\\{ [v]_s^2 \\ | \\ v \\in C_c^\\infty (\\mathbb{R}^n),\\ v_{|_\\Omega} \\geq 1\\} \\,,\n\\end{equation}\nwhere $[v]_s$ is the $\\textit{Gagliardo seminorm}$ of $v$ which is defined by\n\\begin{equation*}\n [v]_s^2 := \\int_{\\mathbb{R}^{2n}} \\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} dx dy \\,.\n\\end{equation*}\nAnalogously, one can define the $\\textit{relative fractional capacity of order s}$ of the couple of sets $(\\Omega,D)$ by\n\\begin{equation} \\label{caps_rel_def}\n {\\rm cap}_s(\\Omega;D) := \\inf\\{ [v]_s^2 \\ | \\ v \\in C_c^\\infty (\\Omega),\\ v_{|_D} \\geq 1\\} \\,.\n\\end{equation}\n\nThe Euler-Lagrange equations associated to \\eqref{caps_def} and \\eqref{caps_rel_def} are both related to the so-called fractional Laplacian of order $s$ (or $s$-Laplacian), which is denoted by $(-\\Delta)^s$ and it is given by \n\\begin{equation*}\n (-\\Delta)^s u(x) := c_{n,s} \\, P.V. \\int_{\\mathbb{R}^n} \\frac{u(x) - u(z)}{|x-z|^{n+2s}} dz,\n\\end{equation*} \nfor $u \\in C^{\\infty}_c (\\mathbb{R}^n)$, where\n\\begin{equation} \\label{cns}\n c_{n,s} = s \\, (1-s) \\, 4s \\pi^{-n\/2} \\frac{\\Gamma(n\/2 + s)}{\\Gamma(2-s)} \n\\end{equation}\n(see for example \\cite{di2012hitchhiker}). \nIt can be proved that ${\\rm cap}_s(\\Omega)$ and ${\\rm cap}_s(\\Omega;D)$ are uniquely achieved by two functions $u_\\Omega \\in H^s(\\mathbb{R}^n)$ and $u_{\\Omega,D} \\in H^s(\\Omega)$,\\footnote{Given an open set $E \\subseteq \\mathbb{R}^n$, we denote by $H^s(E)$ the homogeneous fractional Sobolev space of $E$, which defined as the completion of $C_c^\\infty (E)$ with respect to the Gagliardo seminorm of order $s$.\n} which satisfy\n\\begin{equation}\n\\label{a2s1eq10}\n \\begin{cases}\n (-\\Delta)^s u_\\Omega = 0 \\qquad &\\text{in} \\ \\mathbb{R}^n \\setminus \\overline\\Omega,\\\\\n u_\\Omega = 1 \\qquad &\\text{in} \\ \\overline \\Omega,\\\\\n u_\\Omega(x) \\to 0 \\qquad &\\text{as} \\ |x| \\to +\\infty\\,,\n \\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\\label{a2s1eq12}\n \\begin{cases}\n (-\\Delta)^s u_{\\Omega,D} = 0 \\qquad &\\text{in} \\ A:= \\Omega \\setminus \\overline D,\\\\\n u_{\\Omega,D} = 1 \\qquad &\\text{in} \\ \\overline D,\\\\\n u_{\\Omega,D} = 0 \\qquad &\\text{in} \\ \\mathbb R^n \\setminus \\Omega \\,,\n \\end{cases}\n\\end{equation}\nrespectively. The function $u_\\Omega$ is sometimes called the \\textit{$s$-capacitary potential}.\n\nOverdetermined problems for \\eqref{a2s1eq10} and \\eqref{a2s1eq12} have been considered in \\cite{soave2019overdetermined} where the overdetermined condition is given on the normal \\emph{$s$-derivative} at the boundary, which is assumed to be constant in the spirit of Serrin's overdetermined problem. \n\nIn this paper we consider a somehow discrete version of Serrin's overdetermined condition, and we instead assume that the solution is constant on a surface parallel to the boundary.\\footnote{Regarding problem \\eqref{a2s1eq12}, it is more precise to say that the solution is constant on each connected component of the parallel surface, see Theorem \\ref{a2theorem3} below.} In this setting, our main results can be considered as the generalization of the results in \\cite{ciraolo2021symmetry} to exterior and annular domains.\n\nIn order to clearly state our results, we recall that the Minkowski sum of two sets $A$ and $B$ is defined by\n$$\nA+B = \\{ x + y \\ | \\ x \\in A \\ \\, y \\in B \\}.\n$$\n\n\n\nOur first result deals with solutions of problem \\eqref{a2s1eq10} with the overdermining assumption that the solution is constant on a surface parallel to $\\partial \\Omega$.\n\n\n\n\\begin{theorem}\n\\label{a2s1theorem1}\nLet $\\Omega$ be a bounded domain in $\\mathbb{R}^n$. Let $R>0$ and assume that $G:= \\Omega + B_R$ is such that $\\partial G$ of class $C^1$. Then, there exists a solution $u \\in C^s(\\mathbb{R}^n)$ of \\eqref{a2s1eq10} such that \n\\begin{equation}\\label{a2s1eq16}\n u = c \\qquad \\text{on} \\ \\partial G\n\\end{equation}\nfor some constant $c$ if and only if $G$ and $\\Omega$ are concentric balls and $u$ is radially symmetric.\n\\end{theorem}\n\nWe will prove Theorem \\ref{a2s1theorem1} by using the method of moving planes. Once symmetry is established, one can investigate the quantitative stability result for Theorem \\ref{a2s1theorem1}. The idea is to assume that the overdetermined condition \\eqref{a2s1eq16} is replaced by a weaker condition which implies that the solution is close to a constant on $\\partial G$. In this direction, it is useful to consider the Lipschitz seminorm $[u]_{\\Gamma}$ of $u$ on $\\Gamma=\\partial G$, which is given by\n\\begin{equation*}\n [u]_{\\Gamma} := \\sup_{x,y \\in \\Gamma, \\, x \\neq y} \\frac{|u(x) - u(y)|}{|x-y|}\n\\end{equation*}\nand the parameter\n\\begin{equation}\\label{def:outradius-inradius}\n \\rho (\\Omega) := \\inf \\{ |t - s| \\ | \\ \\exists p \\in \\Omega \\ \\mathrm{such \\ that} \\ B_s(p) \\, \\subset \\Omega \\subset B_t(p) \\} \\,,\n\\end{equation}\nwhich controls how much the set $\\Omega$ differs from a ball (clearly, $\\rho(\\Omega) = 0$ if and only if $\\Omega$ is a ball). \n\nAnother relevant quantity which we need to quantify the stability results is the radius of the touching ball condition. More precisely, given a set $E$ we denote the optimal exterior and interior radii in the touching ball condition by $\\mathfrak{r}_{E}^e$ and $\\mathfrak{r}_{E}^i$, respectively.\n \nHence, our main goal is to obtain quantitative bounds on $\\rho (\\Omega)$ in terms of $[u]_{\\partial G}$, as done in the following theorem.\n\n\\begin{theorem}\n\\label{a2theorem2}\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^n$ with $\\partial \\Omega$ of class $C^2$. Let $R>0$ and let $G =\\Omega + B_R$ be such that $\\partial G$ is of class $C^2$. Let $u \\in C^s(\\mathbb{R}^n)$ be a solution of \\eqref{a2s1eq10}. Then, we have that\n\\begin{equation}\n\\label{a2s1eq8}\n\\rho (\\Omega) \\leq C_* \\, [u]_{\\partial G}^{\\frac{1}{s +2}} ,\n\\end{equation}\nwith $C_* =C_*(n,s,R,\\mathrm{diam}(\\Omega),|\\Omega|,\\mathfrak{r}_{\\Omega}^e)> 0$, where $\\mathrm{diam}(\\Omega)$ and $|\\Omega|$ denote the diameter and the volume of $\\Omega$, respectively, and $\\mathfrak{r}_{\\Omega}^e$ is the radius of the exterior touching ball condition at $\\Omega$. \n\\end{theorem}\n\n\nIn the second part of the article we consider an overdetermined problem involving \\textit{annular sets}. More precisely, let $D,\\Omega \\subset \\mathbb{R}^n$ be bounded open domains such that $\\overline D \\subset \\Omega$, set\n\\begin{equation} \\label{A_def}\nA:= \\Omega \\setminus \\overline{D} \\,,\n\\end{equation}\nand we consider solutions to \\eqref{a2s1eq12}.\nIt is clear that, since $\\partial \\Omega$ and $\\partial D$ do not touch, we have that \n\\begin{equation} \\label{dbar}\n\\overline{d}:= {\\rm dist} (\\overline{D}, \\mathbb{R}^n \\setminus \\Omega) >0 \\,.\n\\end{equation}\nBy choosing a positive parameter $R < \\overline{d}\/2$ we have that the set \n\\begin{equation} \\label{Gamma_RA}\n\\Gamma_R^A := \\{ x \\in A \\ | \\ {\\rm dist} (x,\\partial A) = R \\}\n\\end{equation}\ncan be written as\n\\begin{equation} \\label{Gamma_union}\n\\Gamma_R^A = \\Gamma_R^D \\cup \\Gamma_R^\\Omega\\,,\n\\end{equation}\nwith \\footnote{Notice that $\\Gamma_R^A = \\partial ( (\\Omega^c + B_R) \\setminus (D+B_R))$.}\n\\begin{align*}\n \\Gamma_R^D := \\{ x \\in A \\ | \\ {\\rm dist}(x,\\partial D) = R \\},\\\\\n \\Gamma_R^\\Omega := \\{ x \\in A \\ | \\ {\\rm dist}(x,\\partial \\Omega) = R \\},\n\\end{align*}\nwith $\\Gamma_R^D \\cap \\Gamma_R^\\Omega = \\emptyset$. On each of these hypersurfaces we assume that the solution satisfies the overdetermined condition\n\\begin{equation}\n\\label{a2s1eq15}\n \\begin{aligned}\n u = \\alpha \\qquad &on \\ \\Gamma_R^D,\\\\\n u = \\beta \\qquad &on \\ \\Gamma_R^\\Omega,\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are two positive constants.\n\n\\bigskip\n\nWe have the following symmetry result. \n\n\\begin{theorem}\n\\label{a2theorem3}\nLet $A$ and $\\Gamma^A_R$ be given by \\eqref{A_def} and \\eqref{Gamma_RA}, respectively, where $R$ is such that $\\Gamma^A_R$ is of class $C^1$. \n\nLet $u \\in C^s(A)$ be a solution of \\eqref{a2s1eq12} satisfying the overdetermined conditions \\eqref{a2s1eq15}. \nThen, $D$ and $\\Omega$ are concentric balls and $u$ is radially symmetric.\n\\end{theorem}\n\n\nNow we describe the quantitative stability result that we obtain for Theorem \\ref{a2theorem3}. In this case, we replace the overdetermined condition \\eqref{a2s1eq12} by assuming that the solution has small Lipschitz seminorm on each connected component of $\\Gamma_R^A$. For this reason we define the following deficit \n\\begin{equation} \\label{deficit_A}\n \\mathrm{def}_A(u):= \\max \\{ [u]_{\\Gamma_R^D}, [u]_{\\Gamma_R^\\Omega} \\},\n\\end{equation}\nand we have the following result. \n\n\n\\begin{theorem}\n\\label{a2theorem4}\nLet $A$ and $\\Gamma^A_R$ be given by \\eqref{A_def} and \\eqref{Gamma_RA}, respectively, and assume that $\\partial A$ and $\\Gamma^A_R$ are of class $C^2$.\n\n\nLet $u \\in C^s(\\mathbb{R}^n)$ be a solution of \\eqref{a2s1eq12}. Then\n\\begin{equation}\n\\label{a2s1eq18}\n\\rho(D) + \\rho(\\Omega) \\leq C_* \\mathrm{def}_A(u)^{\\frac{1}{s +2}} \\,\n\\end{equation}\nwith $\\rho$ given by \\eqref{def:outradius-inradius} and $C_*=C_*(n,s,R,{\\rm diam}(\\Omega), |\\Omega|, |D|, \\mathfrak{r}^e_{D}, \\mathfrak{r}^i_{\\Omega}) > 0$, where $\\mathfrak{r}^e_{D}$ and $\\mathfrak{r}^i_{\\Omega}$ are the radius of the uniform exterior touching ball to $D$ and of the interior touching ball to $\\Omega$, respectively. \n\\end{theorem}\n\n\nTheorems \\ref{a2theorem2} and \\ref{a2theorem4} are the main results of this paper, and they are obtained by using a quantitative approach to the method of moving planes, which was originally developed in \\cite{aftalion1999approximate} (see also \\cite{CiraoloVezzoni}, \\cite{ciraolo2016solutions}, \\cite{CMS}, \\cite{ciraolo2018rigidity}, \\cite{ciraolo2021symmetry}). This approach presents many differences when applied in a classical local settings and in a fractional framework, and in this paper we prefer to tackle fractional problems. We mention that all the symmetry and quantitative symmetry results in this paper have their classical local counterpart, which can be still attacked by using the method of moving planes and it will be considered in a future work. \n\nWe finally notice that, in Theorems \\ref{a2s1theorem1}-\\ref{a2theorem4}, we assumed that $\\Omega$ and $D$ are domains and then they are connected. The connectedness assumption is not necessary and it can be easily removed, and hence our results can be extended in that setting. However, this has a cost in managing the notation and it would worsen the presentation and clarity of the paper. For this reason, we preferred to assume that $\\Omega$ and $D$ are connected.\n\n\n\\medskip\n\nThe paper is organized as follows. In Section 2 we present some preliminary notions and results, including a weak maximum principle for $s$-harmonic functions in an unbounded domain. Section 3 is devoted to the results for exterior sets and includes the standard machinery for the method of moving planes. In Section 4 we consider the problems involving annular domains.\n\n\n\n\n\\subsection*{Acknowledgements}\nThe authors have been partially supported by the ``Gruppo Nazionale per l'Analisi Matematica, la Probabilit\\`a e le loro Applicazioni'' (GNAMPA) of the ``Istituto Nazionale di Alta Matematica'' (INdAM, Italy). \n\n\n\n\n\n\\section{Preliminaries and notation}\n\nIn this section we introduce some notation and recall some results which will be useful in the rest of the paper. \n\n\nWe recall that given two functions $u,v \\in H^{s}(\\mathbb{R}^n)$ the $\\textit{Gagliardo seminorm}$ of $u$ is defined as\n\\begin{equation*}\n [u]_s^2 := \\frac{c_{n,s}}{2} \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} \\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx dy,\n\\end{equation*}\nand the scalar product in $H^s(\\mathbb{R}^n)$ between $u$ and $v$ is defined as\n\\begin{equation*}\n \\mathcal{E}(u,v) = \\frac{c_{n,s}}{2} \\bigg( \\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}\\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}} dx dy \\bigg)^{1\/2} \\,,\n\\end{equation*}\nwhere $c_{n,s}$ is given by \\eqref{cns}.\n\n\n\n\nIn order to write a Hopf's boundary point lemma in a quantitative form, it is useful to consider the solution $\\psi_{B_r(x_0)}$ to the fractional torsion problem in a ball of radius $r>0$ centered $x_0$, i.e. $\\psi_{B_r(x_0)}$ satisfies\n\\begin{equation} \\label{torsion_pb} \n\\begin{cases}\n(-\\Delta)^s \\psi_{B_r(x_0)} = 1 & \\text{ in } B_r(x_0) \\\\\n\\psi_{B_r(x_0)} = 0 & \\text{ in } \\mathbb{R}^n \\setminus B_r(x_0)\\,,\n\\end{cases}\n\\end{equation}\nand it is given by \n\\begin{equation} \\label{psi_def}\n\\psi_{B_r(x_0)}(x):= \\gamma_{n,s} (r^2 - |x-x_0|^2)^s_+\n\\end{equation}\nfor any $x \\in \\mathbb{R}^n$, where $\\gamma_{n,s}$ is a constant depending only on $n$ and $s$.\n\n\\begin{lemma}\n\\label{a2s3theorem1}\nLet $\\Omega \\subset \\mathbb{R}^n$ be an open set and let $u \\in C^s (\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s u \\geq 0 \\quad & \\text{in} \\ \\Omega,\\\\\n u \\geq 0 \\quad &\\text{in} \\ \\mathbb{R}^n \\setminus \\Omega \\,.\n \\end{cases}\n\\end{equation*}\nLet $x_0 \\in \\Omega$ and $r>0$ be such that $B_r(x_0) \\subseteq \\Omega$. Let $K\\subset \\mathbb{R}^n$ be a compact set such that \n$$\n|K| > 0 \\,, \\quad {\\rm dist}(K,B_r(x_0)) > 0 \\,,\\quad \\inf_K u > 0 \\,.\n$$ \nThen \\begin{equation*}\n u \\geq C_H \\, \\psi_{B_r(x_0)} \\qquad \\text{in} \\ \\, B_r(x_0)\\,,\n\\end{equation*}\nwhere \n\\begin{equation} \\label{CHopf}\n C_H := c_{n,s} \\frac{|K| \\, \\inf_K u}{(2r + {\\rm dist}(K,B_r(x_0)) + {\\rm diam}(K))^{n+2s}} \\,,\n\\end{equation}\nwith $c_{n,s}$ and $\\psi_{B_r(x_0)}$ given by \\eqref{cns} and \\eqref{psi_def}, respectively.\n\\end{lemma}\n\nLemma \\ref{a2s3theorem1} was already proved in \\cite{greco2016hopf} and \\cite{ros2014dirichlet}. Here, inspired by \\cite{fall2015overdetermined} and \\cite{ciraolo2021symmetry}, we give a proof which allows us to explicitly write the constant $C_H$ given by \\eqref{CHopf}, and to show its dependency on the parameters which are relevant in our problem. This will be useful when we will prove the quantitative results.\n\n\\begin{proof}[Proof of Lemma \\ref{a2s3theorem1}] \nWe consider the barrier function \n$$\nw(x) := \\psi_B (x) + \\delta \\, \\chi_K (x) \\,,\n$$ \nwhere $B=B_r(x_0)$, $\\chi_K$ is the characteristic function of $K \\subset \\mathbb{R}^n$ and $\\delta > 0$ is a constant that will be chosen later.\n\nLet $\\varphi \\in H^s_0 (\\Omega)$ be a nonnegative test function. We have\n\\begin{align*}\n \\mathcal{E} (w,\\varphi) &= \\mathcal{E} (\\psi_B, \\varphi) + \\delta \\, \\, \\mathcal{E} (\\chi_K, \\varphi) = \\int_B \\varphi - \\delta \\, c_{n,s} \\, \\int_K \\int_B \\frac{\\varphi(y)}{|x - y|^{n+2s}} \\, dy \\, dx \\leq \\\\\n &\\leq (1 -\\delta \\, C) \\int_B \\varphi,\n\\end{align*}\nwhich is less or equal than zero if we choose $\\delta \\geq C^{-1}$ with\n\\begin{equation*}\n C = c_{n,s} \\, |K| \\, \\inf_{x \\in K, y \\in B} \\frac{1}{|x - y|^{n+2s}}. \n\\end{equation*}\n\nBy setting \n$$\n\\tau := \\inf_K u \/ \\delta = C \\, \\inf_K u\n$$ \nand applying the weak maximum principle for $s$-harmonic functions to \n$$\nv := u - \\tau \\, w \\,,\n$$ \nwe get that\n\\begin{equation*}\n u \\geq c_{n,s} \\frac{|K| \\, \\inf_K u}{({\\rm diam}(B) + {\\rm dist}(K,B) + {\\rm diam}(K))^{n+2s}} \\, \\psi_B \\quad \\text{in} \\, B,\n\\end{equation*}\nwhich is the desired result. \n\\end{proof}\n\n\n \n\n\nSince our approach is based on the method of moving planes, a particular attention must be given to antisymmetric $s$-harmonic functions. More precisely, we will have to consider functions which are antisymmetric with respect to a hyperplane which can be chosen to be $\\{x_1 = 0\\}$ (up to a translation and rotation). \n\nIn order to list these results, we need to introduce some notation: we set $H^+ := \\{x_1 > 0\\}$, $H^- := \\{ x_1 < 0 \\}$ and $T\\coloneqq \\{ x_1 = 0 \\}$. Let \n$$\n\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n\\,, \\ \\ y \\mapsto y' = (-y_1,y_2,\\dots,y_n) \\,,\n$$ \nbe the reflection with respect to $T$ and, for a given set $E$ we call $E^+:= E \\cap H^+$ and $E^-:= E \\cap H^-$.\n\n\nThe first result is a weak maximum principle for $s$-harmonic antisymmetric functions, which is stated in \\cite[Proposition 3.1]{fall2015overdetermined} on domains that are bounded, although for homogeneous equations this condition is not needed. We report this proposition here and we sketch a proof.\n\n\n\\begin{lemma}[Weak maximum principle for antisymmetric functions]\n\\label{a2s1lemma1}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a compact set and let $u \\in H^s(\\mathbb{R}^n)$ be an antisymmetric (w.r.t. $T=\\{ x_1 = 0 \\}$) solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s u = 0 \\qquad &\\text{in} \\ \\Omega^c,\\\\\n u \\geq 0 \\qquad &\\text{in} \\ \\Omega^+.\n \\end{cases}\n\\end{equation*}\nThen, $u \\geq 0$ a.e. in $H^+$.\n\\end{lemma}\n\n\\begin{proof}\nSince $u$ is $s$-harmonic in $\\Omega^c$ then for every $ \\varphi \\in H^s_0(\\Omega^c)$ we have\n\\begin{equation}\n \\mathcal{E}(u,\\varphi) = 0.\n\\end{equation}\n\nLet $\\varphi = u_- \\chi_{H^+} \\in H^s_0(\\Omega^c)$, where $\\chi_{H^+}$ is the characteristic function of $H^+$. Following the same computations as in \\cite[Proposition 3.1]{fall2015overdetermined} we get \n\\begin{equation}\n 0 \\leq \\mathcal{E}(u,\\varphi) \\leq - \\mathcal{E}(\\varphi,\\varphi) = - [\\varphi]_s^2 \\,,\n\\end{equation}\nwhich immediately implies that $\\varphi=0$ a.e. and hence $u_-=0$ a.e. in $H^+$.\n\\end{proof}\n\nAn analogous weak maximum principle holds for nonnegative functions in $H^s(\\mathbb{R}^n)$. More precisely we have\n\n\\begin{lemma}[Weak maximum principle]\n\\label{a2s1lemma1bis}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a compact set and let $u \\in H^s(\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s u = 0 \\qquad &\\text{in} \\ \\Omega^c,\\\\\n u \\geq 0 \\qquad &\\text{in} \\ \\Omega.\n \\end{cases}\n\\end{equation*}\nThen, $u \\geq 0$ a.e. in $\\mathbb{R}^n$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is analogous to the one of Lemma \\ref{a2s1lemma1}, since it is enough to consider $ \\varphi = u_-$.\n\\end{proof}\n \nAs an immediate consequence we have the following comparison principle for $s$-capacitary functions. \n\n\\begin{corollary} \\label{corol_monot}\nLet $E\\subset F \\subset \\mathbb{R}^n$ be open bounded domains, and let $u_E$ and $u_F$ be the corresponding capacitary functions, i.e. the solutions to \\eqref{a2s1eq10} for $\\Omega=E$ and $\\Omega=F$, respectively. Then we have \n\\begin{equation} \\label{monotonicity}\nu_E \\leq u_F\n\\end{equation}\nin $\\mathbb R^n$.\n\\end{corollary}\n\n\\begin{proof}\nSince $u_E$ is a $s$-capacitary function, from \\cite[Lemmas 2.6 and 2.7]{warma2015fractional} we have that $0 \\leq u_E \\leq 1$ in $\\mathbb{R}^n \\setminus E$. Then, by applying Lemma \\ref{a2s1lemma1bis} to $v:= u_F - u_E$ we obtain the result.\n\\end{proof}\n\n\nFrom Lemma \\ref{a2s1lemma1}, we can also recover a quantitative version of the Hopf lemma for antisymmetric functions as proved in \\cite{ciraolo2021symmetry}, that we recall below.\n\n\\begin{lemma} [Lemma 4.1 in \\cite{ciraolo2021symmetry}]\n\\label{s3lemma2}\nLet $\\Omega$ be an open set in $H^-$ and $B \\subset \\Omega$ a ball of radius $R > 0$ such that $\\mathrm{dist}(B,H^+) > 0$. Let $v \\in C^s(\\Omega)$ be antisymmetric and a solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s v \\ge 0 \\quad & \\text{in} \\ \\Omega,\\\\\n v \\geq 0 \\quad & \\text{in} \\ H^-.\n \\end{cases}\n\\end{equation*}\nLet $K \\subset H^-$ be a set of positive measure such that $\\overline{K} \\subset (H^- \\setminus \\overline{B})$ and $\\inf_K v > 0$. Then we have that\n\\begin{equation}\n\\label{s3eq19}\nv \\geq C \\big[ \\mathrm{dist} (K,H^+) \\, |K| \\, \\inf_K v \\big] \\psi_B \\quad \\text{in} \\ B,\n\\end{equation}\nwith\n\\begin{equation*}\n C:= \\frac{ 2 (n+2s) \\, C(n,s) \\, \\mathrm{dist} (B,H^+)^{n+2s+1}}{(\\mathrm{dist}(B,H^+)^{n+2s}+\\gamma_{n,s} \\, C(n,s) \\, |B| \\, R^{2s}) \\, ( \\mathrm{diam} (B) + \\mathrm{diam}(K) + \\mathrm{dist}(\\mathcal{Q}(K),B))^{n+2s+2}}.\n\\end{equation*}\n\\end{lemma}\n\nIt is clear that Lemma \\ref{s3lemma2} provides a quantitative version of the strong maximum principle for antisymmetric $s$-harmonic functions, which still holds when $\\Omega$ is not bounded, as already noted in \\cite[Proposition 2.1]{soave2019overdetermined}. \n\n\\begin{lemma}[Strong maximum principle for antisymmetric functions] \\label{lemma_SMP}\nLet $\\Omega$ be an open set with $\\Omega \\subset H^-$ and let $v \\in C^s(\\Omega)$ be antisymmetric and a solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s v \\ge 0 \\quad & \\text{in} \\ \\Omega,\\\\\n v \\geq 0 \\quad & \\text{in} \\ H^-.\n \\end{cases}\n\\end{equation*}\nThen, either $v > 0$ in $\\Omega$ or $v \\equiv 0$ in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nFrom the weak maximum principle in Lemma \\ref{a2s1lemma1} we have that $v \\geq 0$ in $\\Omega$. Now assume there exists $x_0 \\in \\Omega$ such that $v(x_0) = 0$ and choose a ball $B$ centered in $x_0$ and such that $\\overline{B} \\subset \\Omega$. Let $K \\subset \\Omega$ be a compact set such that ${\\rm dist}(B,K) > 0$ and $|K| > 0$. If we furthermore choose $B$ and $K$ such that $\\inf_K v > 0$, by applying Lemma \\ref{s3lemma2} we have\n\\begin{equation*}\nv \\geq C \\big[ {\\rm dist} (K,H^+) \\, |K| \\, \\inf_K v \\big] \\psi_B \\quad in \\ B,\n\\end{equation*}\nand in particular $v(x_0) > 0$, which is a contradiction.\n\\end{proof}\n\n\nAnother tool from \\cite{ciraolo2021symmetry} that we will need in our proof is the boundary Harnack inequality for $s$-harmonic antisymmetric functions that we report here for clarity. \n\n\\begin{lemma}[Lemma 2.1 in \\cite{ciraolo2021symmetry}]\n\\label{a2s1lemma5}\nLet $u\\in C^2(B_R) \\cap C(\\mathbb{R}^n)$ be a solution of\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s u = 0 \\quad & \\text{in } B_R,\\\\\n u(x') = - u(x) \\quad &\\text{for every } x \\in \\mathbb{R}^n,\\\\\n u \\geq 0 \\quad & \\text{in } H^+.\n \\end{cases}\n\\end{equation*}\nThere exists a constant $K > 1$ only depending on $n$ and $s$ such that, for every $z \\in B_{R\/2}^+$ and for every $x \\in B_{R\/4}(z) \\cap B_R^+$ we have\n\\begin{equation}\n\\label{s3eq2}\n \\frac{1}{K} \\frac{u(z)}{z_1} \\leq \\frac{u(x)}{x_1} \\leq K \\frac{u(z)}{z_1}.\n\\end{equation}\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\\section{Exterior sets}\nIn this section we consider the exterior overdetermined problem and prove Theorems \\ref{a2s1theorem1} and \\ref{a2theorem2}.\n\n\n\\subsection{The method of moving planes: notation} \\label{subsect_MMP}\nWe introduce some notation in order to exploit the moving planes method. Given $e \\in \\mathbb{S}^{n-1}$, a set $E \\subset \\mathbb{R}^n$ and $\\lambda \\in \\mathbb{R}$, we define\n\\begin{align*}\n &T_\\lambda = T_\\lambda^e = \\{ x \\in \\mathbb{R}^n \\, | \\, x \\cdot e = \\lambda \\} & &\\textrm{a hyperplane orthogonal to } e,\\\\\n &H_\\lambda = H_\\lambda^e = \\{ x \\in \\mathbb{R}^n \\, | \\, x \\cdot e > \\lambda \\} & &\\textrm{the ``positive'' half space with respect to } T_\\lambda \\\\\n &E_\\lambda = E \\cap H_\\lambda & &\\textrm{the ``positive'' cap of } E,\\\\\n &x_\\lambda' = x -2(x\\cdot e - \\lambda) \\, e & &\\textrm{the reflection of } x \\textrm{ with respect to } T_\\lambda,\\\\\n &\\mathcal{Q}= \\mathcal{Q}_\\lambda^e : \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x_\\lambda' & &\\textrm{the reflection with respect to } T_\\lambda.\n\\end{align*}\n\nIf $E \\subset \\mathbb{R}^n$ is an open bounded set with boundary of class $C^1$ then we define \n$$\n\\Lambda_e := \\sup \\{ x \\cdot e \\, | \\, x \\in E \\}\n$$ \nand\n\\begin{equation*}\n \\lambda_e = \\inf \\{ \\lambda \\in \\mathbb{R} \\, | \\, \\mathcal{Q}(E_{\\Tilde{\\lambda}}) \\subset E, \\textrm{for all} \\, \\Tilde{\\lambda} \\in (\\lambda, \\Lambda_e) \\}.\n\\end{equation*}\n\n{F}rom this point on, given a direction $e \\in \\mathbb{S}^{n-1}$, we will refer to $T_{\\lambda_e} = T^e$ and $E_{\\lambda_e} = \\widehat{E}$ as the \\textit{critical hyperplane} and the \\textit{critical cap} with respect to $e$, respectively, and will call $\\lambda_e$ the \\textit{critical value} in the direction $e$. We now recall from \\cite{serrin1971symmetry} that, for any given direction $e$, at least one of the following two conditions holds:\\\\\n\n\\textbf{Case 1} - The boundary of the reflected cap $\\mathcal{Q}^e(\\widehat{E})$ becomes internally tangent to the boundary of $E$ at some point $P \\not \\in T^e$;\\\\\n\n\\textbf{Case 2} - the critical hyperplane $T^e$ becomes orthogonal to the boundary of $E$ at some point $Q \\in T^e$.\n\n\n\n\\subsection{The symmetry result}\n\nWe start with the symmetry result given in Theorem \\ref{a2s1theorem1}. \n\n\\begin{proof}[Proof of Theorem \\ref{a2s1theorem1}]\nLet $e \\in \\mathbb{S}^{n-1}$ be a fixed direction. We recall that we are considering a solution $u \\in C^s(\\mathbb{R}^n)$ of \\eqref{a2s1eq10} satisfying \\eqref{a2s1eq16} and that $G=\\Omega + B_R$, with $\\partial G$ of class $C^1$. \n\nWe apply the method of moving planes described in Subsection \\ref{subsect_MMP} by letting $E=G$. Without loss of generality, we can assume that $\\lambda_e = 0$ (that is, the critical hyperplane $T$ goes through the origin), and we simplify the notation by setting $H^- := \\{ x \\cdot e < 0 \\}$, $\\Omega^- := H^- \\cap \\Omega$ and considering\n\\begin{equation*}\n v(x):= u(x) - u(\\mathcal Q(x)) \\quad \\text{for } \\ x \\in \\mathbb{R}^n,\n\\end{equation*}\nwhere $\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x'$ is the reflection with respect to $T$. We have\n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s v = 0 \\quad &\\text{in} \\ H^- \\setminus \\Omega^- \\\\\n v \\geq 0 \\quad &\\text{in} \\ \\Omega^-\\\\\n v(\\mathcal Q(x)) = - v (x) \\quad &\\text{for every } x \\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation*}\n\n\nBy using Lemma \\ref{a2s1lemma1} we know that $v \\geq 0$ in $H^-$ and then Lemma \\ref{lemma_SMP} tells us that either $v > 0$ in $H^- \\setminus \\Omega^-$ or $v \\equiv 0$ in $\\mathbb{R}^n$. Now we show that if we assume that $v > 0$ in $H^-\\setminus \\Omega^-$ then we obtain a contradiction.\n\n\\textbf{Case 1} - Let $P$ be a critical point on $\\partial G$. Since both $P$ and its reflection $P'$ belong to $\\partial G$ and \\eqref{a2s1eq16} holds, we immediately get\n\\begin{equation*}\n v(P) = u(P) - u(P') = 0,\n\\end{equation*}\nwhich is a contradiction.\\\\\n\n\\textbf{Case 2} - In this case $e$ is tangent to $\\partial G$ at a point $Q \\in \\partial G$, and therefore we have that $\\partial_e v (Q) = 0$. On the other hand, since $Q$ is far away from the boundary $\\partial \\Omega$, we can use Lemma \\ref{a2s1lemma5} to show that $\\partial_e v (Q) < 0$, which is a contradiction.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Almost symmetry in one direction}\nNow we consider the quantitative stability result and prove Theorem \\ref{a2theorem2}. This will be done in two subsequent steps: we first prove the quantitative stability estimate in one direction and then, in the proof of Theorem \\ref{a2theorem2} we will sketch a general idea of how to use the result in one direction to obtain the final quantitative estimate; the proof can be found in details in Section 6 of \\cite{ciraolo2021symmetry}. \n\nWe start by proving a preliminary result which gives the behaviour of the solution to \\eqref{a2s1eq10} close to the boundary. \n\n\\begin{lemma} \\label{lemma_conticino}\nUnder the assumptions of Theorem \\ref{a2theorem2}, let $u$ be a solution of \\eqref{a2s1eq10} and let $v:= 1- u$. For any $r \\leq \\mathfrak{r}_{\\Omega}^e$ we have\n\\begin{equation}\n\\label{a2s3eq35}\n v(x) \\geq C_{cap} \\, ({\\rm dist} (x,\\partial \\Omega))^s \\quad \\text{in} \\ (\\Omega + B_r)\\setminus \\Omega,\n\\end{equation}\nwhere\n\\begin{equation*}\n C_{cap} := \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_{n}}{4} \\, \\frac{r^{n+s}}{(2r + r_0 \\, {\\rm diam} (\\Omega))^{n+2s}}\n\\end{equation*}\nand $r_0 > 0$ is a constant depending on $n$ and $s$.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality, we can assume that the origin $O$ is contained in $\\Omega$ and consider the $s$-capacitary solution $\\tilde{u}$ of the ball $B_{{\\rm diam}(\\Omega)}$ centered at the origin and of radius ${\\rm diam}(\\Omega)$: \n\\begin{equation*}\n \\begin{cases}\n (-\\Delta)^s \\tilde{u} = 0 \\quad &\\text{in} \\ \\mathbb{R}^n \\setminus B_{{\\rm diam}(\\Omega)},\\\\\n \\tilde{u} = 1 \\quad &\\text{in} \\ B_{{\\rm diam}(\\Omega)},\\\\\n \\tilde{u}(x) \\to 0 \\quad &\\text{as} \\ |x| \\to + \\infty.\n \\end{cases}\n\\end{equation*}\nSince $0 \\leq \\tilde{u} \\leq 1$ and $\\tilde{u}$ is radial, non-increasing and continuous (see for instance \\cite[Theorem 1.10]{soave2019overdetermined}, there exists a radius $\\tilde{R} = \\tilde{R}({\\rm diam} (\\Omega)) > 0$ such that\n\\begin{equation*}\n \\tilde{u} < 1\/2 \\quad \\text{in} \\quad \\mathbb{R}^n \\setminus B_{\\tilde{R}}.\n\\end{equation*}\nMoreover, from Corollary \\ref{corol_monot} we have that $\\tilde{u} \\geq u$ in the whole space. From this we get that\n\\begin{equation}\n v = 1 - u \\geq 1 - \\tilde{u} \\geq 1\/2 \\quad \\text{in} \\ \\mathbb{R}^n \\setminus B_{\\tilde{R}}.\n\\end{equation}\n\nWe now choose $K=\\overline{B_{\\mathfrak{r}^e_\\Omega}((\\tilde{R}+\\mathfrak{r}^e_\\Omega) \\,e_1)}$. For $x_0 \\in \\partial \\Omega$, we now apply Lemma \\ref{a2s3theorem1} to $v$ with $B = B_{\\mathfrak{r}^e_\\Omega}(x_0)$ and $K$ and get\n\\begin{equation*}\n v(x) \\geq \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n}{2} \\, \\frac{R^{n+s}}{(4R + {\\rm dist}(K,B))^{n+2s}} \\, (R - |x|)^s \\quad \\text{in} \\ B.\n\\end{equation*}\n\nWe now repeat the same argument on the whole boundary $\\partial \\Omega$ by keeping each time the same fixed set $K$. We notice that in every case we have ${\\rm dist}(K,B) \\leq 2\\tilde{R}$, and by using the previous inequality we obtain \\eqref{a2s3eq35}, where the constant $C_{cap} > 0$ can be written as \n\\begin{equation*}\n C_{cap} := \\frac{c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n}{4} \\, \\frac{R^{n+s}}{(2R + \\tilde{R})^{n+2s}}.\n\\end{equation*}\nIn order to complete the proof, we show how $\\tilde R$ depends on ${\\rm diam}(\\Omega)$. We consider the solution $u_{B_1}$ to the capacitary problem \\eqref{a2s1eq10} with $\\Omega = B_1$, and we set\n$$\nr_0=\\inf\\{ \\, |x| \\ | \\ u_B(x)< 1\/2 \\} \\,. \n$$ \nBy scaling properties, it is clear that $\\tilde R = r_0 \\, {\\rm diam} (\\Omega)$. This completes the proof.\n\\end{proof}\n\nWith this result at hand, we can prove a quantitative estimate which involves the measure of $\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})$.\n\n\\medskip\n\nWe fix a direction $e \\in \\mathbb{S}^{n-1}$. Without loss of generality, we can assume that $e=e_1$ and that the associated critical hyperplane is $T = \\{x_1 = 0\\}$, with $\\mathcal Q: \\mathbb{R}^n \\to \\mathbb{R}^n, x \\mapsto x'$ the reflection with respect to $T$. For the proof of the next lemma we will use the following notation: we set for $t \\geq 0$\n\\begin{equation*}\n \\Omega_t \\coloneqq \\Omega + B_t(0), \\quad \\widehat{\\Omega_t} \\coloneqq \\Omega_t \\cap H^+, \\quad \n \\Omega_t^- \\coloneqq \\Omega_t\\cap H^- \\quad U_t \\coloneqq \\mathcal Q(\\widehat{\\Omega_t}).\n\\end{equation*}\nNote that $G = \\Omega_R$.\n\n\\begin{lemma}\n\\label{a2s3lemma1}\nGiven $P \\in \\overline{U_R}$ with $B=B_{R\/8} (P)$ such that $\\mathrm{dist}(B,\\partial U_0) \\ge R\/8$, for $\\delta > 0$, we have that\n\\begin{equation}\n\\label{a2s3eq39}\n|\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) \\ | \\ \\leq \\Tilde{C} \\, (\\delta^{-(1+s)} v(P) + \\delta),\n\\end{equation}\nwhere $\\Tilde{C} > 0$ is a constant depending only on $n$, $s$, $R$, $\\mathfrak{r}^e_\\Omega$ and $\\mathrm{diam}(\\Omega)$.\n\\end{lemma}\n\n\\begin{proof}\nWe set $K_\\delta := (\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})) \\setminus (E_\\delta \\cup F_\\delta)$,\nwhere\n$$\nE_\\delta := \\mathcal Q (A_\\delta) \\cap (\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})) \\quad \\text{ with } A_\\delta := \\{ x \\in \\Omega^c \\ | \\ {\\rm dist} (x, \\partial \\Omega) < \\delta \\}, \n$$\n$$\nF_\\delta:= \\{ x \\in \\Omega^- \\setminus \\mathcal{Q}(\\widehat{\\Omega}) \\, | \\, \\mathrm{dist}(x, T) < \\delta \\}.\n$$\n\nUsing Lemma \\ref{s3lemma2} with $B := B_{R\/8} (P)$ and $K := K_\\delta$ we obtain\n\\begin{equation*}\n\tv \\geq \\overset{\\star}{C} \\, \\big[ \\mathrm{dist} (K_\\delta ,H^+) \\, | K_\\delta | \\, \\inf_{K_\\delta} v \\big] \\psi_B \\quad \\text{in} \\ B,\n\\end{equation*}\nwhere $\\overset{\\star}{C} > 0$ is an explicit constant depending on $n$, $s$, $R$ and ${\\rm diam}(\\Omega)$. Here we used that, in the present situation, we have $K \\subset \\Omega$ and that ${\\rm dist} (B, U_0) \\le R$.\n\n\nSince $K_\\delta \\subseteq (\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})) \\setminus F_\\delta$, then \n\\begin{equation*}\n\\mathrm{dist} (K_\\delta,H^+) \\geq \\delta .\n\\end{equation*}\n\nWe now point out that in $K_\\delta$ we have $v = u - u' = 1 - u'$; we can therefore apply Lemma \\ref{lemma_conticino} and get\n\\begin{equation*}\n v(x) \\geq C_{cap} (n,s,\\mathfrak{r}^e_{\\Omega}) \\, \\delta^s = C_{cap} \\, \\delta^s \\quad \\text{for every } \\ x\\in K_\\delta.\n\\end{equation*}\n\nMoreover, we have\n\\begin{equation}\n\\label{a2s3eq43}\n|K_\\delta| = |\\Omega^- \\setminus U_R| - |E_\\delta \\cup F_\\delta| \\ge |\\Omega^- \\setminus U_R| - (|E_\\delta| + | F_\\delta |) .\n\\end{equation}\nBy definition of $F_\\delta$, we have that\n\\begin{equation}\n\\label{a2s3eq44}\n\t|F_\\delta| \\le \\mathrm{diam}( \\Omega )^{n-1} \\delta.\n\\end{equation}\nBy using Lemma 5.2 in \\cite{ciraolo2021symmetry}, since $E_\\delta \\subseteq A_\\delta$, we have\n\\begin{equation}\n\\label{a2s3eq45}\n |E_\\delta| \\leq \\left[\\frac{2 n |\\Omega|}{R} \\right] \\delta.\n\\end{equation}\nPutting together \\eqref{a2s3eq43}, \\eqref{a2s3eq44} and \\eqref{a2s3eq45} we get\n\\begin{equation*}\n\t|K_\\delta| \\ge |\\Omega^- \\setminus U_R| - \\tilde{c} \\, \\delta,\n\\end{equation*}\nwhere $\\tilde{c}$ is a positive constant depending on $n$, $\\mathrm{diam}(\\Omega)$ and $\\mathfrak{r}^e_{\\Omega}$.\n\n\\medskip\n\nHence we have proved that\n\\begin{equation*}\n v(P) \\geq \\overset{\\star}{C} \\, C_{cap} \\, (R\/8)^{2s} \\, \\gamma_{n,s} \\delta^{1+s} \\big( |\\Omega^- \\setminus U_R| - \\tilde{c} \\, \\delta \\big) \\,,\n\\end{equation*}\nand, by choosing\n\\begin{equation*}\n \\Tilde{C} := \\max \\left\\lbrace \\frac{8^{2s}}{ C_{cap} \\, R^s \\, \\overset{\\star}{C}} \\, , \\, \\, \\tilde{c} \\right\\rbrace ,\n\\end{equation*}\nwe get the desired inequality \\eqref{a2s3eq39}.\n\\end{proof}\n\nOnce Lemma \\ref{a2s3lemma1} is proved we can follow \\cite{ciraolo2021symmetry} to get the almost symmetry in one direction and, with the same reasoning as in \\cite[Section 6]{ciraolo2021symmetry}, obtain the same quantitative stability estimate required for the proof of Theorem \\ref{a2theorem2}. For this reason we only sketch the main ideas in the following proof.\n\n\\begin{proof}[Proof of Theorem \\ref{a2theorem2}]\nOnce we have inequality \\eqref{a2s3eq39}, we can argue as in Lemma 5.6 of \\cite{ciraolo2021symmetry} to obtain the estimate for the almost symmetry in one direction, namely\n\\begin{equation}\n\\label{a2s3eq49}\n |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) | \\leq \\overline{C} [u]_{\\partial G}^{\\frac{1}{s+2}},\n\\end{equation}\nwhere $\\overline{C}:= \\max\\{ 1, {\\rm diam}(\\Omega), K \\, (R\/2) \\} \\, \\tilde{C}$, with $\\tilde{C}$ as in \\eqref{a2s3eq39} and $K = K(n,s) \\geq 1$ is the constant that appears in the boundary Harnack inequality in Lemma \\ref{a2s1lemma5}.\n\n\\medskip\n\nNow, up to a translation we can assume that the critical hyperplanes with respect to the $N$ coordinate directions $T^{e_j}$ coincide with $\\{ x_j = 0 \\}$ for each $j=1, \\dots , N$, that is, they all intersect at the origin. \n\n\\medskip\n\nThe idea is then the following: for a given direction $e \\in \\mathcal{S}^{n-1}$ we slice the positive cap of $\\Omega$ in (a finite number of) sections depending on the critical value $\\lambda_e$ and consider their measure, namely\n\\begin{equation*}\n m_k := | \\, \\{ x \\in \\Omega \\ | \\ (2k-1) \\, \\lambda_e \\leq x \\cdot e \\leq (2k+1) \\, \\lambda_e \\} \\, |, \\quad \\text{for } k\\geq 1.\n\\end{equation*}\n\nSince $\\Omega$ is bounded, $m_k > 0$ only up to an index $k_0$ which behaves like the inverse of $\\lambda_e$. The key observation is that, by reflecting with respect to the origin and using \\eqref{a2s3eq49}, one has\n\\begin{equation}\n\\label{a2s3eq50}\n m_1 = | \\, \\{ x \\in \\Omega \\ | \\ -\\lambda_e \\leq x \\cdot e \\leq \\lambda_e \\} \\, | \\leq (n+3) \\, \\overline{C} \\, [u]_{\\partial G}^{\\frac{1}{s+2}};\n\\end{equation}\nmoreover, by the moving plane procedure, $m_k \\leq m_1$ for every $k$ up to $k_0$ and therefore one can then write the expression \n\\begin{equation}\n\\label{unaltraequazione}\n |\\Omega_{\\lambda_e}| \\leq \\sum_{k=1}^{k_0} m_k \\leq k_0 \\, m_1 \\leq (n+3) \\, {\\rm diam}(\\Omega) \\, \\overline{C} \\, \\frac{1}{\\lambda_e} \\, [u]_{\\partial G}^{\\frac{1}{s+2}}.\n\\end{equation}\n\nInequalities \\eqref{unaltraequazione} and some further calculations (see section 4 in \\cite{ciraolo2021symmetry}) yield\n\\begin{equation}\n\\label{a2s3eq52}\n |\\lambda_e| \\leq 4 \\, (n+3) \\, \\frac{{\\rm diam}(\\Omega)}{|\\Omega|} \\, \\overline{C} [u]_{\\partial G}^{\\frac{1}{s+2}}.\n\\end{equation}\n\nNow it remains to establish a relationship between $|\\lambda_e|$ and $\\rho(\\Omega)$. We set $\\rho_{min}:= \\min_{z \\in \\partial \\Omega} |z|$, $\\rho_{max}:= \\max_{z \\in \\partial \\Omega} |z|$ and choose $x, y \\in \\partial \\Omega$ such that $|x| = \\rho_{min}$ and $|y| = \\rho_{max}$. We then consider the unit vector\n\\begin{equation*}\n e := \\frac{x - y}{|x - y|}\n\\end{equation*}\nand the corresponding critical hyperplane $T^e$. By construction, we know that ${\\rm dist} (x,T^e) \\geq {\\rm dist} (y,T^e)$ and therefore some simple calculations lead to \n\\begin{equation}\n\\label{a2s3eq53}\n \\rho(\\Omega) \\leq \\rho_{max} - \\rho_{min} = |y| - |x| \\leq 2 \\, {\\rm dist} (0,T^e) = 2 |\\lambda_e|.\n\\end{equation}\n\nCombining \\eqref{a2s3eq52} and \\eqref{a2s3eq53} leads to \\eqref{a2s1eq8}. It is worth pointing out that the new constants appearing in \\eqref{a2s3eq50} and onward only depend on the dimension $n$, the diameter ${\\rm diam}(\\Omega)$ and the volume $|\\Omega|$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Annular sets}\nIn this section we consider annular domains and prove Theorems \\ref{a2theorem3} and \\ref{a2theorem4}. The strategy that we use is still via the method of moving planes and it is similar to the previous one; nevertheless, the method has to be carefully adapted to this situation. We recall that we are considering solutions to \\eqref{a2s1eq12}, where $A=\\Omega \\setminus \\overline D$, with $\\overline D \\subset \\Omega$ bounded open domains.\n\n\n\n\\bigskip\n\nNow for a fixed direction $e$ and a parameter $\\lambda \\in \\mathbb{R}$ we let $T_\\lambda$, $H_\\lambda$, $\\mathcal Q_\\lambda$ be as in the previous section. We now consider \n$$\n\\Sigma_\\lambda := (\\Omega \\cap H_\\lambda) \\setminus \\mathcal Q_\\lambda(\\overline{D})\n$$\nwhich is the cap of the annulus $\\Omega \\setminus D$. Moreover, for a given set $E$, we define \n\\begin{align*}\n &d_{E}:= \\inf \\{ \\lambda \\in \\mathbb{R} \\ | \\ T_\\mu \\cap \\overline{E} = \\emptyset \\}\\\\\n &\\overline{\\lambda}_E := \\inf \\{ \\lambda \\leq d_E \\ | \\ \\text{for every } \\ \\mu > \\lambda, (\\overline{E} \\cap H_\\mu)^\\mu \\subset ( E \\cap H_\\mu^\\mu ) \\ \\text{ and } \\ \\nu(x) \\cdot e > 0 \\ \\forall x \\in T_\\mu \\cap \\partial E \\},\n\\end{align*}\nand the critical parameter $\\overline \\lambda$ is given by\n\\begin{equation*}\n\\overline{\\lambda} := \\max \\{ \\overline{\\lambda}_D, \\ \\overline{\\lambda}_\\Omega \\}.\n\\end{equation*}\n\n\nWe mention that $\\Sigma_\\lambda$ is chosen in such a way that both the function $u$ and its reflection $u'$ are $s$-harmonic, as we will see in the proof of Theorem \\ref{a2theorem3}. We also notice that, thanks to our choices, $\\overline{\\lambda}$ is the critical value for $A$ with respect to the direction $e$, and now the critical position can occur in four possible cases (namely, Cases 1 and 2 in Subsection \\ref{subsect_MMP} for both $D$ and $\\Omega$).\n\n\\medskip\n\nIn order to avoid further technicalities we ask for the domains $D$ and $\\Omega$ to be regular (namely, with boundaries $\\partial G$ and $\\partial \\Omega$ of class $C^2$); the proof works in the same way if we instead assume that $\\partial A$ is just of class $C^1$, and $\\Gamma_R^G$ and $\\Gamma_R^\\Omega$ of class $C^2$.\n\n\\medskip\n\n\n\\subsection{Symmetry result}\n\nWith this setting, we are now ready to give a proof of the symmetry result for annular sets.\n\n\\begin{proof}[Proof of Theorem \\ref{a2theorem3}]\nWe fix a direction $e=e_1$ and reach the critical value $\\overline{\\lambda}$. Without loss of generality, we assume that $T = \\{ e_1 = 0 \\}$ and define the function $w(x) := u(x) - u(x')$ for every $x \\in \\mathbb{R}^n$. To simplify the notation we set $\\mathcal{Q} = \\mathcal{Q}_{\\overline{\\lambda}}$. Our aim is to show that $w$ is actually identically zero in $\\mathcal Q (\\Sigma_{\\overline{\\lambda}})$. This implies that both the function $w$ and the set $A$ itself are symmetric with respect to direction $e$; since the direction $e$ can be chosen arbitrarily, the proof is then complete.\n\nHence we have to show that $w \\equiv 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$. We notice that the function $w$ is antisymmetric with respect to $e=e_1$ and\n\\begin{align*}\n (-\\Delta)^s w(x) = 0 \\qquad \\qquad &\\text{for} \\ x \\in \\mathcal Q(\\Sigma_{\\overline{\\lambda}}),\\\\\n w(x) = u(x) - u(x') = 1 - u(x') \\geq 0 \\qquad \\qquad &\\text{for} \\ x \\in \\overline{D} \\cap \\mathcal Q(\\widehat{\\Omega}),\\\\\n w(x) = u(x) - u(x') = u(x) \\geq 0 \\qquad \\qquad &\\text{for} \\ x \\in \\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}),\\\\\n w(x) = 0 \\qquad \\qquad &\\text{for} \\ x \\in H^- \\setminus \\Omega^-.\n\\end{align*}\nIn particular, the last three inequalities tell us that $w \\geq 0$ in $H^- \\setminus \\mathcal Q(\\Sigma_{\\overline{\\lambda}})$. The weak maximum principle for antisymmetric solutions in Lemma \\ref{a2s1lemma1} implies that $w \\geq 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$; then, from the strong maximum principle in Lemma \\ref{lemma_SMP} we get that either $w > 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$ or $w \\equiv 0$ in $\\mathcal Q(\\Sigma_{\\overline{\\lambda}})$.\n\nIn order to conclude, we notice that in this case we have four possible critical cases, all of which can be treated as in the proof of Theorem \\ref{a2s1theorem1}. The conclusion then follows straightforwardly. \n\\end{proof}\n\n\\subsection{Almost symmetry in one direction}\n\nIn order to prove almost symmetry in one direction for the annular set, we need to make use again of the quantitative Hopf's type lemma (\\cite[Lemma 4.1]{ciraolo2021symmetry}) and adapt it to the current problem. \n\nWe start with a lemma which gives the behaviour of the solution $u$ of \\eqref{a2s1eq12} close to the boundary of the annulus $A$. For this reason, for $t>0$ we define\n\\begin{equation} \\label{At_def}\nA_t = \\{x \\in A :\\ {\\rm dist}(x,\\partial A) > t \\}\\,.\n\\end{equation}\n\nWe start with a simple remark.\n\n\\begin{remark}\nIf $u \\in C^s (\\mathbb{R}^n)$ solves \\eqref{a2s1eq12} with $\\partial A \\in C^1$, then we have\n\\begin{equation}\n\\label{a2s4eq3}\n 0 < u < 1 \\qquad \\text{in} \\ A.\n\\end{equation}\n\\end{remark}\nIndeed, applying the maximum principles for an $s$-harmonic function in $A$ we get that $u$ has to be strictly positive in $A$. By using the same argument for $\\Tilde{u} := 1 - u$ we get the latter part of \\eqref{a2s4eq3}.\n\n\\bigskip\n\nWe have the following lemma.\n\n\n\\begin{lemma}\n\\label{a2s4lemma4}\nUnder the assumptions of Theorem \\eqref{a2theorem4}, let $u$ be a solution of \\eqref{a2s1eq12} and let $r_{ann}:= \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega, \\, \\mathfrak{r}^e_D \\}$. Then\n\\begin{equation}\n\\label{a2s4eq36}\n \\min \\{u, 1 - u \\} (x) \\geq C_{ann} \\, ({\\rm dist} (x,\\partial A))^s \\quad \\text{in} \\ A \\setminus A_{r_{ann}},\n\\end{equation}\nwhere $A_{r_{ann}}$ is given by \\eqref{At_def} and $C_{ann} > 0$ is a constant depending on $n$, $s$, $\\mathfrak{r}^e_{D}$, $\\mathfrak{r}^i_{\\Omega}$, ${\\rm diam}(\\Omega)$ and $|D|$. \n\\end{lemma}\n\n\\begin{proof}\nWe start by proving \\eqref{a2s4eq36} for points which are close to the boundary of $\\partial \\Omega$. Let $R_1 := \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega \\}$; for a given $x_0 \\in \\partial \\Omega$ we choose $\\overline{x} \\in A$ such that $\\partial B_{R_1}(\\overline{x}) \\cap \\partial \\Omega = \\{ x_0 \\}$. By using Lemma \\ref{a2s3theorem1}, with $B = B_R (\\overline{x})$ and $K = \\overline{D}$, some easy computations lead us to\n\\begin{equation*}\n u(x) \\geq \\frac{c_{n,s} \\, \\gamma_{n,s}}{4^{n+2s}} \\frac{|D| \\, R_1^s}{{\\rm diam} (\\Omega)^{n+2s}} \\, (R_1 - |x|)^s \\quad \\text{in} \\, B_{R_1}(\\overline{x}).\n\\end{equation*}\n\nSince $\\partial \\Omega$ is of class $C^2$, then the radius $R_1$ can be chosen uniformly with respect to $\\overline x$ and then the above inequality does not depend on the center $\\overline{x}$. Therefore, we can repeat the same argument on the whole boundary $\\partial \\Omega$ and get\n\\begin{equation}\n\\label{a2s4eq41}\n u(x) \\geq C_{ext} \\, {\\rm dist}(x, \\partial \\Omega)^s \\quad \\text{ for any} \\ x \\in \\{ x \\in A \\ | \\ {\\rm dist} (x,\\partial \\Omega) < R_1 \\},\n\\end{equation}\nwhere \n\\begin{equation*}\n C_{ext} := \\frac{c_{n,s} \\, \\gamma_{n,s}}{4^{n+2s}} \\frac{|D| \\, R_1^s}{{\\rm diam} (\\Omega)^{n+2s}}.\n\\end{equation*}\n\nNow we prove \\eqref{a2s4eq36} for points which are close to the boundary of $\\partial D$. In this case we apply the Hopf's lemma to $v := 1 - u$. We set $R_2:= \\{ \\overline{d}\/4, \\mathfrak{r}^e_D \\}$ and choose $x_\\star \\in \\mathbb{R}^n \\setminus \\Omega$ such that $K:= \\overline{B_{R_2}}(x_\\star)$ is tangent to $\\partial \\Omega$. For each $x_0 \\in \\partial D$, let $\\overline{x} \\in A$ be such that $\\partial B_{R_2}(\\overline{x}) \\cap \\partial \\Omega = \\{ x_0 \\}$. We apply Lemma \\ref{a2s3theorem1} to $B_{R_2}(\\overline{x})$ and $K$ as before and repeat the argument for every boundary point in $\\partial D$. By putting all together we will have \n\\begin{equation}\n\\label{a2s4eq47}\n v(x) \\geq C_{int} \\, ({\\rm dist} (x,\\partial D))^s \\quad in \\, (D + B_{R_2}) \\setminus D,\n\\end{equation}\nwhere\n\\begin{equation*}\n C_{int} := c_{n,s} \\, \\gamma_{n,s} \\, \\omega_n \\, \\frac{R_2^{n+s}}{(4R_2 + {\\rm diam} (\\Omega))^{n+2s}}.\n\\end{equation*}\n\n\\bigskip\n\nWe now see that for $r_{ann} = \\min \\{ \\overline{d}\/4, \\, \\mathfrak{r}^i_\\Omega, \\, \\mathfrak{r}^e_D \\}$ both inequalities \\eqref{a2s4eq41} and \\eqref{a2s4eq47} hold; therefore, by setting\n\\begin{equation*}\n C_{ann}:= \\min \\{ C_{ext}, \\, C_{int} \\},\n\\end{equation*}\ninequality \\eqref{a2s4eq36} holds and the proof is complete.\n\\end{proof}\n\nWe are now ready to state a version of Lemma \\ref{a2s3lemma1} for the annulus, under the assumptions of Theorem \\ref{a2theorem4}. The ball $B$ will be chosen inside of a set where the antisymmetic function $w:= u - u'$ is $s$-harmonic (this time, it will be $Q(\\Sigma_{\\overline{\\lambda}})$); in this case, the compact set $K$ consists of two components, in such a way that we can take into account the symmetric differences between the sets $\\Omega$ and $G$ and their respective reflections via the moving plane method. We define\n\\begin{align*}\n &L := \\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega}) \\\\\n &M := D^- \\setminus \\mathcal Q(\\widehat{D})\\\\\n &\\Tilde{K} := L \\cup (M \\cap \\mathcal Q(\\widehat{\\Omega})).\n\\end{align*}\n\nWe have the following lemma.\n\n\\begin{lemma}\nGiven $P \\in \\mathcal Q(\\Sigma_{\\overline{\\lambda}})$ with $B=B_{R\/8} (P)$ such that $\\mathrm{dist}(B, \\mathcal Q(\\Sigma_{\\overline{\\lambda}})) \\ge R\/8$ and given $\\delta > 0$, we have that\n\\begin{equation}\n\\label{a2s4eq52}\n|\\Tilde{K}| \\leq \\Tilde{C} \\, (\\delta^{-(1+s)} v(P) + \\delta),\n\\end{equation}\nwhere $\\Tilde{C} > 0$ is a constant depending only on $n$, $s$, $R$, $\\mathfrak{r}^e_{D}$, $\\mathfrak{r}^i_{\\Omega}$, ${\\rm diam}(\\Omega)$, $|D|$ and $|\\Omega|$.\n\\end{lemma}\n\n\\begin{proof}\nWe set $K_\\delta := \\Tilde{K} \\setminus (E_\\delta \\cup F_\\delta)$ where \n\\begin{align*}\n &E_\\delta := [ \\, \\{ x \\in \\Omega \\ | \\ {\\rm dist}(x,\\partial \\Omega) < \\delta \\} \\cap (\\Omega^- \\cup \\mathcal Q(\\widehat{\\Omega}) ) \\, ] \\, \\cup \\, [ G^- \\setminus \\mathcal Q(\\widehat{G} + B_\\delta) ) \\, ],\\\\\n &F_\\delta := \\{ x \\in \\Omega \\ | \\ {\\rm dist} (x,H^+) < \\delta \\}.\n\\end{align*}\nWe apply Lemma \\ref{s3lemma2} with $B := B_{R\/8} (P)$ and $K := K_\\delta$ to get \\eqref{s3eq19}, i.e.\n\n\\begin{equation}\n\\label{a2s4eq55}\n\tv \\geq \\overset{\\star}{C} \\, \\big[ \\mathrm{dist} (K_\\delta ,H^+) \\, | K_\\delta | \\, \\inf_{K_\\delta} v \\big] \\psi_B \\quad \\text{in} \\ B.\n\\end{equation}\n\nBy arguing as done for \\eqref{a2s3eq43}, \\eqref{a2s3eq44} and \\eqref{a2s3eq45} we get that\n\\begin{equation}\n\\label{a2s4eq56}\n |\\tilde{K}_\\delta| \\geq |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})| + |G^- \\setminus \\mathcal Q(\\widehat{G})| - \\tilde{c}\\, \\delta \\,,\n\\end{equation}\nand plugging \\eqref{a2s4eq56} into \\eqref{a2s4eq55}, together with Lemma \\ref{a2s4lemma4} and the fact that ${\\rm dist}(\\tilde{K}_\\delta, H^+) \\geq \\delta$ we get \\eqref{a2s4eq52}.\n\\end{proof}\n\nThe way to use Lemma \\ref{a2s4lemma4} to establish almost symmetry in one direction and then prove Theorem \\ref{a2theorem4} is again the one sketched in the proof of Theorem \\ref{a2theorem2}. We just need to highlight some minor differences with the annular case, that we report below.\n\n\\begin{proof}\nThe first goal is to obtain the almost symmetry in one direction from \\eqref{a2s4eq52}. While in the proof of Theorem \\ref{a2theorem2} we need to take into account the two possible critical Cases 1 and 2 for the moving plane method, with the first one being further divided into cases 1a and 1b, now the critical position can be reached for both the set $D$ and the set $\\Omega$, resulting in a total of six possible critical cases. Nonetheless, they are tackled in the same exact way; the only thing that we need to point out is that in each of the critical cases we can write\n\\begin{equation}\n\\label{a2s4eq58}\n v(P) \\leq c_\\star \\max \\{ [u]_{\\Gamma^R_{\\Omega}}, [u]_{\\Gamma^R_{G}} \\} = c_\\star \\, \\mathrm{def}_A(u) \\,,\n\\end{equation}\nwhere $c_\\star := \\max\\{1, {\\rm diam}(\\Omega), K \\, R\/2 \\}$.\nFrom \\eqref{a2s4eq58} we can then recover the inequality\n\\begin{equation}\n |\\Omega^- \\setminus \\mathcal Q(\\widehat{\\Omega})| + |G^- \\setminus \\mathcal Q(\\widehat{G})| \\leq \\overline{C} \\, \\mathrm{def}_A(u)^{\\frac{1}{s+2}},\n\\end{equation}\nwhere $\\overline{C} = c_\\star \\tilde{C}$. The slicing of the two sets can then be performed in the same way, which leads to an estimate of type\n\\begin{equation}\n\\label{a2s4eq60}\n |\\lambda_e| \\leq 4 \\, (n+3) \\frac{{\\rm diam}(\\Omega)}{|\\Omega|} \\, \\overline{C} \\, \\mathrm{def}_A(u)^{\\frac{1}{s+2}},\n\\end{equation}\nwhere now again the bound depends on the seminorms on both of the parallel surfaces. We now only need to make sure that formula \\eqref{a2s3eq53} still applies. Again, for the set $\\Omega$ we define $\\rho_{min}:= \\min_{z \\in \\partial \\Omega} |z|$, $\\rho_{max}:= \\max_{z \\in \\partial \\Omega} |z|$, choose $x, y \\in \\partial \\Omega$ such that $|x| = \\rho_{min}$ and $|y| = \\rho_{max}$ and consider the direction $e= y - x$ up to normalization with its critical hypeplane $T^e$. Since we are now in the annular case $\\overline{\\lambda}^e = \\max \\{ \\overline{\\lambda}^e_D, \\overline{\\lambda}^e_\\Omega \\}$ and therefore the moving plane might stop before reaching the cricial position for the set $\\Omega$ itself. However we can still write\n\\begin{equation}\n\\label{a2s4eq62}\n \\rho(\\Omega) \\leq \\rho_{max} - \\rho_{min} = |y| - |x| \\leq 2 \\, {\\rm dist} (0,T^e) = 2 |\\overline{\\lambda}^e| \\leq 2 \\, |\\overline{\\lambda}^e_\\Omega|.\n\\end{equation}\nCombining \\eqref{a2s4eq60} and \\eqref{a2s4eq62} and repeating the same argument for $G$ lead us to \\eqref{a2s1eq18}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n Consider the space $\\mathbf{L}$ of planar polygons with prescribed edge lengths\\footnote{The space $\\mathbf{L}$ appears in the literature as ``configuration space of a flexible polygon'', or ``configuration space of a polygonal linkage'', or just as ``space of polygons''.} and the oriented area $\\mathcal{A}$ as a Morse function defined on it.\n It is known that generically:\n \\begin{itemize}\n \\item $\\mathbf{L}$ is a smooth closed manifold whose diffeomorphic type depends on the edge lengths \\cite{Farber, MilKap}.\n \\item The oriented area $\\mathcal{A}$ is a Morse function whose critical points are cyclic configurations (that is, polygons with all the vertices lying on a circle), whose Morse indices are known, see Theorem \\ref{Thm_Morse_closed_plane}, \\cite{khipan,panzh, zhu}. The Morse index depends not only on the combinatorics of a cyclic polygon, but also on some metric data. Direct computations of the Morse index proved to be quite involved, so the existing proof comes from bifurcation analysis combined with a number of combinatorial tricks.\n \\item Bifurcations of $\\mathcal{A}$ are captured by cyclic polygons $P$ whose dual polygons $P^*$ have zero perimeter \\cite{panzh}; see also Lemma \\ref{LemmaBifurc}. That is, in a generic one-parametric family of edge lengths a critical point $P$ bifurcates whenever the perimeter of the dual tangential polygon $P^*$ vanishes.\n \\end{itemize}\nThe polygon $P^*$ is \\textit{tangential} (see Definition \\ref{DefTang}), so tangential polygons with zero perimeter play a special role in the framework of flexible polygons and oriented area. The initial motivation of the present paper was to clarify\nthis role.\n\n\n In the paper we consider the following problem: instead of prescribing edge lengths, we prescribe the slopes of the edges. Instead of taking the oriented area as a Morse function, we take the oriented perimeter. We prove:\n \\begin{itemize}\n \\item The space $\\mathbf{S}$ of polygons with prescribed edge slopes is a smooth non-compact manifold (see Theorem \\ref{ThmConfSpace} for its diffeomorphism type).\n \\item The (oriented) perimeter $\\mathcal{P}$ is a Morse function with either zero or two critical points (Theorem \\ref{ThmCritPer}).\n Critical points of $\\mathcal{P}$ are{ tangential} polygons.\n \\item The absence of critical points is captured by existence of tangential polygons with zero perimeter (Corollary \\ref{CorCritPer}). That is, in a generic one-parametric family of slopes critical points disappear whenever the perimeter of the (uniquely defined) tangential polygon vanishes.\n\n \\item Although there are at most two critical points, these are not necessarily maximum and minimum of the perimeter function. The Morse index of a tangential polygon is expressed in Theorem \\ref{ThmMorseTangential}. The proof is based on direct computation of the leading principal minors of the Hessian matrix.\n \\item The Morse index of a tangential polygon depends on the combinatorics of the polygon and the sign of its perimeter only.\n \\item Local projective duality provides an alternative proof of Theorem \\ref{Thm_Morse_closed_plane}, that is, the formula for the Morse index of a cyclic polygon (rel the area function).\n \\end{itemize}\n\n Yet another motivation of this research is projective duality. Oversimplifying, assume that the ambient space of the polygons is the sphere $S^2$.\nThen projective duality takes polygons with prescribed edge lengths to polygons with prescribed angles. It also takes area to a linear function of perimeter, so the critical polygons in the two settings are mutually projectively dual and have related Morse indices.\n(A necessary warning: there exists only a local version of projective duality. However it is sufficient for purposes.)\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nSections 3,4, and 5 are supported by the Russian Science Foundation grant N 14-21-00035.\n\nGaiane Panina is\n supported by the RFBR grant 17-01-00128 and the Program of the Presidium of the Russian\n Academy of Sciences N 01 'Fundamental Mathematics and its Applications'\n under\n grant PRAS-18-01.\n\n We are indebted to Alexander Gaifullin who was the first to point out the vanishing perimeter of a bifurcating polygon.\nWe also thank Mikhail Khristoforov for useful discussions.\n\n\n\\section{Definitions and setups}\n\nA \\textit{polygon} is an oriented closed broken line in the plane. We assume that its vertices are numbered in the cyclic order and thus induce an orientation of the polygon.\n\n\n\n\\begin{definition} \\label{Dfn_area} The \\textit{oriented area} of a polygon $P$ with the vertices \\newline $v_i = (x_i,\ny_i)$ is defined by\n$$2\\mathcal{A}(P) = (x_1y_2 - x_2y_1) + \\ldots + (x_ny_1 - x_1y_n).$$\n\n\nEquivalently, one defines $$\\mathcal{A}(P):=\\int_{\\mathbb{R}^2} w(P,x)dx,$$\n\nwhere $w(P,x)$ is the winding number of $P$ around the point $x$.\n\n\n\n\\end{definition}\n\n\n\\subsection{Polygons with prescribed edge lengths. Oriented area as a Morse function.} Assume that a generic $n$-tuple of positive numbers $(l_1,...,l_n)$ is given.\nThe space of all planar polygons whose consecutive edge lengths are $(l_1,...,l_n)$ (modulo translations and rotations) is called the \\textit{configuration space\nof polygons with prescribed edge lengths.} We denote it by $\\mathbf{L}=\\mathbf{L}(l_1,...,l_n)$.\n\nGenerically, $\\mathbf{L}$ is a smooth closed manifold \\cite{Farber, MilKap}, and\n the oriented area $\\mathcal{A}$ is a Morse function on $\\mathbf{L}$.\n\n\n\n\n\\begin{definition}\n A\npolygon $P$ is \\textit{cyclic} if all its vertices $v_i$\nlie on a circle.\n\n\\end{definition}\n\n\n\n\\begin{theorem}\\label{Thm_crirical_are_cyclic}\\cite{khipan} Generically, $\\mathcal{A}$ is a Morse function.\n At smooth points of the space $\\mathbf{L}$, a polygon $P$ is a critical point of the\noriented area $\\mathcal{A}$ iff $P$ is a cyclic configuration.\n \\qed\n\\end{theorem}\n\n\nBefore we recall a formula for the Morse index of a cyclic configuration from \\cite{zhu}, \\cite{khipan1},\nlet us fix the following notation for a cyclic polygon $P$.\n\n$\\omega_P=w(P,O)$ is the winding number of $P$ with respect to the center $O$ of the circumscribed circle.\n\n$\\alpha_i$ is the half of the angle between the vectors\n$\\overrightarrow{Ov_i}$ and $\\overrightarrow{Ov_{i+1}}$. The angle\nis defined to be positive, orientation is not involved.\n\n\n\n\n$\\varepsilon_i$ is the\norientation of the edge $v_iv_{i+1}$, that is,\n\n $\\varepsilon_i=\\left\\{\n \\begin{array}{ll}\n 1, & \\hbox{if the center $O$ lies to the left of } \\overrightarrow{v_iv_{i+1}};\\\\\n -1, & \\hbox{if $O$ lies to the right of } \\overrightarrow{v_iv_{i+1}}.\n \\end{array}\n \\right.$\n\n\n\n$e(P)$ is the number of positive entries in $\\varepsilon_1,...,\\varepsilon_n$, that is, $e(P)$ is the number of \\textit{positively oriented }edges .\n\n\n\n$\\mu_P=\\mu_P(\\mathcal{A})$ is the Morse index of the function $\\mathcal{A}$ at the point\n$P$. That is, $\\mu_P(\\mathcal{A})$ is the number of negative eigenvalues of the\nHessian matrix $Hess_P(\\mathcal{A})$.\n\n\n\\begin{theorem}\\label{Thm_Morse_closed_plane}\\cite{khipan1}, \\cite{panzh}, \\cite{zhu} Generically,\nfor a cyclic polygon $P$,\n$$\\mu_P( \\mathcal{A})= e(P)-1-2\\omega_P-\\left\\{\n \\begin{array}{ll}\n 0 &\\hbox{if }\\ \\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i>0; \\\\\n 1 & \\hbox{otherwise}.\\qed\n \\end{array}\n \\right.$$\n\\end{theorem}\n\nIn the present paper we give an alternative proof of this theorem: we prove a slightly stronger claim, see Corollary \\ref{RemFinal}.\n\\begin{remark}\n In a continuous one-parametric family of cyclic polygons with non-vanishing edge lengths, $\\mu_P$ changes iff $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i$ vanishes. Although $e(P)$ and $\\omega_P$ can vary, the sum $e(P)-1-2\\omega_P$ is constant.\n\\end{remark}\n\\bigskip\n\n\n\\subsection{Polygons with prescribed edge slopes}\n\nFix $n$ pairwise non-parallel straight lines \\(s_1,\\ldots,s_n\\) in \\(\\R^2\\) passing through the origin and call them \\textit{slope lines}.\n\nEach \\(n\\)-tuple of lines \\(e_1,\\ldots,e_n\\subset \\R^2\\)\n with $e_i$ parallel to $s_i$ yields a polygon $P$ whose consecutive vertices are \\(v_1=e_1\\cap e_2, \\ldots, v_n=e_n\\cap e_1 \\). We will denote this polygon by \\(Q=Q\\left(e_1,\\ldots,e_n\\right)\\) and say that $Q$ is a \\textit{polygon with edge slopes} \\(s_1,\\ldots,s_n\\) .\n\nThe space of all polygons $Q$ (modulo translations)\n with edge slopes \\(s_1,\\ldots,s_n\\),\n is denoted by $\\widetilde{\\mathbf{S}}=\\widetilde{\\mathbf{S}}(s_1,...,s_n)$.\n\n\nThe subspace of $\\widetilde{\\mathbf{S}}(s_1,...,s_n)$ consisting of polygons with $|\\mathcal{A}(Q)|=1$\nis called the \\textit{configuration space\nof polygons with prescribed edge slopes}. We denote it by $\\mathbf{S}=\\mathbf{S}(s_1,...,s_n)$. It splits into a disjoint union \\(\\mathbf{S}_-\\sqcup \\mathbf{S}_+\\) where the index indicates the sign of \\(\\mathcal{A}\\).\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=10 cm]{SPolygon.eps}\n\\caption{Slope lines (left) and a polygon with these edge slopes (right).}\\label{FigSlopes}\n\\end{figure}\n\nNote that: (1) the condition $|\\mathcal{A}(Q)|=1$ means that we factor out dilations; (2) fixing slopes is the same as fixing angles and factoring out rotations.\n\\bigskip\n\n\nNow fix a direction on each of the slope lines $\\vec{s}_i$. Take $Q=Q(e_1,...,e_n)\\in \\mathbf{S}(s_1,...,s_n)$, and orient the lines $e_1,...,e_n$ consistently. Denote the oriented lines by $\\vec{e}_1,\\ldots,\\vec{e}_n$.\n\n\\begin{definition}\\label{DefPerim}\n The perimeter of a polygon $Q\\in \\widetilde{\\mathbf{S}}(s_1,...,s_n)$ is defined as follows:\n\\[\n\\mathcal{P}(Q)=\\sum_{i=1}^n \\sign_Q(i)|v_iv_{i+1}|,\n\\]\n\nwhere \\(\\sign_Q(i)=\\left\\{\n \\begin{array}{ll}\n 1, & \\hbox{if $\\vec{e}_i$ is codirected with $\\overrightarrow{v_iv}_{i+1}$;} \\\\\n -1, & \\hbox{otherwise.}\n \\end{array}\n \\right.\\)\n\n\\end{definition}\n\nThus defined, perimeter may be negative or vanish.\n\n\\section{Topology of the configuration space of polygons with prescribed slopes}\n\n\n\nLet the \\textit{angle \\(\\angle(r,s)\\) between two lines} $r$ and $s$ be the minimal positive angle such that the counterclockwise rotation by \\(\\angle(r,s)\\) takes \\(r\\) to \\(s\\).\n\nAssuming that an $n$-tuple of slope lines \\(s_1,\\ldots,s_n\\) in \\(\\R^2\\) is fixed, set\n\\newline \\(t(s_1,\\ldots,s_n)\\defeq\\sum_{i=1}^{n-1}\\angle(s_i,s_{i+1})+\\angle(s_n,s_1)\\).\n\n\\medskip\n\nThe example for \\(n=3\\) will be useful in the sequel:\n\\begin{example}\\label{triangle}\n\\(t(s_1,s_2,s_3)\\) is either \\(\\pi\\) or \\(2\\pi\\). In the first (respectively, second) case the area of each nondegenerate triangle in \\(\\widetilde{\\mathbf{S}}\\) is negative (respectively, positive).\n\n\\end{example}\n\n\n\\begin{lemma}\n\\label{turn}\n\\(\\,\\)\n\\begin{enumerate}\n\n \\item \\(t(s_1,\\ldots,s_n)\\) takes values in $\\{\\pi,2\\pi, \\dots, (n-1)\\pi\\}$.\n \\item \\(t(s_1,\\ldots,s_n)=t(s_1,\\ldots,s_{n-1})+t(s_1,s_{n-1},s_n)-\\pi\\).\\qed\n\n\\end{enumerate}\n\n\\end{lemma}\n\nThe informal meaning of the following lemma is: the area and perimeter behave additively with respect to homological sum of polygons.\n\\begin{lemma}\n\\label{sum}\n\\(\\,\\)\nFor a polygon \\(Q=Q(e_1,\\ldots,e_n)\\in\\widetilde{\\mathbf{ S}}(s_1,...,s_n)\\), we have\n\\begin{enumerate}\n \\item \\(\\mathcal{A}(Q)= \\mathcal{A}(Q_{1,\\ldots,n})=\\mathcal{A}(Q_{1,\\ldots,n-1})+\\mathcal{A}(Q_{1,n-1,n})\\);\n\t\t\\item \\(\\mathcal{P}(Q)= \\mathcal{P}(Q_{1,\\ldots,n})=\\mathcal{P}(Q_{1,\\ldots,n-1})+\\mathcal{P}(Q_{1,n-1,n})\\),\n\n\\end{enumerate}\nwhere $Q_{i_1,...,i_k}=Q(e_{i_1},...e_{i_k})\\in \\widetilde{\\mathbf{S}}(s_{i_1},...,s_{i_k})$.\\qed\n\\end{lemma}\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{coord.eps}\n\\caption{A summand of $\\mathcal{A}(Q_{1,\\ldots,n})$.}\\label{FigCoordinates}\n\\end{figure}\n\n\n\\begin{theorem}\\label{ThmConfSpace} The configuration space $\\mathbf{S}(s_1,\\ldots,s_n)$ is homeomorphic to disjoint union of products of a sphere and a disc:\n$$\n S^{n-k-2}\\times \\D^{k-1}\\sqcup S^{k-2} \\times \\D^{n-k-1}\n,$$ where $t(s_1,\\ldots,s_n)=k\\pi$.\n\nThe left-hand part corresponds to \\(\\mathbf{S}_-\\) and the right-hand part corresponds to \\(\\mathbf{S}_+\\).\n\\end{theorem}\n\n\\begin{proof}\nWe shall prove that \\(\\mathbf{S}_+\\) is homeomorphic to \\(S^{k-2} \\times \\D^{n-k-1}\\). The proof for \\(\\mathbf{S}_-\\) is analogous.\n\n\nBy Lemma \\ref{sum},\n\\[\n\\mathcal{A}(Q_{1,\\ldots,n})=\\sum_{i=2}^{n-1}\\mathcal{A}(Q_{1,i,i+1})=\\sum_{i=2}^{n-1}\\sign\\Big(t(s_1,s_i,s_{i+1})-\\frac{3\\pi}2\\Big)\\cdot c_i\\cdot \\dist(v_i,e_1)^2,\n\\]\nwhere \\(c_i=c_i(s_1,s_i,s_{i+1})\\) is some positive constant depending only on the edge slopes (see Example \\ref{triangle} and Fig. \\ref{FigCoordinates}). Thus we can parameterize $\\widetilde{\\mathbf{S}}(s_1,...,s_n)$ by \\(\\{x_i=\\sqrt{c_{i+1}}\\dist(v_{i+1},e_1)\\}_{i=1}^{n-2}\\). That is,\n\\[\\mathbf{S}_+=\\{x\\in\\R^{n-2} \\mid \\sum_{j\\in A}x_j^2 - \\sum_{j\\in B} x_j^2 =1\\},\\] where\n\\(|A|=k-1\\), \\(A\\sqcup B=\\{1,\\ldots,n-2\\}\\).\nThis is homeomorphic to \\[\\{x\\in\\R^{n-2} \\mid \\sum_{j\\in A}x_j^2 =1, \\sum_{j\\in B}x_j^2 <1\\}.\\]\nIndeed, for \\(k>1\\) the homeomorphism \\(h: x\\mapsto \\frac{x}{\\sqrt{\\sum_{j\\in A}x_j^2}}\\) is well defined and appropriate, whereas for \\(k=1\\) both sets are empty.\nThe claim follows.\n\\end{proof}\n\n\n\\section{Critical points of the perimeter }\nAssume that an $n$-tuple of directed slope lines \\(\\vec{s}_1,\\ldots,\\vec{s}_n\\) in \\(\\R^2\\) is fixed.\n\n\n\\begin{definition}\\label{DefTang} \\begin{enumerate}\n \\item A polygon \\(Q(e_1,\\ldots,e_n)\\in \\widetilde{\\mathbf{S}}(s_1,...,s_n)\\) is \\textit{tangential} if there exists a circle $\\sigma$ such that\n\n (a) each of $e_i$ is tangent to $\\sigma$, and\n\n\n(b) either $\\sigma$ lies on the left with respect to all of $\\vec{e}_i$, or $\\sigma$ lies on the right with respect to all of $\\vec{e_i }$, see Fig. \\ref{FigSuperscribed}.\n\nIn this case we say that the circle $\\sigma$ is \\textit{inscribed} in $Q$ and write $\\sigma=\\sigma(Q)$.\n \\item By the \\textit{radius} $r=r(\\sigma(Q))$ of the inscribed circle $\\sigma$ we mean the usual radius taken with the sign ''$+$'' if $\\sigma$ lies on the left of $\\vec{e}_i$, and with the sign ''$-$'' otherwise.\n \\end{enumerate}\n\n\n\n\\end{definition}\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=14 cm]{Twosuperscribed.eps}\n\\caption{(a) A tangential polygon $Q$ with the inscribed circle of positive radius. Here $\\mathcal{A}(Q)<0, \\mathcal{P}(Q)<0$.\n(b) The tangential polygon $-Q$ with $\\mathcal{A}(-Q)=\\mathcal{A}(Q)<0, \\mathcal{P}(-Q)=-\\mathcal{P}(Q)>0$. The inscribed circle has negative radius. }\\label{FigSuperscribed}\n\\end{figure}\n\n\\begin{lemma}\\label{APr}\n The area and perimeter of a tangential polytope $Q$ satisfy:\n $$\\mathcal{A}(Q)=\\frac12\\mathcal{P}(Q)\\cdot r(\\sigma(Q)).$$\n\n\\end{lemma}\n\n\\begin{theorem}\\label{ThmCritPer}\n \\(Q\\in\\mathbf{S}\\) is a critical point of \\(\\mathcal{P}\\) iff $Q$ is tangential.\n\\end{theorem}\n\n\n\n\\begin{proof}\n\nLet \\(r_i\\) be the radius of the (uniquely defined) circle inscribed\\footnote{in the sense of Definition \\ref{DefTang}.} in the triangle \\(Q_{1,i+1,i+2}\\), \\ \\(i=1,\\ldots,n-2\\). Denote by \\(p_i\\) the perimeter of the triangle homothetic to $Q_{1,i+1,i+2}$ (that is, with the same edge slopes) whose inscribed radius equals $1$. If $r_i\\neq 0$, \\(p_i\\) is the perimeter of is the triangle\n \\(\\frac{1}{r_i}Q_{1,i+1,i+2}\\).\nThen by Lemma~\\ref{sum} and Lemma~\\ref{APr} we have\n$$\n \\mathcal{P}(Q)=\\sum_{i=1}^{n-2}p_i\\cdot r_i, \\ \\ \\ \\ \\hbox{and}$$\n $$ \\pm1=\\mathcal{A}(Q)=\\frac{1}{2}\\sum_{i=1}^{n-2}p_i \\cdot r_i^2.$$\n\nThe collection of radii gives a coordinate system on $\\widetilde{\\mathbf{S}}$.\nW.l.o.g. we assume \\(r_1>0\\). Locally on $\\mathbf{S}$, the second row implicitly defines a function \\(r_1(r_2,\\ldots,r_{n-2})\\). Let us take \\(j\\)-th partial derivatives of the both rows, \\(j=2,\\ldots,n-2\\).\n $$ \\hbox{Since } \\ 0=\\diffp{\\mathcal{A}}{{r_j}}(Q)=p_1 r_1\\diffp{r_1}{{r_j}}+p_j r_j,\n$$\n\n$$ \\hbox{we have } \\\n \\diffp{r_1}{{r_j}}=-\\frac{p_j r_j}{p_1 r_1} .$$\n$$ \\hbox{Therefore, } \\\n \\diffp{\\mathcal{P}}{{r_j}}(Q)=p_1 \\diffp{r_1}{{r_j}}+p_j=p_j(1-\\frac{r_j}{r_1}). $$\n\n\n\nThus the gradient of the perimeter is zero iff \\(r_1=r_2=\\ldots=r_{n-2}=r\\)\n\\end{proof}\n\nThere exists exactly one (up to a dilation) pair of mutually symmetric tangential polygons $Q$ and $-Q$. For them we have: $\\mathcal{A}(-Q)=\\mathcal{A}(Q),\\ \\mathcal{P}(-Q)=-\\mathcal{P}(Q)$, and $r(-Q)=-r(Q)$, see Fig.~\\ref{FigSuperscribed} for example. If the area is non-zero, scaling gives $|\\mathcal{A}|=1$.\n\n\\begin{corollary}\\label{CorCritPer}\nIf the area of a tangential polygon is zero, there are no critical points of \\(\\mathcal{P}\\) on the configuration space $\\mathbf{S}$.\nOtherwise there are exactly two critical points on the configuration space $\\mathbf{S}$. Either they both lie in $\\mathbf{S}_+$, or they both lie in $\\mathbf{S}_-$.\\qed\n\n\\end{corollary}\n\n\n\\medskip\n\nA configuration space $\\mathbf{S}(s_1,...,s_n)$ with no critical points is called\\textit{ exceptional}.\n\n\\section{Morse index of a tangential polygon}\nNow compute the Hessian matrix of $\\mathcal{P}$ at a critical point. Assume that $Q$ is a tangential polygon. In notation of the previous section, we have:\n\\[\n\\diffp[2]{\\mathcal{P}}{{r_j}}(Q)=p_1\\diffp[2]{r_1}{{r_j}}=p_1\\diffp{}{{r_j}}\\Big(-\\frac{p_j r_j}{p_1 r_1}\\Big)=-p_j\\Big(\\frac1{r_1}-\\frac{r_j}{r_1^2}\\diffp{r_1}{{r_j}}\\Big)=-\\frac{p_j}{r_1}\\Big(1+\\frac{p_j r_j^2}{p_1 r_1^2}\\Big).\n\\]\n\nSince at the critical point all \\(r_i\\) are equal,\n\\[\nH_{jj}(Q)=\\diffp[2]{\\mathcal{P}}{{r_j}}(Q)=-\\frac{p_j}{r p_1}(p_1+p_j).\n\\]\nIn the same way, for \\(j\\neq k\\)\n\\[\n\\diffp{\\mathcal{P}}{{r_j}{r_k}}=p_1\\diffp{r_1}{{r_j}{r_k}}=p_1\\diffp{}{{r_k}}\\Big(-\\frac{p_j r_j}{p_1 r_1}\\Big)=-p_j r_j\\Big(-\\frac1{r_1^2}\\cdot \\diffp{r_1}{{r_k}}\\Big)=-\\frac{p_j r_j}{r_1^2}\\cdot \\frac{p_k r_k}{p_1 r_1}.\n\\]\n\nThus: $$ H_{jk}(Q)=\\diffp{\\mathcal{P}}{{r_j}{r_k}}=-\\frac{p_j p_k}{r p_1}.$$\n\nTo compute its determinant we do the following:\n\\begin{enumerate}\n\\item add all the columns to the first one;\n\\item subtract the first row from the \\(i\\)-th row $(i=2,3,...,n-2)$ taken with the coefficient \\(\\frac{p_{i+1}}{p_2}\\) ;\n\\item subtract all rows from the first row with coefficient \\(\\frac{p_2}{p_1}\\).\n\\end{enumerate}\n\nWe get:\n\n\\begin{equation}\\label{det}\nr^{n-3}\\det(H)=\n\\begin{vmatrix}\n-\\frac{p_2}{p_1}\\Pi & 0 & \\ldots & 0 \\\\\n0 & -p_3 & \\ldots & 0 \\\\\n& & \\ddots \\\\\n0 & \\ldots & 0 & -p_{n-2}\n\\end{vmatrix},\n\\end{equation}\n\nwhere \\(\\Pi=p_1+\\ldots+p_{n-2}=\\frac{\\mathcal{P}(Q)}r\\).\n\n\\begin{definition}\n For an ordered pair of slopes $\\vec{s}_i$ and $\\vec{s}_j$ we say that we have the \\textit{right turn} (\\textit{left turn}, respectively), if $\\vec{s}_j$ is obtained from $\\vec{s}_i$ by a clockwise (counterclockwise, respectively) turn by an angle smaller than $\\pi$, see Fig. \\ref{FigTurn}.\n\n The\\textit{ number of right turns} \\(RT=RT(\\vec{s_1},\\ldots,\\vec{s_n})\\) for a slope collection $(\\vec{s_1},\\ldots,\\vec{s_n})$ is\nthe number of right turns of the pairs $(\\vec{s_1},\\vec{s_2}),(\\vec{s_2},\\vec{s_3}),...,(\\vec{s_{n}},\\vec{s_1}) . $\nThe\\textit{ number of left turns} \\(LT\\) is defined analogously.\n\\end{definition}\n\n\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{Turn.eps}\n\\caption{Right and left turns}\\label{FigTurn}\n\\end{figure}\n\n\n\\begin{theorem}\\label{ThmMorseTangential}\nAssume that for a tangential polygon $Q\\in \\mathbf{S}$, the radius \\(r\\) of the inscribed circle $\\sigma(Q)$ is positive.\n{Then $Q$ is a Morse point, and its Morse index of $\\mathcal{P}$ is equal to}\n\\begin{equation}\\label{index}\n\\mu_Q(\\mathcal{P})= RT-1+2\\omega_Q-\\left\\{\n \\begin{array}{ll}\n 1 &\\hbox{if }\\ \\mathcal{P}(Q)>0; \\\\\n 0 & \\hbox{otherwise}.\n \\end{array}\n \\right.\n\\end{equation}\nIn the case of negative radius it is equal to\n\n\\[\n\\mu_Q(\\mathcal{P})=n-3-\\mu_{-Q}(\\mathcal{P})=LT-1-2\\omega_Q-\\left\\{\n \\begin{array}{ll}\n 1 &\\hbox{if }\\ \\mathcal{P}(Q)>0; \\\\\n 0 & \\hbox{otherwise}.\n \\end{array}\n \\right.\n\\]\n\\end{theorem}\n\n\\begin{example}\n For the polygon depicted in Fig. \\ref{FigSuperscribed} (a), we have $\\omega=0$, $RT=2$, and $m=0$.\n\n\tFor the polygon depicted in Fig. \\ref{FigTanMorse}, we have $\\omega=1$, $RT=1$, and $m=1$.\n\\end{example}\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{tangentialMorse.eps}\n\\caption{This tangential polygon is a saddle critical point of the perimeter.}\\label{FigTanMorse}\n\\end{figure}\n\\begin{proof}\n\nFor now denote the right-hand side of (\\ref{index}) as \\(m(Q)\\) and assume \\(r>0\\).\n\nLet us prove (\\ref{index}) by induction on \\(n\\).\nFor \\(n=3\\) the value of \\(m\\) is always zero, so is the Morse index.\n\nProve the claim for \\(n+1\\) assuming it is true for all the numbers smaller or equal than \\(n\\geq 3\\).\n\nRecall that the number of negative eigenvalues of the matrix equals the number of sign changes in the sequence of its leading principal minors. The \\(k\\)-th leading principal minor of \\(H(Q)\\) is the determinant of \\(H(Q_{1,\\ldots,k+2})\\).\n\nSo we have:\n\\begin{itemize}\n\\item \\(m(Q)=m(Q_{1,\\ldots,n})+1\\) whenever the sign of the determinant (\\ref{det}) is different for \\(n\\) and \\(n+1\\);\n\\item \\(m(Q)=m(Q_{1,\\ldots,n})\\) whenever the sign of the determinant (\\ref{det}) is the same for \\(n\\) and \\(n+1\\).\n\\end{itemize}\n\nThe change of sign of the determinant (\\ref{det}) depends only on the sign of \\(p_{n-1}\\) and the sign change of \\(\\Pi\\).\n\nNote that\n\\[\nRT(s_1,\\ldots,s_{n+1})=RT(s_1,\\ldots,s_n)+RT(s_1,s_n,s_{n+1})-1\n\\]\nand\n\\[\n\\omega_{Q_{1,\\ldots,n+1}}=\\omega_{Q_{1,\\ldots,n}}+\\omega_{Q_{1,n,n+1}}.\n\\]\n\n\nTherefore,\n\n\n\\[\nm(Q_{1,\\ldots,n+1})-m(Q_{1,\\ldots,n})=m(Q_{1,n,n+1})+\\mathbbm{1}_{\\mathcal{P}(Q_{1,n,n+1})>0}-\\mathbbm{1}_{\\mathcal{P}(Q)>0}+\\mathbbm{1}_{\\mathcal{P}(Q_{1,\\ldots,n})>0}.\n\\]\n\nThe first summand is zero by the base of induction. So we get:\n\\[\nm(Q_{1,\\ldots,n+1})-m(Q_{1,\\ldots,n})=\\mathbbm{1}_{p_{n-1}>0}-\\mathbbm{1}_{p_1+\\ldots+p_{n-1}>0}+\\mathbbm{1}_{p_1+\\ldots+p_{n-2}>0},\n\\]\n\nwhich is exactly what we require.\n\nThe case of \\(r<0\\) follows from \\(\\mathcal{P}(-Q)=-\\mathcal{P}(Q)\\).\n\n\n\\end{proof}\n\n\\section{Cyclic polygons and tangential polygons meet}\nTheorem \\ref{Thm_Morse_closed_plane} motivates the following definition:\n\n\\begin{definition}\nA cyclic polygon is a \\textit{bifurcating polygon} if $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i=0$.\n\\end{definition}\nBifurcating polygons correspond to bifurcations of the area function.\n\n\\begin{definition}\nGiven a cyclic polygon $P=\\{v_1,...,v_n\\}$, define its \\textit{dual polygon} $P^*$ (Fig. \\ref{FigRelation}) as a closed broken line with orientations on the edges constructed as follows:\n\n\\begin{enumerate}\n \\item Take the lines $e_1,...,e_n$ tangential to the circle at the points $\\{v_1,...,v_n\\}$.\n \\item Take the intersection points of $e_i$ and $e_{i+1}$.\n \\item Orient each of the lines so that the circle lies to the left of the line.\n\\end{enumerate}\n\n\nThe construction can be easily reversed: given a tangential polygon $Q$, there exists a cyclic polygon $P$ such that $P^*=Q$.\n\\begin{comment}\nHere is the reversing construction:\n\\begin{enumerate}\n \\item Take the tangential points of the edges.\n \\item Connect them into a closed broken line according to the ordering of the slopes.\n\\end{enumerate}\n\\end{comment}\n\\end{definition}\n\n\\begin{lemma}\\label{LemmaBifurc} \\begin{enumerate}\n \\item For a cyclic polygon $P$, the perimeter of $P^*$ (in the sense of Definition \\ref{DefPerim}) equals $\\sum_{i=1}^n \\varepsilon_i \\tan \\alpha_i$.\n \\item In particular, the perimeter of $P^*$ vanishes iff $P$ is a bifurcating polygon.\n \\item The oriented area of $P^*$ vanishes iff $P$ is a bifurcating polygon.\\qed\n \\end{enumerate}\n\\end{lemma}\n\nTo summarize, if a polygon $P$ is a bifurcation of the area, then its dual $Q=P^*$ yields an exceptional configuration space $\\mathbf{S}(s_1,...,s_n)$.\n\n\n\n\\begin{figure}[h]\n\\centering \\includegraphics[width=8 cm]{Relation.eps}\n\\caption{ A cyclic polygon $P$, the polygon $P^*$ (bold), and the projectively dual polygon $P^o$ (dashed).}\\label{FigRelation}\n\\end{figure}\n\n\\section{Shperical polygons, local projective duality, and an alternative proof of Theorem \\ref{Thm_Morse_closed_plane}}\n\n\\subsection{Shperical polygons}\n\nA \\textit{spherical polygon} is an oriented closed broken line lying on the sphere of radius $R$. We always assume that its edges are the unique shortest geodesics, that is, $l_i<\\pi R$, so a spherical polygon is uniquely defined by the (circular) sequence of its vertices.\n\nOne fails to correctly define the area function on the space of spherical polygons with prescribed edge lengths. One also fails to define the space of spherical polygons with prescribed angles together with the perimeter function. However we shall make use of their local versions.\n\nHere is how it goes:\n\n\\begin{definition}\n Let $P_0, Q_0$ be spherical polygons.\n \\begin{enumerate}\n \\item Consider the space of all spherical polygons with the same edge lengths lying close\\footnote{With respect to any reasonable metric} to $P_0$ subject to rotation of the sphere. This space is called the \\textit{local configuration space }\\textit{of spherical polygons with prescribed edge lengths} $\\mathbf{L}^{loc}(P_0)$.\n \\item Elimination of a point from the sphere allows to define the winding numbers for curves in the sphere. So fix a point $\\infty\\notin P_0$ (that is, not lying on the broken line) and define the oriented area of $P\\in \\mathbf{L}^{loc}(P_0)$ as the integral\n $$A(P):=\\int_{S^2} w(P,x)dx.$$\n \\item Analogously, we define the \\textit{local configuration space of spherical polygons with prescribed angles} $\\mathbf{S}^{loc}(Q_0)$. Once we set some fixed orientations on the edges, we have a well defined perimeter function $\\mathcal{P}$ on the space $\\mathbf{S}^{loc}(Q_0)$. Note that the area $\\mathcal{A}$ is constant on $\\mathbf{S}^{loc}(Q_0)$.\n \\end{enumerate}\n\\end{definition}\n\n\n\\begin{proposition}\\label{PropSphCrit}\n Assume that a spherical polygon $P$ fits in a hemisphere not containing $\\infty$. $P$ is a critical point of the area function $\\mathcal{A}$ iff it is a cyclic polygon, that is, its vertices lie on a circle.\n\\end{proposition}\nProof.\n(1) Prove first the statement for polygons with four edges.\nIf a polygon bounds a spherically convex region, then the statement is classical: a cyclic convex $4$-gon exhibits either the maximum or the minimum point of the area, depending on orientation of $P$.\n\n(''If'') Assume that $P$ intersects itself and is a cyclic polygon with vertices $1,2,3,4$. Add a new point $5$ on the circle together with two new bars as is shown in Fig. \\ref{fourgon}. Then $\\mathcal{A}(1234)=\\mathcal{A}(1254)-\\mathcal{A}(2543)$. A local flex of the polygon $P$ induces flexes of the polygons\n$(1254)$ and $(2543)$. Since the latter are critical, the claim follows.\n\n(''Only if'') Assume that $P$ intersects itself and is a critical point of $\\mathcal{A}$, but not a cyclic polygon. Take a circle superscribing $412$ and add a new point $5$ on the circle together with two new bars as we did above. Now $(125 4)$ is a critical polygon, and $(2543)$ is not.\n $\\mathcal{A}(1234)=\\mathcal{A}(1254)-\\mathcal{A}(2543)$ completes the proof.\n\n\n(2) The general case (any number of edges) is obtained by verbatim repeating the reasonings from \\cite{khipan}.\\qed\n\\begin{figure}[h]\n\\centering \\includegraphics[width=4 cm]{Crit4gon.eps}\n\\caption{Notation for Proposition \\ref{PropSphCrit}.}\\label{fourgon}\n\\end{figure}\n\n\n\n\\subsection*{Local duality of $\\mathbf{L}^{loc}(P)$ and $\\mathbf{S}^{loc}(P^o)$. }\nAssume that $P$ is a spherical cyclic polygon. It fits in a hemisphere, and so do all the polygons from $\\mathbf{L}^{loc}(P) $.\nAssume also that a point $\\infty$ lies beyond the hemisphere, so $\\mathcal{A}$ is well-defined on $\\mathbf{L}^{loc}(P) $,\nand $P$ is a critical point.\n\n\\begin{definition}\n\\begin{enumerate}\n \\item In the above setting, define\nthe dual polygon $P^0$:\n\n\\begin{enumerate}\n\\item Fix the hemisphere containing P centered at the center of the superscribed circle.\n \\item Assume that $P=\\{p_1,...,p_n\\}$. Take the big circles $C_1,...,C_n$ projectively dual to the points $\\{p_1,...,p_n\\}$.\n \\item Connect the intersection points $q_i$ of $l_i$ and $l_{i+1}$ lying in the hemisphere by short geodesics.\n \\item Orient each of the lines such that the circle lies on the {left} from each of the lines.\n\\end{enumerate}\n\n \\item Continuously extend the duality to $\\mathbf{L}^{loc}(P) $. The extension is uniquely defined by the condition that edges of the dual polygon lie on dual lines to the vertices of the initial polygon.\n\\end{enumerate}\n\n\n\\end{definition}\n\\begin{lemma}\\label{LemmaRelAP}For a cyclic $P$,\n \\begin{enumerate}\n \\item The polygon $P^o$ is tangential.\n \n \\item \\(\\mu_{P}(\\mathcal{A})=n-3-\\mu_{P^o}(\\mathcal{P}).\\)\n \n\n \\end{enumerate}\n\\end{lemma}\nProof. (1) is straightforward.\n\n Projective duality (on the unit sphere) takes edge lengths of a polygon to the exterior angles of the dual, and vice versa.\n Since\n $$\\mathcal{A}(P)=Const -R\\cdot \\mathcal{P}(P^o),$$\n the claims (2) follows.\\qed\n\n\n\n\\bigskip\n\n\n\\subsection*{Morse indices: planar vs spherical.}\n\nLet $P_0$ be a planar cyclic polygon. It is uniquely defined by the circumscribed circle $\\sigma$ and the ordered sequence of its points. Put the circle $\\sigma$ with the $n$ points on the sphere of radius $R$, provided that $R>r(\\sigma)$. It defines a spherical cyclic polygon $P_0^R$ fitting in the hemisphere centered at the center of $\\sigma$. In turn, the spherical polygon $P_0^R$ gives rise to the local configuration space $\\mathbf{L}^{loc}(P_0^R)$ of polygons with prescribed edge lengths. Clearly, the bigger $R$ is, the smaller is the distortion of edge lengths and angles.\n\n\n\n\\begin{lemma}\\label{LemmaBifurkFree}\nThe Morse index of the spherical polygon $P_0^R$ with respect to the area function $\\mathcal{A}_R$ is the same as the Morse index of $P_0$.\n\\end{lemma}\n\n {Proof.\n\\begin{comment} keeps all the angles of the polygon except for the two last ones, and $\\phi$ keeps the edge lengths except for the last one.\n\nThen in the neighborhood $U(Q_0)$, one has $\\mathcal{P}(\\phi(Q))=\\mathcal{P}(Q)+\\frac{1}{R}\\cdot \\mathbf{R}(Q,R)$, where $\\mathbf{R}$ and its second derivatives are uniformly bounded.\n\\end{comment}\nWe have the one parametric family of spherical polygons and their local configuration spaces $P_0^R,\\mathbf{L}^{loc}(Q_0^R)$. The area function $\\mathcal{A}_R$ is well defined on $\\mathbf{L}^{loc}(P_0^R)$, and $P_0^R$ is its critical point. As the radius $R$ tends to infinity, the polygon $P_0^R$ deforms and tends to $P_0$. Since $P_0^R$ is the unique critical point in the neighborhood, $P_0^R$ does not bifurcate, so its Morse index does not change. Besides, by standard arguments, the Morse index of $P_0^R$ converges to the Morse index of the planar polygon $P_0$. \\qed\n\n\\bigskip\n\n\n\n\\bigskip\nNow we are ready to prove Theorem \\ref{Thm_Morse_closed_plane}. Take a planar cyclic polygon $P$.\nTake $P^R$ for some big $R$. By the above lemma, $\\mu_{P^R}(\\mathcal{A}_R)= \\mu_{P}(\\mathcal{A})$. Gradually make $R$ smaller, such that the circumscribed circle tends to an equator of the sphere. The Morse index stays the same. Now take the projectively dual polygon $Q^R:=(P^R)^o$. It is a small (and therefore, almost planar) tangential polygon.\n\nBy Lemma \\ref{LemmaRelAP}, $$\\mu_{P^R}(\\mathcal{A}_R)=n-3-\\mu_{Q^R}(\\mathcal{P}_R).$$\nReplace $Q^R$ by a planar polygon $Q$. On the one hand, the Morse index $\\mu_{Q}(\\mathcal{P})$ stays the same. On the other hand, we know the Morse index by\nTheorem \\ref{Thm_Morse_closed_plane}. It remains to observe that\n local projective duality maintains the winding number, and\n takes {left turns} to positively oriented edges.\n \\qed\n\nThis approach also gives the following fact which exceeds Theorem \\ref{Thm_Morse_closed_plane}:\n\\begin{corollary}\\label{RemFinal}\nAssume we have a cyclic polygon $P$ such that (1) no two consecutive vertices are antipodal (with respect to the superscribed circle), and (2) the polygon does fit in a straight line. Then\n$P$ is a Morse point of the oriented area function iff it is not a bifurcating polygon.\n\\end{corollary}\nProof. The dual polygon is a non-degenerate Morse point, see Theorem \\ref{ThmMorseTangential}. \\qed\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{secI}Introduction}\nOne of the most studied phenomenon in low-energy nuclear physics, both experimentally and \ntheoretically, is the way nucleonic matter organises itself to support a variety of \nshapes observed in finite nuclei. The occurrence of different shapes, shape coexistence, \nand shape transitions have their origin in the evolution of single-nucleon shell structure \nwith nuclear deformation, angular momentum, temperature and number of valence nucleons. \nThe manifestation of shells is a generic property of finite fermion systems and shell closures, \nin particular, are characteristic of the confining single-particle potential. When nucleons \ncompletely fill a major shell (single or doubly closed-shell nuclei), the relatively large energy gap \nto the next shell stabilises a spherical shape, whereas long-range correlations between \nvalence nucleons in open-shell nuclei drive the nucleus toward deformed (quadrupole, octupole) \nequilibrium shapes. \n\nCoexistence of different shapes in a single nucleus, and shape (phase) transitions as a function of \nnucleon number, present universal phenomena that occur in light, medium-heavy, heavy and \nsuperheavy nuclei, and reflect the organisation of nucleons in finite nuclei \\cite{HW.11,BN.96,SP.08,CJC.10}. \nA unified description of shape evolution and shape coexistence over the entire chart of nuclides \nnecessitates a universal theory framework that can be applied to different mass regions. \nNuclear energy density functionals (EDF) provide an economic, global and accurate \nmicroscopic approach to nuclear structure that can be extended \nfrom relatively light systems to superheavy nuclei, and from the\nvalley of $\\beta$-stability to the particle drip-lines \\cite{BHR.03,VALR.05,LNP.641,SR.07}. \nThis is particularly important for extrapolations to regions far from stability where not enough \ndata are available to determine the parameters of a more local approach such as, for \ninstance, the interacting shell model. \n\nThe basic implementation of the EDF framework is in terms of\nself-consistent mean-field (SCMF) models, in which an EDF is constructed\nas a functional of one-body nucleon density matrices that correspond to\na single product state. Nuclear SCMF models\neffectively map the many-body problem onto a one-body\nproblem, and the exact EDF is approximated by simple \nfunctionals of powers and gradients of ground-state nucleon densities and currents. \nSome of the advantages of the EDF approach are apparent already at the \nSCMF level: an intuitive interpretation of mean-field results in terms of intrinsic \nshapes and single-nucleon states, and the possibility to formulate structure models \nin the full model space of occupied states, with no distinction between core and \nvalence nucleons. \n\nQuantitative studies of low-energy structure phenomena related to shell \nevolution and coexistence usually start from a constrained Hartree-Fock\nplus BCS (HFBCS), or Hartree-Fock-Bogoliubov (HFB) calculation of the\ndeformation energy surfaces with mass multipole moments as\nconstrained quantities. When based on microscopic EDFs or effective\ninteractions, such calculations comprise many-body correlations \nrelated to the short-range repulsive inter-nucleon interaction and \nlong-range correlations mediated by nuclear resonance modes.\nThe result are static symmetry-breaking product many-body states. \nThe basic idea of a deformation energy surface is that, even though the quantum \nmany-body system is determined by a very large number of microscopic states, \nthese can be organised in a collection of basins that are robust to small external \nperturbations \\cite{Laughlin.00}. The basins can be structurally distinct and \ndistant from each other, but they can occur at comparable energies. \n\nThe constrained SCMF method, however, produces semi-classical \ndeformation energy surfaces. The static nuclear mean-field is characterised by the \nbreaking of symmetries of the underlying Hamiltonian -- translational, rotational, particle \nnumber and, therefore, includes important static correlations, e.g. deformations and \npairing. To calculate excitation spectra and electromagnetic transition rates\nit is necessary to extend the SCMF scheme to include collective correlations that\narise from symmetry restoration and fluctuations around the mean-field\nminima. Collective correlations are sensitive to shell effects, display\npronounced variations with particle number and, therefore, cannot be\nincorporated in a universal EDF but rather require an explicit \ntreatment \\cite{BHR.03,NVR.11,LNP.879}. \n\nOn the second level of implementation of nuclear EDFs that takes into\naccount collective correlations through the restoration of broken symmetries\nand configuration mixing of symmetry-breaking product states, the\nmany-body energy takes the form of a functional of all transition\ndensity matrices that can be constructed from the chosen set of\nproduct states. This set is chosen to restore symmetries or\/and to\nperform a mixing of configurations that correspond to specific collective\nmodes using, for instance, the (quasiparticle) random-phase approximation\n(QRPA) or the Generator Coordinate Method (GCM) \\cite{RS.80}.\nThe latter includes correlations related to finite-size fluctuations in a collective\ndegree of freedom and presents the most effective approach for\nconfiguration mixing calculations, with multipole moments used as coordinates \nthat generate the intrinsic wave functions.\n\nMany interesting phenomena related to shell evolution have been investigated \nover the last decade by employing GCM configuration mixing of angular-momentum \nand particle-number projected states based on energy density functionals or effective interactions,\nbut the extension of this method to non-axial shapes and\/or heavy nuclei still presents \nformidable conceptual and computational challenges \\cite{LNP.879,BH.08,RE.10,Yao.10,BABH.14}. \nIn an alternative approach to nuclear collective dynamics that restores rotational symmetry and \nallows for fluctuations around mean-field minima, a collective Hamiltonian\ncan be formulated, with deformation-dependent parameters determined by\nself-consistent mean-field calculations \\cite{RG.87,PR.09}. \nThe dynamics of the collective Hamiltonian is governed by the\nvibrational inertial functions and the moments of inertia \\cite{GR.80}, \nand these functions are\ndetermined by the microscopic nuclear energy\ndensity functional and the effective interaction in the pairing channel. \nFive-dimensional collective Hamiltonian models for quadrupole vibrational and rotational\ndegrees of freedom, with parameters determined by constrained triaxial SCMF calculations based \non the Gogny effective interaction \\cite{Delaroche10}, the Skyrme density functional \\cite{PR.09}, and \nrelativistic density functionals \\cite{NVR.11}, have been developed over the last decade and \napplied in a number of studies of structure phenomena related to shape coexistence and \nshape transitions. \n\nThe present study is based on the framework of relativistic energy density functionals, extended \nto include the treatment of collective correlations using the collective \nHamiltonian model \\cite{NVR.11}. To emphasise the universality of the EDF approach, all \nillustrative calculations performed in this study, from relatively light \nsystems to very heavy nuclei, have been carried out using a single energy density functional --\nDD-PC1 \\cite{Nik.08}. Starting from microscopic nucleon self-energies in nuclear matter, \nand empirical global properties of the nuclear matter equation of state, the coupling parameters \nof DD-PC1 were fine-tuned to the experimental masses of a set of 64 deformed nuclei in the \nmass regions $A \\approx 150 -180$ and $A \\approx 230 - 250$. The functional has been further \ntested in a number of mean-field and beyond-mean-field calculations in different mass \nregions. For the examples considered here, pairing correlations have been taken into account \nby employing an interaction that is separable in momentum space, and is completely determined \nby two parameters adjusted to reproduce the empirical bell-shaped pairing gap in symmetric \nnuclear matter \\cite{Tian_PLB.09}. For the details of the particular implementation of the \nEDF-based collective Hamiltonian used in the present study, \nwe refer the reader to Ref.~\\cite{NVR.11}. \n\\section{\\label{secII}Beyond the relativistic mean-field approximation: collective correlations}\n\nFor a self-consistent description of collective excitation spectra \nand electromagnetic transition rates, the framework of (relativistic) \nenergy density functionals has to be extended to take into account \ncollective correlations in relation to \nrestoration of broken symmetries and fluctuations in collective coordinates. \nBoth types of correlations can be included simultaneously by mixing symmetry-projected states \ncorresponding to different values of chosen collective coordinates. \nThe most effective approach for configuration mixing calculations is the generator \ncoordinate method (GCM), with multipole moments used as coordinates that generate the \nintrinsic wave functions.\nThe GCM is based on the assumption\nthat, starting from a set of mean-field states $\\ket{\\Phi (q)}$ that \ndepend on a collective coordinate $q$, one can build\napproximate eigenstates of the nuclear Hamiltonian:\n\\begin{equation}\n\\ket{\\Psi_\\alpha} = \\int d q {f_\\alpha(q)\\ket{\\Phi (q)}}\\;.\n\\label{GCM-state}\n\\end{equation}\nHere the basis states $\\ket{\\Phi (q)}$ are Slater determinants of single-nucleon states \ngenerated by constrained SCMF calculations. Several advanced implementations of the GCM have been \ndeveloped recently, fully based on the microscopic EDF framework. For the \nrelativistic SCMF approach, in particular, the most advanced model performs \nconfiguration mixing of angular-momentum and particle-number\nprojected wave functions generated by constraints on quadrupole deformations \\cite{Yao.10,Yao.14}\n\\begin{equation}\n|JNZ;\\alpha \\rangle = \\int{dq\\sum_{K}{f_{\\alpha}^{JK} \\hat{P}^J_{MK} \\hat{P}^N\\hat{P}^Z |q\\rangle } },\n\\end{equation}\n\n\\noindent\nwhere $\\alpha = 1,2,\\dots$ denotes different collective states for a given angular momentum $J$,\nand $|q\\rangle \\equiv |\\beta, \\gamma\\rangle$ denotes a set of intrinsic SCMF states with deformation parameters\n$(\\beta,\\gamma)$. \n$\\hat{P}_{MK}^J$ is the angular momentum projection operator, and \nthe operators $\\hat{P}^N$ and $\\hat{P}^Z$ project onto states with \ngood neutron and proton number, respectively. \nThis implementation is equivalent to a seven-dimensional GCM calculation, mixing all five degrees of freedom of the \nquadrupole operator and the gauge angles for protons and neutrons. \nThe weight functions $f^{JK}_{\\alpha}(q)$ in the collective wave function \nare determined from the variational equation:\n\\begin{equation}\n \\delta E^{J} =\n \\delta \\frac{\\bra{\\Psi_\\alpha^{JM}} \\hat{H} \\ket{\\Psi_\\alpha^{JM}}}\n {\\bra{\\Psi_\\alpha^{JM}}\\Psi_\\alpha^{JM}\\rangle} = 0 \\; ,\n\\label{variational}\n\\end{equation}\nthat is, by requiring that the expectation value of the Hamiltonian is\nstationary with respect to an arbitrary variation $\\delta\nf_{\\alpha}^{JK}$. This leads to the Hill-Wheeler-Griffin (HWG)\nintegral equation:\n\\begin{equation}\n \\label{HWEq}\n \\int dq^\\prime\\sum_{K^\\prime\\geq 0}\n \\left[\\mathcal{H}^J_{KK^\\prime}(q,q^\\prime)\n - E^J_\\alpha\\mathcal{N}^J_{KK^\\prime}(q,q^\\prime)\\right]\n f^{JK^\\prime}_\\alpha(q^\\prime)=0,\n\\end{equation}\n where $\\mathcal{H}$ and $\\mathcal{N}$ are the projected GCM\n kernel matrices of the Hamiltonian and the norm, respectively \\cite{BH.08,RE.10,Yao.10,Yao.14}.\n\nMultidimensional GCM calculations involve a number of technical and computational issues \\cite{LNP.879,Yao.14,BABH.14}, \nthat have so far impeded systematic applications to medium-heavy and heavy nuclei. \nCollective dynamics can also be described using an alternative method in which \na collective Hamiltonian is constructed, with deformation-dependent parameters determined from\nmicroscopic SCMF calculations \\cite{RG.87,PR.09}. \nThe collective Hamiltonian can be derived in\nthe Gaussian overlap approximation (GOA)~\\cite{RS.80} to the full\nmulti-dimensional GCM. With the assumption that the GCM overlap kernels can be\napproximated by Gaussian functions, the local expansion of the\nkernels up to second order in the non-locality transforms the\nGCM Hill-Wheeler equation into a second-order differential equation -\nthe Schr\\\"odinger equation for the collective Hamiltonian. \nFor instance, in the case of quadrupole degrees of freedom:\n\\begin{equation}\n\\label{hamiltonian-quad}\n\\hat{H}_{\\rm coll} = \\hat{T}_{\\rm{vib}}+\\hat{T}_{\\rm{rot}}\n +V_{\\rm{coll}} \\; ,\n\\end{equation}\nwhere the vibrational kinetic energy is parameterized by the the mass parameters\n$B_{\\beta\\beta}$, $B_{\\beta\\gamma}$, $B_{\\gamma\\gamma}$\n\\begin{eqnarray}\n\\hat{T}_{\\textnormal{vib}} =-\\frac{\\hbar^2}{2\\sqrt{wr}}\n \\left\\{\\frac{1}{\\beta^4}\n \\left[\\frac{\\partial}{\\partial\\beta}\\sqrt{\\frac{r}{w}}\\beta^4\n B_{\\gamma\\gamma} \\frac{\\partial}{\\partial\\beta}\n - \\frac{\\partial}{\\partial\\beta}\\sqrt{\\frac{r}{w}}\\beta^3\n B_{\\beta\\gamma}\\frac{\\partial}{\\partial\\gamma}\n \\right]\\right.\n \\nonumber \\\\\n +\\frac{1}{\\beta\\sin{3\\gamma}}\\left.\\left[\n -\\frac{\\partial}{\\partial\\gamma} \\sqrt{\\frac{r}{w}}\\sin{3\\gamma}\n B_{\\beta \\gamma}\\frac{\\partial}{\\partial\\beta}\n +\\frac{1}{\\beta}\\frac{\\partial}{\\partial\\gamma} \\sqrt{\\frac{r}{w}}\\sin{3\\gamma}\n B_{\\beta \\beta}\\frac{\\partial}{\\partial\\gamma}\n \\right]\\right\\} \\; ,\n \\end{eqnarray}\n\nthe three moments of inertia $\\mathcal{I}_k$ determine the \nrotational kinetic energy \n\\begin{equation}\n\\hat{T}_{\\textnormal{\\textnormal{\\textnormal{rot}}}} = \n\\frac{1}{2}\\sum_{k=1}^3{\\frac{\\hat{J}^2_k}{\\mathcal{I}_k}} \\; ,\n\\end{equation}\nand $V_{\\rm{coll}}$ is the collective potential that includes zero-point\nenergy (ZPE) corrections. $w$ and $r$ are products of mass parameters \nand moments of inertia, respectively, that specify the volume element in \ncollective space \\cite{Nik.09}. \nThe self-consistent mean-field solution for the \nsingle-quasiparticle energies and wave functions for\nthe entire energy surface, as functions of the quadrupole\ndeformations $\\beta$ and $\\gamma$, provide the microscopic input for \ncalculation of the mass parameters, moments of inertia and the collective \npotential. The Hamiltonian describes quadrupole vibrations,\nrotations, and the coupling of these collective modes.\n \nThe dynamics of the collective Bohr Hamiltonian is governed by the\nvibrational inertial functions and the moments of inertia \\cite{GR.80}.\nFor these quantities either the GCM-GOA \nor the adiabatic approximation to the time-dependent HFB (ATDHFB) expressions \n(Thouless-Valatin masses) can be used. The Thouless-Valatin masses\nhave the advantage that they also include the time-odd components of\nthe mean-field potential and, in this sense, the full dynamics of a\nnuclear system. In the GCM approach these components can only be included if,\nin addition to the coordinates $q_i$, the corresponding\ncanonically conjugate momenta $p_i$ are also taken into account. \nIn many applications, including the collective Hamiltonians considered in the present study, a further\nsimplification is thus introduced in terms of cranking formulas,\ni.e. the perturbative limit for the Thouless-Valatin masses, and the\ncorresponding expressions for ZPE corrections \\cite{Nik.09}. In the present implementation of the \ncollective Hamiltonian model the moments of inertia and \nmass parameters do not include the contributions of time-odd mean-fields (the so \ncalled dynamical rearrangement contributions) and, to a certain extent, this breaks \nthe self-consistency of the approach \\cite{Hin.12}. \n\nThe diagonalization of the collective Hamiltonian gives the \nenergy spectrum $E_\\alpha^I$ and the corresponding eigenfunctions\n\\begin{equation}\n\\label{wave-coll}\n\\Psi_\\alpha^{IM}(\\beta,\\gamma,\\Omega) =\n \\sum_{K\\in \\Delta I}\n {\\psi_{\\alpha K}^I(\\beta,\\gamma)\\Phi_{MK}^I(\\Omega)}\\; , \n\\end{equation}\nthat are used to calculate various observables, for instance the E2 reduced\ntransition probabilities. The shape of a nucleus can be characterized in a qualitative way\nby the expectation values of invariants $\\beta^2$, $\\beta^3\\cos{3\\gamma}$, as well as their\ncombinations.\n\nNuclear excitations characterised by quadrupole and octupole vibrational and rotational degrees of\nfreedom can be simultaneously described by considering quadrupole and octupole collective coordinates\nthat specify the surface of a nucleus $R=R_0\\left[1+\\sum_\\mu{\\alpha_{2\\mu}Y^*_{2\\mu}\n + \\sum_\\mu{\\alpha_{3\\mu}Y_{3\\mu}^*} } \\right]$. In addition, when axial symmetry is imposed, \n the collective coordinates\n can be parameterized in terms of two deformation parameters $\\beta_2$ and $\\beta_3$, and three Euler\n angles $ \\Omega\\equiv(\\phi,\\theta,\\psi)$. After quantization the collective Hamiltonian takes the form\n\\begin{eqnarray}\n{\\hat H}_{\\rm coll} &= -\\frac{\\hbar^2}{2\\sqrt{w{\\cal I}}}\n \\left[\\frac{\\partial}{\\partial\\beta_2}\\sqrt{\\frac{{\\cal I}}{w}}B_{33}\\frac{\\partial}{\\partial\\beta_2}\n -\\frac{\\partial}{\\partial\\beta_2}\\sqrt{\\frac{{\\cal I}}{w}}B_{23}\\frac{\\partial}{\\partial\\beta_3} \n -\\frac{\\partial}{\\partial\\beta_3}\\sqrt{\\frac{{\\cal I}}{w}}B_{23}\\frac{\\partial}{\\partial\\beta_2}\\right.\\nonumber\\\\\n&\n \\left. +\\frac{\\partial}{\\partial\\beta_3}\\sqrt{\\frac{{\\cal I}}{w}}B_{22}\\frac{\\partial}{\\partial\\beta_3}\\right.\n +\\frac{\\hat{J}^2}{2{\\cal I}}+{V}(\\beta_2, \\beta_3),\n \\label{eq:CH2}\n\\end{eqnarray}\nwhere the mass parameters $B_{22}$, $B_{23}$ and $B_{33}$, and the moment of inertia\n$\\mathcal{I}$, are functions of the quadrupole $\\beta_2$ and octupole $\\beta_3$ deformations.\n$w=B_{22}B_{33}-B_{23}^2$. \n\nJust as in the case of the quadrupole five-dimensional collective Hamiltonian, the moments of inertia \nare\ncalculated from the Inglis-Belyaev formula:\n\\begin{equation}\n\\label{Inglis-Belyaev}\n\\mathcal{I}= \\sum_{i,j}{\\frac{| \\langle ij |\\hat{J} | \\Phi \\rangle |^2}{E_i+E_j}}\\;,\n\\end{equation}\nwhere $\\hat{J}$ is the angular momentum along the axis perpendicular to the symmetric axis, \nthe summation runs over\nproton and neutron quasiparticle states\n$|ij\\rangle=\\beta^\\dagger_i\\beta^\\dagger_j|\\Phi\\rangle$, and\n$|\\Phi\\rangle$ represents the quasiparticle vacuum. The quasiparticle\nenergies $E_i$ and wave functions are determined by SCMF calculations of \ndeformation energy surfaces with constraints on the quadrupole and octupole \ndeformation parameters. \nThe mass parameters associated with the collective \ncoordinates $q_2=\\langle\\hat{Q}_{2}\\rangle$ and $q_3=\\langle\\hat{Q}_{3}\\rangle$\nare calculated in the cranking approximation, as well as the vibrational and rotational \nzero-point energy corrections to the collective energy surface \\cite{GG.79}. \n\\section{\\label{secIII}Evolution of shapes and coexistence in $\\bm{N=28}$ isotones}\nNuclei with closed major proton and\/or neutron shells are usually characterised by spherical \nequilibrium shapes. However, this is not necessarily the case for nuclei away from \nthe $\\beta$-stability line in which energy spacings between single-particle levels \ncan undergo considerable changes \nwith the number of neutrons and\/or protons. This can result in\nreduced spherical shell gaps, modifications of shell structure,\nand in some cases spherical magic numbers may disappear.\nThe reduction of a spherical shell closure is associated with the\noccurrence of deformed ground states and, in a number of cases,\nwith the phenomenon of shape coexistence. Because of the low density of \nsingle-particle states close to the Fermi surface, in relatively light nuclei coexistence \noccurs only when proton and neutron density distributions favour different \nequilibrium deformations, e.g. prolate {\\em vs} oblate shapes. \nHere we consider the well known example of neutron-rich $N=28$ isotones, which \nexhibit rapid shape variations and shape coexistence. \n\nIn Fig.~\\ref{fig:n28_pes} we plot the self-consistent triaxial \nquadrupole deformation energy surfaces of $N=28$ isotones \\cite{Li.11}. \nThe equilibrium shape of the doubly-magic nucleus $^{48}$Ca is, of course, \nspherical but the $N=28$ spherical shell is strongly reduced in the isotones \nwith a smaller number of protons. This leads to rapid transitions between deformed equilibrium \nshapes and shape coexistence in $^{44}$S. \nThe energy surface of $^{46}$Ar is \nsoft both in $\\beta$ and $\\gamma$ directions, with a shallow extended minimum\nalong the oblate axis. Only four protons away from the doubly magic $^{48}$Ca,\nin $^{44}$S the self-consistent mean-field calculation predicts a coexistence of \nprolate and oblate minima separated by a rather low barrier ($ < 1$ MeV). \nFor $^{42}$Si the energy surface displays a deep oblate minimum at\n$(\\beta,\\gamma)=(0.35, 60^\\circ)$, whereas a prolate equilibrium minimum \nat $(\\beta,\\gamma)=(0.45, 0^\\circ)$ is predicted in the very neutron-rich nucleus $^{40}$Mg.\nSimilar results for the quadrupole deformation energy surfaces\nwere also obtained in self-consistent Hartree-Fock-Bogoliubov (HFB) studies \nbased on the finite-range and density-dependent Gogny interaction D1S~\\cite{Delaroche10,RE.11}.\n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.42]{Figs\/N=28\/pes_ar46.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.42]{Figs\/N=28\/pes_s44.eps} \\\\\n\\includegraphics[scale=0.42]{Figs\/N=28\/pes_si42.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.42]{Figs\/N=28\/pes_mg40.eps} \n\\end{tabular}\n\\caption{\\label{fig:n28_pes} Self-consistent triaxial\nquadrupole constrained energy surfaces of $N=28$ isotones in the $\\beta-\\gamma$ plane\n($0\\le \\gamma \\le 60^0$). For each nucleus energies are normalized with respect to the\nbinding energy of the global minimum. The contours join points on\nthe surface with the same energy, and the spacing between \nneighbouring contours is 0.5 MeV.}\n\\end{center}\n\\end{figure}\n\nRapid transitions between equilibrium shapes in a chain of isotones (isotopes) \nare governed by the evolution of the shell structure of single-nucleon\norbitals. In particular, one or more close-lying deformed minima can develop \nas a result of the occurrence of gaps or regions of low single-particle level density\naround the Fermi surface at finite deformation. In the present analysis we illustrate \nthis phenomenon with the example of shape coexistence in $^{44}$S.\n\nThere are many experimental indications for the transitional nature of $^{44}$S:\nthe low-lying $0_2^+$ and $2_1^+$ excited states, the large value of the reduced \ntransition probability $B(E2, 2_1^+ \\to 0_1^+)$~\\cite{Grevy.05,Glasmacher.97}, \nthe monopole strength $\\rho^2(E0, 0_2^+ \\to 0_1^+)$, and the \nreduced transition probability $B(E2, 2_1^+ \\to 0_2^+)$ determined in a recent \nexperiment~\\cite{Force.11}. Based on these results, shell model calculations and\na simple two-level mixing model have shown that $^{44}$S exhibits \na shape coexistence between a prolate ground state and a spherical $0_2^+$ excited state.\nOn the other hand, spectroscopic calculations based on the self-consistent mean-field \napproach indicate a coexistence of prolate and\noblate shapes in $^{44}$S~\\cite{RE.11,Li.11}.\nIn a very recent shell model study several inconsistencies in previous interpretations\nof data for $^{44}$S have been resolved~\\cite{Chevrier.14}. Using quadrupole invariants\nto determine axial and triaxial shape parameters from a shell model calculation, \na pronounced triaxial\nshape for the ground state and a slightly more pronounced prolate shape for the excited\n$0_2^+$ state have been predicted. It has been shown that these states display a \nlarge overlap in the $(\\beta,\\gamma)$\nplane. Higher-lying members of the ground-state band show a tendency towards prolate deformation,\nwhereas strong fluctuations have been predicted in the band built on top of the state $0_2^+$.\nAn isomeric state $4_1^+$ was recently observed in $^{44}$S~\\cite{Santiago-Gonzalez.11},\nand the nature of this state has been explored in a shell model study~\\cite{Utsuno.15} which has\nshown that this state most probably corresponds to a two-quasiparticle $K=4$ configuration. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/N=28\/s44_levels.eps}\n\\caption{\\label{fig:s44_levels}Proton (upper panel) and neutron (lower panel) \nsingle-nucleon energy levels of $^{44}$S, as functions of the\ndeformation parameters along a closed path in the $\\beta - \\gamma$ plane.\nSolid (blue) curves correspond to levels with negative parity, and\n(red) dashed curves denote positive-parity levels.\nThe dot-dashed (green) curves corresponds to the Fermi levels.\nThe panels on the left and right display prolate ($\\gamma =0^\\circ$) and oblate\n ($\\gamma =60^\\circ$) axially-symmetric single-particle levels, respectively.\n In the middle panel the proton and neutron levels are\n plotted as functions of $\\gamma$ for a fixed value \n $|\\beta|=0.35$. }\n\\end{center}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:s44_levels} we plot the proton (upper panel) and neutron (lower panel) single-particle levels in the canonical basis for $^{44}$S. \nSolid (blue) curves correspond to levels with negative parity, and dashed (red) curves denote positive-parity levels. \nThe dot-dashed (green) curves correspond to \nthe Fermi levels. The proton and neutron levels are plotted as functions of the deformation parameters along a closed \npath in the $\\beta$-$\\gamma$ plane. The panels on the left and right display prolate \n($\\gamma = 0^\\circ$) and oblate ($\\gamma = 60^\\circ$) axially symmetric single-particle levels, respectively. \nIn the middle panel of each figure the neutron and proton levels are plotted as functions of $\\gamma$ \nfor a fixed value of the axial deformation $|\\beta| = 0.35$ which corresponds to the position of the prolate mean-field minimum \nin $^{44}$S isotope (cf. Fig.~\\ref{fig:n28_pes}). Starting from the spherical configuration, we follow the single-nucleon levels on a \npath along the prolate axis up to the approximate position of the minimum (left panel), then for this fixed value of $\\beta$ the path from \n$\\gamma = 0^\\circ$ to $\\gamma=60^\\circ$ (middle panel) and, finally, back to\nthe spherical configuration along the oblate axis (right panel). \nAxial deformations with $\\gamma=60^\\circ$ are denoted by negative values of $\\beta$. \nThis figure illustrates the principal\ncharacteristics of structural changes in neutron-rich $N=28$ nuclei: the near degeneracy\nof the $d_{3\/2}$ and $s_{1\/2}$ proton orbitals, and the reduction of the size of the\n$N=28$ shell gap \\cite{Sorlin.10}. Between the doubly magic $^{48}$Ca and $^{44}$S the \nspherical gap $N=28$ decreases from 4.73 MeV to 3.86 MeV, respectively. \nConsequently, the largest gap\nbetween neutron states at the Fermi surface \nis located on the oblate axis (lower panel of Fig.~\\ref{fig:s44_levels}), \nand we also notice the increased density of single-neutrons levels close to the\nFermi surface at $\\gamma \\approx 30^\\circ$ which leads to formation of \nthe potential barrier in the triaxial region.\nFor the protons (upper panel of Fig.~\\ref{fig:s44_levels}), the largest gap\nis located on the prolate axis. The competition between pronounced \nproton-prolate and neutron-oblate energy gaps is at the origin\nof the coexistence of deformed shapes in $^{44}$S.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=28\/s44_spectrum.eps}\n\\caption{\\label{fig:s44_spec}The theoretical excitation spectrum of $^{44}$S (left), compared to \ndata~\\cite{Glasmacher.97,Force.11,Santiago-Gonzalez.11} (right). The\n$B(E2)$ values are in units of $e^2$fm$^4$. The comparison of\nthe monopole transition strength $\\rho^2(E0;0_2^+ \\to 0_1^+)$ \nwith the experimental value is also included. }\n\\end{figure}\n\nStarting from constrained self-consistent solutions of the relativistic \nHartree-Bogoliubov (RHB) equations at each point on the \nenergy surfaces (Fig.~\\ref{fig:n28_pes}), we calculate \nthe mass parameters $B_{\\beta \\beta}$, $B_{\\beta \\gamma}$, $B_{\\gamma \\gamma}$,\nthe three moments of inertia $\\mathcal{I}_k$, as well as the zero-point energy corrections, \nthat determine the collective Hamiltonian~(\\ref{hamiltonian-quad}).\nThe diagonalization of the resulting Hamiltonian yields the excitation energies and reduced transition\nprobabilities. Physical observables are calculated in the full configuration space and there are no \neffective charges in the model. In Fig.~\\ref{fig:s44_spec} we display the low-energy spectrum of $^{44}$S \nin comparison to available data for the excitation energies, reduced electric quadrupole\ntransition probabilities $B(E2)$ (in units of $e^2$fm$^4$), and the electric monopole transition\nstrength $\\rho^2(E0;0_2^+ \\to 0_1^+)$. The model reproduces both the excitation energy and\nthe reduced transition probability $B(E2; 2_1^+ \\to 0_1^+)$ for the first excited\nstate $2_1^+$. The theoretical value for $B(E2;0_2^+ \\to 2_1^+)$ is also in good agreement with\nthe data. However, the calculated excitation energy of the $0_2^+$ state is much \nhigher than the experimental counterpart, and the monopole\ntransition strength $\\rho^2(E0;0_2^+ \\to 0_1^+)$ overestimates the\nexperimental value considerably. This indicates that there is probably more mixing between the \ntheoretical states $0_1^+$ and $0_2^+$ than what can be inferred from the data. We also \nnote that very recently the low-lying state $4_1^+$ has been interpreted \nas a $K = 4$ isomer dominated by the two-quasiparticle configuration \n$\\nu \\Omega^\\pi = 1\/2^- \\otimes \\nu \\Omega^\\pi = 7\/2^-$ \\cite{Utsuno.15}, a configuration \nnot included in our collective model space.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_0_2.eps} \\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_3_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_2_3.eps} \\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=28\/s44_wave_4_4.eps}\n\\end{tabular}\n\\caption{\\label{fig:s44_waves} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the lowest collective states of $^{44}$S.}\n\\end{center}\n\\end{figure}\n\nTo illustrate the degree of configuration mixing and shape coexistence \nin $^{44}$S, in Fig.~\\ref{fig:s44_waves} we plot the\nprobability density distributions for the lowest three states in the ground-state band (left), \nthe (quasi) $\\gamma$-band (middle), and the excited band built on the state $0_2^+$ (right). \nFor a given collective state, the probability distribution\nin the $(\\beta,\\gamma)$ plane is defined as\n\\begin{equation}\n\\rho_{I\\alpha}(\\beta,\\gamma) = \\sum_{K \\in \\Delta I}{ \\left| \\psi_{\\alpha K}^I(\\beta,\\gamma)\\right|^2\\beta^3 },\n\\label{eq:probability}\n\\end{equation}\nwith the summation over the allowed set of values of the projection $K$ of the angular \nmomentum $I$ on the body-fixed symmetry axis, and \nwith the normalization\n\\begin{equation}\n\\int_0^\\infty{\\beta d\\beta \\int_0^{2\\pi}{ \\rho_{I\\alpha}(\\beta,\\gamma)|\\sin{3\\gamma}|d\\gamma }}=1.\n\\end{equation}\n\n\\begin{table}\n\\caption{\\label{tab:s44_K_spec}Percentage of the $K=0$ and $K=2$ components (projection of the\nangular momentum on the body-fixed symmetry axis) for the collective wave functions of the\nthree lowest $2^+$ states in $^{44}$S, and the corresponding spectroscopic quadrupole moments\n(in $e$fm$^2$).}\n\\centering\n\\begin{tabular}{cccc}\n\\hline\\hline\n & $K = 0$ & $K = 2$ & $Q_{spec}$ \\\\ \\hline\n $2_1^+$ & $89\\%$ & $11\\%$& $-10.8$\\\\\n $2_2^+$ & $21\\%$ & $79\\%$ & $8.2$\\\\\n$2_3^+$ & $78\\%$ & $22\\%$ & $-7.3$\\\\\n \\hline \\hline\n\\end{tabular}\n\\end{table}\n\nThe probability distribution of the ground state $0_1^+$ displays a deformation $|\\beta| \\ge 0.3$,\nextended in the $\\gamma$ direction from the prolate $\\gamma=0^\\circ$ to the oblate\n$\\gamma=60^\\circ$ shape. The $\\gamma$ softness of the $0_1^+$ state reflects the \nground-state mixing of configurations based on the prolate and oblate minima of the \nmean-field potential (cf. Fig.~\\ref{fig:n28_pes}).\nStates of the ground-state band with higher angular momenta are progressively concentrated\non the prolate axis and the average $\\beta$ deformation gradually increases because of \ncentrifugal stretching. Although the $0_2^+$ state is predominantly prolate, one notices \noblate admixtures and,\nconsequently, a relatively large overlap between the wave functions of $0_1^+$ and \n$0_2^+$. The mixing between these states leads to a pronounced level repulsion which \nis probably the cause for the too high excitation energy of the \ntheoretical state $0_2^+$. \nThe low-lying $0_2^+$ state at $1.365$ MeV and the monopole strength \n$\\rho^2(E0;0_2^+ \\to 0_1^+)=8.7(7)\\times 10^{-3}$ have been regarded as signatures of\nprolate-spherical shape coexistence in $^{44}$S~\\cite{Force.11}. \nHowever, recent studies have re-examined the structure of $^{44}$S~\\cite{Chevrier.14},\nemphasizing the effect of the triaxial degree of freedom on the low-lying excitation \nstructure. The probability distribution of \n$2_3^+$ is concentrated on the prolate axis and the transition strength $B(E2;2_3^+\\to 0_2^+)$\nis comparable to $B(E2;2_1^+\\to 0_1^+)$. \nFor the three lowest $2^+$ levels, in Table~\\ref{tab:s44_K_spec} we list the percentage of the\n$K=0$ and $K=2$ components in the collective wave functions, together with the spectroscopic\nquadrupole moments. The wave functions of the states $2_1^+$ and $2_3^+$ are dominated by\nthe $K=0$ components, and the spectroscopic quadrupole moments indicate prolate configurations.\nIn contrast the positive quadrupole moment of $2_2^+$ state points to a predominantly oblate\nconfiguration, while the $\\approx 80\\%$ contribution of the $K=2$ component in the wave function\nshows that this state is the band-head of a (quasi) $\\gamma$-band. One also notices the\nclose-lying doublet $3_1^+$ and the $4_2^+$, characteristic for a $K=2$ band in a $\\gamma$-soft \npotential.\n\nThe example considered in this section and similar calculations reported recently have shown that \nthe EDF approach provides an accurate microscopic interpretation of the reduction of the $N=28$\nspherical energy gap in neutron-rich nuclei, and a quantitative description of the evolution\nof shapes in $N=28$ isotones in terms of single-nucleon orbitals as functions of the\nquadrupole deformation parameters $\\beta$ and $\\gamma$. In particular, \nthe formation of the oblate neutron and prolate proton gaps in $^{44}$S, illustrated in Fig.~\\ref{fig:s44_levels}, \nis at the origin of the predicted shape coexistence, in very good agreement with recent data.\n\n\\section{\\label{secIV}Lowest $\\bm{0^+}$ excitations in $\\bm{N \\approx 90}$ rare-earth nuclei}\n\nRare-earth nuclei with neutron number $N\\approx 90$ present some of the best examples \nof rapid shape evolution and shape phase transitions \\cite{HW.11,CJC.10,CMC.07}. Employing a \nconsistent framework of structure models (GCM, quadrupole collective Hamiltonian) based on \nrelativistic energy density functionals, in a series of studies \\cite{NVL.07,LNV.09,Li.09} we \nanalysed microscopic signatures of ground-state shape phase transitions in this \nregion of the nuclear mass table. Phase transitions in equilibrium shapes of atomic nuclei correspond\nto first- and second-order quantum phase transitions (QPT) between\ncompeting ground-state phases induced by variation of a non-thermal\ncontrol parameter (number of nucleons) at zero temperature. In general, one observes \na a gradual evolution of shapes with the number of nucleons, and these transitions \nreflect the underlying modifications of shell structure and interactions between valence nucleons. \nA phase transition, on the other hand, is characterised by a significant variation of one or more order\nparameters as functions of the control parameter. Even though in systems composed\nof a finite number of particles phase transitions are actually smoothed out, in many cases clear \nsignatures of abrupt changes of structure properties are observed. A number of experiments \nover the last two decades, as well as many theoretical studies of deformation energy surfaces and \nalso direct computation of observables related to order parameters, have shown that \ntwo-neutron separation energies, isotope shifts, energy gaps between the ground state and the \nexcited vibrational states with zero angular momentum,\nisomer shifts, and monopole transition strengths, exhibit sharp discontinuities at neutron number\n$N=90$. \n\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/sm152_spec_corr.eps}\n\\caption{\\label{fig:sm152_spec}The theoretical excitation spectrum of $^{152}$Sm \n(left), compared to data~\\cite{data}. The intraband \nand interband B(E2) values (thin solid arrows) are in Weisskopf units(W.u.), \nand (red) dashed arrows denote E0 transitions with the \ncorresponding $\\rho^2(E0) \\times 10^3$ values.}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/gd154_spec_corr.eps}\n\\caption{\\label{fig:gd154_spec} Same as in the caption to Fig.~\\ref{fig:sm152_spec} \nbut for $^{154}$Gd.}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[scale=0.65]{Figs\/N=90\/dy156_spec.eps}\n\\caption{\\label{fig:dy156_spec}Same as in the caption to Fig.~\\ref{fig:sm152_spec} \nbut for $^{156}$Dy.}\n\\end{figure}\n\nIn the present study we focus on the low-lying $0^+$ excitations in the deformed \n$N=90$ isotones and examine the mixing between the lowest $K=0$ bands. Traditionally the \nfirst excited $0^+$ level in deformed nuclei has been interpreted as a $\\beta$-vibrational state, \nwith the associated $K=0$ rotational $\\beta$-band. However, many experimental studies \nhave shown that most of the $0^+_2$ excitations are not $\\beta$ vibrations. In the \nexhaustive review of properties of the lowest-lying $0^+$ states in deformed rare-earth \nnuclei \\cite{Garr.01}, it was emphasized that there is no {\\it a priori} reason to associate \nthe $\\beta$ vibration with the lowest $0^+$ excited state. A set of properties was suggested that \nthe first excited $0^+$ level should exhibit in order to be labelled as a $\\beta$ vibration. \nAmong those: $B(E2; 0^+_\\beta \\to 2^+_1)$ values of $12 - 33$ W.u. or conversely \n$B(E2; 2^+_\\beta \\to 0^+_1)$ of $2.5 - 6$ W.u., and $\\rho^2(E0;{0^+_2 \\to 0^+_1}) \\times 10^3$ values of $85-230$.\nIn our microscopic analysis of order parameters in nuclear quantum phase transitions \\cite{Li.09}, \nin particular for the Nd isotopic chain, it was shown that \nthe excitation energies of both $0^+_2$ and $0^+_3$ exhibit a pronounced dip at\n$N=90$, which can be attributed to the softness of the potential with respect to $\\beta$\ndeformation in $^{150}$Nd. The calculated monopole transition strengths \nexhibit a pronounced increase toward $N=90$, and the \n$\\rho^2(E0;{0^+_2 \\to 0^+_1})$ values remain rather large\nin the deformed nuclei $^{152,154,156}$Nd, a behaviour \ncharacteristic for an order parameter at the point of first-order QPT.\n\nIn Figs.~\\ref{fig:sm152_spec} - \\ref{fig:dy156_spec} we plot the theoretical low-energy \nspectra of the $N=90$ nuclei: $^{152}$Sm, $^{154}$Gd and $^{156}$Dy, in comparison \nwith data~\\cite{data}. The ground-state bands, lowest $K=0$ and $K=2$ bands \nare compared to their experimental counterparts: excitation energies, intraband \nand interband $B(E2)$ values, and $E0$ transition strengths. The theoretical spectra \ncomprise eigenstates of the five-dimensional quadrupole collective Hamiltonian~(\\ref{hamiltonian-quad}), \nbased on triaxial SCMF solutions obtained with the density functional DD-PC1 and \na finite-range pairing force separable in momentum space. No additional parameters \nare adjusted to data, and transition rates have been calculated with bare \ncharges, that is, $e_p = e$ and $e_n =0$. \n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_1.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_1.eps}\\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_2.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_2.eps}\\\\\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_waves_0_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_waves_0_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_waves_0_3.eps}\\\\\n\\end{tabular}\n\\caption{\\label{fig:N=90_0_states} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the lowest collective $0^+$ states of $^{152}$Sm, $^{154}$Gd and $^{156}$Dy.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/sm152_wave_2_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/gd154_wave_2_3.eps}&\n\\hspace{-1cm}\\includegraphics[scale=0.3]{Figs\/N=90\/dy156_wave_2_3.eps}\\\\\n\\end{tabular}\n\\caption{\\label{fig:N=90_2_3_states} Probability distributions Eq.~(\\ref{eq:probability})\nin the $\\beta-\\gamma$ plane for the band-heads of the $K=2$ $\\gamma$-bands \nof $^{152}$Sm, $^{154}$Gd and $^{156}$Dy.}\n\\end{center}\n\\end{figure}\n\nFor the ground-state bands the theoretical excitation energies and $B(E2)$ values for \ntransitions within the band are in very good agreement with data, except for the fact that the \nempirical moments of inertia are systematically larger than those calculated with the\ncollective Hamiltonian. This is a well known effect of using the simple Inglis-Belyaev \napproximation for the moments of inertia, and is also reflected in the excitation energies \nof the excited $K=0$ and $K=2$ bands \\cite{DGL.10}. The wave functions, however, are not \naffected by this approximation and we note that the model \nreproduces both the intraband and interband $E2$ \ntransition probabilities. The $K=2$ $\\gamma$-bands are predicted at somewhat \nhigher excitation energies compared to their experimental counterparts, and this is most \nprobably due to the potential energy surfaces being too stiff in $\\gamma$. \nThe deformed rare-earth $N=90$ isotones are characterised by very low $K=0$ bands \nbased on the $0^+_2$ states. In $^{152}$Sm, for instance, this state is found \nat 685 keV excitation energy, considerably below the $K=2$ $\\gamma$-band. Nevertheless, \nthis state has been interpreted as the band-head of the $\\beta$-band \\cite{Garr.01,DGL.10}. \n\nIn $^{152}$Sm the excited $K=0$ band is calculated at moderately higher energy \ncompared to data, while the agreement with experiment is very good for $^{154}$Gd and $^{156}$Dy. \nWe note that a very similar excitation spectrum for $^{152}$Sm was also obtained \nwith the collective Hamitonian based on the D1S Gogny interaction \\cite{DGL.10}. \nParticularly important for the present study are the $E0$ transitions between the two lowest \n$K=0$ bands, and the $B(E2; 0^+_2 \\to 2^+_1)$ and $B(E2; 2^+_2 \\to 0^+_1)$ values. \nThe available data are very accurately reproduced by the calculation and, in particular \nfor $^{152}$Sm, the $E0$ transition strengths and $B(E2)$ values seem to match the \ncriteria for a $\\beta$-vibrational state \\cite{Garr.01}. \n\nThe $E0$ transitions strengths reflect the degree of mixing between the two lowest $K=0$ bands, \nand Figs.~\\ref{fig:sm152_spec} - \\ref{fig:dy156_spec} show that the theoretical values that \ncorrespond to transitions between the eigenstates of the collective Hamiltonian reproduce \nthe empirical $\\rho^2(E0)$ values. The structure of the low-lying $0^+$ states is \nanalysed in Fig.~\\ref{fig:N=90_0_states}, in which the probability density distributions \nare plotted in the $\\beta-\\gamma$ plane for the three lowest collective \n$0^+$ states of $^{152}$Sm, $^{154}$Gd and $^{156}$Dy. We note that the probability \ndistributions for these states are concentrated on the prolate axis $\\gamma = 0^\\circ$, \nin contrast to the band-heads of the $K=2$ $\\gamma$-bands, for which the probability \ndensity distributions are shown in Fig.~\\ref{fig:N=90_2_3_states}. The dynamical \n$\\gamma$-deformations of the latter clearly point to the $\\gamma$-vibrational nature \nof these states. The average values of the deformation parameter \n$\\beta$ for the collective ground-state wave functions of $^{152}$Sm: $<\\beta> = 0.32$, \n$^{154}$Gd: $<\\beta> = 0.31$, and $^{156}$Dy: $<\\beta> = 0.30$, correspond to the \nminimum of the respective deformation energy surface. The corresponding values for the first \nexcited $0^+$ states are: $<\\beta> = 0.33$ for $^{152}$Sm, $<\\beta> = 0.34$ for \n$^{154}$Gd, and $<\\beta> = 0.37$ for $^{156}$Dy. For a pure harmonic $\\beta$-vibrational \nstate one expects that the average deformation is the same as for the ground-state, that \nthe ratio of $\\Delta \\beta$ values for $0^+_\\beta$ with respect to $0^+_1$ is $\\sqrt{3}$, and that \nthe probability density distribution displays one node at $<\\beta>_{\\rm g.s.}$ \nand two peaks of the same amplitude. The ratio of $\\Delta \\beta$ values for $0^+_2$ with respect to \n$0^+_1$ is: 1.6 for $^{152}$Sm, 1.53 for $^{154}$Gd, and 1.42 for $^{156}$Dy. Considering \nall these quantities, it appears that the best candidate for the $\\beta$-vibrational state is \n$0^+_2$ in $^{152}$Sm. However, even in this case the probability distribution does not \ndisplay two peaks of equal amplitude, and the shift to larger deformation is more \npronounced in $^{154}$Gd and $^{156}$Dy. Note that for the latter two nuclei the calculated \nexcitation energy of the $0^+_2$ states are in even better agreement with data. The \nprobability distributions for the $0^+_3$ levels, plotted in the third row of \nFig.~\\ref{fig:N=90_0_states}, indicate the development of a second node and third peak, \nthat is, the appearance of two-phonon states. One notices, however, the mixing \nwith states based on $\\gamma$ vibrations, which becomes even more pronounced \nfor higher lying $0^+$ states, not shown in the figure. An experimental exploration \nof a possible occurrence of multiple (two) phonon intrinsic collective excitations \nin $^{152}$Sm did not find evidence for two-phonon $K^\\pi = 0^+$ quadrupole \nvibrations \\cite{Kulp.08}. In fact, it has been argued that an emerging pattern of repeating \nexcitations built on $0^+_2$, similar to those based on the ground state, \nshows that $^{152}$Sm is an example of shape coexistence \\cite{Garr.09}.\n \n\nThe simple analysis presented in this section illustrates the complex structure of \nexcited $0^+$ levels in deformed nuclei, and the difficulties in classifying these \nstates as simple collective vibrational states, that is, as one and two-phonon \n$\\beta$ vibrations. A more quantitative theoretical investigation should involve \nadditional effects not included in our collective Hamiltonian model, such as \nare the coupling between shape oscillations and pairing vibrations \\cite{PP.93} and, \nin general, the coupling between collective and intrinsic two-quasiparticle \nexcitations \\cite{Ber.11}, that can lower the collective energy levels \nand improve the agreement with data \\cite{data,Garr.01}. In particular, a more \nadvanced model that includes coupling between collective and intrinsic two-quasiparticle \nexcitations can be used to analyse excited rotational bands \nbased on pairing isomers, such as those identified in \n$^{154}$Gd \\cite{Kulp.03} and $^{152}$Sm \\cite{Kulp.05}.\n\n\n\\section{\\label{secV}Quadrupole and octupole shape transition in Thorium}\n\nMost deformed medium-heavy and heavy nuclei display quadrupole equilibrium shapes, \nbut there are also regions of the mass table \nin which octupole deformations (reflection-asymmetric, pear-like shapes) occur.\nReflection-asymmetric shapes are characterized by the presence \nof negative-parity bands, and by pronounced electric dipole and octupole\ntransitions. In the case of static octupole deformations, for instance, the lowest \npositive-parity even-spin states and the negative-parity odd-spin states form an\nalternating-parity band, with states connected by the enhanced\nE1 transitions. In a simple microscopic picture strong octupole correlations \narise through a coupling of orbitals near the Fermi surface with quantum numbers \n($l$, $j$) and ($l+3$, $j+3$). This leads \nto reflection-asymmetric intrinsic shapes that develop either dynamically \n(octupole vibrations) or as static octupole equilibrium deformations \\cite{BN.96,BW.15}.\nFor example, in the case of $N\\approx 134$ and $Z\\approx 88$\nnuclei in the region of light actinides, the coupling of the neutron orbitals based on \n$g_{9\/2}$ and $j_{15\/2}$, and that of the proton single-particle\nstates arising from $f_{7\/2}$ and $i_{13\/2}$, can give rise to octupole mean-field deformations. \n\nAn interesting phenomenon are simultaneous quadrupole and octupole shape transitions. \nIn a series of recent studies \\cite{Nom.13,Li.13,Nom.14} we have analyzed the evolution of \nquadrupole and octupole shapes using a consistent microscopic framework based on \nrelativistic EDFs. In thorium isotopes, in particular, the calculated triaxial quadrupole and \naxial quadrupole-octupole energy surfaces, and predicted \nobservables (excitation energies, isotope shifts of charge radii, electromagnetic\ntransition rates) point to the occurrence of a simultaneous phase\ntransition between spherical and quadrupole-deformed prolate shapes,\nand between non-octupole and octupole-deformed shapes, with $^{224}$Th being\nclosest to the critical point of the double shape phase transition \\cite{Li.13}. \n\n\\begin{figure}[]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th220_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th222_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th224_pes.eps} \\\\\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th226_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th228_pes.eps}\n\\includegraphics[scale=0.275]{Figs\/Th-PES\/th230_pes.eps}\n\\end{tabular}\n\\caption{\\label{fig:th_pes} Axially symmetric energy surfaces of the isotopes\n $^{220-230}$Th in the $(\\beta_{2},\\beta_{3})$ \n plane. The contours join points on the surface with the same energy\n and the energy difference between neighboring contours is\n 1 MeV. Positive (negative) values of $\\beta_{2}$ correspond to prolate (oblate) \n configurations. Energy surfaces are symmetric with respect to the\n $\\beta_{3}=0$ axis.}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:th_pes} displays the deformation energy surfaces in the plane \nof axial quadrupole and octupole deformation parameters \n($\\beta_{2}$, $\\beta_{3}$) for the isotopes $^{220 - 230}$Th. \nThis isotopic chain exhibits an interesting structural evolution, visible already \nat the SCMF level. A rather soft energy surface is calculated for $^{220}$Th with the \nminimum at $(\\beta_{2}, \\beta_{3})\\approx (0, 0)$, and this will give rise to quadrupole \nvibrational excitation spectra. Quadrupole deformation becomes more pronounced in $^{224}$Th, \nand one also notices \nthe emergence of octupole deformation. The energy minimum is found in the \n$\\beta_{3}\\neq 0$ region, located at $(\\beta_{2}, \\beta_{3})\\approx (0.1, 0.1)$ . \nFrom $^{224}$Th to $^{228}$Th the occurrence of a rather strongly marked \noctupole minimum is predicted. Starting from $^{228}$Th, the minimum \nbecomes softer in the octupole $\\beta_{3}$ direction. An octupole-soft surface, \nalmost completely flat in $\\beta_{3}$ for $\\beta_{3} \\leq 0.3$,\nis calculated for $^{232}$Th. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/Th-PES\/states_th.eps}\n\\caption{\\label{fig:th_spec}Excitation energies of low-lying yrast positive-parity (left)\nand negative-parity (right) collective states of $^{224-232}$Th. Lines and symbols denote the \n theoretical and experimental \n \\cite{data} levels, respectively.}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:th_spec} we analyse the systematics of energy spectra of the positive-parity \nground-state band ($K^{\\pi}=0^{+}$) (left) and the lowest negative-parity ($K^\\pi =0^-$) sequences \n(right) in $^{224-232}$Th. The theoretical values calculated using the quadrupole-octupole collective \nHamiltonian~(\\ref{eq:CH2}) are shown in comparison to available data \\cite{data}. The excitation energies of \npositive-parity states systematically decrease with mass number, reflecting the increase of \nquadrupole collectivity. $^{220,222}$Th exhibit a quadrupole vibrational structure, whereas \npronounced ground-state rotational bands with\n$R_{4\/2}=E(4^{+}_{1})\/E(2^{+}_{1})\\approx 3.33$ are calculated in\n$^{226-232}$Th. For the lowest negative-parity bands\nthe excitation energies display a parabolic \nstructure centered between $^{224}$Th and $^{226}$Th. \nThe approximate parabola of $1^{-}_{1}$ states has a minimum \nat $^{226}$Th, in which the octupole deformed minimum \nis most pronounced. Starting from $^{226}$Th the energies of negative-parity \nstates systematically increase and the band becomes more \ncompressed. A rotational-like collective band based on the octupole vibrational\nstate, i.e., the $1^{-}_{1}$ band-head, develops. The parabolas of negative-parity \nstates calculated with the quadrupole-octupole Hamiltonian are in qualitative agreement \nwith data, although the minima are predicted to occur at $^{228}$Th rather than $^{226}$Th. \nNote, however, that all levels shown in Fig.~\\ref{fig:th_spec} are below 1 MeV excitation \nenergy, so that the differences between calculated and experimental levels are rather \nsmall, especially considering that no parameters were adjusted to data. The approximations \ninvolved in the calculation of the Hamiltonian parameters (perturbative cranking for the \nmass parameters and the Inglis-Belyaev formula for the moments of inertia) determine \nthe level of quantitative agreement with experiment. We note that the theoretical \nB(E2) values for transitions within the ground state bands are in agreement \nwith availabe data, while the calculated $B(E3; 3^-_1\\to 0^+_1)$: \n61 W.u. for $^{230}$Th and 41 W.u. for $^{232}$Th, are somewhat larger than \nthe experimental values: 29(3) W.u. for $^{230}$Th and 24(3) W.u. for $^{232}$Th \\cite{Kib.02}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{Figs\/Th-PES\/th226-single-particle.eps}\n\\caption{\\label{fig:th226-single-particle} $^{226}$Th single-neutron (upper panel) and single-proton\n(lower panel) levels in the canonical basis as functions of the deformation\nparameters. The path follows the quadrupole deformation parameter\n$\\beta_2$ up to the position of the equilibrium minimum $\\beta_2=0.2$, with \nthe constant octupole deformation $\\beta_3=0$ (left panels). For $\\beta_2=0.2$ \nthe panels on the right display the single-nucleon energies from $\\beta_3=0$ to $\\beta_3=0.3$. \nDashed curves denote the position of the Fermi level at each deformation.}\n\\end{center}\n\\end{figure}\n\nLet us consider in more detail the structure of $^{226}$Th which, at the SCMF level, exhibits a nice \nexample of coexistence of axial quadrupole and octupole minima. The energy \ndifference between the local minimum at $\\beta_3 = 0$ and the equlibrium minimum \nat $\\beta_3 = 0.16$ is 1.8 MeV. We have also carried out a constrained triaxial \nquadrupole SCMF calculation which confirms that the quadrupole minimum is indeed at \n$\\gamma = 0^\\circ$, that is, axial prolate. The microscopic origin of coexistence \nof the two minima becomes apparent from the dependence of the single-nucleon\nlevels on the two deformation parameters. Figure \\ref{fig:th226-single-particle}\ndisplays the single neutron and proton levels\nof $^{226}$Th along a path in the $\\beta_2 - \\beta_3$ plane. Starting from the spherical configuration,\nthe path follows the quadrupole deformation parameter $\\beta_2$\nup to the position of the equilibrium minimum $\\beta_2 = 0.2$, with the\noctupole deformation parameter kept constant at zero value. Then,\nfor the constant value $\\beta_2 = 0.2$, the path continues from\n$\\beta_3 = 0$ to $\\beta_3 = 0.3$. The necessary condition for the occurrence of \nlow-energy octupole collectivity is the presence of pairs of orbitals near the Fermi level \nthat are strongly coupled by the octupole interaction. In the panels on the left of Fig.~\\ref{fig:th226-single-particle} \nwe notice states of opposite parity that originate from the spherical levels $g_{9\/2}$ and $j_{15\/2}$ for neutrons, \nand $f_{7\/2}$ and $i_{13\/2}$ for protons. The total energy can be related to the level density \naround the Fermi surface, that is, a lower-than-average density of single-particle levels results in extra \nbinding. Therefore, the local quadrupole minimum seen on the axial energy surface \nof $^{226}$Th reflects the $\\beta_2$-dependence of the levels of the Nilsson diagram \nfor $\\beta_3 = 0$. \nFor the levels in the panels on the right of Fig.~\\ref{fig:th226-single-particle} \nparity is not conserved, and the only quantum number that \ncharacterises these states is the projection of the angular momentum on the symmetry axis. \nThe octupole minimum, rather soft along the $\\beta_3$-path in $^{226}$Th, \nis attributed to the low density of both proton and neutron states close to the \ncorresponding Fermi levels in the interval of deformations around $\\beta_3 = 0.16$.\n\nFully microscopic analyses of coexistence of quadrupole and octupole shapes and, in general, the \nevolution of octupole correlations in heavy nuclei, will be particularly important for future experimental \nstudies of reflection-asymmetric shapes using accelerated radioactive beams \\cite{Gaffney.13}, \nand in searches for new symmetry violating interactions beyond the standard model \\cite{BW.15}. \n\\section{\\label{secVI}Summary}\n\nThe framework of nuclear energy density functionals provides an intuitive and yet accurate \nmicroscopic interpretation of the evolution of single-nucleon shell structure and the related \nphenomena of deformations, shape transitions and shape coexistence. Self-consistent mean-field \ncalculations of deformation energy surfaces produce symmetry-breaking many-body states \nthat include important static correlations such as deformations and pairing. Intrinsic \nshapes that correspond to minima on the deformation energy surface are determined by the \nstatic (constrained) collective coordinates. Dynamical correlations are included by \ncollective structure models that restore symmetries broken by the static mean field \nand take into account quantum fluctuations of collective variables.\nThe microscopic input \nfor the generator coordinate method or the collective Hamiltonian model are completely \ndetermined by the choice of the energy density functional and pairing interaction. These models \ncan be used to calculate observables that characterise the evolution and eventual \ncoexistence of different shapes: low-energy excitation spectra, electromagnetic transition rates, \nchanges in masses (separation energies), isotope and isomer shifts, and that can be directly \ncompared to data. \n\nUsing a single relativistic energy density functional and a finite-range pairing interaction \nseparable in momentum space, in the present study we have analysed illustrative examples \nof diverse phenomena related to evolution of shell structure: shape transition and \ncoexistence in neutron-rich $N=28$ isotones, the structure of\nlowest $0^+$ excitations in deformed $N = 90$ rare-earth nuclei,\nand quadrupole and octupole shape transitions \nin thorium isotopes. Spectroscopic properties have been calculated using the \nfive-dimensional quadrupole and axial quadrupole-octupole collective Hamiltonian models. \nThe very good agreement between theoretical predictions and available data, and especially \nthe fact that very different regions of the mass table could be considered with no \nneed to adjust or fine-tune model parameters to specific data, demonstrate that the approach \nbased on universal density functionals is a method of choice for studies of shape transitions \nand coexistence over the entire table of nuclides, including regions of exotic short-lived \nnuclei far from stability. \n\nFuture developments of structure methods based on nuclear energy density functionals \ninclude a number of major challenges. One of the most important and certainly most \ndifficult is the construction of a consistent set of approximations for the exchange-correlation \nenergy functional. In the context of phenomena discussed in the present analysis, \nit would be very interesting to try to develop microscopic functionals that, in addition to the \ndependence on ground-state densities and currents composed of occupied Kohn-Sham \norbitals, also include a dependence on unoccupied orbitals. This is particularly important \nfor studies of the evolution of shell structure and modification of gaps in nuclei far from \nstability and\/or superheavy nuclei. For quantitative comparison with available \nspectroscopic data and predictions in new regions of the chart of nuclides, \naccurate and efficient algorithms have to be developed that perform a complete \nrestoration of symmetries broken by self-consistent mean-field solutions for general \nquadrupole and octupole shapes. Deformation-dependent parameters of \ncollective quadrupole (-octupole) Hamiltonians (vibrational inertial functions and moments \nof inertia) need to be determined using methods that go beyond the simple cranking \nformulas and include the full dynamics of a nuclear system. Finally, theoretical studies \nof low-energy spectroscopic properties that characterise shape coexistence will also have to \ninclude uncertainty estimates, quantify theoretical errors and evaluate correlations \nbetween observables. \n\n\n\\ack{\nThe authors acknowledge discussions with Bing-Nan Lu, P. Marevi\\'{c}, K. Nomura, P. Ring and J. M. Yao.\nThis work was supported in part by the Chinese-Croatian project \"Universal models of exotic\nnuclear structure\", the Croatian Science Foundation under the project \"Structure and Dynamics\nof Exotic Femtosystems\" (IP-2014-09-9159), and the NSFC project No. 11475140.}\n\\bigskip\n\\bigskip\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Conclusion and Future Work}\nThis paper presented a method for conducting online merge planning with awareness of the latent behaviors of vehicles on the main lane. We modeled the behavior of agents on a freeway using the Cooperative Intelligent Driver Model and demonstrated accurate estimation of the latent cooperation level using a particle filter. Our estimate of the cooperation level informed a behavior-aware POMDP model that was solved online for a merging policy. The performance of the proposed strategy closely matched the results achieved by baseline strategies that assumed full knowledge of other agents' latent characteristics. Furthermore, our driving strategy greatly outperformed a rule-based driving strategy that did not consider the cooperation level of other agents. Although the application described in this paper is specific to freeway merge scenarios, the parameter estimation and online planning framework is quite general and can easily be extended to other driving scenarios or robotic applications.\n\nThis work assumed that the latent behavior of vehicles was time-invariant; future work will relax this assumption to consider agents that change their behavior during a driving scene. \nFurthermore, the safety and efficiency of a merging ego vehicle will be improved by quantifying the position and intention uncertainty of agents on a freeway. Uncertainty-aware models of freeway agents will inform the construction of probabilistically safe ego trajectory envelopes.\n\\section{Driver Models}\n\nRule-based driver models can provide reasonable approximations of microscopic driver behavior with a relatively small number of parameters, making them ideally suited to the online parameter estimation framework. In this section we discuss the ubiquitous IDM \\cite{treiber2000congested} and an IDM extension for merge scenarios, the Cooperative IDM \\cite{bouton2019cooperation}. We also present approaches for learning driver model parameters.\n\n\\subsection{Intelligent Driver Model}\nThe Intelligent Driver Model is an adaptive cruise control model that governs longitudinal vehicle motion. The IDM specifies a mathematical model that balances an agent's desire to travel at a particular speed with the need to maintain adequate separation distance between a leader vehicle. \nGiven a current speed $v(t)$, a distance headway $d(t)$, and a desired speed $v_{\\mathrm{des}}$, the IDM computes a desired acceleration according to\n\n\\begin{equation}\n {a_{{\\mathrm{IDM}}}} = {a_{\\max }}\\left( {1 - {{\\left( {\\frac{{v(t)}}{{{v_{{\\mathrm{des}}}}}}} \\right)}^4} - {{\\left( {\\frac{{{d_{{\\mathrm{des}}}}}}{{d(t)}}} \\right)}^2}} \\right).\n\\end{equation}\nThe desired separation distance $d_{\\mathrm{des}}$ is defined as\n\\begin{equation}\n {d_{{\\mathrm{des}}}} = {d_{{\\mathrm{min}}}} + \\tau \\cdot v(t) - \\frac{{v(t) \\cdot r(t)}}{{2\\sqrt {{a_{{\\mathrm{max}}}} \\cdot {b_{{\\mathrm{pref}}}}} }}.\n\\end{equation}\n\nModel inputs include the minimum allowable separation distance $d_{\\mathrm{min}}$; the minimum allowable separation time $\\tau$; the acceleration limit $a_{\\mathrm{max}}$; and the deceleration limit $b_{\\mathrm{pref}}$.\n\n\n\n\\subsection{Cooperative Intelligent Driver Model}\n\nThe Cooperative Intelligent Driver Model is an extension of the IDM developed by \\citet{bouton2019cooperation} for merge scenarios. The model governs the longitudinal acceleration of a vehicle on the main lane while considering the time to reach the merge point ($TTM$) for a merging agent. The C-IDM is parameterized by the IDM parameters as well as the \\textit{cooperation level} $c \\in [0,1]$ that determines the extent to which the driver cooperates with the merging vehicle. A cooperation level of $c = 1$ corresponds to a \\textit{cooperative} agent that yields to the merging vehicle, while a cooperation level of $c = 0$ corresponds to a \\textit{non-cooperative} agent that ignores the merging vehicle. \n\nConsider the merge scenario shown in \\cref{fig:cidm_setup} with an ego vehicle $V_{\\mathrm{E}}$, a leading agent on the main lane $V_{\\mathrm{L}}$, and a trailing agent on the main lane $V_{\\mathrm{T}}$. Let $TTM(V_{i})$ represent the time required for the $i$th actor to reach the specified merge point. C-IDM outputs the longitudinal acceleration for the trailing agent according to the following rule:\n\n\\begin{itemize}\n \\item If $TTM(V_{\\mathrm{E}}) < c \\times TTM(V_{\\mathrm{T}})$, then $V_{\\mathrm{T}}$ follows IDM while considering the projection of the ego on the main lane as its leading vehicle.\n \\item If $TTM(V_{\\mathrm{E}}) \\geq c \\times TTM(V_{\\mathrm{T}})$ or if no merging vehicle exists, then $V_{\\mathrm{T}}$ follows IDM while considering the next actor on the main lane, $V_{\\mathrm{L}}$, as its leading vehicle.\n\\end{itemize}\n\\begin{figure}[h!]\n\\centering\n\\include{figs\/cidm_setup}\n\\label{fig:cidm_setup}\n\\end{figure}\n\n\\subsection{Rule-Based Driver Model Parameter Estimation}\n\nRule-based driver models are governed by a relatively small number of parameters, which can be estimated in either an \\textit{offline} or an \\textit{online} fashion. Offline methods typically aggregate driving data in the training dataset and thus lose information on individual traffic participants. However, such methods can use arbitrarily large computation times and dataset sizes to maximize accuracy. Conversely, online methods must operate with constraints on computation time and data availability but capture the behavior of individual drivers. Offline strategies for rule-based driver model parameter estimation include constrained nonlinear optimization \\cite{lefevre2014comparison}, the Levenberg-Marquardt algorithm \\cite{morton2016analysis}, and genetic algorithms \\cite{kesting2008calibrating}. Online strategies include recursive Bayesian filters such as the Extended Kalman filter \\cite{monteil2015real} and the particle filter \\cite{bhattacharyya2021hybrid, buyer2019interaction, schulz2018interaction}. We employ a particle filter to conduct latent parameter estimation due to its ability to represent complex belief states.\n\\section{Experiments}\n\nIn this work, we consider a freeway merge scenario as visualized in \\cref{fig:cidm_setup}. An ego vehicle---representing either an autonomous agent or a human-driven vehicle with a guardian software system---seeks a gap on the main freeway lane. The vehicles are positioned such that the projection of the ego vehicle is in front of the trailing agent. Furthermore, we induce a conflict between the ego and the trailing agent by selecting initial velocities that place them on an intercept trajectory at the merge point. \\Cref{tab:experimental_setup} shows key values for the POMDP reward functions. The POMDP was solved online using partially observable Monte Carlo planning with observation widening (POMCPOW) \\cite{sunberg2018online}.\n\n\\begin{table}\\caption{\\label{tab:experimental_setup} POMDP Reward Function Parameters}\n\\centering\n\\begin{tabular}{@{}lr@{}} \\toprule\n \\textbf{Parameter} & \\textbf{Value} \\\\\n \\midrule\n $\\lambda_1$ (speed deviation) & $1$ \\\\\n $\\lambda_2$ (control effort)& $1$ \\\\\n $\\lambda_3$ (collision) & $100$ \\\\\n $d_{\\mathrm{safety}}$ & $\\SI{15.0}{\\meter}$ \\\\\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\vspace{-2.5mm}\n\\end{table}\n\n\\subsection{Benchmark Methods}\nWe tested our approach for learning the cooperation parameter against two other POMDP-based planning approaches that make \\textit{a priori} assumptions about the trailing agent's cooperation level: first, a cautious approach where the trailing agent is always assumed to be non-cooperative (e.g., the ego always assumes $c = 0.0$), and second, a risky approach where the trailing agent is always assumed to be cooperative (e.g., the ego always assumes $c = 1.0$). We also benchmarked against a scenario wherein the ego acceleration was governed by the stochastic IDM (SIDM)---an extension of the IDM that outputs noisy acceleration values \\cite{treiber2017intelligent}. We reproduced the merge scene shown in \\cref{fig:merge_scene} using the Applied Intuition simulation engine and ran 500 trials for each driving strategy; the trailing agent was assigned a ground truth cooperation level of $c = 0.0$ in 250 trials and a ground truth cooperation level of $c = 1.0$ in the other 250 trials.\n\\subsection{Filtering}\n\\Cref{fig:parameter_estimation} shows two examples of cooperation level parameter estimation drawn from the batch of experiments. The red line indicates the true cooperation level of the trailing agent while the blue line represents the ego's estimate of the cooperation level. The light blue regions show the 90\\% empirical error bounds. The parameter estimate is initialized to $c=0.5$ with diffuse error bounds. The ego vehicle observes the trailing agent either slow down to facilitate a merge (\\cref{fig:est_coop}) or maintain its current speed while ignoring the ego (\\cref{fig:est_noncoop}) and updates its estimate of the cooperation level accordingly. \\Cref{fig:parameter_estimation} indicates that the ego is able to quickly identify whether a trailing agent is behaving in a generally cooperative or uncooperative manner. The fact that the parameter estimates do not converge exactly to the true value is not unexpected; since the C-IDM relates the time to merge of the ego and the trailing agent with an inequality, there are in general a range of cooperation levels that will yield the same behavior from the trailing agent. A visualization of parameter estimation during merge scenarios was created using Applied Intuition simulation tools and can be found on the lab video channel at \\url{https:\/\/youtu.be\/KnM3azGH_Sg}.\n\n\\begin{figure*}[h]\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\input{figs\/EST_COOP}\n \\caption{Sample cooperation level estimation when the trailing agent is fully cooperative, e.g., $c = 1.0$}\n \\label{fig:est_coop}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\input{figs\/EST_NONCOOP}\n \\caption{Sample cooperation level estimation when the trailing agent is fully non-cooperative, e.g., $c = 0.0$}\n \\label{fig:est_noncoop}\n \\end{subfigure}\n \\hfill\n\\caption{Examples of online latent parameter estimation for the trailing agent's cooperation level. The estimate of $c$ quickly approaches the true value, indicating that the ego is aware of the trailing agent's cooperation level as it nears the merge point.}\n\\label{fig:parameter_estimation}\n\\vspace{-2.5mm}\n\\end{figure*}\n\n\\subsection{Performance Results}\nWe count the occurrences of hard braking and divide by the total number of trials to obtain the hard brake rate across all experimental configurations. We also measure \\textit{time-to-collision} (TTC), which is a safety metric, and \\textit{time-to-merge}, which indicates how efficient a merge maneuver is.\nHigher TTC values are associated with safer driving scenes \\cite{minderhoud2001extended}, while excessively high time-to-merge values could be indicative of the \\textit{freezing robot problem} wherein the robot freezes in place to avoid a collision \\cite{trautman2010unfreezing}.\n\n\\Cref{tab:hard-brake-rate} displays the rate of hard brake incidents across all driving methods. Hard brake rates remain predictably low across experiments with a cooperative trailing agent since the trailing agent avoids conflict with the merging ego by considering its projection on the main lane. When the trailing agent is non-cooperative, an upper bound on performance is given by the POMDP-based planner that assumes full knowledge of the true cooperation level of $c = 0.0$. Our proposed method for learning $c$ and the strategy of fixing $c = 0.0$ complete all trials without a hard brake incident. Planning with a fixed $c = 1.0$ expresses overconfidence in the desire of the vehicle on the main lane to cooperate and results in a hard brake rate of over $65\\%$. The SIDM driving strategy, which does not consider the cooperation level of the vehicle on the main lane, must conduct hard braking on every trial to avoid a collision.\n\n\\begin{table}\\caption{\\label{tab:hard-brake-rate}Hard Brake Rate}\n\\centering\n\\begin{tabular}{@{}lcr@{}} \\toprule\n \\textbf{Driving Strategy} & \\multicolumn{2}{c}{\\textbf{Hard Brake Rate (\\%)}} \\\\\n \\midrule\n \n & $c_T = 1.0$ & $c_T = 0.0$ \\\\\n \\midrule\n Learned $c$ & $0.0$ & $0.0$\\\\ \n Fixed $c = 0.0$ & $0.0$ & $0.0$ \\\\\n Fixed $c = 1.0$ & $0.4$ & $65.2$\\\\\n SIDM & $4.0$ & $100.0$\\\\\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\vspace{-2.5mm}\n\\end{table}\n\\Cref{fig:metrics_a1} presents the results for the trials with a cooperative trailing agent. The POMDP-based driving strategies achieve average TTC values greater than $10$ seconds. Many SIDM trials also achieve high TTC values; however, the SIDM trials that result in hard brake incidents are associated with small TTC values. Our strategy for learning $c$ achieves comparable results to the planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 1.0$. Note that assuming non-cooperative behavior does not underperform in these trials since the scenario is designed to induce conflict. In fact, assuming a non-cooperative agent leads to smaller time-to-merge values as the ego accelerates to reach the gap ahead of the trailing agent, as shown in \\cref{fig:ttm_a1}.\n\\begin{figure*}[h]\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\centering\n \\input{figs\/MIN_TTC_A1}\\caption{Distribution over minimum time-to-collision values achieved in trials with a cooperative agent.}\n \\label{fig:min_ttc_a1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\centering\n \\input{figs\/TTM_A1}\\caption{Distribution over time required to merge in trials with a cooperative agent.}\n \\label{fig:ttm_a1}\n \\end{subfigure}\n \\hfill\n\\caption{Experimental results given a cooperative agent on the main lane. Our strategy for learning $c$ achieves comparable results to the nominal planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 1.0$.}\n\\label{fig:metrics_a1}\n\\end{figure*}\n\\Cref{fig:metrics_a0} displays the results for the trials with a non-cooperative trailing agent. Both the SIDM and fixed $c = 1.0$ methods suffered from elevated hard brake rates and correspondingly low average TTC values. Our proposed method and the nominal benchmark obtained by fixing $c = 0.0$ achieved higher TTC values as shown in \\cref{fig:min_ttc_a0}. An explanation for the difference in TTC values is clear from \\cref{fig:ttm_a0}. Our proposed approach and the nominal baseline experience decreased time-to-merge values as they accelerate into the gap in front of the trailing agent. The POMDP-based driving strategy that assumes cooperative behavior takes more time to merge and thus encounters a conflict with the trailing agent at the merge point. The SIDM approach does not consider the behavior of the trailing agent at all and thus takes the longest time to merge on average.\n\\begin{figure*}[h]\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\centering\n \\input{figs\/MIN_TTC_A0}\\caption{Distribution over minimum time-to-collision values achieved in trials with a non-cooperative agent.}\n \\label{fig:min_ttc_a0}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\centering\n \\input{figs\/TTM_A0}\\caption{Distribution over time required to merge in trials with a non-cooperative agent.}\n \\label{fig:ttm_a0}\n \\end{subfigure}\n \\hfill\n\\caption{Experimental results given a non-cooperative agent on the main lane. Our strategy for learning $c$ achieves comparable results to the nominal planner that assumes full knowledge of the trailing agent's true cooperation level of $c = 0.0$.}\n\\label{fig:metrics_a0}\n\\vspace{-2.5mm}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nEnsuring safe interactions between automated and human traffic participants is a crucial step to making autonomous driving in urban environments a reality. Autonomous vehicles (AVs) must reason about the evolution of highly stochastic driving scenarios to provide protection for human passengers. Predicting the behavior of other traffic participants is an essential step to modeling scene evolution. However, forecasting human behaviors is a challenging task, as myriad psychological and physiological factors determine a traffic participant's driving tendencies \\cite{bhattacharyya2020online}. Furthermore, urban driving scenarios induce complex multi-agent interactions as traffic participants seek to complete individual objectives. For example, vehicles attempting to merge on a freeway must identify a gap in the traffic, while vehicles on the freeway may exhibit cooperative or non-cooperative behavior depending on their own latent characteristics. Consequently, reasoning about the intentions and future actions of agents is important for multi-agent driving interactions such as merging \\cite{bouton2019cooperation}, intersection navigation \\cite{bouton2017belief}, and freeway driving \\cite{sunberg2017value}.\n\n\\begin{figure}[t]\n\\centering\n\\include{figs\/merge_scene}\n\\label{fig:merge_scene}\n\\end{figure}\n\n\nHuman driving behavior can be modeled using \\textit{data-driven} or \\textit{rule-based} methods. Data-driven models are trained on expansive driving datasets and are highly parameterized to capture nuanced driver characteristics. For example, generative adversarial networks and imitation learning techniques have been used to construct expressive models of human driving behavior \\cite{bhattacharyya2020modeling}. However, data-driven approaches are \\textit{black-box} in nature and difficult to interpret. Furthermore, data-driven models might induce dangerous behavior in driving scenes that do not appear in the training dataset \\cite{bhattacharyya2020online}. An alternative to data-driven models is rule-based driver models such as the Intelligent Driver Model \\cite{treiber2000congested}. Rather than extracting driving characteristics from data, rule-based models specify a mathematical model using a set of rules derived from expert knowledge. Although such models rely on assumptions of human driving characteristics and can fail to generalize across a range of scenarios, they are highly interpretable and provide useful insights into microscopic human driving behavior. \\citet{brown2020taxonomy} present an extensive survey of human driver models.\n\nOnce the parameters of a driver model have been specified, the behavior of traffic participants can be forecasted and input to a decision-making system such as open-loop planing, or \\textit{model predictive control} \\cite{brown2017safe, leung2018infusing}. Game-theoretic frameworks \\cite{wang2021game} and reinforcement learning \\cite{ma2021reinforcement, bouton2019cooperation} have also been used for automotive decision making, but these strategies are computationally expensive. Partially observable Markov decision processes (POMDPs) are well-suited for AV decision making because they account for uncertainty in the environment and latent behaviors of traffic participants, and have been used with great success in the freeway driving context \\cite{sunberg2017value, sunberg2020improving}. In this work, we model the merge scenario shown in \\cref{fig:merge_scene} as a POMDP and solve the POMDP online for a driving policy. \\textit{Online} driving strategies choose the next action using real-time sensor information, while \\textit{offline} driving strategies plan all actions in advance. \n\nSeveral papers have investigated interaction-aware planning for merge scenarios. \\citet{ward2017probabilistic} considered merging on freeways and T-junctions; they fit parameters to an extension of the IDM using a nonlinear least-squares procedure. \\citet{bouton2019cooperation} and \\citet{ma2021reinforcement} conducted reinforcement learning-based techniques to learn merging policies. These works represent offline strategies for parameter estimation; we instead pursue an online strategy to capture the idiosyncrasies of individual drivers in real-time. The strategy presented by \\citet{sunberg2020improving} is most similar to our approach; however, they only consider lane change merging. We investigate freeway merging, which has been qualitatively described as one of the riskiest and complex driving maneuvers frequently encountered in day-to-day driving \\cite{ward2017probabilistic}.\n\nIn this paper, we perform online parameter estimation for a rule-based driver model---the Cooperative Intelligent Driver Model (C-IDM) \\cite{bouton2019cooperation}---that explicitly models the cooperation level of vehicles on a main freeway lane encountering a merging vehicle. The estimated cooperation level informs an online POMDP-based planner that executes a merge maneuver with awareness of latent driver behaviors. Our specific contributions are:\n\\begin{itemize}\n\\item We formulate the latent behavior prediction task as a recursive Bayesian estimation problem and estimate the cooperation level of vehicles using a particle filter.\n\\item We use the estimates of latent driver behaviors to inform an online POMDP-based planner which controls the merging vehicle. We thus account for two levels of uncertainty, as we propagate uncertainty over latent behavior through the planning framework.\n\\item We evaluate the safety and efficiency of our driving strategy in a high-fidelity automotive simulator and benchmark our planning strategy against methods that do not model latent behaviors or rely on \\textit{a priori} assumptions about agent behavior. \n\\end{itemize}\n\n\\section{Methodology}\n\nWe model the merge scenario as a partially observable Markov decision process, which can be defined by the tuple $(\\mathcal{S},\\mathcal{A},T,R,\\mathcal{O},Z,\\gamma)$, with $\\mathcal{S}$ the state space, $\\mathcal{A}$ the action space, $T$ the transition model, $R$ the reward function, $\\mathcal{O}$ the observation space, $Z$ the observation likelihood function, and $\\gamma$ the discount factor. An agent in state $s \\in \\mathcal{S}$ takes action $a \\in \\mathcal{A}$ and transitions to a successor state $s^{\\prime}$ according to the transition model $T(s^{\\prime},s,a) = \\mathrm{Pr}(s^{\\prime} \\mid s, a)$. The agent receives a reward according to the reward function $R(s,a,s^{\\prime})$. Rather than observing the true state, the agent receives a potentially imperfect observation $o \\in \\mathcal{O}$ at each time step. The likelihood of receiving observation $o$ is governed by $Z(o,s,a) = \\mathrm{Pr}(o \\mid s, a)$. Finally, the discount factor $\\gamma$ can be adjusted to make the agent more or less myopic. We model the merge scenario with the following definitions:\n\n\\subsection{The State Space}\n\nThe state of a given driving scene,\n\\begin{equation*} s=(V_{E},\\ V_{T},\\ c_{T},\\ V_{L})\\in \\mathcal{S}, \\end{equation*}\nconsists of the observable states $V_{E}$, $V_{T}$, $V_{L}$ of the ego, trailing agent, and lead agent, respectively, and the latent cooperation level $c_{T}$ which governs the C-IDM for the trailing agent. The observable vehicle state\n\\begin{equation*} V_{i}=(x_{i},\\ y_{i},\\ v_{i},\\ \\dot{v}_{i},\\ \\theta_{i}) \\end{equation*}\nconsists of the $i$th vehicle's position $(x_{i},\\ y_{i})$ on the simulator map, longitudinal velocity $v_{i}$, longitudinal acceleration $\\dot{v}_{i}$, and heading angle $\\theta_{i}$.\n\n\\subsection{The Action Space}\n\nAt each time step, the ego takes an action $a \\in \\mathcal{A}$ which consists of a longitudinal jerk value. Planning occurs in jerk space to produce smooth velocity profiles. Jerk values are restricted to the range $\\SI{-0.6}{\\meter\/\\second^3} \\leq a \\leq \\SI{0.6}{\\meter\/\\second^3}$ to prevent unrealistic or unsafe motion.\n\n\\subsection{The Transition Model}\n\nThe transition model is implicitly defined by a generative model, $s^{\\prime} \\sim T(s,a)$, which generates successor states from the distribution $\\mathrm{Pr}(s^{\\prime} \\mid s, a)$. Vehicles on the main lane propagate according to one-dimensional point mass dynamics:\n\\begin{align*} & {v^{t + 1}} = {v^t} + {\\dot{v}^t}\\Delta t \\\\ & {x^{t + 1}} = {x^t} + {v^t}\\Delta t + \\frac{1}{2}{\\dot{v}^t}\\Delta {t^2} \\end{align*}\n\nLateral speed is assumed to be zero. The trailing agent's acceleration is governed by the C-IDM, while the lead agent moves with constant velocity. The ego vehicle propagates according to a state transition function \n\\begin{align*} & {\\dot{v}^{t + 1}} = {\\dot{v}^t} + \\ddot{v}^t\\Delta t \\\\ & {v^{t + 1}} = {v^t} + {\\dot{v}^t}\\Delta t \n\\end{align*}\nThe applied jerk $\\ddot{v}^t$ is the action $a$ extracted at the current time step. We assume that the ego travels a distance ${v^t}\\Delta t$ along the merge lane curve during each discrete time step.\n\n\\subsection{The Reward Function}\n\nAutonomous vehicles must balance multiple and occasionally competing objectives such as safety, control effort, and rider comfort. We construct the following reward function to promote safe and efficient merge maneuvers:\n\\begin{equation} R(s)= -\\lambda_1 \\Vert v_{E} - v_{\\text{ref}}\\Vert - \\lambda_2 \\Vert \\dot{v}_{E} \\Vert - \\lambda_3 \\mathbf{1}_{b_{\\text{hard}}} \\end{equation}\nwhere $v_{\\text{ref}}$ is the ego vehicle's desired reference velocity and $\\mathbf{1}_{b_{\\text{hard}}}$ is an indicator function of hard braking:\n\\begin{equation*}\n\\mathbf{1}_{b_{\\text{hard}}} = \\begin{cases}\n{1}&{\\text{if}\\;\\text{$\\Vert(x_E,\\ y_E) - (x_T,\\ y_T) \\Vert < d_{\\mathrm{safety}} $}}\\\\\n{0}&{\\text{otherwise}}\\\\\n\\end{cases}\n\\end{equation*}\nwhere $d_{\\mathrm{safety}}$ is a user-defined safety threshold. We define \\textit{hard braking} as an uncomfortably abrupt deceleration that can be applied to prevent a collision. The hard brake rate is thus a proxy metric for AV safety and rider comfort across simulations. The weights $\\lambda_i$ can be adjusted to modify the ego's preferences for safety, efficiency, and control effort. \n\n\\subsection{Online Parameter Estimation}\n\n\nGiven a sequence of observations of the trailing agent's position, we wish to obtain a distribution over the latent cooperation level $c_T$. We assume that the cooperation level is time-invariant during the brief interaction between the trailing agent and the merging ego. Our proposed approach frames the task of estimating latent driver behavior as a state estimation problem and solves the inference problem using recursive Bayesian estimation. The general form of the recursive Bayesian update equation is \n\\begin{equation}\n p(\\theta_T \\mid \\mathbf{y}_{1:t}) = \\frac{p(\\mathbf{y}_t \\mid \\theta_T)p(\\theta_T \\mid \\mathbf{y}_{1:t-1})}\n {\\int_{\\theta_T} p(\\mathbf{y}_t \\mid \\theta_T)p(\\theta_T \\mid \\mathbf{y}_{1:t-1}) \\mathrm{d}\\theta_T} \\text{.}\n \\label{eqn:recursive_state_estimation}\n\\end{equation}\n\nBecause \\cref{eqn:recursive_state_estimation} cannot be solved analytically \\cite{bhattacharyya2020online}, we use particle filtering to provide an approximate solution to the inference problem. \\Cref{alg:particle_filtering} presents our strategy for estimating the cooperation level of the trailing agent. The particles are initially sampled from a uniform distribution and are then propagated according to the POMDP transition model. To prevent particle deprivation, we add noise $d_{c_T}$ sampled from a discrete uniform distribution $D = \\mathcal{U}(\\{-0.05,\\ 0,\\ 0.05 \\})$.\n\n\\begin{algorithm}[htb]\n\n\\setstretch{1.2}\n\n\\textbf{Input:} Initial particle set $\\Theta$, ego $V_E$, trailing agent $V_T$, leading agent $V_L$, POMDP model $\\mathcal{P}$ \\\\\n$c_{T}^{0} \\leftarrow 0.5$ (initialize cooperation parameter)\\\\\n\\For{$t \\gets 0, 1, \\ldots, N\\ \\mathrm{(time \\ steps)}$}{\n $x_T^{t+1} \\leftarrow \\mathrm{true \\ position \\ of \\ trailing \\ agent}$\\\\\n \\For{$i \\leftarrow 1, 2, \\ldots, K\\ \\mathrm{(particles)}$}{\n \\uIf{$TTM(V_{\\mathrm{E}}) < c \\times TTM(V_{\\mathrm{T}})$}{\n $\\dot{v}_{T}^{t+1} \\leftarrow$ IDM with ego projection as leader\\\n }\n \n \n \n \\Else{\n $\\dot{v}_{T}^{t+1} \\leftarrow$ IDM with lead agent as leader\\\n }\n Propagate vehicles according to $\\mathcal{P}.T(s^{\\prime}, a, s)$ \\\\\n $w_{i} \\gets Z\\left(x_T^{t+1} \\ | \\ x_{T,i}^{t+1}\\right)$\\\\\n $d_{c_T} \\sim D$ (dithering)\\\\\n \n $c_{T}^{t+1} \\leftarrow c_{T}^{t} + d_{c_{T}}$, clamped between $0$ and $1$\n }\n \n $\\Theta \\leftarrow \\mathrm{Resample \\ from \\ } \\Theta \\propto \\left[ w_{1}, \\ldots, w_{K}\\right]$\n}\\caption{Online Parameter Estimation}\n\\label{alg:particle_filtering}\n\\end{algorithm}\n\\section{Related Work}\n\nThe Intelligent Driver Model (IDM), developed by \\citet{treiber2000congested}, is a rule-based adaptive cruise control model. The IDM is collision-free by construction and used for longitudinal motion modeling. It has inspired numerous extensions to capture microscopic driver behaviors. The Enhanced IDM prevents excessive breaking reactions due to lane changes, resulting in more relaxed and natural driving behavior \\cite{kesting2010enhanced}. The Foresighted Driver Model is a risk-aware driver model that considers hazards such as obstacles and roadway curves \\cite{eggert2015foresighted}. The Stochastic IDM attempts to model the stochasticity of human drivers by randomly sampling an output acceleration from a Gaussian distribution \\cite{treiber2017intelligent}. In this work, we are concerned with freeway merge scenarios and thus consider the Cooperative Intelligent Driver Model proposed by \\citet{bouton2019cooperation}. The C-IDM uses a simple deterministic model to represent the behavior of vehicles on the main lane in the presence of a merging vehicle. Both the The C-IDM and its foundational IDM framework are presented in more detail in the next section.\n\nRule-based driver models are governed by a relatively small number of parameters, which can be estimated in either an \\textit{offline} or an \\textit{online} fashion. Offline methods typically aggregate driving trajectory data in the training dataset and thus lose information on individual traffic participants. However, such methods can use arbitrarily large computation times and dataset sizes to maximize accuracy. Conversely, online methods must operate with constraints on computation time and data availability but capture the behavior of individual drivers. Offline strategies for rule-based driver model parameter estimation include constrained nonlinear optimization \\cite{lefevre2014comparison}, the Levenberg-Marquardt algorithm \\cite{morton2016analysis}, and genetic algorithms \\cite{kesting2008calibrating}. Online strategies include Bayesian changepoint detection \\cite{galceran2017multipolicy} and recursive Bayesian filters such as the Extended Kalman filter \\cite{monteil2015real} and the particle filter \\cite{bhattacharyya2021hybrid, buyer2019interaction, schulz2018interaction}. We employ a particle filter to conduct latent parameter estimation due to its ability to represent complex belief states.\n\nSeveral papers have investigated interaction-aware planning for merge scenarios. \\citet{ward2017probabilistic} consider merging on freeways and T-junctions; they fit parameters to an extension of the IDM using a nonlinear least-squares procedure. \\citet{bouton2019cooperation} and \\citet{ma2021reinforcement} conduct reinforcement learning-based techniques to learn merging policies. These works represent offline strategies for parameter estimation; we instead pursue an online strategy to capture the idiosyncrasies of individual drivers in real-time. The online parameter strategy presented by \\citet{sunberg2020improving} is most similar to our approach; however, they only consider merging in lane change scenarios. We investigate freeway merging, which has been qualitatively described as one of the riskiest and complex driving maneuvers frequently encountered in day-to-day driving \\cite{ward2017probabilistic}.\n\n\n\\section*{Acknowledgments}\nToyota Research Institute (TRI) provided funds to assist the authors with their research, but this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. This research was supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656518. We thank Applied Intuition for support with the simulation platform.\n\n\\renewcommand*{\\bibfont}{\\footnotesize}\n\\printbibliography\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGenome assembly is one of the most complex and computationally exhaustive tasks in genomics projects~\\cite{PadovanideSouza2018}. Additionally, it is among the most important tasks which allow genome sequences analyses. The main goal of genome assembly is to reconstruct a genome using an assembler software, that does it by the analysis of several small fragments coming from a genome --- commonly referred to as \\emph{target genome}~\\cite{Li2011}. These fragments, named \\emph{reads}, are obtained from an equipment, called the DNA sequencer.\n\nA high-performance assembler is something highly desired among researchers, as it will imply more accurate genomes, allowing researchers to reach a better understanding of the traits and functions of living organisms~\\cite{Manzoni2016}. As a result, the knowledge acquired from these whole genomes produces positive impacts in several fields, such as medicine, biotechnology and biological sciences.\n\nA DNA sequencer is a machine responsible for the initial, but fractional, reading of the genetic code of living organisms~\\cite{Paszkiewicz2010}. However, the genomes of most organisms --- even microorganisms --- are too long for being read in a single run in the sequencer~\\cite{Heather2016}. To surpass this limitation, a technique called Shotgun is applied, consisting of cutting up the genome into small pieces and producing small fragments of DNA which can be entirely (or mostly) read by the sequencer, representing the corresponding genetic information in text sequences (the \\emph{reads})~\\cite{Rodriguez2012}. \n\nSince DNA molecules are formed by sequential pairs of complementary nucleotides (each of them composed by Adenine-Thymine or Guanine-Cytosine), \\emph{reads} represent only a single nucleotide from each pair, sequentially written as a character in the text. In biological terms, we can say that nucleotides of only one (out of two) strand of each DNA fragment is read. The reading of each nucleotide is represented by its corresponding initial (A, C, G or T). Thus, the number of characters in each \\emph{read} is commonly referred to as base pairs (or $bp$). \n\nThe genome of an organism is the sequence of all nucleotides from its DNA molecules, represented by letters (A, C, T, G). Each nucleotide isolated does not represent any relevant biological information, however, when we put them all together, the corresponding sequences provide a deep knowledge about this organism. Within the organism's genome, for example, there are (among other information) the species genes. Genes are continuous fragments of the genome whose nucleotide sequences define species traits and behaviors (e.g. the human eye color)~\\cite{Portin2017}. A single \\emph{read}, however, generally cannot represent the complete information from even a gene, thus genome assembly is commonly required to obtain the whole genome~\\cite{Heather2016}.\n\nGenome sequencing technology defines the maximum number of base pairs will be read from each DNA fragment~\\cite{Heather2016}, which, in turn, defines the size of the produced \\emph{reads} and directly affects the number of \\emph{reads} produced during the sequencing process. As genome assembly is a computational task aiming to order \\emph{reads} in an attempt to reconstruct the original DNA sequence, the number of \\emph{reads} and their sizes directly impact the complexity of the assembly process --- the more and smaller the \\emph{reads}, the higher the complexity for assembling them.\n\nFrom a computational perspective, given a set, $R={read_1,read_2,...,read_n}$, with $n$ small text sequences, called \\emph{reads}, the assembly problem is defined as the search for the superstring that originated the $n$ reads of $R$ by analysing the existing overlaps between them.\n\nGenome assembly is usually carried out by assemblers adopting \\emph{de novo} strategy and\/or the comparative strategy. The comparative approach is relatively simpler and computationally treatable, however it requires a previously assembled genome as reference (e.g. the genome of a similar species) to guide the assembly process by comparing the produced \\emph{reads} with the reference genome, meanwhile the \\emph{de novo} approach presents no such dependence~\\cite{Ji2017}.\n\n\\emph{De novo} strategy is particularly important given that only a small number of reference genomes are currently available, compared to the number of existing and non-sequenced genomes - it is estimated that the vast majority of microorganisms' genomes are still unknown~\\cite{Wong2020}. However, although \\emph{de novo} assembly approach allows the assembly of new genomes without requiring a reference genome, it is considered a highly complex combinatorial problem, falling into the theoretically intractable class of computational problems, referred to as NP-hard~\\cite{Medvedev2007}.\n\nIn computer science, the commonly applied strategies for \\emph{de novo} genome assembly process are based on heuristics and graphs, and they are known as \\emph{Greedy}, \\emph{Overlap-Layout-Consensus (OLC)}, and \\emph{De Bruijn graph}~\\cite{Pop2009}. For instance, in OLC strategy, each \\emph{read} is represented as a node in a graph (named overlap graph) and edges represent the overlap between \\emph{reads}. Thus, the reconstructed genome corresponds to the \\emph{reads} along the path traversing all the nodes. This algorithm corresponds to the Hamiltonian path algorithm. Another computational formulation for genome assembly is to find the shortest common superstring (SCS) formed by the \\emph{reads} --- which can also be polynomially reduced to the Travelling Salesman Problem (TSP)~\\cite{Springer2007}.\n\nRegardless the difficulties and limitations, current \\emph{de novo} assemblers are capable of producing acceptable solutions. However, the use and application of \\emph{de novo}-based assemblers normally require specific bioinformatics knowledge in order to correctly set configurations and parameters for the assembler. Nevertheless, optimal results are not always guaranteed --- given the high complexity enclosed to this theme~\\cite{Gurevich2013}.\n\nDespite the great contributions of the current assemblers for developing scientific discoveries on the genomics analyses of organisms, genome assembly is not yet a fully solved problem. So, it is very important to continue the development of new and more robust assemblers, in order to assemble DNA sequences faster and with improved accuracy~\\cite{bocicor2011}. This challenge has been the aim of numerous currently ongoing researches, which apply computational techniques to genomics in the search for better solutions, including the use of machine learning (ML)~\\cite{PadovanideSouza2018}.\n\nAlthough machine learning is an alternative approach to heuristics for dealing with high complex computational problems (as it is the case of NP-hard problems), few approaches apply machine learning for dealing with the assembly problem. According to the literature review presented in Souza et al.~\\cite{PadovanideSouza2018}, the few scientific investigations evaluating the application of machine learning techniques for genome assembly were recently published, and only one study dated to the end of the previous century while 12 others date from later. For comparison, in a similar review~\\cite{Henrique2019}, Henrique et al. reported 140 publications applying machine learning techniques into the problem of financial credit risk.\n\nWith the recent access to computational advances --- including increased processing and storage power of computers, the investigation of machine learning application for complex and high-scale computational problems has started to increase in the scientific community and some good results have been reported~\\cite{LeCun2019}. This larger resources availability has also allowed the return of reinforcement learning application for these problems~\\cite{Botvinick2019}. \n\nReinforcement learning (RL), alongside supervised learning and unsupervised learning, is a basic machine learning paradigm that works with intelligent agents which take actions in an environment that represents a task. Ideally, this agent is supposed to solve this task if it is able to learn how to maximize the rewards received from its actions taken~\\cite{sutton2018}.\n\nAlthough scarcely applied in machine learning development, RL has shown some surprisingly positive results, especially for games~\\cite{Botvinick2019}. Sutton et. al, by the way, recently suggested in \\cite{Silver2021} that RL, unlike supervised and unsupervised learning, may be the path to the development of general artificial intelligence. Two great (and commonly cited) examples of the application of reinforcement learning are the training of agents for playing the classic board game Go, as well as several Atari games, showing better results when compared to those from previous approaches and showing superior performances even when compared to trained humans~\\cite{Mnih2015,Silver2017,Vinyals2019}.\n\nHowever, RL successful applications are predominant in problems that rely on accurate environment simulators, such as games, where the rules and environments are known and allow the development of simulations for intensive training of intelligent agents~\\cite{Nian2020}. Despite the importance of games to both society and computer science, there is a growing expectation, and some initial efforts have been made, for extending the success of RL in games into real world problems --- for which it is generally impossible or impracticable to create the required simulators that would provide appropriate training for the agents~\\cite{dulac2019}.\n\nSuch low use of RL in real world problems is also observed in the specific scenario of the genome assembly, where only a single study can be found, proposed by Bocicor et al.~\\cite{bocicor2011, PadovanideSouza2018}. This approach, which is also the object of this study and will be hence forward referred to as seminal approach, proposes the use of an episodic trained agent (i.e. whose training has been divided into episodes) applying the Q-learning reinforcement learning algorithm for learning the correct order of a set of \\emph{reads} and, consequently, for reaching the corresponding genome.\n\nOne of the most valued characteristics of the reinforcement learning is that it allows the agent's autonomy for learning. For example, in supervised learning, a great deal of intervention is required (usually by human specialists) during the learning process, given that all information provided for the machine is previously and properly labeled. In reinforcement learning, the agent learns through the consequences of its successive failures and successes.\n\nThis is particularly useful for solving tasks whose solutions extrapolate the human knowledge and capacity, such as genome assembly. The assembly complexity starts from the assembler's choice --- as assemblers using similar strategies may produce different results~\\cite{Vollmers2017, vanderWalt2017} -- and extends until the assembler's configuration following the user's decisions. Obtaining intelligent and trained agents by reinforcement learning is important in this scenario as it could eliminate the need for human specialists.\n\nAnother relevant aspect for the application of reinforcement learning is the capacity of agents to deal with large volumes of data and extract new rules associated with the main task, which were not explicitly provided before and, in some cases, were not identified by humans. In games, as above mentioned~\\cite{Mnih2015, Silver2017}, agents were able to play new games on their own - without human supervision - and in some cases, they were able to outperform the best known human players.\n\nConsidering that Q-learning algorithm requires a Markov decision process definition with established parameters of states and actions, together with a reward system to be achieved by the agent at each action in every state~\\cite{sutton2018}, the problem was modeled by the authors~\\cite{bocicor2011} through a space of states capable of representing all possible \\emph{read} permutations, so that only one initial state there existed and, in each state, there is one action to be taken by the agent for each \\emph{read} in the pool to be assembled.\n\nFollowing these definitions, from the graph theory perspective, the proposed states space for $n$ \\emph{reads} can be visualized as a complete $n$-ary tree, with height equal to $n$, as the set of states presents only one initial state and forms a connected and acyclic graph~\\cite{cormen2009}. Thus, we can reach the number of existing states in the states space, represented by Equation \\ref{eq:04}.\n\n\\begin{equation}\n\\label{eq:04}\n\\textrm{number of states}=\\frac{n^{n+1}-1}{n-1}\n\\end{equation}\n\nThe proposed reward system depends, first, on the type of the state reached after taking each action, which can be an absorbing or a non-absorbing state. An absorbing state, once reached, does not allow any other state to be reached~\\cite{grinstead2012,sutton2018}. The authors have defined that each state requiring $n$ actions to be reached (being $n$ the number of \\emph{reads} to be assembled) is an absorbing state. \n\nA small and constant reward (e.g. 0.1) was assigned as reward for actions reaching non-absorbing states. This reward was also set to every action leading to absorbing states where repeated \\emph{reads} were used to achieve them. Finally, actions leading to other absorbing states (the final states) produce a reward corresponding to the sum of overlaps between all pairs of consecutive \\emph{reads} used to reach these states.\n\nFor your better understanding, Figure \\ref{fig:02} presents a simple example of a space of states for a set of only 2 \\emph{reads}, identified as A and B. In this example, we can observe the existence of a single initial state, two actions associated with non-absorbing states and four absorbing states (highlighted in black), achieved after taking two any actions.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=6cm,keepaspectratio]{fig2}\n\\caption{Example of state space for a set of two \\emph{reads}, here referred to by A and B.}\n\\label{fig:02}\n\\end{figure}\n\nIn this figure two of the absorbing states are highlighted by the letter $X$. These states, unlike the previous ones, are final states, as they are the only ones in the space of states reached directly from the initial state without repeated actions --- one is achieved from actions referred to as \\emph{read A} and \\emph{read B}, respectively, and the other from the actions of \\emph{read B} and, then, \\emph{read A}. \n\nThe Smith-Waterman algorithm (SW) was applied for obtaining the overlaps between pairs of reads, which were added for obtaining the rewards of actions that led to final states~\\cite{smith1981}. The sum of overlaps when reaching a final state $s$, here referred to as Performance Measure (PM), is described in Equation \\ref{eq:03}, where $read_s$ corresponds to the sequence of reads associated with the actions taken for achieving $s$. In an optimal solution, repeated reads tend to completely overlap, as the pairs reach maximum PM.\n\n\\begin{equation}\n\\label{eq:03}\nPM(s) = \\sum_{i=1}^{n-1}sw(read_s[i], read_s[i+1])\n\\end{equation}\n\nBy using these definitions, the seminal approach produced attractive results, however, it has been evaluated by the authors against only two small sets of \\emph{reads}, one with 4 \\emph{reads} with less than 10$bp$ and the second with 10 \\emph{reads} of 8$bp$ each. These \\emph{reads} were obtained by simulating the sequencing process, assuming as the target genome only a small fragment (a microgenome) of the real genome of the bacterium \\emph{Escherichia coli}.\n\nIn order to perform a scalability analysis of the seminal approach, Xavier et al.~\\cite{xavier2020} evaluated the performance of this approach against 18 datasets. These 18 datasets were produced following the same simulation methods. The first dataset corresponds to the set of 10 \\emph{reads} with 8$bp$ from the seminal approach, originated from a 25$bp$ microgenome. From this same microgenome and from a 50$bp$ microgenome, a total of 17 new datasets were generated (8 from the minor microgenome and 9 from the major one) each containing 10, 20 or 30 \\emph{reads}, with 8$bp$, 10$bp$ or 15$bp$.\n\nAlmost all definitions made by Bocicor et al. were replicated by Xavier et al. in this analysis, but they experimentally set $\\alpha$ and $\\beta$ to $0.8$ and $0.9$, respectively, and the space of actions was slightly reduced, so that actions associated with \\emph{reads} that has already been taken previously were removed from the available actions. In the states space depicted in Figure \\ref{fig:02}, for example, the leftmost and rightmost leaves (i.e. absorbing states) would be removed after this change. Although this change reduces the number of states, the size of the space of states continues to grow exponentially, as we can observe in Equation \\ref{eq:08}.\n\n\\begin{equation}\n\\label{eq:08}\n\\textrm{number of states}=\\sum_{i=0}^{n}\\frac{n!}{(n-i)!}\n\\end{equation}\n\nThis study confirmed the previously positive results found in the seminal approach with the first dataset, however, as the dataset size increases the performance of the seminal approach decreases considerably, reaching the target microgenome in only 2 out of 17 major datasets. According to the authors, such bad results may be related to the high cost required by the agent to explore such a vast states space --- as seen in Equation \\ref{eq:04}, this space grows exponentially --- and also from the failures in the proposed reward system.\n\nIn order to continue the investigation of the application of reinforcement learning in the genome assembly problem and targeting the current challenge of applying the reinforcement learning into real-world problems~\\cite{dulac2019}, here we propose to analyze the limits of reinforcement learning into a real-world problem, which is also a key problem for the development of science. This analysis was carried out by evaluating the performance of strategies that are complementary to those previously studied, and that could be incorporated into the seminal approach for obtaining improved genome assembly.\n\n\\section{Methods}\n\nIn this study, 7 experiments were evaluated against the seminal approach --- referred to as approaches 1.1, 1.2, 1.3, 1.4, 2, 3.1, and 3.2 --- and their methodologies are described in the next 3 subsections. In each approach, as in the seminal approach, the main goal is to reach an agent trained by reinforcement learning capable of identifying the correct order of \\emph{reads} from a sequenced genome. Figure 2 illustrates this proposal, where the set of \\emph{reads} from the sequencing process is initially represented as the agent's interaction environment; the agent learns the order of \\emph{reads} by observing the environment's current state and the rewards produced from the successive actions taken --- until it (ideally) reaches the correct order for the analyzed \\emph{reads}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig7}\n\\caption{Illustration that demonstrates the application of reinforcement learning in the genome assembly problem. The set of \\emph {reads}, which are obtained in random order by the sequencer, is represented computationally by a reinforcement learning environment. Through successive interactions with the environment, caused by taking actions, the agent ideally learns the correct order of \\emph {reads} --- thus allowing the target genome to be reached.}\n\\label{fig:07}\n\\end{figure}\n\nThese approaches will consider the findings of the scalability analysis from the previously mentioned work of Xavier et. al. Efforts were then made for improving the reward system adopted in the seminal approach --- especially in the approaches described in the next subsection - and to optimize the agent's exploration --- in the approaches described in the last two subsections.\n\n\\subsection*{Approaches 1: Tackling sparse rewards}\n\nIn approaches 1, the reward system, originally defined by Equation 3, has been improved for preventing that permutations of \\emph{reads} that are inconsistent, in terms of alignment, produce high reward values for the agent. This is an undesirable behavior as the agent's learning process is based on maximizing the accumulated rewards. Therefore, ideally, high rewards are expected to be associated with high quality responses.\n\n\\begin{equation}\n\\label{eq:01}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM(s') & \\mbox{if s' is a final state},\\\\\n0.1 &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\nThis observed inconsistency stem from the \\emph{Smith-Waterman} (SW) algorithm, chosen for calculating the local alignment between two given sequences. With this algorithm, a numeric score is calculated to represent the major alignment size (even if partial) between two sequences. However, the SW algorithm has no constraint for the order between sequences. Therefore, it does not previously consider that a sequence (in our case, a \\emph{read}) must be aligned either left or right to the other one.\n\nIn the case of the genome assembly problem, where the alignment between subsequent \\emph{reads} is expected to follow an order, the overlap score obtained from the SW algorithm may induce the agent to find \\emph{reads} permutations with high overlap values in pairs, however, without presenting a consecutive alignment (suffix-prefix) between \\emph{reads} in the set.\n\nIn Figure \\ref{fig:03} we can observe this type of inconsistency through an example, which presents two different permutations for a set of \\emph{reads} obtained from a given (and known) microgenome --- identified by letters ranging from A to J. The first permutation is formed, in this order, by the \\emph{reads} A, B, C, D, E, F, G, H, I, and J, and, according to SW algorithm, the accumulated overlap (aka PM) is $40.34$. Such score was obtained by summing the maximum overlap, calculated by SM algorithm, between \\emph{reads} A and B with the maximum overlap between \\emph{reads} B and C, and so on. Table \\ref{tab:04} presents the maximum overlap between all pairs of subsequent \\emph{reads} in this permutation.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccccccccc}\n\\hline\n\\textbf{A-B} & \\textbf{B-C} & \\textbf{C-D} & \\textbf{D-E} & \\textbf{E-F} & \\textbf{F-G} & \\textbf{G-H} & \\textbf{H-I} & \\textbf{I-J} \\\\\n\\hline\n4 & 5 & 4 & 5 & 5 & 3.67 & 4 & 2.67 & 7 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:04}Maximum overlap between each pair of subsequent \\emph{reads} of the permutation A-B-C-D-E-F-G-H-I-J, obtained by using the SW algorithm (using $match=1.0$, $mismatch=-0.33$, and $gap=-1.33$), thus producing a cumulative overlap of $40.34$.}\n\\end{table}\n\nThe aforementioned permutation (from A to J) corresponds to the optimal permutation, because the union of subsequent \\emph{reads} through the maximum exact overlap between the suffix of the previous read and the prefix of the next read produces exactly the target microgenome (provided at the bottom of the figure as the output genome).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig5b}\n\\caption{Illustration showing that the measure used as a reward to train agents does not produce maximum values for optimal outputs in some cases. Above, we have an optimal permutation of \\emph{reads}, for which the PM is $40.34$ and whose corresponding genome is equal to the target genome itself; and, below, we have another permutation whose output differs from the target genome, but the corresponding PM is greater than the PM of the optimal permutation.}\n\\label{fig:03}\n\\end{figure}\n\nWe can see in Table \\ref{tab:04} that the overlap between \\emph{reads} I and J is equal to 7, and it is easy to understand such value when we observe the alignment of the corresponding \\emph{reads} in Figure \\ref{fig:03} --- we have 7 letters matching between the sequences (i.e. GACACCC). However, it is not that trivial to understand why the SW algorithm reached a maximum overlap of 4 letters between \\emph{reads} A and B --- since the way they are aligned in Figure \\ref{fig:03} suggests only 1 overlapping letter (i.e. A). We will return to this case and explain it further later.\n\nSimilarly, we can analyze the second permutation, formed by the \\emph{reads} H, G, F, E, C, B, D, A, J, and I, respectively, is not optimal, as the corresponding output genome (obtained using the same read union procedure mentioned above) is quite different (and also larger) from the target genome. However, for this permutation, the SW algorithm presents an accumulated overlap of $43.02$, which is greater than that obtained for the optimal permutation and can be achieved by summing up the overlaps between each pair of subsequent \\emph{reads}, presented in Table \\ref{tab:05}.\n\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular}{cccccccccc}\n\\hline\n\\textbf{A-B} & \\textbf{B-C} & \\textbf{C-D} & \\textbf{D-E} & \\textbf{E-F} & \\textbf{F-G} & \\textbf{G-H} & \\textbf{H-I} & \\textbf{I-J} \\\\\n\\hline\n4 & 3.67 & 5 & 6 & 5 & 4.01 & 4.67 & 3.67 & 7 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:05}Maximum overlap between each pair of subsequent \\emph{reads} of the permutation H-G-F-E-C-B-D-A-J-I, obtained by using the SW algorithm (using $match=1.0$, $mismatch=-0.33$, and $gap=-1.33$), thus producing a cumulative overlap of $43.02$.}\n\\end{table}\n\nWe can see in Table \\ref{tab:05} that the overlapping score calculated by SW algorithm considering \\emph{reads} I and J is 7; but, again, it may be not clear to understand it, because the way they are aligned in Figure \\ref{fig:03} suggests no overlapping between them. The reason SW algorithm produces an overlapping score equals to 7 is that such algorithm does not constraint an order between the pair of sequences. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=4cm,keepaspectratio]{fig9}\n\\caption{Illustration to emphasize that the overlap score calculated by SW algorithm does not take in to account the relative order of the pair of sequences.}\n\\label{fig:09}\n\\end{figure}\n\nIn that case, it will return the maximum overlap, whether obtained considering \\emph{read} I to the left of read J, or vice versa --- and, as we mentioned above, in the first permutation, we have a 7-letter overlap when considering that \\emph{read} I is to the left of \\emph{read} J. This is the same reason why we have, even in the first permutation, a 4-letter overlap between \\emph{reads} A and B, because, considering \\emph{read} B is to the left of A (and not in reverse), it is possible to achieve a 4-letter overlap between the sequences (i.e. TAAC), as illustrated in Figure \\ref{fig:09}.\n\nAs the general goal of reinforcement learning agents is to learn actions that \\emph{maximize} the accumulated rewards, at the ending of the training, it is expected that the agent learns sequences of actions whose rewards are the highest possible. Thus, it is common and expected that the reward system will yield maximum rewards for optimal solutions. So that, using PM score as a reward for training, may be ineffective for some datasets. \n\nAnother aspect to be considered in the reward system of the seminal approach is that the agent is able to receive high rewards only when it takes actions leading to sparse final states in the states space. That is, considering that the applied training is episodic, in each training episode, the agent receives a non-constant and high reward for only 1 of the $n$ actions taken.\n\nGiven that, the reward system was adjusted in 4 different ways, in order to explore two aspects: (a) the use of overlap score that considers the relative order of \\emph{reads} and\/or (b) the use of dense rewards. These new reward systems are referred here by approaches 1.1, 1.2, 1.3 and 1.4, as follows.\n\nAs proposed in the seminal approach, the reward system of the 1.1 approach defines that actions leading to final states produce a bonus reward (of 1.0) that is added to another numerical overlap score between all subsequent \\emph{reads} used since the initial state. Thus, these actions produces a reward corresponding to the sum of the normalized overlap score (ranging from 0 to 1) of each pair of \\emph{reads}, taking into account the relative order of them.\n\nStill, every action that leads to a non-final state produces a constant and low reward (0.1). Equation \\ref{eq:05} formalizes the reward system for Approach 1.1, with $PM_{norm}(s')$ representing the normalized overlap between the \\emph{reads} used to reach the $s'$ (see more information on the normalized overlap calculation in Section 2 of the supplementary material).\n\n\\begin{equation}\n\\label{eq:05}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM_{norm}(s') + 1.0 & \\mbox{if s' is a final state},\\\\\n0.1 &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\nDespite the use of overlap score that considers the order of the \\emph{reads} in approach 1.1, it is susceptible to the sparse rewards problem --- as well as in the seminal approach. Although it often produces a small, constant and usually positive reward, and not a zero-value reward as traditionally applied by sparse reward systems, only few and sparse state-action pairs would produce higher rewards.\n\nWe can observe in both systems (from Equations \\ref{eq:01} and \\ref{eq:05}) that there is no rewards provided during learning process to guide the agent towards its goal (since any \\emph{read} incorporated would produce a reward $0.1$). Thus, the agent's learning process would depend exclusively on the sparse actions taken during the exploration of this vast space of states.\n\nConsidering that the agent's learning process tends to take longer in environments suffering from the sparse reward problem~\\cite{trott2019}, the proposed changes in approaches 1.2, 1.3 and 1.4 mainly focused on improving this aspect and, for this, higher rewards, previously obtained only at the end of the episode, were distributed for each action taken in each episode.\n\nThus, these approaches \u2013-- in addition to making the reward system dense, instead of sparse as originally proposed \u2013-- focused on reducing or eliminating the occurrence of inconsistencies, which, from the genome assembly perspective, would allow permutations of unaligned \\emph{reads} to produce maximum accumulated rewards.\n\nEquations \\ref{eq:07}, \\ref{eq:06} and \\ref{eq:02} represent, respectively, the reward systems for approaches 1.2, 1.3 and 1.4 --- so that $ol_{norm}(s,s')$ represents the normalized overlap between the two subsequent \\emph{reads} that allowed to reach state $s$ and then $s'$.\n\n\\begin{equation}\n\\label{eq:07}\nr(s,a, s')=PM_{norm}(s')\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:06}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nPM_{norm}(s') + 1.0 & \\mbox{if s' is a final state},\\\\\nol_{norm}(s, s') &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:02}\nr(s,a, s')=\\left\\{\n\\begin{array}{ll}\nol_{norm}(s, s') + 1.0 & \\mbox{if s' is a final state},\\\\\nol_{norm}(s, s') &\\mbox{otherwise}\n\\end{array}\n\\right.\n\\end{equation}\n\n\\subsection*{Approach 2: Pruning-based action elimination}\n\nOne of the great challenges for applying reinforcement learning in real-world problems is the high dimensionality of the states space that must be explored by the agents~\\cite{zahavy2018,dulac2020}. The Q-learning algorithm is especially susceptible to the dimensionality curse, as the number of states (as well as the number of actions) directly affects the data structure required for the agent's learning ~\\cite{wiele2020}. Given that the number of actions in each state directly affects the number of states in the seminal proposal, eliminating actions are a strategy for dealing with the high dimensionality~\\cite{zahavy2018}.\n\nTo reduce even more the states space proposed by the seminal approach, a heuristic procedure was applied to eliminate actions where states reached directly or indirectly had already been fully explored and where the maximum cumulative reward achieved is smaller than the cumulative reward obtained by taking any other action available in the state. \n\nIn Figure \\ref{fig:04}, we have an example of this action elimination in the given states space considering 3 reads. Looking again at the changed states space as a tree --- so removing actions associated with \\emph{reads} that have already been used, we see 16 states in this illustration, 6 of them are absorbing states (in the base of the tree) and also correspond to final states following the proposed modeling approach.\n\nFor better understanding the pruning process, note that 3 out of 6 final states are highlighted in black, while the remaining states are in gray and white. Black states correspond to explored final states (i.e. already visited by the agent). Gray states, such as the one reached by taking action $a$ in the initial state, represent the states in which all children states have been fully visited in the learning process. Finally, white states (final or not) are those not yet explored and\/or that have unexplored children --- e.g. the initial state, where one child is not explored and the other one is partially explored.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig3}\n\\caption{Illustration of the pruning procedure for a state space referring to the assembly of 3 \\emph{reads}, referred by \\emph{a}, \\emph{b} and \\emph{c}. The generic pruning procedure is defined in detail by Algorithm \\ref{alg:prunning}}\n\\label{fig:04}\n\\end{figure}\n\nWhen reaching an unexplored final state, such as the rightmost final state in Figure \\ref{fig:04}, the accumulated rewards achieved since the initial state is maintained and propagated for its predecessors, maintaining only the highest value propagated for the children. Each reward is represented by integer numbers within the states in the figure. Thus, each non-final state stores the highest accumulated reward achieved from it during the training process.\n\nBased on this information, it is possible to prune irrelevant actions, those actions that, after taken, do not produce the maximum accumulated reward. This type of action can be found in action \\emph{a} of the initial state in Figure \\ref{fig:04}. Note that all possible achievable states after taking this action have been explored and the maximum cumulative reward obtained is 6, obtained from consecutively taking the actions \\emph{a}, \\emph{b}, \\emph{c}. Also note that action \\emph{c} in the initial state, even that not fully explored, is capable of producing a higher reward, equals to 8, and obtained by taking the actions \\emph{c}, \\emph{a} and \\emph{b}, in that order. \n\nThus, when the agent first goes through the sequence of states corresponding to the actions \\emph{c}, \\emph{a} and \\emph{b}, the pruning mechanism propagates the maximum reward value up to the initial state and, at that moment, it cuts the action \\emph{a} from the initial state. The pseudo code presented in Algorithm \\ref{alg:prunning} presents the procedure to update pruning process when the last explored final state (referred to as $state$) is reached obtaining the corresponding accumulated reward achieved (referred to as $newReward$).\n\n\\begin{algorithm}\n\n\t\\caption{Pruning's algorithm}\\label{alg:prunning}\n\t\\begin{algorithmic}[1]\n\t\t\\Procedure{Prune}{$state:treeNode,newReward:float$}\n\n \\If{$state \\not= null$ \\textbf{and} $(state.unseen$ \\textbf{or} $newReward > state.maxReward)$}\\Comment{$state.unseen$ starts $true,\\forall_{state}$}\n \\State $state.unseen \\gets false$\n \\State $state.maxReward \\gets newReward$\n \\If{$state.final$}\n \\State\n \\textsc{PruneUselessChildren}($state$)\\Comment{i.e. prune all fully explored children where $maxReward < newReward$ }\n \\EndIf\n \\State \\textsc{Prune}($state.parent, newReward$)\n \\EndIf\n\n\t\t\\EndProcedure\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection*{Approaches 3: Evolutionary-based exploration}\n\nIn this proposal, the potential for mutual collaboration between reinforcement learning and evolutionary computing was investigated --\u2013 by applying the elitist selection of the genetic algorithm ~\\cite{Baluja95, Konar2005} --- to optimize the exploration of the states space. For assessing the individual contribution of the genetic algorithm in this hybrid proposal, this approach has been divided into two smaller approaches, referred to as approaches 3.1 and 3.2 and presented in the following subsections.\n\n\\subsubsection*{Approach 3.1: Evolutionary-aided reinforcement learning assembly}\n\nThe strategy of applying $\\epsilon$-greedy --- and its variations --- is a classic solution for expanding the exploration of agents trained by the Q-learning algorithm, as it allows a broader initial exploration, achieving the optimal policy once the states space has been sufficiently explored~\\cite{sutton2018}. However, the existing trade-off between exploitation and exploration remains a major and challenging problem for reinforcement learning in high-dimensional environments~\\cite{gimelfarb2020, peterson2019}.\n\nSearching for a more efficient exploration process and also considering the good performance of genetic algorithms in a similar genome assembly approach carried out by Oliveira et al.\\cite{oliveira2017}, here, the interaction between reinforcement learning and evolutionary computation was introduced into the exploration process.\n\nThis approach is based on the traditional operation of the Q-learning algorithm. However, in each Q-learning episode, the sequence of actions is stored, and at the end of the episode it is transformed into a chromosome of an initial population, that later will evolve. This procedure is presented in Figure \\ref{fig:05}, where we can see the list of actions made in each episode being used for constructing a new individual.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig4}\n\\caption{Illustration of the proposed interaction between reinforcement learning and genetic algorithm. At each reinforcement learning episode, the actions taken by the agent are converted into the chromosome (having each action as a gene) of an individual of the initial population of the genetic algorithm, whose size $n$ is predefined. After $n$ episodes (and thus $n$ individuals in the initial population), this population evolves for an also predefined number of generations through the genetic algorithm. Then, the most adapted individual of the last generation is obtained. In the end, that individual's chromosome genes are used as actions in the next episode of reinforcement learning.}\n\\label{fig:05}\n\\end{figure}\n\nNew chromosomes are inserted into the initial population until the amount of chromosomes reaches the predefined and expected size for this population. At this point, agent training is interrupted and $m$ genetic generations are carried out \u2013-- being $m$ also predetermined (for details see Section 4 of supplementary material) and applying the normalized sum of overlaps between \\emph{reads} as the adaptive function --- the same applied in Equation \\ref{eq:02} and detailed in Section 2 of the supplementary material.\n\nAfter $m$ generations, according to the objective function, the most fit individual is used for conducting the next episode in the agent's reinforcement learning training, hitherto interrupted. As each gene of the individual's chromosome corresponds to one possible action, the complete gene sequence will contain distinct successive actions to be taken by the agent in the current episode, producing then a mutual collaboration between reinforcement learning and the genetic algorithm --- the initial populations of the genetic algorithm are produced by reinforcement learning and, as a counterpart, the results from the evolution of the genetic algorithm is introduced in an episode of reinforcement learning.\n\nFor your better understanding of all the aforementioned approaches, considering they share the common basis of the Q-learning algorithm, Figure \\ref{fig:06} presents a flowchart where we can see the proposed updates for these approaches. The elements highlighted in gray represent the procedures and conditions of Q-learning algorithm --- which, briefly, is based on the action choice, action taking and the update of Q-values in the Q-table during several episodes, all of them starting in an initial state and ending when an absorbing state is reached.\n\nAs a consequence, these elements (in gray) fully represent Approaches 1.1, 1.2, 1.3, and 1.4, as the proposal is focused on replacing the reward system proposed by the seminal approach. Approaches 2 and 3.1 emerge as complementary to this common basis of procedures with specific procedures, namely the insertion of the dynamic pruning mechanism in approach 2 --- represented by the dashed element with double edges --- and the introduction of mutual contribution with the genetic algorithm to improve the agent's exploration performance --- represented by the dashed elements with simple edges. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig6}\n\\caption{Flowchart representing Approaches 1.1, 1.2, 1.3, 1.4, 2 and 3.1, so that Approaches 1.1, 1.2, 1.3, and 1.4 are defined by the elements in gray, Approach 2 by the dashed element with double edges and Approach 3.1 by the dashed elements with single border.}\n\\label{fig:06}\n\\end{figure}\n\n\\subsubsection*{Approach 3.2: Evolutionary-based assembly}\n\nFinally, to estimate the contribution of the genetic algorithm in Approach 3.1, which applies a mutual collaboration between reinforcement learning and the genetic algorithm, the genetic algorithm assembling performance was evaluated separately, following the same configurations set of the previous approach, but this time, adopting as starting population a set of individuals whose chromosomes were built from random permutations without repetition of \\emph{reads}.\n\n\\subsection*{Datasets}\n\nTo assess the performance of all approaches (including seminal approach), in addition to the 18 datasets proposed and made available by Xavier et al.~\\cite{xavier2020}, 5 novel datasets derived from other microgenomes extracted from the genome of the previous studies~\\cite{bocicor2011, xavier2020} were complementarily created here. These microgenomes are not arbitrary genome fragments, as the previously used microgenomes (which had 25$bp$ and 50$bp$), but represent larger fragments of previously annotated genes from the corresponding organism (i.e. \\emph{E. coli}). As the datasets are simulated data, it was considered that there are no cycles in the genome, thus being a limitation of the approach. The experiments were then carried out on 23 datasets, whose microgenomes sizes, number of \\emph{reads} and sizes of \\emph{reads} are presented in Table \\ref{tab:03} --- the last 5 lines correspond to the 5 datasets derived from genes.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{$\\mu$gen.} & \\textbf{\\# of} & \\textbf{read} & \\textbf{Gym} \\\\\n\\textbf{size} & \\textbf{reads} & \\textbf{size} & \\textbf{environment name} \\\\\n\\hline\n25 & 10 & 8 & GymnomeAssembly\\_25\\_10\\_8-v2 \\\\\n25 & 10 & 10 & GymnomeAssembly\\_25\\_10\\_10-v2 \\\\\n25 & 10 & 15 & GymnomeAssembly\\_25\\_10\\_15-v2 \\\\\n50 & 10 & 8 & GymnomeAssembly\\_50\\_10\\_8-v2 \\\\\n50 & 10 & 10 & GymnomeAssembly\\_50\\_10\\_10-v2 \\\\\n50 & 10 & 15 & GymnomeAssembly\\_50\\_10\\_15-v2 \\\\\n25 & 20 & 8 & GymnomeAssembly\\_25\\_20\\_8-v2 \\\\\n25 & 20 & 10 & GymnomeAssembly\\_25\\_20\\_10-v2 \\\\\n25 & 20 & 15 & GymnomeAssembly\\_25\\_20\\_15-v2 \\\\\n50 & 20 & 8 & GymnomeAssembly\\_50\\_20\\_8-v2 \\\\\n50 & 20 & 10 & GymnomeAssembly\\_50\\_20\\_10-v2 \\\\\n50 & 20 & 15 & GymnomeAssembly\\_50\\_20\\_15-v2 \\\\\n25 & 30 & 8 & GymnomeAssembly\\_25\\_30\\_8-v2 \\\\\n25 & 30 & 10 & GymnomeAssembly\\_25\\_30\\_10-v2 \\\\\n25 & 30 & 15 & GymnomeAssembly\\_25\\_30\\_15-v2 \\\\\n50 & 30 & 8 & GymnomeAssembly\\_50\\_30\\_8-v2 \\\\\n50 & 30 & 10 & GymnomeAssembly\\_50\\_30\\_10-v2 \\\\\n50 & 30 & 15 & GymnomeAssembly\\_50\\_30\\_15-v2 \\\\\n381 & 20 & 75 & GymnomeAssembly\\_381\\_20\\_75-v2 \\\\ \n567 & 30 & 75 & GymnomeAssembly\\_567\\_30\\_75-v2 \\\\ \n726 & 40 & 75 & GymnomeAssembly\\_728\\_40\\_75-v2 \\\\ \n930 & 50 & 75 & GymnomeAssembly\\_930\\_50\\_75-v2 \\\\ \n4224 & 230 & 75 & GymnomeAssembly\\_4224\\_230\\_75-v2 \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:03}Data sets used in the experiments. The first column shows the size (in $bp$) of the microgenome used to generate the \\emph{reads} of each set; the second column shows the number of \\emph{reads} generated; the third column shows the size of the generated \\emph{reads}; and the fourth column shows the name of the environment built for each set in the OpenAI Gym toolkit (accessible in \\url{http:\/\/github.com\/kriowloo\/gymnome-assembly}).}\n\\end{table}\n\nAs each of these datasets correspond to a reinforcement learning environment, an environment for each of them was created in the OpenAI Gym toolkit~\\cite{brockman2016}, in order to share such reinforcement learning challenges in a simple way. These environments are provided in \\url{http:\/\/github.com\/kriowloo\/gymnome-assembly} (see section 1 of the supplementary material for additional technical information) and use the reward system proposed in Approach 1.4. The identification names of each environment are presented in the last column of Table \\ref{tab:03}. The seminal reward system is also implemented and available \u2013-- for running it, use the version 1, replacing \\emph{v2} by \\emph{v1} in the environment name field.\n\nTwo experiments were then carried out for evaluating the approaches. In each experiment, 20 successive runs of each evaluated approach was performed for all the 23 existing datasets; totalizing 460 runs per approach. Given that each approach has different levels of complexity, the real execution time for each approach was considered for comparing them. To reduce the interference of external factors in execution time, all experiments were individually and sequentially performed in the same station (with Ubuntu 16.04 in an AWS EC2 instance of the \\emph{r5a.large} type, dual core, with 16GB RAM and 30GB of storage).\n\nIn the first experiment, here referred to as \\emph{Experiment A}, the objective was to verify the impact of progressively including the previously described strategies. For this, the performance of the seminal approach was evaluated (according to \\cite{bocicor2011}) against approaches 1.1, 1.2, 1.3, 1.4 (which modify the reward system), 2 (which includes the pruning dynamic) and 3.1 (which uses the AG as an complementary factor).\n\nIn the second experiment, referred to as \\emph{Experiment B}, the main objective was to compare the performance of the new RL-based approaches against the performance of the AG alone. Therefore, in addition to Approaches 1.1, 1.2, 1.3, 1.4, 2 and 3.1, the approach 3.2 (which explores GA alone) was performed in an equivalent time.\n\nFor performance measure in each experiment, two percentage measures were calculated, called distance-based measure (DM) and reward-based measure (RM). Evaluations of \\emph{de novo} assembly are commonly performed using proper metrics, such as the N50~\\cite{Bradnam2013}. These metrics were created because, as previously indicated, \\emph{de novo} assemblies are not supported by a reference genome. This way, in some scenarios, it is not possible to accurately assess the results obtained from the assemblers \u2013-- as the optimal output is unknown. \n\nHere, although a \\emph{de novo} assembler is evaluated, its assessment environment is restricted and the target genomes are known, and this scenario allows the use of specific (and exact) evaluations, such as the DM and RM metrics. \n\nDM considers that a run was successful when the consensus sequence resulting from the orders of \\emph{reads} produced in that run is exactly identical to the expected sequence. RM, however, consider any run as successful when the proposed order of \\emph{reads} presents the corresponding sum of PM$_{norm}$ higher or equal to the sum of PM$_{norm}$ from the optimal \\emph{reads} sequence (for details, see Section 3 of the supplementary material).\n\n\\section{Results}\n\nThe results obtained from \\emph{Experiment A} are presented in Table \\ref{tab:01}, where it is possible to observe that the seminal approach, although consuming the longest running time (23 hours and 34 minutes), also presented the lowest average performances, obtaining an optimal response in only 16.96\\% of the runs (i.e. 78 out of 460 executions) in terms of distance between the produced and the expected genome (DM) and 21.30\\% (98 out of 460) in terms of maximum reward (RM). This difference is based on the previously mentioned inconsistency of the proposed reward system, which allowed non-optimal permutations to produce maximum accumulated rewards.\n\nBeyond that, following the updates in the rewards system, the DM and MR performances in Approaches 1.2, 1.,3, and 1.4 surpassed those of the previous approach, and they also consumes about 4 hours less (19 hours and 38 minutes of execution time). In such experiments the part of the gains are due to the improved agent's performance in one of the sets where the sum of rewards for the optimal permutation of \\emph{reads} were not maximum in the previous reward system (as presented in Figure \\ref{fig:03}).\n\nDespite the gains obtained from the updated reward system, it was possible to note, based on the results, that the previously mentioned inconsistencies were not completely resolved. In some of the datasets the agent reached and even surpassed the maximum expected accumulated rewards, however, without obtaining the target genome. \n\nA minor improvement is observed after the application of the approach 2, with the incorporation of the agent's dynamic pruning, presenting a performance slightly superior to that of the agent in Approaches 1, thus requiring approximately one hour less of processing. The great highlight for this comparison, however, is presented in Approach 3.1, which, while benefiting from the environment exploration improved by AG, has reached a significantly improved result in a reduced amount of time for execution.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Experiment A} & \\textbf{Average} & \\textbf{Average} &\\textbf{Total} \\\\\n\\textbf{(Approach)} & \\textbf{DM} & \\textbf{RM} &\\textbf{runtime} \\\\\\hline\nSeminal & 16.96\\% & 21.30\\% & 23h34m \\\\ \n1.1 & 9.57\\% & 13.70\\% & 19h38m \\\\ \n1.2 & 18.48\\% & 21.30\\% & 19h38m \\\\ \n1.3 & 20.00\\% & 24.35\\% & 19h38m \\\\ \n1.4 & 20.43\\% & 24.78\\% & 19h38m \\\\ \n2 & 20.65\\% & 25.00\\% & 18h41m \\\\ \n3.1 & \\textbf{73.91\\%} & \\textbf{80.87\\%} & 17h03m \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:01}Results of Experiment A, which compares the performances of trained agents with different reinforcement learning strategies. The performances of each approach are expressed using distance-based (DM) and reward-based (RM) metrics (see \\emph{Methods} for details).}\n\\end{table}\n\nIn order to analyze the contribution of each technique for the gain of performance in the approach 3.1, a second experiment was performed, here referred to as \\emph{Experiment B}, to compare the performance of RL-based approaches (approaches 1.4, 2 and 3) and the isolated use of AG, but using random initial populations (approach 3.2). The results of Experiment B are presented in Table \\ref{tab:02}.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Experiment B} & \\textbf{Average} & \\textbf{Average} &\\textbf{Total} \\\\\n\\textbf{(Approach)} & \\textbf{DM} & \\textbf{RM} &\\textbf{runtime} \\\\ \\hline\n1.4 & 13.91\\% & 17.61\\% & 01h36m \\\\ \n2 & 12.39\\% & 16.30\\% & 01h36m \\\\\n3.1 & 14.78\\% & 14.78\\% & 01h42m \\\\\n3.2 & \\textbf{87.83\\%} & \\textbf{95.65\\%} & 01h34m \\\\\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:02}Experimental performances considering similar running times (RT). Performances were expressed using Distance-based Measure (DM) and Reward-based Measure (RM) (see \\emph{Methods})}.\n\\end{table}\n\n\nGiven the remarkable performance of Approach 3.2, Experiment B applied as reference the time taken by the AG to find an optimal solution in terms of RM for 22 out of the 23 datasets used (i.e. $95.65\\%$), which corresponds to 1 hour and 34 minutes. Given this results, it is evident the superiority of the results obtained from Approach 3.2 when compared to Approaches 1.4, 2 and 3.1.\n\nFinally, after verifying the dominance of Approach 3.2, an additional experiment was carried out, aiming to verify the limits of this approach. It considers the dataset for which no optimal response was achieved. This dataset corresponds to the reads from the largest gene (4Kbp) selected for the production of simulated datasets.\n\nFor that experiment, the running time was considerably increased, reaching approximately 38 hours of running --- surpassing the total time allocated for running all datasets in the other experiments, considering about 30 minutes per run (against less than 2 minutes for the same dataset in the tests run for approach 3.2 of \\emph{Experiment B}).\n\nAlthough with an expressive increase in execution time, no optimal solution has yet been reached for any of the 20 runs of this dataset. However, as presented Figure \\ref{fig:01}, through violinplots, it is possible to observe a consistent gain in performance, both in terms of DM (where longer runs had smaller distances than most distances reached by shorter runs) and RM (which was higher accumulated rewards in all longer time runs).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig1c}\n\\caption{Violin plots demonstrating the performances obtained by the GA in experiments with short (1h34m) and long (37h58m) execution times in terms of sums of DM and RM. Gray dots represent the distance\/reward obtained for all runs; the black line in the middle indicates interquartile range; and the violin curves shows the distribution density, where the wider the section, the greater the probability of the observations take the corresponding value.}\n\\label{fig:01}\n\\end{figure}\n\n\\section{Discussion}\n\nAs mentioned, genome assembly is among the most complex problems confronted by computer scientists within the context of genomics projects, regardless the importance of its results for scientific development. This complexity, in computational terms, allocates the problem of finding an optimal permutation of sequenced \\emph{reads} and reaching the target genome into a class of problems called NP-hard which comprises the most difficult problems in computer science~\\cite{roughgarden2020}.\n\nThis high complexity is particularly expressed in the vast space of states required for representing the assembly problem into the modeling proposed by the seminal approach and, consequently, into the approaches proposed here. To achieve the optimal solution in sets of \\emph{reads} of only 30 sequences in the seminal approach, for example, the agent should explore a states space composed of approximately $2e44$ states~\\cite{bocicor2011} (this number exceeds the estimated number of stars in the universe and corresponds to the number of sand grains on Earth). It is also worth noting that, in real-world scenarios, it is common to sequence millions of \\emph{reads} per genome --- which increases even more the corresponding states space. So, applying RL combined with a heuristic is a strategy to deal with the complexity of the problem, since it is not feasible to map all the state space. Therefore, this approach seeks map actions into states that tend to maximize their reward, thus decreasing the computational complexity of the problem.\n\nThe approaches proposed in this study aimed to expand the agent's learning based on two difficulties observed in the seminal approach for applying reinforcement learning into the genome assembly problem: (1) the reward system and (2) the agent's exploration strategies. Both the updates in the rewards system and the incorporation of new explorative strategies improved the agent's learning performance, as demonstrated by the results.\n\nThe definition of extrinsic rewards is one of the most challenging tasks for the construction of environments suitable for the agent's learning. The updates into the reward systems proposed here favored the agent's improved learning. However, it is not yet an optimal solution for the problem, as some non-optimal solutions are still present in datasets that produced maximum accumulated rewards. This would justify the occurrence of RM percentages higher than DM percentages in several experiments.\n\nThe dynamic pruning mechanism showed a discreet improvement for the learning process. However, the relationship between the additional processing cost resulting from the application of this mechanism and the benefit obtained from its implementation did not indicate a reasonable net gain from its use as bypass for the problem emerging from the high dimensionality of the space of states.\n\nIn this sense, the application of the RL strategy combined with the AG, in the hybrid approach, presented a much more attractive performance for supporting the exploration process in the long term. This combination was proved to be advantageous, probably given the dimensionality curse encountered by the Q-learning algorithm, as a strong AG support was observed for the agent while conduct the RL exploration.\n\nDespite the performance improvements, it remains evident the insufficiency of the approaches when applied to real-world scenarios. This insufficiency is more evident in the experiments performed with the largest dataset analyzed, with a size that corresponds to a gene of approximately $4Kbp$. Despite being the largest dataset employed, this gene remains much smaller than the smallest genomes from living organisms. None of the applied approaches presented an optimal response in this scenario, not even when applying the isolated genetic algorithm for a time that was largely longer than the time applied for the tests with the other datasets.\n\nThis finding, together with the superior results obtained from the GA alone, allow us to conclude on the infeasibility of applying the Q-learning algorithm to solve the genomes assembly problem in search for an optimal \\emph{reads} permutation, as originally proposed in the seminal approach.\n\nHowever, given the absence of approaches in the literature for tackling the problem through reinforcement learning and also considering the optimistic results obtained from RL, especially when combined to deep learning, further investigations on the applicability of reinforcement learning are required, including the use of different modeling approaches and algorithms.\n\nConsidering the importance of reproducing scientific studies and with the special intention of supporting future investigations, all experiments performed in this study, as well as the reinforcement learning environments applied to simulate the assembly problems, are available at \\url{https:\/\/osf.io\/tp4zj\/?view_only=8bd12290d9d4480f95bc2c6e49c73ff7}, and are open for reuse (for details on how to reproduce the experiments see Section 5 of the supplementary material).\n\nOne of the major challenges for applying reinforcement learning to real-world problems is the low sample efficiency of the algorithms~\\cite{yang2018}; and it is not different in the genome assembly problem treated here. From the time required by the agent trained by the Q-learning algorithm to reach an optimal solution, it is possible to perceive a high need for numerous interactions with the data. Considering that real-world inputs are even bigger than those experimentally applied here, obtaining a sample efficient algorithm for the problem would be required for supporting the development of solutions applicable into real-world problems.\n\nOne aspect that directly interferes with the agent's sample efficiency is optimizing the exploration of the space of states. In this sense, the application of techniques that remove duplicate \\emph{reads} --- due to repeats --- and the use of an intrinsic motivation could be alternatively investigated to bypass the exploration problem, given the high dimensionality of the proposed space of states~\\cite{barto2012, yang2018}.\n\nAs previously mentioned, it is important to continue the investigation for updating and replacing the Q-learning algorithm, as well as the use of distributed approaches and\/or algorithms using eligibility traces. Although simple, Q-learning is a very powerful reinforcement learning method for solving tasks and remains applied for real-world applications~\\cite{Chakole2021, Abdulhai2003}. As is the case with other RL methods, however, its use in real-world problems where the space of states is too vast is not recommended. The use of approximating methods for the Q function may prove opportune in such cases, especially given the promising results obtained in this study through the combination of RL with deep learning~\\cite{fjelland2020}.\n\nAlthough it faces some obstacles for being implemented into commercial applications, it is reasonable to consider the recent achievements of deep reinforcement learning when applied to games, which presents an equivalent computational problem to that of the genome assembly~\\cite{Vinyals2019,Silver2017,fjelland2020}. One of the great challenges for production equivalent proposals for the assembly problem is that several promising works apply convolutional artificial networks, which use images as inputs.\n\nThe transformation of real problems into games is a possibility for reusing the promising technology developed focusing on games~\\cite{Reis2020}. In this sense, the modeling of the genome assembly problem through a game can work as an alternative representation of the problem and may reduce the space of states explored by the agent. Representing the assembly problem as a maze with multiple targets\/objectives could be an example for such modeling approach. \n\nOne of the main benefits of representing the problem as a game is the reduction of the space of actions, which, in the proposed modeling approach, increases with the number of \\emph{reads}. Mnih et. al~\\cite{Mnih2015}, for example, were capable of achieving a common deep reinforcement learning architecture capable of resolving several Atari games, being remarkable that the best performances were achieved in games with fewer actions, as in the game \\emph{Breakout}, which requires only two actions (move right or left). In this game the agent was able to learn, from its performance and without instruction, that producing a gap is a valid strategy for optimizing its results (illustrated in Figure \\ref{fig:08}).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=8cm,keepaspectratio]{fig8}\n\\caption{Example of application of deep reinforcement learning in games where the agent was able to find strategies to optimize accumulated rewards. Here, the agent was able to discover that caving a tunnel could maximize rewards gain (images adapted from a video demonstration of learning progress in ~\\cite{Mnih2015}).}\n\\label{fig:08}\n\\end{figure}\n\nThe use of \\emph{Graph Embedding} may act as an option of modeling approach allowing the use of deep reinforcement learning without requiring the conversion of the problem into an image \u2013-- especially when considering that the genome assembly problem may be represented through a graph for optimizing the problem, in the shape of the Traveling Salesman Problem (TSP)~\\cite{Cook2012, Zhenyu2011}. As presented by Vesselinova et. al, numerous studies investigated the application of deep reinforcement learning for solving graph problems, including TSP~\\cite{Vesselinova2020}.\n\nFinally, one other aspect to be considered before the adoption of reinforcement learning into the genome assembly problem is the generalization of the agent\u00b4s learning --- a major challenge for the use of RL in real-world problems~\\cite{Ponsen2010}. As designed into the RL environment for the genome assembly problem, the learning acquired by the agent when assembling a set of \\emph{reads} will hardly be applied for the assembly of a new set.\n\n\\section{Additional information}\n\nAll data generated or analysed during this study are included in this published article. Supplementary material and reproduction codes are available at \\url{https:\/\/osf.io\/tp4zj\/?view_only=8bd12290d9d4480f95bc2c6e49c73ff7}.\n\n\\section{Competing interests}\nThe authors declare no competing interests.\n\n\\section{Author contributions statement}\n\nK.P., R.X and R.A. conceived the experiments, K.P. and R.X. conducted the experiments, K.P., R.A., A.C. and A.R. analysed the results. All authors reviewed the manuscript. \n\n\\section{Acknowledgments}\nThis study was financed in part by the Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior \u2013 Brasil (CAPES) \u2013 Finance Code 001.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}