diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfxzp" "b/data_all_eng_slimpj/shuffled/split2/finalzzfxzp"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfxzp"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n Slipping contact between solids is responsible for various important dynamic phenomena, such as dynamic jamming, friction-induced vibrations, brake squeal, stick-slip and sprag-slip oscillations. For many engineering applications, it is crucial to understand and predict the onset of these phenomena, as they are associated with impulsive contact forces, mechanical damage, unwanted noise and intensive wear of the interacting surfaces \\citep{ibrahim1994friction,sinou2007mode,butlin2013friction,mo2013effect,le2013friction,kruse2015influence}. \n \nMoving solids with contact interactions are often modelled as rigid multi-body systems with unilateral point contacts subject to Coulomb friction. Even within this simple framework, it is an open question what kind of dynamic phenomena may occur during slip motion and how they can be  predicted. The only exception is the dynamics of one single point contact in two dimensions, which has been analyzed in detail by several authors. Such systems may undergo a singularity called dynamic jam during slip \\citep{Genot1999}, in addition to the classical slip-stick, slip-reversal and slip-liftoff transitions. The same systems may also exhibit other forms of contact dynamics such as self-excited bouncing motion (a.k.a inverse chattering) \\citep{paper2} and impulsive contact forces or \"impact without collision\" \\citep{LeSuanAn_book,Zhao15} albeit transition into such modes does not occur during slip motion.\n\nThe goal of the present paper is to review these results and to investigate systematically the generic transitions and singularities of  multicontact systems. We perform a detailed analysis of systems with two slipping point contacts in two dimensional physical space, and many new phenomena are discovered. Our new findings include\n\\begin{itemize}\n\\item the sudden onset of impulsive contact forces (i.e. impact without collision or IWC) blocking slip motion\n\\item the onset of self-excited oscillations during slip, which may either remain microscopic, or they may grow exponentially to a macroscopic scale\n\\item convergence to two different types of codimension 2 singular manifolds where the dynamics becomes ill-defined within rigid body theory. One of these singularities is analogous to dynamic jamming, whereas the other one does not have an analogue in the single-contact case.\n\\end{itemize}\nWe then turn to systems with arbitrary number of point contacts and highlight some general properties of the dynamic jamming singularity.\n\nThere are two classical ways to deal with the piecewise smooth character of rigid unilateral contacts and Coulomb friction. The appealing mathematical framework of complementarity problems and measure differential inclusions offers a unified approach \\citep{Leine_book,Brogliato1999} or one can  distinguish between various contact modes (slip, stick and separation) each of which implies different smooth behavior. Since our goal includes predicting and understanding transitions between various contact modes, it is a natural choice here to follow the contact mode-based approach. \n\nIt is well-known that rigid body theory cannot be developed into a complete and consistent modeling framework in the presence of contacts. Solution inconsistency and indeterminacy have been known for a long time \\citep{Jellet1872,Painleve1895}, and many examples of this phenomenon have been found and studied more recently \\citep{champneys2016painleve}. In addition,  the solution may become ill-defined at attractive singular points \\citep{Genot1999,szalai2014nondeterministic}. The limitations of rigid models can often be resolved by contact regularization: the rigid contacts are replaced by compliant ones, and the behavior of the regularized system is studied in the quasi-rigid limit $\\epsilon\\rightarrow 0$ of a compliance parameter $\\epsilon$. We will make extensive use of contact regularization in the present paper.\n\nIn Sec. 2, we introduce our notation (Sec. 2.1) and review relevant results of the contact mode-based approach and of contact regularization wherein we follow the previous works \\citet{champneys2016painleve} and \\citet{varkonyi2017dynamics}. We also introduce a conceptual model system in Sec. 2.4 (inspired by a similar model system from \\citet{nordmark2017dynamics}), which is used to illustrate certain phenomena by numerical simulation as well as to construct examples of other phenomena analytically. \n\nSec. 3 focuses on the transitions of systems with slipping contacts. Existing results regarding the case of $n=1$ point contact are reviewed first, and we also formulate a theorem, which shows how and to what extent these systems stay away from those states where rigid models become ill-defined. We then turn towards systems with $n=2$ point contacts where a detailed analysis of transitions is given. Finally, we  discuss the case of general $n$, and a theorem related to dynamic jamming is generalized to this case. The paper is closed by a Discussion section, and by an appendix where some technical proofs are presented. \n\n\n\n\n\n\n\n\\section{Instantaneous behaviour of systems with sliding contacts}\n\n\n\\subsection{General formulation}\nWe consider the motion of an autonomous  mechanical system consisting of one or more rigid elements in two dimensions, subject to any number of ideal constraints (i.e. bilateral, frictionless connections) as well as $n\\geq 1$ unilateral point contacts. We assume Coulomb friction with given coefficients of friction at all contact points. We use lowercase letters for scalars and vectors, whereas capital letters denote matrix quantities. The state of the system is characterized by a vector of generalized coordinates $q$ as well as the generalized velocities $\\dot q$.\nThe continuous dynamics of such a system is given by a differential equation of the form\n\\begin{align}\n\\ddot{q}=f(q,\\dot q)+G_T(q)\\lambda_T+G_N(q)\\lambda\n\\label{eq:basiceq}\n\\end{align}\nwhere $\\lambda\\in\\mathbb{R}^n$ is a column vector of $n$ non-negative normal contact forces $\\lambda_i$ and $\\lambda_T$ is a similar vector containing tangential contact forces. The contact forces are determined by unilateral contact constraints and by the friction model. (Note that the equation above is not valid during impacts, since then the contact forces are impulsive.)\n\nLet $x_i(q)$ and $z_i(q)$ denote tangential and normal coordinates of the point contacts such that $z_i=0$ corresponds to a closed contact and $z_i>0$ to an open one, and $z_i=\\dot z_i=\\dot x_i=0$ means stick. \n\nWe will examine slip motion, which means that contacts are initially closed with nonzero tangential velocity. We assume without loss of generality that $\\dot x_i>0$ for all $i$. Then, the tangential contact forces $\\lambda_{t,i}$ become dependent on the normal forces $\\lambda_{i}$ by Coulomb's law\n$$\n\\lambda_{t,i}=-\\mu_i(q)\\lambda_i\n$$\n(where $\\mu_i$  is known and velocity-independent). In this case, \\eqref{eq:basiceq} can be written as\n\\begin{align}\n\\ddot{q}=f(q,\\dot q)+G(q)\\lambda\n\\label{eq:basiceq2}\n\\end{align}\n\nIf the functions $x_i(q)$, $z_i(q)$, $f(q,\\dot q)$, and $G(q)$ are known, then one determine two sets of equations of the form\n\\begin{align}\n\\ddot x&=a(q,\\dot q)+K(q)\\lambda\n\\label{eq:xdyn}\\\\\n\\ddot z&=b(q,\\dot q)+P(q)\\lambda\n\\label{eq:zdyn}\n\\end{align}\ndescribing the dynamics of the contact points.\nHere, $z=[z_1,z_2,...,z_n]^T$, $x=[x_1,x_2,...,x_n]^T$. The vectors $a,b\\in\\mathbb{R}^n$ and the $n$-by-$n$ matrices $K,P$ depend on the system in question. $P$ is sometimes referred to as the Delassus matrix of the system. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=4cm] {rod.pdf}\n\\caption{Rod slipping at one endpoint}\\label{fig:rod}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example:} in the case of a  rigid rod (Fig. \\ref{fig:rod}) of length $2l$, mass $m$, radius of inertia $\\rho$ under gravitational force $mg$, with one endpoint in unilateral contact with a flat, horizontal surface, we can use the generalized coordinates $q=[x_c,z_c,\\phi]^T$. The equations of motion are\n\\begin{align}\n\\ddot x_c&=-m^{-1}\\mu\\lambda\\\\\n\\ddot z_c&=-g+m^{-1}\\lambda\\\\\n\\ddot\\phi&=m^{-1}\\rho^{-2}l\\lambda(-\\cos\\phi+\\mu\\sin\\phi)\n\\end{align}\nand the contact coordinates can be expressed as\n\\begin{align}\nx&=x_c+l\\cos\\phi\\\\\nz&=z_c-l\\sin\\phi\n\\end{align}\n The Lagrange equations and the kinematics of the rod then imply \\citep{Genot1999}\n\\begin{align}\na&=-l\\dot\\phi^2\\cos\\phi\\\\\nb&=l(\\dot\\phi^2\\sin\\phi-g)\\label{eq:rodb}\\\\\nK&=m^{-1}\\left[-1+l^2\\rho^{-2}(-\\mu\\sin^2\\phi+\\cos\\phi\\sin\\phi)\\right]\\\\\\\\\nP&=m^{-1}\\left[1+l^2\\rho^{-2}\\cos^2\\phi-\\mu\\cos\\phi\\sin\\phi\\right]\\label{eq:rodP}\\\\\n\\end{align}\nwhere $\\mu$ is the coefficient of friction . Note that $a,b,K,P$ are all scalars, for now $n=1$.\n\nIn what follows, we denote the $j^{th}$ column vector of $K,P$ by $k_j$ and $p_j$ and the $i^{th}$ element of vectors $a,b,k_j,p_j$ by $a_i,b_i,k_{ij},p_{ij}$.\nWe drop arguments like $t,q(t),\\dot q(t)$ for brevity. We will use a lower index $_{init}$ to denote initial conditions at some initial time $t_{init}<0$ and index $_{0}$ for quantities evaluated at $t=0$. Typically, we will choose initial conditions in such a way that transitions from slip occur at $t=0$. \n\nSlip motion requires an initial state $q_{init},\\dot q_{init}$ inducing $z_{init}=\\dot z_{init}=0$, $\\dot x_{init}>0$ (where the inequality should be satisfied by all elements of vector $x$). In this situation, each contact point may undergo sustained slip in the positive $x_i$ direction (contact mode S) or lift-off (contact mode F). Stick and negative slip  are not possible since $\\dot x_{init}>0$. Each one of these contact modes has a corresponding equality and an inequality constraint:\n\\begin{align}\n\\ddot z_i=0\n\\label{eq:slip-equality}\\\\\n\\lambda_i\\geq 0\n\\label{eq:slip-inequality}\n\\end{align}\n for S mode and \n \\begin{align}\n\\lambda_i=0\n\\label{eq:liftoff-equality}\\\\\n\\ddot z_i\\geq 0\n\\label{eq:liftoff-inequality}\n\\end{align}\nfor F mode. The contact mode of a full system can be represented by an $n$-letter word from the two-letter alphabet $\\{S,F\\}$. We can check the feasibility of all $2^n$  contact modes by solving \\eqref{eq:zdyn} together with the $n$ equality constraints, and by testing the inequality constraints of the contact modes. The fact that we may find multiple or no feasible contact mode is known as Painlev\\'e paradox \\citep{champneys2016painleve}. This kind of limitation of rigid models is the primary motivation for contact regularization techniques introduced below.\n\n\\subsection{Contact regularization}\n\nThe non-uniqueness and non-existence of solution associated with rigid models is often resolvable by contact regularization techniques. A regularized contact model allows small penetrations $z_i<0$ at the contact points and assumes a certain relation between the contact force and the penetration. Here, we will use a standard, linear, unilateral viscoelastic contact law \n\\begin{align}\n\\lambda_i=\n\\begin{cases}\n0 & if\\;\\;  z_i>0\\\\\n\\max(0,-\\epsilon^{-2}k_iz_i-\\epsilon^{-1}\\nu_i\\dot z_i) & if\\;\\;  z_i \\leq 0\n\\end{cases}\n\\label{eq:contactlaw}\n\\end{align}\nwhere, $k_i$ and $\\nu_i$ are scaled stiffness and damping coefficients, whereas $\\epsilon$ is a scaling factor. Now we will say that a contact is closed if $\\lambda_i>0$ (as opposed to $z_i=0$ in the rigid case). Note that the contact model is smooth as long as the contact is closed. \n\n\nThen, the analysis of a rigid model is replaced by the quasi-rigid limit $\\epsilon\\rightarrow 0$ of the regularized model, which often yields deeper insight into the behavior of the system. The regularized model is a slow-fast system in which $a,b,P,K$ evolve on a slow time scale, and the internal dynamics of contacts ($z,\\dot z$) represents the fast subsystem. In Section 2 of the paper, we will focus exclusively on the fast dynamics and we do not even specify  the full slow-fast dynamics or the reduced problem. Then in Section 3, we discuss sudden transitions of the fast dynamics induced by the slow dynamics.\n\nWhen all $n$ contact points are closed ($\\lambda_i>0$ for $i=1,2,...,n$), then  the equations \\eqref{eq:zdyn} and \\eqref{eq:contactlaw} induce the fast dynamics:\n\\begin{align}\ng' = \n\\begin{bmatrix}\nO_n & I_n \\\\\n -PK & -PN  \n\\end{bmatrix}\ng+\n\\begin{bmatrix}\no_n,\\\\b  \n\\end{bmatrix}\n\\label{eq:contactdynamics_generaln}\n\\end{align}\nwhere $'$ represents differentiation with respect to fast time $\\tau=\\epsilon^{-1}t$; $O_n$ and $I_n$ are zero and identity matrices of size $n\\times n$, $o_n$ is a column vector of $n$ zeros, and\n\\begin{align}\ng=\n\\epsilon^{-2}\\begin{bmatrix}\nz\\\\\nz'  \n\\end{bmatrix},\nK=\n\\begin{bmatrix}\nk_1&0&...&&0\\\\\n0&k_2&0&...&0\\\\\n\\vdots&&&&\\\\\n0&...&&0&k_n \n\\end{bmatrix},\nN=\n\\begin{bmatrix}\n\\nu_1&0&...&&0\\\\\n0&\\nu_2&0&...&0\\\\\n\\vdots&&&&\\\\\n0&...&&0&\\nu_n \n\\end{bmatrix}\n\\end{align}\nNote that $g$ is a rescaled vector depending on those variables, which represent motion in the directions of the contact normals. Such a reduction is possible because the fast dynamics of $g$ decouples from all other components of the dynamics including the dynamics of $x$ and its time derivative. \n\nWhen the regularized model is used, we identify slip motion with motion during which all contacts are closed and the fast dynamics remains stationary at the invariant point \n$$\ng=-\\begin{bmatrix}\nO_n & I_n \\\\\n -PK & -PN  \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\no_n,\\\\b  \n\\end{bmatrix}\n$$\nof the linear system \\eqref{eq:contactdynamics_generaln}. Such an invariant point may be asymptotically stable or unstable depending on the eigenvalues of the coefficient matrix. Hence, regularization shows that certain modes involving slipping contacts are unstable and do not occur in practice, which can eliminate some forms of solution non-uniqueness. \n\n\n\nAs we point out later, contact regularization has other benefits as well. Most importantly, we will encounter situations, when the invariant point is asymptotically unstable and  $z_i$ diverges towards $-\\infty$. Such an event induces rapidly increasing contact force $\\lambda_i$. We will identify this kind of behaviour with the impact without collision (IWC) phenomenon of rigid models. Detailed analysis of dynamics induced by diverging contact forces (similar to \\citet{Zhao15} in the case of $n=1$) is however beyond the scope of this work.   \n\n\n\\subsection{Systems with a single point contact}\\label{sec:1contact-instantaneous}\nIn the case of $n=1$\\citep{champneys2016painleve}, $a,b,P,K,\\lambda$ are scalars. Slip motion means that \n\\begin{align}\n\\lambda=-b\/P\n\\label{eq:lambda-1contact}\n\\end{align}\naccording to \\eqref{eq:zdyn} and \\eqref{eq:slip-equality}. Then the feasibility condition \\eqref{eq:slip-inequality} is satisfied if $b$ and $P$ have opposite signs. Similarly, liftoff is feasible if $b>0$. Since both $b$ and $P$ may have any sign, either one, both or none of the two modes may be feasible (Table \\ref{table:1contact}). The issue of non-existence is resolved by considering a third type of solution involving impulsive contact forces. Whenever $P<0$, an impact without collision (IWC) may occur: an impact occurs, which causes instantaneous jump in tangential velocity into stick or reverse slip \\citep{LeSuanAn_book,Zhao15,hogan2017regularization}. The regularized contact model \\eqref{eq:contactlaw} predicts rapid divergence of $\\lambda$ to $\\infty$ in such situations, which is consistent with the assumption of an IWC in the $\\epsilon\\rightarrow 0$ limit.\n\nStability analysis using regularized contact model reveals that slip is stable if and only if $P>0$, yet stability analysis is not able to eliminate non-uniqueness if $P<0<b$.\\\\\n\n\\textbf{Example:} the parameters $b,P$ of the sliding rod (Fig.\\ref{fig:rod}) given by \\eqref{eq:rodb},\\eqref{eq:rodP} may have either sign. In particular, $b$ is always negative if $\\dot\\phi$ is small, but may become positive for large $\\dot\\phi$, whereas $P$ is always positive for small $\\mu$ and may have either sign if $\\mu$ is large. \n\n\\begin{table} \n\\centering\n\\caption{Feasibility, and stability of contact modes and IWC in the case of one point contact}\n\\label{table:1contact}\n\\begin{tabulary}{0.6\\textwidth}{ccc}\n\\hline\n& b\\textless 0 & b\\textgreater 0                                                            \\\\ \n\\hline\nP\\textless 0    & IWC         & \\begin{tabular}{c}IWC\\\\ liftoff\\\\ slip (unstable)\\end{tabular}   \n\\\\ \\hline\nP\\textgreater0 & liftoff     & slip (stable)                                                             \\\\ \\hline\n\\end{tabulary}\n\\end{table}\n\n\\subsection{Systems with two point contacts}\\label{sec:2contact-instantaneous}\n\nIn the case of 2 contacts, there are 4 possible contact modes: SS, FF, SF, and FS. The feasibility and stability analysis of these systems was recently investigated by  \\citet{varkonyi2017dynamics}. The analysis reveals that the most important model parameters are the angle $\\beta$ between vector $b$ and the unit vector $[1,0]$ (measured in the direction of $[0,1]$), as well as the analogous angles $\\gamma_1$ and $\\gamma_2$ associated with the vectors $p_1$, $p_2$ (Figure \\ref{fig:gammabeta}). The feasibility of the four contact modes depends exclusively on these three parameters. For example, FF motion (liftoff at both points) is feasible if $b_1,b_2>0$, i.e. if both points accelerate in the absence of contact forces away from the contact surfaces. This can also be expressed as $0\\leq\\beta\\leq\\pi\/2$. Contact mode FS is feasible if we have positive contact force at point 2, and positive normal acceleration at point 1 i.e. if\n\\begin{align}\n-b_2\/p_{22}>0 \\label{eq:FS1}\\\\\nb_1-p_{21}b_2\/p_{22}>0 \\label{eq:FS2}\n\\end{align}\nAs we know from Sec. \\ref{sec:1contact-instantaneous}, the stability of FS additionally requires $p_{22}>0$. These conditions depend exclusively on the angles $\\gamma_1,\\beta$ (Fig. \\ref{fig: 2contact feasibility}(a)). The feasibility and stability conditions of SF are analogous.\n\nFor simultaneous slip at both contact points, the contact force is determined by \\eqref{eq:zdyn}:\n\\begin{align}\n\\lambda=-P^{-1}b\n\\label{eq:lambda-2contact}\n\\end{align}\nThe feasibility condition \\eqref{eq:slip-inequality} is satisfied exactly when $-b$ is in the cone spanned by $p_1$ and $p_2$, which can be expressed in terms of the angles $\\beta,\\gamma_1,\\gamma_2$ as illustrated by Fig. \\ref{fig: 2contact feasibility}(b). The figure also shows the region of stability obtained by eigenvalue analysis. This region will soon be analyzed in more detail (see Fig. \\ref{fig:SSrange} below) \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6cm] {gammabeta.pdf}\n\\caption{Definition of the angles $\\gamma_i$ ($i=1,2$) and $\\beta$.}\\label{fig:gammabeta}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=13cm] {feasibility.pdf}\n\\caption{Feasibility and stability of the contact modes SF (a) and SS (b). Grey areas or volumes show feasibility and hatching marks regions where motion is also stable. } \\label{fig: 2contact feasibility}\n\\end{center}\n\\end{figure}\n\nThe feasibility of impulsive contact forces (IWC) has also been examined in \\citet{varkonyi2017dynamics}. It was found that there are three different types of IWC since point 1, point 2 or both points may exhibit impulsive contact forces. Unlike in the case of $n=1$, contact regularization does not yield uniform answer with respect to the feasibility of IWCs: in certain cases, the feasibility of the IWC may depend on the stiffness and dissipation coefficients $k_1,k_2,\\nu_1,\\nu_2$ of the regularized contact model.\n\nThe stability analysis of the contact modes  via contact regularization yields the following results:\n\\begin{enumerate}\n\\item the stability of the SF and FS modes is  determined by $\\gamma_1,\\gamma_2$ and it is independent of any other parameter. The same is true for SS in a large range of $\\gamma_1,\\gamma_2$\n\\item in certain ranges of $\\gamma_1,\\gamma_2$, the stability of the SS mode may also depend on the model parameters $k_1,k_2,\\nu_1,\\nu_2$\n\\item stability of any mode is independent of $\\beta$\n\n\\end{enumerate}\n\nFinally, \\citep{varkonyi2017dynamics} shows that the regularized model predicts unusual forms of contact dynamics in some ranges of $\\gamma_1,\\gamma_2,\\beta$, including limit cycles of the fast dynamics, and self-excited exponentially growing oscillations. A comprehensive list of such dynamic phenomena is not known.\n\n\\subsection{An illustrating example}\\label{sec:numericexample}\nWe will illustrate several types of dynamic behavior by analytical investigation and numerical simulation of a conceptual model system with $n=2$ contacts. Our model captures the algebraic structure of the equations of motion, but does not correspond to a specific mechanical system. \n\nWe have seen that $P$ is a function of $q$ whereas $b$ may additionally depend on $\\dot q$. Similarly to the equations \\eqref{eq:xdyn}-\\eqref{eq:zdyn} developed for the dependent variables $x$ and $z$, we can also develop a set of expressions for $\\dot P$ and $\\dot b$ by using \\eqref{eq:basiceq2} and the chain rule. In the present paper, we do not explicitly construct such equations for any system, but we note that they can always be written as\n\\begin{align}\n\\dot P = A_1(q,\\dot q)\\\\\n\\dot b =  \\alpha_2(q,\\dot q)+ A_3(q,\\dot q)\\lambda \n\\end{align}\nwhere the functions $A_1,A_3\\in\\mathbb {R}^{2\\times 2}$, $\\alpha_2\\in\\mathbb {R}^{2}$ are system-specific. For example, the  values of $A_1,\\alpha_2,A_3$ corresponding to a frictional impact oscillator originally introduced by \\citet{leine2002} are shown in Appendix B of \\citet{nordmark2017dynamics}.\n\nFor the sake of illustration, we will now consider the case when $A_1,\\alpha_2,A_3$ are constants:\n\\begin{align}\n\\dot P = \\alpha_1\\label{eq:dotP}\\\\\n\\dot b =  \\alpha_2+ A_3\\lambda \\label{eq:dotb}\n\\end{align}\nThis model system can be combined with the rigid contact model, where we have $\\lambda=0$ in F mode and $\\lambda$ is given by \\eqref{eq:lambda-2contact} in S mode. Then the equations \\eqref{eq:dotP}-\\eqref{eq:dotb} decouple from all other equations of motion, thus we are able to investigate the dynamics of $P$ and $b$ in isolation. This system will be used to demonstrate the onset of some phenomena analytically. \nAlternatively, we can also combine \\eqref{eq:dotP}-\\eqref{eq:dotb} with the regularized contact model. Then, the equations \\eqref{eq:zdyn}, \\eqref{eq:contactlaw}, \\eqref{eq:dotP} and\\eqref{eq:dotb} decouple from all other equations of motion. We will use this system to illustrate our findings by a series of numerical simulations (Fig. \\ref{fig:sim1}-Fig. \\ref{fig:sim4}). In each figure, we show the time history of $z_1$ and $z_2$ with circles denoting liftoff and landing at one of the contact points. In some of the figures, the time history of the angles $\\beta$, $\\gamma_i$ is also shown. The model parameters and initial conditions used in these simulations are given in the appendix.   \n\n\\section{Generic transitions from slip}\nThe main goal of the paper is to identify and to give a qualitative description of generic transitions and singularities of a system, which is initially in slip mode. Clearly, such events occur when the actual contact mode of the system becomes either unfeasible or unstable. In what follows, we begin with the $n=1$ case, where existing results are reviewed and some new results are also presented. This is followed by detailed analysis for $n=2$ and some results for general $n$.\n\n\n\\subsection{Transitions of systems with a single point contact}\n\nAssume now that a system with a single point contact undergoes slip motion with nonzero velocity\n and this mode of motion is feasible and stable. As we know from Sec. \\ref{sec:1contact-instantaneous}, this means \n\\begin{align}\n\\dot x_1&>0\n\\label{eq:1pointslidingcondition1}\\\\\nb&<0<P\n\\label{eq:1pointslidingcondition2}\n\\end{align} \nSince $\\dot x_1,b$ are functions of all state variables ($q$ and $\\dot q$) and $P$ depends on $q$, all three variables vary over time.\n\nA sudden transition occurs, if one or several of the conditions \\eqref{eq:1pointslidingcondition1}-\\eqref{eq:1pointslidingcondition2} break down. There are two trivial transitions. First, reaching the border $\\dot x_1=0$ means transition to stick or slip reversal. Moreover,\\citet{nordmark2017dynamics} show using regularized contacts that this transition is in some cases immediately followed by exponentially growing bouncing motion of $z$, which is termed by those authors \\textit{reverse chatter}. \nThe second simple option is reaching the \\textit{liftoff manifold} $b=0$, where slip becomes infeasible and the contact lifts of. It is tempting to declare that crossing $P=0$ (where slip again becomes infeasible) is also a generic event. Nevertheless $P=0$, which we will refer to the  \\textit{\\Pain manifold}, corresponds to a singularity of the contact force according to \\eqref{eq:lambda-1contact}. What happens instead was uncovered first by \\citet{Genot1999}. They studied the problem of a sliding rod, and found that as $P$ approaches 0, the large contact force induces large acceleration of the rod. The value of $b$ is a function of the velocity $\\dot q$ whereas $P$ is independent of it. In the case of the rod, $b$ rapidly increases in response to large contact forces and thus it may approach $b=0$. Thereby the system may converge to the codimension-2 manifold given by $b=P=0$. We will refer to this as the \\textit{G\\'enot-Brogliato manifold}. Here, the contact force is left ill-defined by\\eqref{eq:lambda-1contact}. Local analysis of the system near the \\GB manifold reveals that it attracts an open set of initial conditions, and the contact force diverges to infinity, which inspired the name \\textit{dynamic jamming} of this phenomenon. What happens beyond that point is a subtle question, which was only partially uncovered by using asymptotic analysis and geometric singular perturbation techniques \\citep{nordmark2017dynamics,kristiansen2017canard}.\n\nThe local analysis of the rod was extended in unpublished work by Nordmark, Dankowicz, and Champneys (see also Fig. 13 in \\citet{champneys2016painleve}) to general systems with a single point contact. They found that the G\\'enot-Brogliato manifold is often attractive and the contact force may or may not diverge to infinity when approaching it. \n\nThese local results suggest that the $P=0$ boundary is never crossed by  systems with $n=1$ away from the \\GB manifold. Nevertheless a global extension of this result has never been published. Here we give a formal proof of this property:\n\n\\begin{theorem}\nAssume that a mechanical system  with a single point contact is initially in slip mode, in a state with $b<0<P$, furthermore $P$ is Lipschitz with respect to $q$ and $K$, $a$, $b$ are continuous functions of their arguments. Then slip motion of the system may not reach a state where $ P = 0 <  b$\n\\label{thm: no_painleve_1_contact}\n\\end{theorem}\n\nThe proof of this statement is based on the observation that crossing the boundary would induce a $1\/x$ type singularity in the contact force, which is non-integrable. This implies that the contact force would eliminate slip motion for any finite initial tangential velocity. The detailed proof is given below.\n\\begin{proof} [Proof by contradiction:] \n\nAssume that the system is launched at $t_{init}<0$ and a state with $P=0> b$ is reached at time $t=0$.\n \n\nThe velocity $\\dot q$ is a continuous function of time during impact-free motion, hence $|\\dot q|$ has a global maximum $\\dot q_{max}$ over the closed interval $t\\in[t_{init},0]$. This means that the following bound applies to $P$:\n\\begin{align}\n0<P(q(t))<\\mathcal{L}|q(t)-q_{0}| \\leq \\mathcal{L}\\dot q_{max}(-t)\n\\label{eq:Pbound}\n\\end{align}\nwhere $\\mathcal L$ is a Lipschitz constant of $P$. This relation shows that the contact force \\eqref{eq:lambda-1contact} undergoes an $f(x)=1\/x$ type singularity.\n\nNext we investigate how the tangential velocity of the contact point varies in response to the singular contact force. The tangential acceleration of the contact point can be expressed according to \\eqref{eq:xdyn}, and \\eqref{eq:lambda-1contact} as\n\\begin{align}\n\\ddot x&=a+K\\lambda\\\\\n&=a-KP^{-1}b\n\\end{align}\nThis can be rearranged as\n\\begin{align}\nP\\ddot x=Pa-Kb\n\\label{eq:tanaccformula}\n\\end{align}\nimplying\n\\begin{align}\n(P\\ddot x)|_{t=0}&=\n\\underbrace{P_0}_0a_0-K_{0}b_{0} <0\n\\end{align}\nwhere the index $_0$ refers to values at $t=0$. The last inequality holds because $K_{0}<0$ follows from $P_{0}=0$, since any external force must cause an acceleration, whose scalar product with the force vector is positive (unless the force is directly opposed by a hard constraint). \n\nThe right-hand side of \\eqref{eq:tanaccformula} is continuous, thus there exists a time\n$t_1$ sufficiently close to $0$ such that for all $t_1\\leq t\\leq 0$, $P\\ddot x$ is strictly negative with some finite bound $E$:\n\\begin{align}\nP\\ddot x<-E<0\n\\end{align}\n\nThen, we can combine this with \\eqref{eq:Pbound} to obtain \n\\begin{align}\n\\ddot x&<-P^{-1}E\\\\\n     & < -\\frac{E} {\\mathcal{L}\\dot q_{max}}(-t)^{-1}\n\\end{align}\nThis bound involves a non-integrable singularity, which means that the variation of $\\dot x$ in the time interval $[t_1,0]$ is unbounded. In other words, for any positive sliding velocity at $t_1$, $\\dot x = 0$ is reached for some $t<0$. Then a slip-stick transition occurs, contradicting the initial assumption that the system slips until $t=0$.\n\\end{proof}\n\nWe can summarize the results of this subsection as follows. There are four generic transitions from slip: transition to stick or slip reversal (possibly followed by inverse chattering); lift-off; and the dynamic jamming singularity. Crossing the \\Pain manifold $P=0$ is impossible, i.e. such systems never evolve into states suffering from Painlev\\'e paradox except for the codimension 2 \\GB manifold. Notice that the last result required the Lipschitz property of $P$. This is violated if the friction coefficient jumps abruptly, in which case a system may slip into a state subject to \\Pain paradox away from the \\GB manifold.\n\n\\subsection{Transitions of  systems with two point contacts}\nAs before, we assume that the system under consideration is in slip state initially with nonzero tangential velocity at both contact points, furthermore slip motion is feasible and stable. In the next subsection, we find all generic boundaries at which feasibility and stability can be lost. Then the remainder of the section focuses on the question: what happens after reaching each of these boundaries?\n\n\\subsubsection{Feasibility and stability boundaries}\nReaching zero tangential velocity is an obvious violation of feasibility. In addition to that, detailed analysis of \\citet{varkonyi2017dynamics} shows that we have feasibility and stability if the system parameters $\\beta$, $\\gamma_1$, $\\gamma_2$ are in  a certain range (Fig. \\ref{fig:SSrange}). This range is bounded by 5 smooth surfaces as follows:\n\\begin{enumerate}\n\\item two planar \\textit{liftoff boundaries} L1, L2 given by $\\beta=\\gamma_2$ (L1) and $\\beta=\\gamma_1$ (L2). One of the contact forces $\\lambda_1$ and $\\lambda_2$ becomes zero at each of these surfaces.\n\\item the \\textit{Painlev\\'e boundary} P given by $|\\gamma_1-\\gamma_2|=\\pi$ where matrix $P$ is singular, and the (degenerate) cone spanned by $p_1$ and $p_2$ is a full line. Here, the contact forces \\eqref{eq:lambda-1contact} become infinitely large.\n\\item the stability boundaries S1, S2, which are curved surfaces composed of straight lines parallel to the $\\beta$ axis. The coefficient matrix of \\eqref{eq:contactdynamics_generaln} has a pair of purely imaginary eigenvalues along these surfaces, and thus slip motion becomes unstable. The exact locations of these surfaces depend on parameters $|p_i|,k_i,\\nu_i$ but they always lie within the boundaries shown in Fig. \\ref{fig:L1parts}. \n\\end{enumerate}\nThere are three additional curves along the boundary, which deserve special attention as they are associated with singular behaviour:\n\\begin{enumerate}\n\\item the line sections GB1,GB2 are at the intersections of L1 and L2 with P. As we will see, these are analogous to the G\\'enot-Brogliato manifolds of the single-contact case.\n\\item the line L12 is at the intersection of the liftoff manifolds L1 with L2 and it is given by $\\beta=\\gamma_1=\\gamma_2$. Along L12, $P$ is singular and the cone spanned by $p_1$ and $p_2$ is a half-line. \n\\end{enumerate}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=8cm] {SSregion.pdf}\n\\caption{The range of angles $\\beta$,$\\gamma_1$, $\\gamma_2$ where the SS contact mode is feasible and stable. Note that the intersection of the coordinate axes is shifted from 0 for better visibility. The exact location of the curved surfaces S1, S2 depends on $|p_1|, |p_2|$, $k_1$, $k_2$, $\\nu_1$, and $\\nu_2$ The figure shows the case $|p_1|=|p_2|=k_1=1$, $k_2=10$, $\\nu_1=\\nu_2=0$. For more information about possible variations of the stability boundaries, see Fig. \\ref{fig:L1parts}.}\\label{fig:SSrange}\n\\end{center}\n\\end{figure}\n\nSlip motion persists until one of the feasibilility or stability conditions is violated. In what follows, we analyze the consequences of crossing each of the boundaries listed above.\n\n\n\\subsubsection{Reaching zero tangential velocity}\nThe system may undergo slip-stick transition or slip reversal, if the tangential velocity at one or both contact points drops to zero. Note that simultaneous zero crossing is generic in the presence of certain kinematic constraints, as for example in the case of a single rigid object with two point contacts. Similarly to the case of a single point contact \\citep{paper2}, it is likely that the regularized contact model sometimes predicts self-excited bouncing motion resembling inverse chattering in this situation, however we do not examine this possibility in the present paper.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=9cm] {L1parts.pdf}\n\\caption{Projection of Fig. \\ref{fig:SSrange} to the $\\gamma_1-\\gamma_2$ plane (dark grey area), possible locations of the projection if parameters $|p_i|,k_i,\\nu_i$ are varied (light grey area), and subdivision of the L1 surface (solid lines and two-digit numbers).}\\label{fig:L1parts}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Crossing L1} \nAs we have seen, the contact force $\\lambda_1$ becomes zero at L1, i.e. contact point 1 lifts off while point 2 continues to slip. We can now test the feasibility and stability of the FS mode. The result depends on where the L1 surface was crossed. Fig. \\ref{fig:L1parts} shows a subdivision of L1 into regions marked by 2-digit numbers. The numbers refer to a classification scheme introduced by \\citet{varkonyi2017dynamics}. The meaning of the labels is not explained here. Within ranges 23,12,13,41,42,31, the feasibility conditions \\eqref{eq:FS1}-\\eqref{eq:FS2} are satisfied, and we also have $p_{22}>0$, i.e. slip motion on contact 2 is feasible and stable (see Sec. \\ref{sec:1contact-instantaneous}). Thus, we conclude that the system undergoes an SS$\\rightarrow$FS contact mode transition. This scenario is illustrated by numerical simulation in Fig. \\ref{fig:sim1}(a).\n\nIn contrast, if L1 is crossed in region 14 or 46, then we have $p_{22}<0<b_{22}$, implying that slip at contact point 2 is feasible but unstable. According to Table \\ref{table:1contact}, the fast dynamics of point 2 may evolve either towards liftoff or towards an IWC (as long as contact 1 is separated). The unstable slip mode of point 2 corresponds to a saddle point of the fast contact dynamics, and its stable manifold separates the basins of attractions of liftoff and IWC. In order to determine, what happens to our system at this point, one should find the dynamic state of point 2 when point 1 lifts off to see within which basin of attraction it lies. Unfortunately, this simple approach is inconclusive here since it is straightforward to show that contact point 2 is exactly at the saddle point (which is part of the separatrix between liftoff and IWC). Hence, the analysis of the fast dynamics itself does not enable us to decide, in which direction the system evolves from here. The indeterminacy can perhaps be removed by considering the full slow-fast system. The full system evolves initially along the critical manifold corresponding to motion in SS mode. This manifold is probably off the separatrix between liftoff and IWC in the full slow-fast system. Nevertheless such a detailed analysis is beyond the scope of the present paper. Instead, we perform  numerical simulations using the system introduced in Sec. \\ref{sec:numericexample}. Our results suggest evolution towards liftoff whenever the L1 surface is crossed in region 14, and towards IWC in region 46 (Fig. \\ref{fig:sim1}(b)-(c)). Proving or refuting that the opposite scenarios (liftoff in region 46 and IWC in 14) are impossible remains an open question.\n\n In those two cases, which did occur in the simulations, the fate of the system can be deduced analytically:\n\\begin{itemize}\n\\item If point 2 evolves towards liftoff and we are in region 14, then the gradually decreasing contact force $\\lambda_2$ and the negative signs of $p_{21},p_{22}$ imply that both contact points have increasing normal accelerations and eventually contact point 2 also lifts off. Then,  $b_{1},b_2>0$ imply that the FF mode persists, i.e. the system undergoes SS$\\rightarrow$FF transition.\n\\item If point 2 evolves towards IWC, and we are in region 46, then $p_{21}>0$ implies that the large contact force  lifts up contact 1, i.e. it remains in separation. In this case an IWC develops at point 2, while point 1 is in F mode. IWC means that slip motion at point 2 stops rapidly, after which contact 2 may jump into separation with nonzero velocity\n\\end{itemize}  .\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {caseL1egyeb.pdf}}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {case14.pdf}}\n\\subfigure[]{\\includegraphics [width=0.32\\textwidth] {case46.pdf}}\n\\caption{Crossing the L1 surface in numerical simulation of the model system introduced in Sec. \\ref{sec:numericexample}.  L1 is crossed at $t=0$ in region 13 (a), 14 (b), or 46 (c). The diagrams show $z_1$(solid curve) and $z_2$ (dashed curve) versus time $t$. Circles mark the liftoff ($\\lambda=0$) of the contact points. The corresponding model parameters and initial conditions are given in the appendix.\n}\n\\label{fig:sim1}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nThe results related to region 46 are particularly interesting, because Theorem \\ref{thm: no_painleve_1_contact} and other results from \\citet{champneys2016painleve} imply that a system with a single point contact never exhibits an IWC inside the \\Pain regime ($P<0$). Nevertheless we see here that the analogous statement does not hold for systems with 2 point contacts. That is, we uncovered a generic mechanism, which does not occur in previously studied simple model systems.  \n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_01.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_01_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_02_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_03_detail.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_04.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{caseS1_04_detail.pdf}}\n\\caption{Crossing the S1 surface in numerical simulation of the model system.  Panel (a) shows inverse chattering  and (b) is a magnified detail of the same diagram. (c) is an example of simulatneous liftoff at both contact points, whereas (d) shows liftoff at point 1 accompanied by the onset of an IWC at point 2. (e) shows converges to a microscopic limit cycle of the fast subsystem and (f) is a magnified detail of the same diagram. \n}\n\\label{fig:sim2}\n\\end{figure}\n\n\n\\subsubsection{Crossing S1}\nWe have seen that the system matrix of \\eqref{eq:contactdynamics_generaln} has a pair of purely imaginary eigenvalues at S1. As the system crosses the stability boundary, the invariant point of the regularized fast dynamics corresponding to the SS mode becomes unstable and $z_1$ and $z_2$ exhibit gradually growing harmonic oscillations. Soon, one of the two points will lift off. Beyond liftoff, the regularized contact law switches between the two cases of \\eqref{eq:contactlaw} and it is not immediately clear what is going to happen. It can be shown that the FS and SF contact modes are either inconsistent or unstable, and FF is inconsistent except for a relatively small region within the S1 surface where $0<\\beta<\\pi\/2$. During systematic numerical simulations, we found 4 different behaviors (Figure \\ref{fig:sim2}): \n\\begin{enumerate}\n\\item rapidly growing oscillations with repeated impacts and lift-off at both contact points ((Figure \\ref{fig:sim2}(a-b))). The figure shows a regular oscillating pattern where the amplitude of the oscillation grows roughly by a factor of 10 in each cycle of the oscillation and thus only the last two cycles are visible. \n\\item transition to FF mode ((Figure \\ref{fig:sim2}(c))). The figure shows that $z_1$ and $z_2$ become positive and continue to grow rapidly.\n\\item IWC at point 1, while point 2 lifts off ((Figure \\ref{fig:sim2}(d))) The figure shows that $z_1$ becomes positive and continues to grow rapidly, while $z_2$ simultaneously diverges towards minus infinity.\n\\item limit cycle of the fast dynamics involving repeated liftoff and reestablishment of the contacts. This behavior  tends to occur when $0<\\gamma_2<\\pi\/2$ and $\\beta+\\pi$ is close to $\\gamma_2$. The results depicted in Figure \\ref{fig:sim2}(e-f) show oscillations of slowly growing amplitude (rather than exact limit cycles), since a finite value of $\\epsilon$ was used in the simulation. \n\\end{enumerate}\nIt is possible that other qualitatively different phenomena may also occur.\nThe first scenario listed above  appears to an external macroscopic observer as self-excited bouncing motion of increasing amplitude, i.e. inverse chattering \\citep{paper2}. The last one appears as sliding combined with sustained microscopic, high-frequency vibration. Similar phenomena are known to lie behind brake squeal \\citep{kinkaid2003automotive}.\n\n\n\n\n\\subsubsection{Crossing L2 or S2}\nThese scenarios are equivalent of the cases of crossing L1 and S1 with the only difference being that the roles of contact point 1 and 2 are reversed, furthermore the roles of regions 14 and 31, of 13 and 41, and of 23 and 46 are also reversed.\n\n\n\\subsubsection{Crossing P}\nMatrix $P$ is singular along the surface P, hence \\eqref{eq:lambda-2contact} shows that the contact forces diverge to infinity unless vector $b$ is a linear combination of the column vectors $p_i$ (which is satisfied at the GB1,GB2 lines). This property of surface P, and lines GB1, GB2 makes them similar to the \\Pain manifold and the \\GB manifold in the case of 1 point contact. This similarity suggests that a generalization of Theorem \\ref{thm: no_painleve_1_contact} is true for 2 contacts as well. Indeed we will present a general theorem for system with arbitrary $n$ in Sec. \\ref{sec:general n}, which implies that P is never reached away from the GB1, GB2 curves. The codimension 2 curves are investigated below.\n\n\\subsubsection{Attractive codimension 2 manifolds}\n\nWe have seen that the contact force has singularities at the codimension 2 manifolds L12, GB1, and GB2. In other words, the right-hand sides of the governing equations are discontinuous at these points. In such situations, the solution may be non-unique forward and\/or backward in time. As a consequence, it is theoretically possible that any of these co-dimension 2 manifolds can be reached in finite time from an open set of initial conditions. In what follows, we construct conceptual examples to demonstrate that all of these manifolds may be reached from generic initial conditions, and we also investigate the consequences of such a transition.\n\n\\textbf{Crossing L12 (double liftoff singularity):} L12 is at the intersection of 2 liftoff boundaries, which inspires the name proposed for this phenomenon. We consider the system \\eqref{eq:dotP}-\\eqref{eq:dotb} with ideally rigid contacts. Then, by \\eqref{eq:lambda-2contact} the system becomes\n\\begin{align}\n\\dot P = A_1\\label{eq:dotP2}\\\\\n\\dot b =  \\alpha_2- A_3P^{-1}b \\label{eq:dotb2}\n\\end{align}\nThe dynamics of $P$ is determined uniquely by the choice of\n\\begin{align}\nA_1=\\left[\n\\begin{matrix}\n-1&1\\\\\n0&0\n\\end{matrix}\n\\right]\n,P_0=\\left[\n\\begin{matrix}\n0&0\\\\\n1&1\n\\end{matrix}\n\\right]\n\\label{eq:alpha1P0}\n\\end{align}\nThis choice implies that the angle $\\gamma_1$ grows while $\\gamma_2$ decreases and both angles become equal to $\\pi\/2$ at $t=0$. \nIn addition, we choose the values\n\\begin{align}\n\\alpha_2=\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n,A_3=\\left[\n\\begin{matrix}\n2&-2\\\\\n0&0\n\\end{matrix}\n\\right]\n\\label{eq:alpha23}\n\\end{align}\n\nAssume now that the system has an initial state at some $t_{init}<0$ where  the system is initially in SS mode and the system parameters are within the region of feasibility and stability of SS, i.e.:\n\\begin{align}\n\\gamma_1<\\beta+\\pi<\\gamma_2\n\\label{eq:SSfeasible-stable}\n\\end{align}\nwhere $\\beta$ and $\\gamma_i$ are the angles illustrated by Fig. \\ref{fig:gammabeta}. As we approach $t=0$, the gap between the $\\gamma_1$ and $\\gamma_2$ angles shrinks to 0. Nevertheless,\n\\begin{lemma}\n For any initial condition satisfying \\eqref{eq:SSfeasible-stable} at $t_{init}<0$, the same condition is not violated as long as $t<0$ and thus the system remains in SS mode until reaching the L12 manifold at $t=0$. \n \\label{lem:L12}\n\\end{lemma}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=10cm] {lemma.pdf}\n\\caption{Illustrations of the statements of Lemma \\ref{lem:L12} (a) and Lemma \\ref{lem:GB1} (b). Solid curves indicate the evolution of the angles $\\gamma_i$. The dashed curves show the evolution of the angle $\\beta+\\pi$ for various initial conditions. The SS mode is feasible and stable in the grey areas. Filled circles mark the attractive singular points.}\n\\label{fig:lemmas}\n\\end{center}\n\\end{figure}\nIn the proof we show that whenever $\\beta=\\gamma_1$, the resulting contact force causes $\\beta$ to increase faster then $\\gamma_1$, whereas in the case of $\\beta=\\gamma_2$, the resulting contact force makes $\\beta$ decrease faster than $\\gamma_2$. This mechanism keeps $\\beta$ in the shrinking interval $(\\gamma_1,\\gamma_2)$ until the L12 manifold is reached at $t=0$ (Fig. \\ref{fig:lemmas}(a)). The detailed proof, given in the appendix, borrows ideas from  \\citet{Genot1999}. We apply a singular rescaling of time and show that the L12 manifold turns into an attractive invariant point of the rescaled dynamics. We note that the proof of Lemma \\ref{lem:L12} does not rely on the specific values of $A_1,A_3,\\alpha_2$ chosen above, hence the mechanism illustrated by this example may occur in many systems.\n\nAt the L12 manifold, the contact force $\\lambda$ becomes undefined due to the singularity of $P$, and the dynamics of $b$ given by \\eqref{eq:dotb2} becomes ill-defined. The indeterminacy may be resolvable by contact regularization, nevertheless such an analysis is beyond the scope of the present work. For the sake of illustration, we show examples of numerical simulation using regularized contact in Fig. \\ref{fig:sim3}. The results confirm that angle $\\beta$ remains in the shrinking interval $(\\gamma_1,\\gamma_2)$ until the singularity is reached. After crossing the singularity contact 1 lifts off while 2 remains in slip state for some initial conditions. Shortly thereafter, contact 1 becomes active again, and it exhibits rapidly increasing contact force (indicating the onset of an IWC), while contact 2 lifts off. For other initial conditions, contact 2 lifts off and contact one enters an IWC directly. This behavior appears to be qualitatively similar to one of the possible scenarios during the dynamic jamming singularity of the slipping rod reported by \\citet{nordmark2017dynamics}.\n\n\n\\textbf{Crossing GB1 (dynamic jam):} we again consider the system introduced in Sec. \\ref{sec:numericexample} with rigid contacts. This time, we choose \n\\begin{align}\nA_1=\\left[\n\\begin{matrix}\n-1&-1\\\\\n0&0\n\\end{matrix}\n\\right]\n,P_0=\\left[\n\\begin{matrix}\n0&0\\\\\n-1&1\n\\end{matrix}\n\\right]\n,\n\\alpha_2=\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n,\nA_3=\\left[\n\\begin{matrix}\n0.5&0.5\\\\\n0&0\n\\end{matrix}\n\\right]\n\\label{eq:alpha1P0masodik}\n\\end{align}\nThen, the angle $\\gamma_1$ decreases while $\\gamma_2$ increases, and we have $\\gamma_1=\\gamma_2+\\pi=3\\pi\/2$ at $t=0$. Furthermore,\n\\begin{lemma}\n For any initial condition such that the system is in SS mode at time $t_{init}<0$ and $0<\\beta+\\pi<\\gamma_2$, the SS mode persists until $t=0$, where the angle $\\beta$ converges to $3\\pi\/2$, i.e. the system reaches the GB1 manifold.\n \\label{lem:GB1}\n \\end{lemma}\nThe proof is based on the observation that large $\\lambda$ causes $b_1$ to converge towards 0 rapidly (Fig. \\ref{fig:lemmas}(b)). Technically, the proof is very similar to that of Lemma \\ref{lem:L12} and it is again given in the appendix.\n\nUpon reaching the GB1 or GB2 manifold, the contact force again becomes ill-defined due to the singularity of $P$. Systematic investigations of what happens after this point is beyond the scope of the paper. Based on lessons learned from the single-contact case \\citep{nordmark2017dynamics,kristiansen2017canard}, we expect that the regularized system exhibits liftoff, or impulsive contact forces. Numerical simulations (Fig. \\ref{fig:sim4}) are consistent with our findings and expectations. Depending on the initial conditions, solution trajectories converge to $\\beta=\\pi\/2$ or $3\\pi\/2$ at $t=0$, i.e. the system is attracted by the GB1 or by the GB2 manifold.In the first case, contact 1 lifts off, whereas both ones lift off in the second. Further simulations (not shown) indicated that impulsive contact forces are possible, furthermore the initial liftoff is often followed by additional mode transitions, similarly to the results of Fig. \\ref{fig:sim3} and to numerical results of \\citet{nordmark2017dynamics} in the case of $n=1$.  \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6.5cm] {L12szog.pdf}\n\\includegraphics [width=6.5cm] {L12y.pdf}\n\\caption{Numerical simulation of the double liftoff singularity. Left: $\\beta+\\pi$ (dashed curves) and $\\gamma_i$ (solid curves) versus time for four different initial conditions. Right: $z_1$ (solid curve) and $z_2$ (dashed curves) in the same simulations. \t\n}\n\\label{fig:sim3}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics [width=6.5cm] {GB1szog.pdf}\n\\includegraphics [width=6.5cm] {GB1y.pdf}\n\\caption{Numerical simulation of dynamic jamming. Left: $\\beta+\\pi$ (dashed curves) and $\\gamma_i$ (solid curves) versus time for 8 different initial conditions. Right: $z_1$ (solid curves) and $z_2$ (dashed curves) in the same simulations.\n}\n\\label{fig:sim4}\n\\end{center}\n\\end{figure}\n\n\\subsection{Dynamic jamming of general systems} \\label{sec:general n}\n\nWe have seen previously that the boundary of the region of feasibility and stability of slip motion includes the codimension-1  \\Pain surfaces where $P=0$ ($n=1$) or $detP=0$ ($n=2$). Nevertheless general points of these surfaces are inpenetrable to the systems. In addition, we have also seen that some codimension-2 surfaces with $detP=0$ may become attractive in turn. In what follows, we generalize Theorem \\ref{thm: no_painleve_1_contact} to systems with arbitrary number of point contacts in two dimensions and thus show that the inpenetrability of the \\Pain surfaces is a general property in contact mechanics of planar systems. \n\nAssume that a mechanical system  with $n$  point contacts initially undergoes slip motion at all contact points with $\\dot x_i>0$. Similarly to the case $n=2$, slip motion must be feasible, which means that $-b$ must be in the cone spanned by the column vectors of $P$.\n\nFirst we define the \\Pain manifold as follows:\n\\begin{definition}\n The \\Pain manifold in the state space consists of those points where $P$ is singular and the cone spanned by column vectors of $P$ is a full $(n-1)$-dimensional space.\n\\end{definition}\n\nIf we approach a generic point of the \\Pain manifold in state space transversally, the cone spanned by the column vectors of $P$ gradually grows and converges to an $n$-dimensional half-space. When crossing the manifold, the cone collapses to an $n-1$ dimensional space and then it turns inside out, i.e. it becomes a half-space again. Because of this property, slip motion is feasible on exactly one side of the \\Pain manifold, hence the \\Pain manifold may form a boundary of the range of feasible and stable slip motion in state space.\n\nNext, we define the \\GB manifold as follows:\n\\begin{definition}\nThe \\GB manifold is a submanifold of the Painlev\\'e manifold including those points where $b$ is contained in the $n-1$ dimensional space spanned by the column vectors of $P$.\n \\end{definition}\nIt is easy to show that by approaching the \\GB manifold in a direction transversal to the \\Pain manifold, one  of the contact forces in slip mode dictated by \\eqref{eq:lambda-1contact}  converges to 0. Hence, the \\GB manifold contains those points within the \\Pain manifold, which are at the verge of liftoff (in full analogy with the $n=1$ case).\n\nNow we can state the following theorem: \n\\begin{theorem}\nIf $P$ is Lipschitz with respect to $q$ and $a,b,K$ are continuous function of their arguments, then the system never reaches the Painlev\\'e manifold except for points of the \\GB manifold. \n\\label{thm: no_painleve_n_contact}\n\\end{theorem}\n\nThe proof is highly similar to that of Theorem \\ref{thm: no_painleve_1_contact} but involves some additional technical steps. We present the proof in the Appendix.\n\n\n\n\\section{Conclusions}\n\nIn this paper, we investigated singularities and transitions of mechanical systems with frictional point contacts during slip motion. We found that the presence of 2 (or more) point contacts induces several dynamic phenomena, which are not possible in previously studied single-contact systems. Slip motion may terminate due to destabilization or loss of feasibility. In both cases, various transitions may occur, including lift-off at one or both contact points, self-excited microscopic limit cycle oscillations or exponentially growing macroscopic oscillations, as well as impact without collisions. The last three transitions are not possible in the single-contact case. The list of phenomena is not comprehensive, as other types of behavior may be possible. On the other hand, the author believes that the patterns found by this study occur frequently in systems with any number of contacts and thus they are of practical interest.\n\nWe also demonstrated that two different types of singularity are generic in the case of $n=2$. The first one occurs when a system converges to a codimension-2 manifold, where the associated matrix $P$ is singular and simultaneously the system is at the boundary of lift-off at one contact point.  This singularity is essentially identical to the previously uncovered \\emph{dynamic jamming} singularity of single-contact systems first described by \\citet{Genot1999}. Based on earlier results, we expect that the contact force may or may not diverge to infinity, and passing the singularity may be followed by impulsive contact forces or lift-off at one point. A second type of singularity occurs when the system converges to another codimension-2 manifold, which is at the boundary of liftoff at both contact points. The double liftoff singularity is a novel phenomenon. Here, the contact force does not diverge to infinity, nevertheless the system shows indeterminacy at the point of passing the singularity. Either one of the contact points may lift off, while the other contact may continue to slip or exhibit impulsive contact forces. The underlying mechanism and the induced indeterminacy are both similar to the well-known two-fold singularity of Filippov system \\citep{Springerbook}. \n\n\nThe analysis was based on the assumption of a picewise linear, regularized contact model with two parameters ($k_i,\\nu_i$). The phenomena uncovered in the paper did not rely on choosing special values of the parameters $k_i,\\nu_i$ of the model, hence they appear to be generic. The assumption of linearity was motivated by its simplicity, but it is not crucial either. In the case of a nonlinear contact model (such as Hertz law), the stability of contact modes can be investigated after linearization, and the result of the analysis is often independent of the presence or the exact form of nonlinearity \\citep{champneys2016painleve}. Hence we believe that the uncovered phenomena also occur in systems with nonlinear contacts.  \n\nThis paper leaves many questions open for future work. Our aim was to provide a general overview of generic transitions, and we skipped the detailed analysis of these transitions. Open questions include\n\\begin{itemize}\n\\item detailed description of the dynamics after crossing the liftoff manifold within region 14 and 46\n\\item comprehensive list of transitions for $n=2$ (or more) contacts after crossing the stability boundaries and conditions under which they occur\n\\item general characterization of the double lift-off and dynamic jamming singularities for nonlinear systems with $n=2$ (or more) contacts\n\\end{itemize}\nAlso, we did not present real examples of the most interesting transitions and singularities. It was demonstrated that these singularities occur in a conceptual model, nevertheless finding real-world examples will be subject of future work.\n\n\\begin{acknowledgement}\nThe author thanks two anonymous reviewers for their useful suggestions. This work was supported by Grant 124002 of the National Research, Development, and Innovation Office of Hungary.\n\\end{acknowledgement}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{s1}\n\n\n\nAn important open problem in Loop Quantum Gravity (LQG)~\\cite{alrev,crbook,crlrr,ttbook} is to obtain a well \ndefined method of perturbatively computing its dynamics. The covariant approach given by Spin Foam Models\n(SFM)~\\cite{perezrev,jb-BF,crbook} provides an avenue to obtain such a method, however there are still several open issues regarding its precise\n relation to the Hamiltonian theory.  \n\nIf SFM and canonical LQG  are to be the covariant and canonical descriptions of a single quantum theory of gravity, one\nshould be able to derive one from the other. We are still far from rigorously establishing such connection,\nhowever important progress has arisen in recent years. These include, in the {\\small covariant $\\to$ canonical}  direction,\nthe derivation of the LQG Hilbert space as well as the spectra of geometrical\noperators~\\cite{newlook,engleper,dingrov}, and, in the {\\small canonical $\\to$ covariant}  direction, the extension of the EPRL\namplitude to arbitrary valent spin foams vertices thereby allowing general histories of graphs~\\cite{kkl}. \nThe latter direction leads also to the picture of regarding spin foams as spin networks histories~\\cite{rr}. \nThis interpretation however, has not gone beyond the heuristic level (as far the full four dimensional theory \nis concerned; in the three dimensional case the connection between the canonical and covariant descriptions is well established~\\cite{np}). \n\nRecently, the {\\small canonical $\\to$ covariant}  direction was analyzed at the symmetry reduced level of homogeneous and isotropic\ncosmology~\\cite{ach1,ach2}. Here we extend that analysis and consider a non-isotropic cosmological model. The\nadditional degrees of freedom allow for a richer discussion than in the isotropic case. In particular, besides\nthe `vertex expansion' present in the Friedman-Robertson-Walker (FRW) case, there are additional sums over the\nextra parameters, which are interpreted as giving a `colouring' of the graph, thus strengthening the analogy with spin foams.\n\nThe aim here is to obtain a `sum over histories' description within \nLoop Quantum Cosmology (LQC)~\\cite{mblrr}. Our construction is then different from that of `spinfoam cosmology' \\cite{rv1,brv},\n where cosmological amplitudes are obtained using SFM as the starting point. An interesting question which we do not address here\n is whether there is any precise relation among the two constructions. Let us also mention that the idea of looking for a \\mbox{{\\small  canonical}-{\\small  covariant} }\n connection at the homogeneous level has appeared before in the context of Plebanksi theory~\\cite{npv}; such approach could shed light\n into the previous question of relating  `cosmological spinfoams' with `spinfoam cosmologies'.\n\nThe paper is organized as follows. In Section~\\ref{s2} we introduce the model we will work with, namely the quantum Bianchi~I\ncosmology with a massless scalar field, as obtained by Ashtekar and Wilson-Ewing in~\\cite{awe}. In Section~\\ref{s3} we construct\nthe sum over histories description of the model. We outline  how individual amplitudes are to be calculated and illustrate the\nprocedure for a simple history. In Section \\ref{s4} the vertex expansion of the FRW~\\cite{ach1} model is recovered by integrating out\n the anisotropies of our vertex expansion. We finish the paper with a discussion in Section~\\ref{s5}.\n\n\\section{Loop Quantum Cosmology of the Bianchi~I model} \\label{s2}\n\nWe are interested in the Bianchi~I cosmological model, which is the \nsimplest non-isotropic homogeneous cosmology, coupled with a massless scalar field $\\phi$.\nAs in the isotropic case, one can fix a fiducial 3-metric $ds_0^2$ and choose Cartesian coordinates on the spacial slice\n such that ${\\rm d}s_0^2= {\\rm d}z_1^2 + \n{\\rm d}z_2^2 + {\\rm d}z_3^2$. The physical 3-metric is then determined by three  directional scales factors, $a_1,a_2$ and $a_3$,\n\\begin{equation}\n{\\rm d}s^2 =  a_1^2 ~ {\\rm d}z_1^2 + a_2^2 ~ {\\rm d}z_2^2 + a_3^2 ~ {\\rm d}z_3^2~.\n\\end{equation}\nThe Hamiltonian analysis requires one to choose a fiducial cell $\\mathcal{V}$, for which we take the rectangular prism  $0 \\leq z_i\n\\leq L_i$, for each direction $i=1,2,3$. The physical volume of the cell is then given by $V= |a_1 a_2 a_3| L_1 L_2 L_3$.\nNote that the choice of the fiducial cell $\\mathcal{V}$ (i.e.\\ the choice of $L_1$, $L_2$ and $L_3$)\n is arbitrary, and one has to ensure that physical results are insensitive to that choice (see \\cite{awe} for further discussion).\n\nWhen one goes to the quantum theory \\cite{awe}, it is convenient to work with a new set of variables, $(\\lambda_1,\\lambda_2,v)$,\ndefined by \n\\begin{eqnarray} \\label{lambda}\n\\lambda_1 & := & \\frac{\\sgn(a_1) \\sqrt{|a_2 a_3| L_2 L_3}}{(4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o )^{1\/3}}~, \\\\\n\\lambda_2 & := & \\frac{\\sgn(a_2) \\sqrt{|a_3 a_1| L_3 L_1}}{(4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o )^{1\/3}}~, \\\\\nv & := & \\frac{\\sgn(a_1 a_2 a_3) |a_1 a_2 a_3| L_1 L_2 L_3}{2 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o } = 2 \\lambda_1 \\lambda_2 \\lambda_3.\n\\end{eqnarray}\nHere $\\ell_o$ is the square root of the `area gap' $\\Delta =4\\sqrt{3}\\pi\n\\gamma\\, {\\ell}_{\\rm Pl}^2$, $\\gamma$ is the Barbero-Immirizi parameter, and $\\lambda_3$ is defined in a similar way as $\\lambda_1$ and $\\lambda_2$. In\nthis representation, the gravitational Hilbert space $\\H_{\\rm kin}^{\\rm grav}$ consists of functions\n$\\Psi(\\lambda_1,\\lambda_2,v)$ with support on a\ncountable number of points and with finite norm $||\\Psi||^2 :=\n\\sum_{\\lambda_1,\\lambda_2,v}\\,|\\Psi(\\lambda_1,\\lambda_2,v)|^2 <\\infty$. The matter Hilbert space is the standard\none: $\\H_{\\rm kin}^{\\rm matt} = L^2(\\mathbb{R}, \\textrm{d}\\phi)$. The total kinematical Hilbert space is thus a tensor product $\\H_{\\rm kin}=\\H_{\\rm kin}^{\\rm grav} \\otimes\n\\H_{\\rm kin}^{\\rm matt}$ and, as usual in LQC, the dynamics of the system are encoded in the constraint equation\n\\begin{equation}  -\\,C \\Psi \\equiv \\partial^2_\\phi \\Psi + \\Theta \\Psi = 0~,\n\\end{equation}\n\nwhere $\\Theta$ is a symmetric operator that acts on the gravitational part. As in \\cite{awe}, we restrict attention to the\n`positive octant' ($v,\\lambda_1,\\lambda_2 \\geq 0$). The action of $\\Theta$ takes the form,  \n\\begin{align} \\label{qham6} \\left(\\Theta \\Psi \\right)(\\lambda_1,\\lambda_2,v) =& \\frac{\\pi G}\n{4}\\sqrt{v}\\Big[(v+2)\\sqrt{v+4}\\,\\Psi^+_4(\\lambda_1,\\lambda_2,v) - (v+2)\\sqrt\nv\\, \\Psi^+_0( \\lambda_1,\\lambda_2,v)\\nonumber \\\\& -(v-2)\\sqrt v\\,\n\\Psi^-_0(\\lambda_1,\\lambda_2,v) + (v-2)\n\\sqrt{|v-4|}\\,\\Psi^-_4(\\lambda_1,\\lambda_2,v)\\Big], \\end{align}\nwhere $\\Psi^\\pm_{0,4}$ are defined as:\n\\begin{align} \\label{qham7}\\Psi^\\pm_4(\\lambda_1,\\lambda_2,v)=& \\:\\Psi\n\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,\nv\\pm4\\right)+\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_1,\\lambda_2,\nv\\pm4\\right)\\nonumber\\\\& +\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1,\n\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,v\\pm4\\right)+\\Psi\n\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1, \\lambda_2,v\\pm4\\right)\\nonumber\n\\\\&+\\Psi\\left(\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,\nv\\pm4\\right)+\\Psi\\left(\\lambda_1,\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,v\\pm4\\right),\n\\end{align}\nand\n\\begin{align} \\label{qham8} \\Psi^\\pm_0(\\lambda_1,\\lambda_2,v)=\n& \\:\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1, \\frac{v}{v\\pm2}\\cdot\\lambda_2,v\\right)\n+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_1,\\lambda_2,v \\right)\\nonumber \\\\&\n+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_1,\\frac{v\\pm2}{v}\\cdot\\lambda_2,v\\right)+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_1,\\lambda_2,v\\right)\\nonumber\n\\\\& +\n\\Psi\\left(\\lambda_1,\\frac{v}{v\\pm2}\\cdot\\lambda_2,v\\right)+\\Psi\\left(\\lambda_1,\\frac{v\\pm2}{v}\n\\cdot\\lambda_2,v\\right)\\, .\\end{align}\nAs noted in \\cite{awe}, the operator preserves the subspaces ${\\cal H}_\\epsilon \\subset \\H_{\\rm kin}^{\\rm grav}$ of states whose support lies on the lattice\n\\begin{equation}\nv = \\epsilon + 4 \\mathbb{Z}~.\n\\end{equation}\nThese superselection sectors have the same form as in the isotropic case \\cite{acs}, and, as done there, we will restrict\nto the physically interesting $\\epsilon = 0$ sector (the one that contains the classically singular value  $v=0$).  The space we\nfinally work with is  the space of vectors with support on the positive octant and the $\\epsilon=0$ lattice, which we denote\nby ${\\cal H}^{+++}_{0}$.\n\nWe now introduce a different representation of the space ${\\cal H}^{+++}_{0} \\subset \\H_{\\rm kin}^{\\rm grav}$, by changing coordinates\n$(\\lambda_1,\\lambda_2,v) \\to (n,{\\bf x})$ where\n\\begin{eqnarray} \\label{eq:intertwiner}\nn  & := & \\frac{1}{4} v \\in \\mathbb{N}~, \\\\\n{\\bf x}   =  (x_1,x_2) & := & (\\log \\lambda_1,\\log \\lambda_2) \\in \\mathbb{R}^2~.\n\\end{eqnarray}\nIn this representation, states are described by functions $\\Psi(n,{\\bf x})$, which again have support on a countable\nnumber of points and have a finite norm $||\\Psi||^2 = \\sum_{{\\bf x},v}\\,|\\Psi({\\bf x},v)|^2<\\infty$.\nThey represent the components of the state in a\n$\\{ |n,{\\bf x} \\rangle \\}$ basis, which is characterized by eigenvalues of  $\\widehat{\\lambda}_1,\\widehat{\\lambda}_2$ and\n$\\widehat{V}=4 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o \\widehat{\\lambda}_1\\widehat{\\lambda}_2\\widehat{\\lambda}_3$ as follows \n\\begin{eqnarray}\n\\widehat{V} |n,{\\bf x} \\rangle & = & 8 \\pi \\gamma {\\ell}_{\\rm Pl}^2 \\ell_o n |n,{\\bf x} \\rangle~, \\\\\n\\widehat{\\lambda}_i |n, {\\bf x} \\rangle & = & \\log x_i |n,{\\bf x} \\rangle , \\quad i=1,2~,\n\\end{eqnarray}\nwith their normalization given by the product of Kronecker deltas:\n\\begin{equation}\n\\langle n', {\\bf x}'| n,  {\\bf x} \\rangle = \\delta_{n,n'} \\delta_{{\\bf x},{\\bf x}'}.\n\\end{equation}\n\nIn the subsequent sections, we will regard ${\\cal H}^{+++}_{0}$ as a tensor product of the `volume' factor and the `anisotropy' factor:\n\\begin{eqnarray}\n{\\cal H}^{+++}_{0} & = &{\\cal H}_V \\otimes {\\cal H}_\\lambda~, \\\\\n|n, {\\bf x} \\rangle & = & |n \\rangle \\otimes | {\\bf x} \\rangle~.\n\\end{eqnarray}\nWithin this splitting, $\\Theta$ is expressed as a sum of the tensor product of operators acting on ${\\cal H}_V$ and ${\\cal H}_\\lambda$,\n\\begin{equation} \\label{theta}\n\\Theta = \\sum_n | n+1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n+1\\right) n} + | n \\rangle \\langle n| \\otimes \\Theta_{nn} +\n | n-1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n-1\\right) n}~.\n\\end{equation}\nThe form of the operators acting on the anisotropy factor is quite simple: they are composed of translations in the\n ${\\bf x}$ plane of lengths\n\\begin{equation} \\label{stepsize}\na^\\pm_n  :=\\log \\frac{2n\\pm 1}{2n}~,\n\\end{equation}\nwhich depend on the volume $n$. If we write the operator generating the translation by $a_n$ in the $x_1$ direction as\n\\begin{equation}\n\\left(e^{i a_n p_1} \\Psi \\right) (n, x_1 ,x_2):= \\Psi(n, x_1 +a_n ,x_2)~,\n\\end{equation}\nand similarly for translations in the $x_2$ direction, the operators acting on ${\\cal H}_\\lambda$ take the form\n\\begin{equation} \\label{diag}\n \\Theta_{n n}   =   2 \\pi G \\left( n (2n + 1)\\left[ \\cos a^+_n p_1 + \\cos a^+_n p_2 + \\cos (a^+_n p_2-a^+_n p_1)\n\\right]+n(2n-1) [ a^+_n \\to a^-_n ] \\right)~,\n\\end{equation} \n\\begin{equation} \\label{offdiag}\n\\Theta_{n\\pm1\\; n}  =   - \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)  \\left( e^{-i a^\\pm_n p_1} + e^{i a^\\pm_{n\\pm 1\/2} p_1}\n+ e^{-i a^\\pm_n p_1} e^{i a^\\pm_{n\\pm 1\/2} p_2} + p_1 \\leftrightarrow p_2 \\right)~.\n\\end{equation}\nOne can easily verify that the action of Eq.~(\\ref{theta}), when written in terms of the original representation, reproduces \nEq.~(\\ref{qham6}).\n\nLet us conclude this section by mentioning a remarkable property of the $\\Theta$ operator. As found in \\cite{awe}, one can recover the\nFRW cosmology by `integrating out' the anisotropies of the Bianchi I model. Specifically, it was shown that there is a\nprojection map from the Bianchi~I space to the FRW space defined by\n\\begin{equation}\n\\sum_{{\\bf x}} \\Psi(n,{\\bf x};\\phi) = \\Psi^\\text{FRW} (n ; \\phi)~,\n\\end{equation}\nin which the $\\Theta$ operator is mapped to the $\\Theta^\\text{FRW}$ operator of the FRW model\\footnote{See~\\cite{Nelson:2009yn}\nfor an alternative projection that produces isotropic states, but not the $\\Theta^\\text{FRW}$ associated\nwith the $\\nu$-quantization procedure.} namely, \n\\begin{equation}\n\\sum_{{\\bf x}} \\Theta \\Psi(n,{\\bf x} ; \\phi) = \\Theta^\\text{FRW} \\Psi^\\text{FRW} (n ; \\phi).\n\\end{equation}\nWe will later see how this projections holds order by order in the vertex expansion.\n\n\n\n\n\\section{Sum over histories} \\label{s3}\n\nAs in \\cite{ach1,ach2}, the natural object on which to construct a sum over histories expansion is the physical inner product between\n `initial' $| [n_i, {\\bf x}_i, \\phi_i] \\rangle$ and `final' $| [n_f,{\\bf x}_f, \\phi_f] \\rangle$ physical states. This inner product is constructed \nfrom a group averaging formula involving the kinematical states $|n_i, {\\bf x}_i, \\phi_i \\rangle$ and $|n_f, {\\bf x}_f, \\phi_f \\rangle$,\n\\begin{equation} \\label{physip-0}\n ([n_f,{\\bf x}_f, \\phi_f] , [n_i, {\\bf x}_i, \\phi_i] ) =\n 2 \\langle n_f, {\\bf x}_f, \\phi_f | \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\,e^{i \\alpha C} \\ |{p}_{\\phi}|\\, | n_i, {\\bf x}_i,\\phi_i \\rangle,\n\\end{equation}\n(the $|{p}_{\\phi}|$ term is there so that the normalization agrees with the one used in \\cite{ach2}).\nAs in the FRW case \\cite{ach1,ach2}, a  key\nsimplification comes from the fact that the constraint $C$ is a\nsum of two commuting pieces that act separately on $\\H_{\\rm kin}^{\\rm matt}$ and\n$\\H_{\\rm kin}^{\\rm grav}$. Consequently, the integrand of Eq.~(\\ref{physip-0}) splits into a matter and gravitational factors:\n\\begin{equation} \\label{integrandsplit}\n2 \\langle n_f, {\\bf x}_f, \\phi_f |  \\,e^{i \\alpha C} \\ |{p}_{\\phi}|\\, | n_i, {\\bf x}_i,\\phi_i \\rangle  = 2\\,\\langle \\phi_f |\ne^{i \\alpha p_{\\phi}^2 }\\, |p_{\\phi}|  | \\phi_i \\rangle \\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} | n_i, {\\bf x}_i \\rangle.\n\\end{equation}\nThe matter part can be easily evaluated as,\n\\begin{equation}\n\\label{aphi} 2\\,\\langle \\phi_f |  e^{i \\alpha\np_{\\phi}^2 }\\, |p_{\\phi}|  | \\phi_i \\rangle\\,=\\, 2\\,\\textstyle{\\int} \\frac{\\textrm{d} p_\\phi}{2 \\pi}\ne^{i \\alpha p_\\phi^2}\\, e^{i p_\\phi (\\phi_f-\\phi_i)}\\, |p_\\phi|~.\n\\end{equation}\nThe non-triviality of Eq.~(\\ref{physip-0}) lies in the gravitational part, $\\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} | n_i, {\\bf x}_i \\rangle$. \n Following the strategy depicted in \\cite{ach1}, we will express such term as a sum over histories. This can be achieved by observing that\n the term has the form of a matrix element of a fictitious evolution operator $e^{-i \\alpha \\Theta}$, with $\\Theta$ playing the role of\n Hamiltonian and $\\alpha$ that of time. \n\nOnce the gravitational factor is written as a sum over histories, the idea is to perform the integral over $\\alpha$ for each history\n separately, obtaining at the end a sum over histories expansion of the physical inner product. These steps will be discusses in the\n following subsections.\n\n\n\n\\subsection{Sum over histories for the gravitational amplitude}\n\n\nTo construct a `sum over histories' expansion of the gravitational amplitude $\\langle n_f, {\\bf x}_f | e^{-i \\alpha \\Theta} |\nn_i, {\\bf x}_i \\rangle$, one would proceed with a Feynman-like procedure of dividing the `time' $\\alpha$ into $N$ steps of length\n$\\epsilon = \\alpha\/N$, inserting a complete basis in between each factor, and finally taking the $N \\to \\infty$ limit. In~\\cite{ach2}\nit was shown (in the FRW context, but the result is generic for any discrete labeled basis) that the resulting limit is equivalent to a specific perturbative expansion of the `evolution' operator under study. We will use this result here to construct the sum over histories directly from the perturbation series.\n\nThe starting point in such a derivation is to write the fictitious Hamiltonian $\\Theta$ as an `unperturbed part'\n$\\Theta_0 $ plus a `perturbation' $\\Theta_1$,\n\\begin{equation}\\label{eq:diag_offdiag}\n\\Theta=\\Theta_0 + \\Theta_1~.\n\\end{equation}\n\nIn a spin network\/spin foam picture, the above splitting would correspond to a graph preserving piece, $\\Theta_0$, plus the\n remaining graph changing part, $\\Theta_1$. In our case, we choose to interpret the label $n$ as containing the information \nof the `graph' , and the remaining ${\\bf x}$ label as the colouring of the graph. Thus, $\\Theta_0$ and $\\Theta_1$ are respectively\n diagonal and off-diagonal in $n$. In the tensorial notation used in Eq. (\\ref{theta}), these operators are given by,\n\\begin{eqnarray} \n\\Theta_0 & = & \\sum_n   | n \\rangle \\langle n| \\otimes \\Theta_{n n}~,  \\\\\n\\Theta_1 & = & \\sum_n | n+1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n+1\\right) n} + | n-1 \\rangle \\langle n| \\otimes \\Theta_{\\left(n-1\\right) n}~.\n\\end{eqnarray}\n\nThe construction now follows as in the FRW case \\cite{ach2}, where the same label was used to trigger the transitions.\nUsing standard perturbation theory in the interaction picture,  the transition amplitude is be written as\n\\begin{align} \\label{exp1-1}\n\\langle n_f,{\\bf x}_f | \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i \\rangle & =  \\langle n_f,{\\bf x}_f| \\bigg[ \\sum_{M=0}^{\\infty} (-i)^M\n \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M}\n\\ldots \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1  \\nonumber \\\\\n   & e^{-i (\\alpha-\\tau_M)\\Theta_0} \\Theta_1 e^{-i (\\tau_M-\\tau_{M-1})\\Theta_0} \\Theta_1 \\ldots \ne^{-i (\\tau_2-\\tau_{1})\\Theta_0} \\Theta_1  e^{-i \\tau_{1}\\Theta_0} \\bigg]  | n_i, {\\bf x}_i\\rangle~.\n\\end{align}\nThe $M$-th term of the sum generates all histories with $M$ transitions. These histories are obtained by inserting $M-1$\nidentities in the form $\\mathbf{1}=\\sum_{n_m} \\left(| n_m \\rangle \\langle n_m | \\right) \\otimes \\mathbf{1}_{\\lambda}$, $m=1,\\ldots,\nM-1$ next to each $\\Theta_1$ factor. This results in a sum over a sequence of volumes $(n_0,n_1,\\ldots,n_M)$ (with $n_0 \\equiv n_i$\nand $n_M \\equiv n_f$ held fixed), given by\n\\begin{equation} \\label{sohg}\n\\langle n_f,{\\bf x}_f| \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i\\rangle = \\sum_{M=0}^\\infty \\Big[\\sum_{n_{M-1},\\ldots,n_{1}}\n\\langle {\\bf x}_f | A(n_M,\\ldots,n_0;\\alpha) |{\\bf x}_i \\rangle \\Big]~,\n\\end{equation}\nwhere\n\\begin{align} \\label{amphg}\n& A(n_M,\\ldots,n_0;\\alpha)  :=  (-i)^M \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M} \\ldots  \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1 \\;  A(n_M,\\ldots,n_0;\n\\tau_M,\\ldots,\\tau_1;\\alpha)~, \\\\ \n& A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)  :=  A_{n_M}(\\alpha-\\tau_M) V_{n_M n_{M-1}}A_{n_{M-1}}(\\tau_M-\\tau_{M-1}) \\ldots\nV_{n_1 n_0}A_{n_0}(\\tau_1)~,\n\\label{amphgt} \n\\end{align} \nwith  $A_n\\left(\\tau\\right)$ and $V_{n'n}$  defined as,\n\\begin{eqnarray} \n&A_n(\\tau) &:= e^{-i \\tau \\Theta_{nn}}  \\label{A-1} \\\\ \n&V_{n' n}& : =  \\left\\{ \\begin{array}{ll} \\Theta_{n' n} \\label{V-1}\n& \\quad n' \\neq n \\\\\n0 & \\quad n' = n.\n\\end{array}\\right.\n\\end{eqnarray}\n\nNote that all the factors in Eqs.~(\\ref{amphg} - \\ref{V-1}) are operators on ${\\cal H}_\\lambda$ , whilst the actual amplitude,\nEq.~(\\ref{sohg}), involves matrix elements of the operator defined by Eq.~(\\ref{amphgt}). Note also that the only\nsequences entering in the sum are such that $n_{m}=n_{m-1} \\pm 1, m=1,\\ldots,M$. In particular, for $M$ fixed, there\nis a finite number of terms.\n\nThe construction above has the same form as in the FRW case. The distinction however lies on the fact that there are\nadditional degrees of freedom, given by the anisotropies ${\\bf x}$. However, in the description given so far, intermediate\nanisotropies do not appear since they are implicitly `summed over'. To make these additional sums explicit, we insert\nidentities in the form $\\mathbf{1}_\\lambda= \\sum_{{\\bf x}_m}| {\\bf x}_m \\rangle \\langle {\\bf x}_m |$  and \n$\\mathbf{1}_\\lambda= \\sum_{{\\bf y}_m}| {\\bf y}_m \\rangle \\langle {\\bf y}_m |$ to the right and left of the $A_{n_m}(\\tau_{m+1}-\\tau_m)$ operators in\nEq.~(\\ref{amphgt}). The gravitational amplitude (\\ref{exp1-1}) then takes the form,\n\n\\begin{align}\\label{diagram}\n  & \\langle n_f,{\\bf x}_f| \\ e^{-i \\alpha \\Theta}\\ | n_i, {\\bf x}_i\\rangle = \\nonumber \\\\ \n& \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0,\\ldots,{\\bf y}_{M-1}}}\n A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ; \\alpha)\n\\end{align}\nwhere now we have, on top of the `graph history' sum (given by the volume sequence), a sum over all possible `colourings' for\n each `graph history'. The amplitude for such history is given by\n\\begin{eqnarray} \\label{amphg2}\n&&A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ; \\alpha) :=\\\\\n&&\\nonumber  (-i)^M \\int_{0}^{\\alpha} \\textrm{d} \\tau_{M} \\ldots  \\int_{0}^{\\tau_2}\\! \\textrm{d} \\tau_1 \\;  A({\\bf y}_M, n_M, {\\bf x}_M,\n {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0  ;\n\\tau_M,\\ldots,\\tau_1;\\alpha)~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{ampanis}\n&&A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0  ;\n\\tau_M,\\ldots,\\tau_1;\\alpha) :=\\\\\n&&\\nonumber  A_{n_M {\\bf y}_M {\\bf x}_M }(\\alpha-\\tau_M)V_{n_M {\\bf x_M} n_{M-1} {\\bf y}_{M-1}} A_{n_{M-1} {\\bf y}_{M-1}\n {\\bf x}_{M-1}}(\\tau_M-\\tau_{M-1}) \\ldots\nV_{n_1  {\\bf x_1} n_{0} {\\bf y}_{0}}A_{n_0}(\\tau_1)~,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&A_{n {\\bf y} {\\bf x}}(\\tau) & := \\langle {\\bf y} | A_n(\\tau) | {\\bf x} \\rangle \\label{A-2} \\\\\n&V_{n' {\\bf x} n {\\bf y}} &:= \\langle {\\bf x}| V_{n'n} | {\\bf y} \\rangle \\quad \\label{V-2}.\n\\end{eqnarray}\nNote that, in spite of their appearance, the sums over intermediate anisotropies in (\\ref{diagram}) are well defined.\nAlthough in principle the anisotropies can take any real value, in practice only a countable subset of the real numbers is\ninvolved in the sum (the amplitude vanishes elsewhere).  More details on this point are given in Section~\\ref{histamp}.\n\n\n\n\\subsection{Vertex expansion of the physical inner product}\n\nWe now use the above construction to obtain an expansion for the physical inner product, Eq.~(\\ref{physip-0}). First, one\nrewrites the integrand of Eq.~(\\ref{physip-0}) as in Eq.~(\\ref{integrandsplit}) and then the gravitational factor\nis written using the expansion given in Eq.~(\\ref{diagram}).\nOne then interchanges the integral over $\\alpha$ with the sums over $M$ and the intermediate labels, to arrive at a `sum over histories' expansion of the\nphysical inner product,\n\\begin{eqnarray} \n ([n_f,{\\bf x}_f, \\phi_f] , [n_i, {\\bf x}_i, \\phi_i] ) & = & \n \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0,\\ldots,{\\bf y}_{M-1}}}A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0) \\nonumber \\\\\n&=& \\sum_{M=0}^\\infty  \\sum_{n_{M-1},\\ldots,n_{1}} A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f) \\label{physip-1}\n\\end{eqnarray}\nwhere\n\n\\begin{align} \\label{amppath}\n A({\\bf y}_M, n_M, {\\bf x}_M,  {\\bf y}_{M-1}, n_{M-1}, & \\ldots,  {\\bf y}_0, n_0, {\\bf x}_0) :=2\\,\\textstyle{\\int} \\frac{\\textrm{d} p_\\phi}{2\\pi} \\;\n e^{i p_\\phi (\\phi_f-\\phi_i)}\\, |p_\\phi| \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\, \\\\\n & e^{i \\alpha p_\\phi^2} A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0 ;\n\\alpha)~.\\nonumber \n\\end{align}\n\\begin{figure}\n \\includegraphics{spin-network_final.eps}\n \\caption{\\label{fig:1} The Bianchi~I model can be written in terms of three parameters $(n, x_1, x_2)$. $n$ dictates the\nvolume, whilst $x_1$ and $x_2$ are analogous to the spin labels\nof LQG and are represented here as two edges. Note that this is not intended to be a true LQG graph,\nrather it is a pictorial description of the degrees of freedom that our states have.} \n\\end{figure}\nPictorially, we can represent the expansion as follows. First, we represent the gravitational ket $| n, {\\bf x } \\rangle$\nas depicted in Fig.~\\ref{fig:1}. A `history' with one transition in $n\\rightarrow \\bar{n}$ is then represented in Fig.~\\ref{fig:2}.\n\n\n\\begin{figure}\n \\includegraphics{spin-foam_final.eps}\n \\caption{\\label{fig:2} The history of the spin network state i.e.\\ the spin foam analogue of our vertex expansion,\nwith $M=1$ is represented pictorially. The initial state has volume $n$ and anisotropies $x_i$. The state then `evolves', keeping $n$ constant but allowing the anisotropies to vary. Eventually there is a transition to a new volume, $\\bar{n}$. The anisotropy before and after the transition are $y_i$ and $\\bar{x}_i$ respectively. The final state has labels $\\bar{n}$ and $\\bar{y}_i$. \n The dashed lines indicate that the `spin' labels ($x_i$) evolve along each of the constant volume pieces. The amplitude for such process, in the notation of Eq. (\\ref{amppath}), is given by $A(\\bar{{\\bf y}}, \\bar{n}, \\bar{{\\bf x}},  {\\bf y}, n, {\\bf x})$.}\n\\end{figure}\n\nNote that here the analogue\nto spin foams is not exact, since in a spin foam, the spin labels on each face of the triangulation are constant. In the expansion\nderived here there is non-trivial dynamics for the `spins', ${\\bf x}_i$, even with a fixed `graph' $n$. In general there are\ntwo distinct labels for a face, those at the beginning, ${\\bf x}_i$ and those at the end, ${\\bf y}_i$, and  ${\\bf y}_i \\neq\n{\\bf x}_i$. Also, in a \nspin foam there is no restriction on the spins across a vertex, whereas in our case ${\\bf y}_i$ and ${\\bf x}_{i+1}$\n(the `spin' labels on either side of the $(i+1)^{\\rm th}$ vertex ) are very closely related, by the form of\nEq.~(\\ref{offdiag}). Thus, although the analogue is not complete, the form of the expansions are qualitatively the\nsame. \n\n\n\\subsection{Histories amplitudes} \\label{histamp}\n\nWe now discuss how path amplitudes appearing in Eq.~(\\ref{physip-1}) can be calculated and illustrate the procedure\nin the simplest case.  The first step is to evaluate the matrix element $\\langle {\\bf x}_f |A(n_M,\\ldots,n_0; \\tau_M,\n\\ldots, \\tau_1;\\alpha)|{\\bf x}_i \\rangle$ of the operator defined in Eq.~(\\ref{amphgt}).  This operator consist of compositions\n of operators $A_n(\\tau)$ and $V_{n' n}$, given respectively in Eq.~(\\ref{A-1}) and Eq.~(\\ref{V-1}), which themselves are\nconstructed from translations in the  ${\\bf x}$ plane.  Because of the translation invariance, we will consider the matrix\nelements between  $\\langle {\\bf x}|$ and $|0 \\rangle$; the original matrix element is then recovered by the substitution\n${\\bf x} \\to {\\bf x}_f-{\\bf x}_i$.\n\nLet us discuss the structure of the operators in more detail.  As already noted, $V_{ n' n}$ vanishes unless $n'=n\\pm 1$, in which\ncase it is given by Eq.~(\\ref{offdiag}).  It consist of an overall factor times six simple shifts involving lengths of $a^\\pm_n$ and\n $a^\\pm_{n \\pm 1\/2}$.  The $A_n(\\tau)$ operator is the exponentiation of ($-i \\tau$ times) the operator given in\nEq.~(\\ref{diag}).  It can be factored into two terms,\n\\begin{equation}\nA_n(\\tau)=A^+_n(\\tau)A^-_n(\\tau)~,\n\\end{equation}\nwhere\n\\begin{equation}\nA^\\pm_n(\\tau)=e^{-i \\tau 2 \\pi G \\, n (2n \\pm 1)\\left[ \\cos a^\\pm_n p_1 + \\cos a^\\pm_n p_2 + \\cos (a^\\pm_n p_2-a^\\pm_n p_1) \\right]}~,\n\\end{equation}\nwhich only involves shifts with step-size $a^\\pm_n$.  The total operator $A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)$ is then a product\n of operators involving shifts of lengths $a^\\pm_j$ with $j=n_m$ or $j=n_m \\pm 1\/2$ ($m=0,\\ldots,M$).\n As a result the matrix element between $\\langle {\\bf x} |$ and $|0 \\rangle $ vanishes unless\n${\\bf x}$ lies in the `lattice' generated by the $a^\\pm_j$ steps. The computation simplifies by selecting among the\n$a^\\pm_j$'s, a set of independent (i.e. incommensurate) lengths that generate the `lattice'. \n\nLet us illustrate the situation by considering the simplest path, namely the $M=0$ case. In this case the amplitude is given by\n the matrix element $\\langle {\\bf x} |A_n(\\tau) |0 \\rangle$.   As already noticed, this operator contains two step-sizes, $a^+_n$ and\n $a^-_n$, and so the nontrivial matrix elements occur whenever ${\\bf x}$\nlies in the lattice~\\footnote{Notice that the lattices we will referring to are not the usual ones (as for instance a square\n lattice $a \\mathbb{Z}^2 \\subset \\mathbb{R}^2$) where points form a grid. Rather they fill in the entire plane. For instance one can show that\n the points of the lattice defined by   Eq.~(\\ref{lattice}) form a dense subset of $\\mathbb{R}^2$ (because $a^+_n$ and $a^-_n$ are\n incommensurate numbers).},\n\\begin{equation} \\label{lattice}\n{\\bf x} = {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\ ; \\qquad   {\\bf k}^+ = (k^+_1,k^+_2) \\in \\mathbb{Z}^2, \\; {\\bf k}^- = (k^-_1,k^-_2)\\in \\mathbb{Z}^2~.\n\\end{equation}\nFrom the definition of $a^+_n$ and $a^-_n$, Eq.~(\\ref{stepsize}), one can check that these two numbers are always incommensurate,\nand so they form an independent set of generators of the lattice.  That is to say, a point in the lattice defined by Eq.~(\\ref{lattice}) is\nuniquely decomposed into its $a^+_n$ and $a^-_n$ components, i.e., if  ${\\bf k}^+ a^+_n+{\\bf k}^- a^-_n = {\\bf l}^+ a^+_n+\n{\\bf l}^- a^-_n$ then ${\\bf k}^\\pm={\\bf l}^\\pm$. \n\nThus, the kets $|{\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\rangle$, form a basis of the subspace of vectors which give a\nnon-vanishing matrix element.  This space has the structure of a tensor product of two copies of $L^2(\\mathbb{Z}^2)$,\n\\begin{equation}\n|{\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n \\rangle~= |{\\bf k}^+ \\rangle \\otimes |{\\bf k}^- \\rangle \n\\end{equation}\nwhere $|{\\bf k}^+ \\rangle$ and $|{\\bf k}^- \\rangle$ are viewed as basis of two abstract $L^2(\\mathbb{Z}^2)$ spaces.\n\nViewed in this way, the $A^\\pm_n(\\tau)$ operators act separately on each $L^2(\\mathbb{Z}^2)$ factor:\n\\begin{equation} \\label{mea}\n\\langle {\\bf x}= {\\bf k}^+ a^+_n+{\\bf k}^- a^-_n |A_n(\\tau) | 0 \\rangle = \\langle {\\bf k}^+|A^+_n(\\tau)|0 \\rangle \\langle {\\bf k}^-|A^-_n(\\tau)|0 \\rangle~.\n\\end{equation}\nEach factor is the matrix element of a (translation invariant) operator in a single $L^2(\\mathbb{Z}^2)$ space, and so can\nbe evaluated by taking the Fourier transform.  Using Eq.~(\\ref{diag}) one finds \n\\begin{equation} \\label{apm}\n\\langle {\\bf k}^\\pm|A^\\pm_n(\\tau)|0 \\rangle=\\int_0^{2 \\pi}\\frac{\\textrm{d}{\\theta_1}}{2 \\pi} \\int_0^{2 \\pi}\\frac{\\textrm{d}{\\theta_2}}{2 \\pi}\ne^{i k^\\pm_1 \\theta_1+i k^\\pm_2 \\theta_2}e^{-i 2\\pi G n(2n \\pm 1) \\tau \\left(\\cos \\theta_1 + \\cos \\theta_2 + \\cos\n(\\theta_2-\\theta_1) \\right)}~.\n\\end{equation}\n\nIn the general case of a path with $M$ transitions and $g$ independent generators, the space of vectors giving a non-vanishing matrix elements will have now the structure of a tensor\nproduct of $g$ copies of $L^2(\\mathbb{Z}^2)$. Because the vertex $V_{n' n}$ is the sum of six terms, one will generically have a\ntotal of $\\sim6^M$ terms, each of them involving $g$ Fourier integrals of the type described above.  The expressions for\nthese integrals can be directly read off from Eqs.~(\\ref{diag}) and (\\ref{offdiag}). \n\n\nAfter the evaluation of the matrix element $\\langle {\\bf x}_f |A(n_M,\\ldots,n_0; \\tau_M,\\ldots, \\tau_1;\\alpha)|{\\bf x}_i \\rangle$,\n one has to perform the $\\tau$ integrals in Eq.~(\\ref{amphgt}), and the $p_\\phi$ and $\\alpha$ integrals in Eq.~(\\ref{amppath}).\nIn our $M=0$ case, there are no $\\tau$ integrals to perform, and the\n $\\alpha$ integral can be done if interchanged with the Fourier integrals of Eq.~(\\ref{apm}). This gives a Dirac delta, which\n in turn allows one to evaluate the integral over $p_\\phi$. The result is,\n\\begin{equation}\nA(n;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f)=\\int_D \\frac{\\textrm{d}^4\\theta}{(2\\pi)^4} e^{i \\theta_1  k^+_1+\\theta_2 k^+_2+\\theta_3\nk^-_1+\\theta_4  k^-_2} e^{i (\\phi_f-\\phi_i) \\sqrt{2 \\pi G n}\\sqrt{(2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)h(\\theta_3,\\theta_4)}}~,\n\\end{equation}\nwhere ${\\bf x}_f-{\\bf x}_i={\\bf k}^+ a^+_n+{\\bf k}^- a^-_n$, \\,  $h(\\theta_1,\\theta_2)=\\cos \\theta_1 + \\cos \\theta_2 + \\cos \n(\\theta_2-\\theta_1)$\nand the domain of integration is \n\\begin{equation}\nD= \\{(\\theta_1,\\theta_2,\\theta_3,\\theta_4) \\in [0,2\\pi)^4 \\; \/ \\, (2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)h(\\theta_3,\\theta_4) > 0 \\}.\n\\end{equation}\n\n\nSo far we have discussed histories amplitudes of the form $A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f)$, where intermediate\n anisotropies are already summed over. Let us now discuss briefly how one would compute the amplitude for an individual\n`colouring' of such a history, $A({\\bf y}_M, n_M, {\\bf x}_M, {\\bf y}_{M-1}, n_{M-1}, \\ldots, {\\bf y}_0, n_0, {\\bf x}_0)$.\nThe building blocks  in this case are the amplitudes $A_{n {\\bf y} {\\bf x}}(\\tau)$ and $V_{n' {\\bf x} n {\\bf y}}$,\nEq.~(\\ref{A-2}) and Eq.~(\\ref{V-2}). The first  amplitude coincides with the $M=0$ case discussed above. We thus have that\n $A_{n {\\bf y} {\\bf x}}(\\tau)$ vanishes unless ${\\bf y}- \n{\\bf x}={\\bf k}^+ a^+_n+{\\bf k}^- a^-_n$ in which case it is given by Eqs.~(\\ref{mea}) and (\\ref{apm}). On the other hand, the value\nfor $V_{n' {\\bf x} n {\\bf y}}$  can be easily read off from its definition:\nit vanishes unless $n'=n\\pm 1$ and ${\\bf y}- {\\bf x}$ lies in one of the following six points, $\\{(a^{\\pm}_n,0),(a^{\\pm}_{n\\pm 1\/2},0),\n(a^{\\pm}_n,a^{\\pm}_{n\\pm 1\/2}),(0,a^{\\pm}_n),(0,a^{\\pm}_{n\\pm 1\/2}),(a^{\\pm}_{n\\pm 1\/2},a^{\\pm}_n) \\}$, in which case it takes the value\n$- \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)$. After multiplying by the remaining factors in Eq.~(\\ref{ampanis}), one has to perform the\nsame integrals as previously, in this case given in Eq.~(\\ref{amphg2}) and Eq.~(\\ref{amppath}).\n\n\\subsection{Vacuum case}\n\nThe presence of matter entered only at the very end of the construction. If we did not have matter at all, we could still follow the\nsame  procedure and arrive at the vacuum equivalent of Eq.~(\\ref{physip-1}),\n\\begin{equation} \n ([n_f,{\\bf x}_f] , [n_i, {\\bf x}_i] ) = \\sum_{M=0}^\\infty \\Big[ \\sum_{n_{M-1},\\ldots,n_{1}} A^{\\text{vacuum}}(n_M,\\ldots,\nn_0;{\\bf x}_i,{\\bf x}_f) \\Big]~.\n\\end{equation}  \nThe difference between the matter and vacuum cases lies only in the form of the amplitudes, which in the vacuum case are formally\ngiven by,\n\\begin{equation} \\label{ampvac}\nA^{\\text{vacuum}}(n_0,\\ldots,n_M;{\\bf x}_i,{\\bf x}_f)= \\textstyle{\\int}_{-\\infty}^{\\infty}\n \\textrm{d}{\\alpha} \\,  \\langle {\\bf x}_f | A(n_0,\\ldots,n_M;\\alpha) |{\\bf x}_i \\rangle~.\n\\end{equation}\nThese amplitudes can then be evaluated following the strategy given in the previous section, the only difference being the\nabsence of the final integral over $p_\\phi$. It is not obvious whether the integral in Eq.~(\\ref{ampvac}) converges for all\n paths thus giving a meaningful expansion. Nevertheless, by looking at the generic behaviour of these integral, one finds\nsome evidence that it may converge. For instance, in the constant volume ($M=0$) path, Eq.~(\\ref{ampvac}) gives\n\\begin{equation}\nA^{\\text{vacuum}}(n;{\\bf x}_i,{\\bf x}_f)=\\frac{1}{2 \\pi G n}\\int \\frac{\\textrm{d}^4\\theta}{(2\\pi)^4} e^{i \\theta_1  k^+_1+\n\\theta_2  k^+_2+ \\theta_3  k^-_1+\\theta_4  k^-_2} \\delta\\left( (2n + 1)h(\\theta_1,\\theta_2)+ (2n - 1)\nh(\\theta_3,\\theta_4) \\right)~\n\\end{equation}\nwhich is clearly finite (at least for $n>0$). This is to be contrasted with the vacuum FRW case~\\cite{ach2}, and the\nexample in~\\cite{rv2}, where a regulator is required in order to render finite the otherwise divergent amplitudes,\neven in this $M=0$ case. Further study of the vacuum case is in progress~\\cite{Ed_Adam}. \n\n\n\\section{Projection to FRW}\\label{s4}\nAt the end of section~\\ref{s2}, we commented on the projection from  Bianchi~I to FRW. We show here that\nwhen such projection is done at the level of the vertex expansion, Eq~(\\ref{physip-1}), one recovers the FRW vertex\nexpansion of \\cite{ach2}.\n\nThe structure in both cases is almost identical. One has a  sum over volume sequences $(n_M,\\ldots,n_0)$, and each amplitude\nis constructed by first obtaining a gravitational amplitude, and then performing the group averaging and scalar field\nintegrations. Thus, all that remains is to show that the amplitude $\\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)\n | {\\bf x}_i \\rangle$ given in Eq.~(\\ref{amphgt}) projects to the corresponding FRW one,\n\\begin{align} \\label{ampfrw}\nA^\\text{FRW}(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)= & \\; e^{-i(\\alpha - \\tau_M) \\Theta^\\text{FRW}_{n_M n_M}}\\,\\,\n \\Theta^\\text{FRW}_{n_M n_{M-1}}\\,\\,\n\\times \\nonumber\\\\\n&\\ldots\\,\\, e^{-i(\\tau_2-\\tau_1) \\Theta^\\text{FRW}_{n_1 n_1}}\\,\\,\n \\Theta^\\text{FRW}_{n_1 n_{0}}\\,\\, e^{-i\\tau_1 \n\\Theta^\\text{FRW}_{n_0 n_0}}~,\n\\end{align}\nwhen summing over all possible values of ${\\bf x}_f$. \n\nTo show this, it is convenient to explicitly write the intermediate anisotropies in the amplitude $\\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\n\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_i \\rangle$. As before, this is done by introducing complete basis $\\mathbf{1}_\\lambda= \\sum_{{\\bf x}_m}\n| {\\bf x}_m \\rangle \\langle {\\bf x}_m |$  and  $\\mathbf{1}_\\lambda= \\sum_{{\\bf y}_m}| {\\bf y}_m \\rangle \\langle {\\bf y}_m |$ to the right and left\nof the $A_{n_m}(\\tau_{m+1}-\\tau_m)$ operator in Eq.~(\\ref{amphgt}). Calling ${\\bf x}_f \\equiv {\\bf y}_M$ and ${\\bf x}_i \\equiv {\\bf x}_0$\nwe have,\n\\begin{align}  \\label{amphgtl}\n   &\\sum_{{\\bf y}_M} \\langle {\\bf y}_M |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_0 \\rangle  = \n \\sum_{\\substack{{\\bf x}_1,\\ldots,{\\bf x}_M\\\\{\\bf y}_0, \\ldots,{\\bf y}_{M}}}\n  \\langle {\\bf y}_M | A_{n_M}(\\alpha-\\tau_M) |{\\bf x}_M \\rangle  \\nonumber \\\\ \n   & \\langle {\\bf x}_M | V_{n_M n_{M-1}} |{\\bf y}_{M-1}  \\rangle \\langle {\\bf y}_{M-1}|A_{n_{M-1}}(\\tau_M-\\tau_{M-1})|{\\bf x}_{M-1}\n  \\rangle \\ldots \\langle {\\bf x}_1 |  V_{n_1 n_0}|{\\bf y}_{0} \\rangle \\langle {\\bf y}_{0}|A_{n_0}(\\tau_1)|{\\bf x}_0 \\rangle~.\n\\end{align}\nWe now use the translation invariance of the operators to write each matrix elements in Eq.~(\\ref{amphgtl}) as $\\langle {\\bf x}'| f |\n{\\bf x} \\rangle = \\langle {\\bf x}'-{\\bf x}| f | 0 \\rangle$ where $f$ is either the $A$ or $V$ operators. We then change the\nsummation variables to ${\\bf y}'_m={\\bf y}_m-{\\bf x}_m$ and ${\\bf x}'_m={\\bf x}_m-{\\bf y}_{m-1}$.\nThe different summations in Eq.~(\\ref{amphgtl}) then decouple giving,\n\\begin{align} \\label{amphgtl2}\n   &\\sum_{{\\bf y}_M} \\langle {\\bf y}_M |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_0 \\rangle  =  \\left(\\sum_{{\\bf y}'_M}\n  \\langle {\\bf y}'_M | A_{n_M}(\\alpha-\\tau_M) |0 \\rangle\\right) \\left(\\sum_{{\\bf x}'_M} \\langle {\\bf x}'_M | V_{n_M n_{M-1}} |0  \\rangle \\right) \n  \\nonumber \\\\   &  \\left(\\sum_{{\\bf y}'_{M-1}} \\langle {\\bf y}'_{M-1}|A_{n_{M-1}}(\\tau_M-\\tau_{M-1})|{\\bf x}_{M-1} \\rangle \\right) \\ldots\n  \\left(\\sum_{{\\bf x}'_1} \\langle {\\bf x}'_1 | V_{n_1 n_0}|0 \\rangle \\right) \\left(\\sum_{{\\bf y}'_0} \\langle {\\bf y}'_{0}|A_{n_0}(\\tau_1)|0\n \\rangle\\right)~.\n\\end{align}\nComparing Eq.~(\\ref{amphgtl2}) and Eq.~(\\ref{ampfrw}) we see that our task reduces to showing that\n $\\sum_{\\bf x} \\langle {\\bf x} | A_{n}(\\tau) |0 \\rangle = e^{-i\\tau \\Theta^{FRW}_{n n}}$ and $\\sum_{\\bf x}\n\\langle {\\bf x} | V_{n' n} |0 \\rangle = \\Theta^{FRW}_{n' n}$. That this\n is so can be seen as a direct consequence of the result in~\\cite{awe}. Let us nevertheless show it explicitly.\n\nFor the $V$ term, it suffices to look at $V_{n\\pm1 n}$. The operator, given\nin Eq.~(\\ref{offdiag}) consist in an overall constant times six different translations in the ${\\bf x}$ plane. Each term thus\nwill pick up a single ${\\bf x}$ from the sum. For instance, the first term gives a nonzero value only for ${\\bf x}=(a^\\pm_n,0)$,\nin which case it gives a contribution of $- \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1)$. We then conclude that\n\\begin{equation} \\label{vfrw}\n\\sum_{{\\bf x}} \\langle {\\bf x}| V_{n\\pm1 n} |0 \\rangle = - 6 \\pi G \\sqrt{n(n \\pm 1)}(2n \\pm 1) = \\Theta^{FRW}_{n\\pm 1 n}~,\n\\end{equation}\nas required.\n\nFor the $A_n(\\tau)$ term we have\n\\begin{align} \\label{afrw}\n& \\sum_{\\bf x} \\langle {\\bf x} | A_{n}(\\tau) |0 \\rangle = \\nonumber \\\\\n & \\sum_{{\\bf k}^+,{\\bf k}^- \\in \\mathbb{Z}^2} \\langle {\\bf k}^+|A^+_n(\\tau)|0 \\rangle \\langle {\\bf k}^-|A^-_n(\\tau)|0 \\rangle = \\nonumber \\\\\n& e^{-i \\tau 24 \\pi G n^2} = e^{-i \\tau \\Theta^{FRW}_{n n}}~,\n\\end{align}\nwhere in going from the first to second line, we used Eq.~(\\ref{mea}). In going from the second\nto third line, we used Eq.~(\\ref{apm}) and the identity $\\sum_{k \\in \\mathbb{Z}} e^{i k \\theta}\/2\\pi = \\delta(\\theta)$\nto directly evaluate the Fourier integrals. Using Eqs.~(\\ref{vfrw}) and (\\ref{afrw}) we then have\n\\begin{equation}\n\\sum_{{\\bf x}_f} \\langle {\\bf x}_f |  A(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha) | {\\bf x}_i \\rangle = A^\\text{FRW}(n_M,\\ldots,n_0;\\tau_M,\\ldots,\\tau_1;\\alpha)~,\n\\end{equation}\nwhich implies\n\\begin{equation}\n\\sum_{{\\bf x}_f} A(n_M,\\ldots,n_0;{\\bf x}_i,{\\bf x}_f;\\phi_i,\\phi_f) =  A^\\text{FRW}(n_M,\\ldots,n_0;\\phi_i,\\phi_f)~.\n\\end{equation}\nThus we see that the vertex expansion for our Bianchi model, Eq.~(\\ref{physip-1}) projects down to the vertex expansion\nof the FRW model, order by order.\n\n\n\\section{Discussion}\\label{s5}\n\nRecently it has been shown~\\cite{ach1} that one can take the Loop Quantum version of FRW cosmology\nand expand it as a sum over volume transition of amplitudes compatible with given initial and final states i.e.\\\nthat the cosmological model of Loop Quantum Gravity can be re-written in terms of a sum over amplitudes, analogous to\nthe spin foam approach. This sum over transition amplitudes is produced as a perturbation expansion of the\nconstraint operator of LQC, thus linking perturbative dynamics of LQC to (the analogue of) spin foams.\nThis analogue provided a useful link between the two theories, however because there is only\none dynamic parameter in FRW cosmologies -- the volume -- the system has no analogue of the spin labels. In this\npaper we have extended the approach of~\\cite{ach1} to the Bianchi~I cosmological model, which, in addition to volume,\nhas anisotropic degrees of freedom. We have shown that it is again possible to expand the dynamics of the\nmodel in terms of sums of amplitudes over volume transitions compatible with initial and final states. The additional\nanisotropic degrees of freedom of this model are analogous to the spin labels of spin networks, thus significantly\nimproving the analogue to spin foams.\n\nThe analogue remains at a formal level however, because one cannot directly associate the amplitudes with the \nchanging of an underlying spin network. Despite this the association of the anisotropic degrees of freedom with\nthe spin labels is well motivated by the fact that in LQC they give the area (of the fiducial cell), which is\nprecisely the role played by the spin labels (and edges) in a spin network. In addition to showing that the resulting\nsummation over `spin' labels is finite, we show that the projection to the FRW system occurs order by order\nin the expansion, thus recovering the results of~\\cite{ach1}.\n\nFinally, although spin foams are typically taken to have spin changes only at vertices, it is generally\nexpected that spin dynamics in the absence of graph changing vertices will play an important role in the final\ntheory~\\cite{freidel}. More precisely, that the action of the full constraint is non-trivial, even in the absence of vertices\nand hence that the amplitude for each vertex-free segment of the spin foam will be non-diagonal in the spin labels. In\nthe analogue produced here we show that indeed the `spin' changing amplitude is non-trivial, even in the absence of volume\nchanging `vertices'. Thus our full expansion is the analogue of a generalization of spin foams, allowing for\n`spin' dynamics. \n\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank Abhay Ashtekar and Edward Wilson-Ewing for illuminating discussions.\nThis work was supported in part by NSF grants PHY0748336,\nPHY0854743, The George A.\\ and Margaret M.~Downsbrough Endowment and\nthe Eberly research funds of Penn State.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Cholesky decomposition - Introduction}\nFor a positive-definite symmetric matrix Cholesky decomposition provides a unique representation in the form of $\\Lb\\Lb^T$, \nwith a lower triangular matrix $\\Lb$ and the upper triangular $\\Lb^T$. \nOffered by a convenient $O(n^3)$ algorithm,\nCholesky decomposition is favored by many for expressing the covariance matrix~\\cite{Pourahmadi2011}. \n The matrix $\\Lb$ itself can be used to transform independent normal variables into dependent multinormal~\\cite{Moonan57} which is particularly useful for Monte Carlo simulations. \n\nExplicit forms of $\\Lb$ are known for limited correlation structures such as the equicorrelated~\\citep[pp. 104]{Tong1990}, tridiagonal~\\cite{MiwaHayterKuriki2003}, and the multinomial~\\cite{TanabeSagae1992}. The general correlated case is typically computed by using spherical parametrizations~\\cite{PinheiroBates1996,RapisardaBrigoMercurio2007,RebonatoJackel2007,MittelbachMatthiesenJorswieck2012}, a multiplicative ensemble of trigonometric functions of the angles between pairs of vectors. Others may use Cholesky matrix~\\cite[pp. 49]{CookeEtAl2011} that utilizes the multiplication of partial correlations. \n \nIn this paper, we will present two explicit parametrizations of Cholesky factor for a positive-definite correlation matrix. \nBoth parametrizations offer a preferable, simpler alternatives to the multiplicative forms of spherical parametrization and partial correlations. \nIn Section~\\ref{sec:1stL} we show that the nonzero elements of Cholesky factor are the \\textit{semi-partial correlation coefficients}\n\\[\n  \\rhob_{ij(1,\\ldots,i-1)} \n= \\frac{\\rho_{ij} - \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i}\n          {\\sqrt{1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}} ,\n\\]\nwhere $\\Rb_{i-1}^{-1}$ is the inverse of the correlation matrix $\\Rb_i = (\\rho_{kj})_{k,j=1}^{i-1}$, $\\rhob_i^{*j} = (\\rho_{1j}, \\rho_{2j}, \\ldots, \\rho_{i-1,j})$ and $\\rhob_i = \\rhob_i^{*i}$. The order of the $\\rhob_{ij(1,\\ldots,i-1)}$s is determined by Cholesky factorization, and the notations are borrowed from Huber's trivariate discussion of semi-partial correlation in regression~\\cite{Huber1981}.\n In Section~\\ref{sec:2ndL} we uncover that the squares, $\\rhob_{ij(1,\\ldots,i-1)}^2$, are equivalent to the differences between two successive ratios of determinants, \nand we use this equivalence to construct the second parametrization for $\\Lb$. In Section~\\ref{sec:gencov} we extend the representation of $\\Lb$ to the structure of a covariance matrix, \nand in Section~\\ref{sec:welldefL} we study two inequality conditions that are essential for the positive-definiteness of $\\Lb\\Lb^T$.\nWe conclude this paper by offering two possible applications, one for each of the suggested forms. \nIn Section~\\ref{sec:ttest} we present a simple $t$-test that employs the semi-partial correlation structure for testing the dependence of a single variable upon a set of multivariate normals. \nIn Section~\\ref{sec:randomcorr} we utilize the second parametrization to design a simple algorithm for the generation of random positive-definite correlation matrices. \nWe end the paper with the simple case of generalization of random AR(1) correlation in Section~\\ref{sec:AR1}.\n\n\\section{The first parametrization for Cholesky factor}\\label{sec:1stL}\nLet $\\Rb_n = (\\rho_{ij})_{ij=1}^n$ be a positive-definite correlation matrix, for which each sub-matrix $\\Rb_k = (\\rho_{ij})_{ij=1}^k$ is positive-definite. \nLet also $\\Lb = (l_{ij})_{ij = 1}^n$ be Cholesky factor of $\\Rb$, \n$|\\Rb|$ be the determinant of $\\Rb$, $\\Rb^{-1}$ its inverse, and $\\rhob_i^{*j} = (\\rho_{1j}, \\rho_{2j}, \\ldots, \\rho_{i-1,j})$ for $j\\ge i$, so $\\rhob_i \\equiv \\rhob_i^{*i}$. \n To simplify writing also set $\\Rb_0^{-1} \\equiv 1$. \n The first representation of $\\Lb$ will use \n the semi-partial correlations $l_{ji} = \\rhob_{ij(1,\\ldots,i-1)} = \\frac{\\rho_{ij}-\\rhob_i^{*j}\\Rb_{i-1}^{-1}\\rhob_i^T}{\\sqrt{1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}}$, $i\\le j$,\n\\begin{equation}\\label{chol:L1}\n     \\Lb = \\left( \\begin{array}{ccccc}\n       1 & 0 & 0 & \\cdots & 0 \\cr\n       \\rho_{12} & \\sqrt{1 - \\rho_{12}^2} & 0 \n& \\cdots & 0\\cr \n       \\rho_{13} & \\frac{\\rho_{23} - \\rho_{12}\\rho_{13}}{\\sqrt{ 1 - \\rho_{12}^2}} & \n         \\sqrt{1 -  \\rhob_3 \\Rb_2^{-1} \\rhob_3^T} & \\cdots & 0\\cr  \n\\vdots & \\vdots & \\vdots \n& \\ddots & 0 \\cr\n\\rho_{1n} &  \n     \\frac{\\rho_{2n} - \\rho_{12}\\rho_{1n}}{\\sqrt{1 - \\rho_{12}^2}} &  \n           \\frac{\\rho_{3n} - \\rhob_3^{*n} \\Rb_2^{-1} \\rhob_3^T}{\\sqrt{1 -  \\rhob_3 \\Rb_2^{-1} \\rhob_3^T}} & \n      \\cdots & \\sqrt{1 -  \\rhob_n \\Rb_{n-1}^{-1} \\rhob_n^T} \n\\end{array}\\right)\n\\end{equation}\nFor $i=j$, we have $\\rho_{ii(1,\\ldots,i-1)} = \\sqrt{1-\\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T}$, and $1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T > 0$ for positive-definite $\\Rb_i$. \nSome may recognize $1 - \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T$ as the \\textit{schur-complement} \nof the matrix $\\Rb_{i-1}$ inside $\\Rb_i$ from the formula for computing the determinant of $\\Rb_i$, using the block matrix $\\Rb_{i-1}$~\\citep[pp. 188]{Harville1997},\n\\begin{equation}\\label{eq:schur_det}\n   |\\Rb_i|  =  \\left|\n \\begin{array}{cc} \\Rb_{i-1} & \n \\rhob_i^T \\\\ \\rhob_i & 1\\end{array}\n  \\right| \n   = |\\Rb_{i-1}| (1-\\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T).\n\\end{equation} \n \nTo show that $\\Rb_n = \\Lb\\Lb^T$ we introduce Theorem~\\ref{thm:CholSums}.  \n\\begin{thm}\\label{thm:CholSums}\nFor $i \\ge 1$ and $n \\ge j \\ge i+1$,\n\\begin{equation}\\label{Q_recursive}\n    \\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T  \n           = \\sum_{k=1}^i \\rho_{ki(1,\\ldots ,k-1)} \\cdot \\rho_{kj(1,\\ldots ,k-1)} . \n\\end{equation}\n\\end{thm} \nBy the virtue that Cholesky factor of a positive-definite matrix has a unique representation, Theorem~\\ref{thm:CholSums} will serve as a general proof for the form (\\ref{chol:L1}). \nSome may recognize the Eq.~(\\ref{Q_recursive}) in Theorem~\\ref{thm:CholSums} as the inner-product used for the familiar algorithm of Cholesky Decomposition~\\citep[pp. 235]{Harville1997}:\n\\[\n   l_{ii} = \\left(1 - \\sum_{k=1}^{i-1} l_{ik}^2\\right)^{1\/2} \n   \\mbox{ and } \n   l_{ji} = \\left(\\rho_{ij} - \\sum_{k=1}^{i-1} l_{jk} l_{ik} \\right)\/l_{ii} .\n\\]\nSurprisingly, the equality in Theorem~\\ref{thm:CholSums} seems to be unknown or neglected. \nThe proof for Theorem~\\ref{thm:CholSums} will be given in~\\ref{sec:thm1proof}, and will be heavily based on the recursive arguments of the Lemma~\\ref{lem:recursive}: \n\\begin{lem} \\label{lem:recursive}\nFor $i \\ge 1$ and $n \\ge j \\ge i+1$,\n\\begin{equation}\\label{lem:recursive_eq}\n       \\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T  \n     =\n       \\rhob_i^{*j}\\Rb_{i-1}^{-1} (\\rhob_i^{*i+1})^T  \n     + \\frac{ (\\rho_{i,i+1} - \\rhob_i^{*i+1} \\Rb_{i-1}^{-1} \\rhob_i^T)\n              (\\rho_{ij} - \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i^T)\n      }\n       {1 -  \\rhob_i \\Rb_{i-1}^{-1} \\rhob_i^T} . \n\\end{equation}\n\\end{lem}\nThe proof for Lemma~\\ref{lem:recursive} will be given in~\\ref{sec:lemproof}.\n\n\\textbf{Remark.} Lemma~\\ref{lem:recursive} is useful to illustrate the recursive nature of the computation of the part $\\rhob_{i+1}^{*j}\\Rb_i^{-1} \\rhob_{i+1}^T$. \nAs an exercise, we suggest to verify that $\\rhob_2^{*j}\\Rb_1^{-1} \\rhob_2^T = \\rho_{1j}\\rho_{12}$, and then to compute $\\rhob_3^{*j}\\Rb_2^{-1} \\rhob_3^T$ by using Eq.~(\\ref{lem:recursive_eq}).\nThe result should be equal to $\\rhob_3^{*j}\\Rb_2^{-1} \\rhob_3^T  = \\rho_{13} \\rho_{1j}  + \\rho_{23(1)}\\rho_{2j(1)}$ as claimed by Theorem~\\ref{thm:CholSums}.\n \n \n\n\n \\section{The second parametrization for Cholesky factor}\\label{sec:2ndL}\nThe second parametrization for $\\Lb$ will follow directly from~(\\ref{chol:L1}) by applying Lemma~\\ref{lem:ab} \nthat claims an equivalence between semi-partial correlation coefficients to the difference between two successive schur-complements.\n\\begin{lem}\\label{lem:ab}\nLet us use the above notations and the positive-definiteness assumptions as before, and define $\\Rb_i^{*j} \\equiv  \\left( \\begin{array}{cc} \\Rb_{i-1} & (\\rhob_i^{*j})^T \\\\ \\rhob_i^{*j} & 1 \\end{array}\\right)$.\nThen for $j \\ge i+1 \\ge 3$ the difference between two successive ratios of determinants(or schur-complements) is\n\\begin{equation}\\label{eq:lem2}\n  |\\Rb_i^{*j}|\/|\\Rb_{i-1}| - |\\Rb_{i+1}^{*j}|\/|\\Rb_i|  \n     = \\left(\\rho_{ij}- \\rhob_i^{*j} \\Rb_{i-1}^{-1} \\rhob_i^T \\right)^2\n  |\\Rb_{i-1}|\/|\\Rb_i|.\n\\end{equation}\n\\end{lem}\nThe proof for Lemma~\\ref{lem:ab} will be found in~\\ref{sec:lemproof}.  \nSetting $s_{ij} \\equiv sign(\\rho_{ij(1,\\ldots,i-1)})$, it is possible to write Cholesky factor~(\\ref{chol:L1}) by the equivalent form\n\\begin{equation}\\label{chol:L2}\n    \\left( \\begin{array}{ccccc} \n             1 & 0 & 0 & \\cdots & 0 \\cr\n            s_{12}\\sqrt{1-\\frac{|\\Rb_2|}{1}} & \\sqrt{|\\Rb_2|} & 0 & \\cdots & 0\\cr\n            s_{13}\\sqrt{1-\\frac{|\\Rb_2^{*3}|}{1}} & s_{23}\\sqrt{\\frac{|\\Rb_2^{*3}|}{1} - \\frac{|\\Rb_3|}{|\\Rb_2|}} &  \\sqrt{\\frac{|\\Rb_3|}{|\\Rb_2|}} \n& \\cdots & 0\\cr \n\\vdots & \\vdots & \\vdots & \\ddots & 0 \\cr\ns_{1n}\\sqrt{1-\\frac{|\\Rb_2^{*n}|}{1}} &  \n       s_{2n}\\sqrt{\\frac{|\\Rb_2^{*n}|}{1} - \\frac{|\\Rb_3^{*n}|}{|\\Rb_2|}} &  \n       s_{3n}\\sqrt{ \\frac{|\\Rb_3^{*n}|}{|\\Rb_2|} -  \\frac{|\\Rb_4^{*n}|}{|\\Rb_3|}} & \n      \\cdots & \\sqrt{\\frac{|\\Rb_n|}{|\\Rb_{n-1}|}  }\n\\end{array}\\right)\n\\end{equation}\n\n\\subsection{The extension to nonsingular covariance matrix}\\label{sec:gencov}\nThe Cholesky factor~(\\ref{chol:L2}) can be easily extended to the general structure of nonsingular covariance when we replace each ratio $\\sigma_j^2|R_i^{*j}|\/|R_{i-1}|$ by its equivalent $|\\Sigmab_i^{*j}|\/|\\Sigmab_{i-1}|$.\nWe summarize this result into Theorem~\\ref{thm:Chlsky}.   \n\\begin{thm} \\label{thm:Chlsky}\nLet $\\Sigmab_n$ be a nonsingular covariance matrix with entries $\\sigma_{ij} = \\sigma_i\\sigma_j \\rho_{ij}$. \nLet $\\sigmab_i^{*j} \\equiv \\left(\\sigma_{1j},\\sigma_{2j}, \\cdots, \\sigma_{i-1,j}\\right)$, \nset $|\\Sigmab_1^{*j}|\\equiv \\sigma_j^2$, $|\\Sigmab_0| \\equiv 1$,\nand $\\Sigmab_i^{*j} \\equiv \\left( \\begin{array}{cc} \\Sigmab_{i-1} & \\left(\\sigmab_i^{*j}\\right)^T\n \\cr \\sigmab_i^{*j} \n & \\sigma_i^2 \\end{array}\\right)$.\nThen, Cholesky factor $\\Lb = (l_{ij})_{i,j=1}^n$ for $\\Sigmab_n$ is given by $l_{ji} = 0$ (for $j < i$), and by  \n\\[\n      l_{ji} = \n      \\left\\{\\begin{array}{ll}\n           sign(\\rho_{ij(1,\\ldots,i-1)}) \\sqrt{|\\Sigmab_i^{*j}|\/|\\Sigmab_{i-1}| \n                 - |\\Sigmab_{i+1}^{*j}|\/|\\Sigmab_i|} & \\mbox{ if } j >  i \\\\\n           \\sqrt{|\\Sigmab_i|\/|\\Sigmab_{i-1}|}          &  \\mbox{ if } i = j            \\end{array}\\right. .\n    \n\\]\n\\end{thm}  \n\\textbf{Remark.} Mathematically, Theorem~\\ref{thm:Chlsky} might as well describe Cholesky factor for arbitrary nonsingular symmetric matrix, when allowing the entries of $\\Lb$ to be complex numbers. \n\n\n\\subsection{Two order conditions on the magnitudes of sub-determinants essential to a well defined $\\Lb$}\\label{sec:welldefL}\nWe start with the well known order relation between the magnitudes of successive determinants~\\cite[pp. 525]{Yule1907,StuartOrdArnold2010}\n\\begin{equation}\\label{det:order}\n    1 \\ge |\\Rb_2| \\ge |\\Rb_3| \\ge \\cdots \\ge |\\Rb_{n-1}| \\ge |\\Rb_n| > 0.\n\\end{equation}\n One may alternatively see the order (\\ref{det:order}) as a direct result of Eq.~(\\ref{eq:schur_det}) when adding equal signs to account for $(\\rhob_i = 0)$'s.\nTo the order~(\\ref{det:order}) we will add a second order that arises from the positivity of the right-hand side of the Eq.~(\\ref{eq:lem2}) and seems to be rather new. \nFor $j=2,3, \\ldots,n$,\n\\begin{equation}\\label{detratio:order}\n    1 \\ge |\\Rb_2^{*j}| \\ge |\\Rb_3^{*j}|\/|\\Rb_2|  \n                   \\ge \\cdots \\ge  \n    |\\Rb_{j-1}^{*j}|\/|\\Rb_{j-2}|\\ge  \n    |\\Rb_j|\/|\\Rb_{j-1}| > 0 .\n \\end{equation}\n It is possible to view the order relations (\\ref{det:order}) and (\\ref{detratio:order}) as posing necessary and sufficient conditions for the correlation matrix $\\Rb = \\Lb \\Lb^T$ to be positive-definite. \nSince both order relations follow from the positive-definiteness property of $\\Rb$, and on the other hand, any failure to satisfy any of the determinant ordering in (\\ref{det:order}) or (\\ref{detratio:order}) will lead to ill defined $\\Lb$. \nIn Section~\\ref{sec:randomcorr} we shall show how to use the conditions (\\ref{det:order}) and (\\ref{detratio:order}) to generate positive-definite random correlation structures. \n\n\n\n\n\\section{Application I  - A simple $t$-test for linear dependence}\\label{sec:ttest}  \nAs a first application we will establish a procedure for testing the linear dependence of a single variable upon other variables by employing the first parametrization of Cholesky factor. \nLet $\\xb = (\\xb_1,\\cdots,\\xb_p)$ be a matrix of $N$ samples from a $p$-variate random variable that is multinormally distributed. Assume that $N>p$ and let $\\hat\\Rb = (r_{ij})_{i,j=1}^p$ be the estimated correlation sample matrix and $\\{r_{ij(1,\\ldots,i-1)}\\}_{i \\le j}$ be the nonzero elements of Cholesky factor for $\\hat\\Rb$. \nSuppose that, for some $k<p$, we wish to test the linear dependence of $\\xb_p$ upon $\\xb_1,\\xb_2,\\cdots,\\xb_k$. \nConsider the test \n\\[ \n  H_{0k} : \\rho_{1p} = \\rho_{2p(1)} = \\cdots = \\rho_{kp(1,2,\\ldots,k-1)} = 0\n\\] \nagainst the alternative \n\\[\nH_{1k} : \\rho_{ip(1,2,\\ldots,i-1)} \\ne 0 \\mbox{ for some } i<k.\n\\]\nUnder $H_{0k}$, the estimator $r_{kp(1,2 , \\ldots ,k-1)}$ forms a simple $t$ statistic~\\cite{Morrison2004}\n\\[\n  T_{kp:N} = \\frac{\\sqrt{N-k} \\cdot r_{kp(1,2 , \\ldots ,k-1)}}\n                 {\\sqrt{1 - r_{kp(1,2 , \\ldots ,k-1)}^2}} \n                 {H_{0k} \\atop \\sim } \\; t_{N-k}.\n\\] \nThe hypothesis $H_{0k}$ can be rejected, at level $\\alpha$ ($0<\\alpha <1$), if $|T_{kp:N}|>t_{\\alpha\/2,N-k}$. \nOne may further establish a sequential testing procedure that searches for the largest $k$ for which $H_{0k}$ can be rejected. \n\n\\textbf{Remark.} We leave it to the reader to verify that the null hypotheses $H_{0k}$ is equivalent to $\\rho_{1p} = \\rho_{2p} = \\cdots = \\rho_{kp} = 0$.   \n \n\\section{Application II - Generating realistic random correlations}\\label{sec:randomcorr}\nThe problem of generating random correlation structures is well discussed at the literature~\\cite{MarsagaliaOlkin84,Joe2006,MittelbachMatthiesenJorswieck2012}.\nHowever, in practice, many of the suggested procedures are not so easy to apply~\\cite{Holmes1991}, and when applied, some typically fail to provide a sufficient number of realistic correlation matrices~\\cite{BohmHornik2014}. \nMore recent algorithms for the generation of random correlations either utilize a beta distributio\n~\\cite{Joe2006,LewandowskiEtAl2009}, \nor employ uniform angular values~\\cite{RapisardaBrigoMercurio2007,RebonatoJackel2007,MittelbachMatthiesenJorswieck2012}.\nThe algorithm we will suggest in this section will be considerably simple. \nIt will be based on uniform values that are assigned to reflect the ratios $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i\\le j}$ which constitute the parametrization~(\\ref{chol:L2}). \nThe order of the values of $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i \\le j}$ will be chosen to preserve the ordering in (\\ref{det:order}) and (\\ref{detratio:order}), to ensure the positive-definiteness of $\\Lb\\Lb^T$. \nWe will start by choosing $n-1$ random uniform values in $(0,1]$, that will be further assigned by their size to reflect the determinants $\\{|\\Rb_j|\\}_{j=2}^n$, as directed by (\\ref{det:order}). \nThe diagonal of $\\Lb$ will be constructed from the ratios $\\{|\\Rb_j|\/|\\Rb_{j-1}|\\}_{j=1}^n$.  \nThen, for each row of $\\Lb$, $j$, an additional set of $j-2$ random uniform values will be chosen to serve as the ratios $\\{|\\Rb_i^{*j}|\/|\\Rb_{i-1}|\\}_{i=2}^{j-1}$, organized to keep the order as in (\\ref{detratio:order}).\nThe signs $s_{ij}$ will be chosen to be $(-1)^{Bernoulli(0.5)}$ and the matrix $\\Lb$ will be computed according to (\\ref{chol:L2}).  \n\n\\begin{enumerate}[Step 1.]\n\\item[\\textbf{Algorithm}] \\textbf{for generating realistic random correlation matrix $\\Lb\\Lb^T$}.\n\\item{\\textbf{Diagonals:}} Choose $n-1$ random uniform values from the interval $(0,1]$. \nOrder them in decreasing order, $U_{(2)} \\ge U_{(3)} \\ge \\cdots \\ge U_{(n)}$, \n and set $l_{11} = 1$, $l_{22}=\\sqrt{U_{(2)}}$, and $l_{jj} = \\sqrt{U_{(j)}\/U_{(j-1)}}$ for $j=3,\\ldots,n$;\n\\item{\\textbf{Rows:}} Repeat this step for each $j = 2,3, \\ldots, n-1$. Set $U_{(1)}^{(j)}\\equiv 1$ and $U_{(j)}^{(j)}\\equiv l_{jj}^2$.\n For $j\\ge 3$, choose $j-2$ more (additional) random uniform values $\\{U_i^{(j)}\\}_{i=2}^{j-1}$ inside $[l_{jj}^2,1]$ and sort them in decreasing order $U_{(2)}^{(j)} \\ge \\cdots \\ge U_{(j-1)}^{(j)}$.\nCompute $l_{ji} = \\sqrt{U_{(i)}^{(j)} - U_{(i+1)}^{(j)}}$ for $i=1,2,\\ldots,j-1$;\n\\item{\\textbf{Signs:}} Choose $n(n-1)\/2$ Bernoulli$(0.5)$ values $B_{ij}$'s for $j>i$, and multiply each $l_{ji}$ by $s_{ij} = (-1)^{B_{ij}}$;\n\\item Compute $\\Rb$ by $\\Lb \\Lb^T$ to obtain the actual correlation structure.\n\\end{enumerate}\n\n\n\\subsection{Generating random AR(1) structures}\\label{sec:AR1}\nWe end this paper by revealing the simple form of Cholesky factor for the $AR(1)$ structure. The AR(1) correlation matrix is defined by $\\rho_{ij} = \\rho^{|i-j|}$, \nand it is possible to verify that $|\\Rb_n| = (1-\\rho^2)^{n-1}$, $\\rhob_i^{*j} = \\rho^{j-i}\\rhob_i$ and     \n\\[ \n  \\rho_{ij} - \\rhob_i^{*j}\\Rb_{i-1}^{-1}\\rhob_i^T \n  = \\rho^{j-i}\\left(1-\\rhob_i\\Rb_{i-1}^{-1}\\rhob_i^T\\right) \n  = \\rho^{j-i} |\\Rb_i|\/|\\Rb_{i-1}|.\n\\]\nCholesky factor for the AR(1) structure enjoys the simple form: \n\\[\nl_{ji} = \\left\\{\\begin{array}{cc}\n\\rho^{j-1} & j \\ge i=1 \\cr\n\\rho^{j-i}\\sqrt{1-\\rho^2} & j \\ge i \\ge 2\n\\end{array}\\right. \n\\] \nHence, for any choice of $\\rho$, $|\\rho| < 1$, it is possible to transform the standard normals $(X_i)_{i=1}^n$ into the autocorrelated normals \n\\[\n  Y_i = \\rho^{i-1} X_1 + \\sqrt{1-\\rho}\\sum_{k=2}^i \\rho^{i-k} X_k, \n      \\qquad i = 1, \\ldots,n.\n\\] \n\n\\section*{Acknowledgments} I am thankful to the Editor and the Reviewers for their helpful discussion and their literature suggestion. Special thanks to Professor Sen for encouraging me to submit this paper and for his resourceful suggestion to add a test of hypotheses for the application part. Thanks to Professor Speed for introducing me to Yule's paper, and to Dr. Batista for her proofreading.  \n\n\\section*{References}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section*{Appendix \\theappndx: #1} }\n\n\\def\\ba#1{\\begin{array}{#1}}\n\\def\\end{array}{\\end{array}}\n\\def\\beq#1{\\begin{equation}\\label{#1}}\n\\def\\end{equation}{\\end{equation}}\n\\newcommand{\\ab}[1]{\\left\\langle{#1}\\right\\rangle}\n\\newcommand{\\mathrm{const}\\,}{\\mathrm{const}\\,}\n\\newcommand{\\mathrm{dist}\\,}{\\mathrm{dist}\\,}\n\\newcommand{\\displaystyle}{\\displaystyle}\n\\newcommand{\\mathbb{R}}{\\mathbb{R}}\n\\newcommand{\\mathbb{N}}{\\mathbb{N}}\n\\newcommand{\\mathbb{Q}}{\\mathbb{Q}}\n\\newcommand{\\mathbb{Z}}{\\mathbb{Z}}\n\\newcommand{\\mathbf{e}}{\\mathbf{e}}\n\\newcommand{\\mathbf{x}}{\\mathbf{x}}\n\\newcommand{\\mathbf{r}}{\\mathbf{r}}\n\\newcommand{\\mathbf{y}}{\\mathbf{y}}\n\\newcommand{\\mathbf{z}}{\\mathbf{z}}\n\\newcommand{\\varepsilon}{\\varepsilon}\n\\newcommand{\\tilde\\alpha}{\\tilde\\alpha}\n\\newcommand{\\tilde\\gamma}{\\tilde\\gamma}\n\\newcommand{\\tilde\\xi}{\\tilde\\xi}\n\\newcommand{\\dlnxi}{\\lambda}\n\\newcommand{\\alpha_{\\infty}}{\\alpha_{\\infty}}\n\\newcommand{X^{0}}{X^{0}}\n\\newcommand{X^{\\!\\times}}{X^{\\!\\times}}\n\\newcommand{t^{0}}{t^{0}}\n\\newcommand{t^{\\ell}}{t^{\\ell}}\n\\newcommand{t^{r}}{t^{r}}\n\\newcommand{\\stackrel{\\text{\\scriptsize\\rm def}}{=}}{\\stackrel{\\text{\\scriptsize\\rm def}}{=}}\n\n\\newcommand{X^*}{X^*}\n\\newcommand{X^{**}}{X^{**}}\n\n\n\\newcommand{g_{\\mathrm{asym}}}{g_{\\mathrm{asym}}}\n\n\\newcommand{\\color{black}\\scriptsize}{\\color{black}\\scriptsize}\n\\newcommand{\\color{blue}\\scriptsize}{\\color{blue}\\scriptsize}\n\n\\newcommand{\\eop}{\\hfill$\\Box$\n  \n\\usepackage{pgf,tikz,pgfplots}\n\\usepackage{mathrsfs}\n\\usetikzlibrary{arrows}\n\n\\title{Precise asymptotics with log-periodic term in an elementary optimization problem}\n\n\\author{Sergey Sadov\\footnotemark[1]\\footnote{E-mail: serge.sadov@gmail.com}}\n\\date{}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nThe function $\\inf_n nx^{1\/n}$ has the asymptotics $eu+e d^2(u)\/(2u)+O(1\/u^2)$ as $x\\to\\infty$, where\n$u=\\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. \n\nFirst,\nthe curves $y=nx^{1\/n}$ can be written parametrically\nas $\\log x=nt$, $y=nt$. In general, let \n$(u_n(t),v_n(t))$ be a family of parametric\ncurves  with asymptotics \n$u_n=n p_1(t)+q_1(t)+r_1(t)\/n+O(1\/n^2)$ and \n$v_n=n p_2(t)+q_2(t)+r_2(t)\/n+O(1\/n^2)$.\nSuppose the function $p_1(t)\/p_0(t)$ has a unique \nnondegenerate minimum in the parameter domain. \nIt is shown that the asymptotics of their lower envelope\n$v(u)=\\inf_{n,t} v_n(t)$ while $u=u_n(t)$, has the asymptotics of the form $v(u)=a_0 u+a_1+\\Phi(u)\/u+O(1\/u^2)$, where $\\Phi(\\cdot)$ is an affinely transformed function $d^2(\\cdot)$.\n\nSecond, note that $nx^{1\/n}$ is the minimum\nof the sum $t_1+t_2\/t_1+\\dots+t_{n}\/t_{n-1}$ subject to the constraint $t_n=x$. We consider a similar\nasymptotic problem for the sums $t_1+t_2\/(t_1+1)+\\dots+t_n\/(t_{n-1}+1)$. \nLet $F_n(x)$ is the minimum value of the $n$-term sum under the constraint $t_n=x$. Define $F(x)=\\inf_n F_n(x)$. We show that $F(x)=eu-A+e d^2(u+b)\/(2u)+O(1\/u^2)$, $u=\\log x$, with certain numerical constants $A$ and $b$. We present alternative forms of this optimization problem, in particular, \na ``least action'' formulation. Also we find the asymptotics $F_n^{(p)}(x)=e\\log n-A(p)+O(1\/\\log n)$\nfor the function arising from the sums with denominators of the form $t_j+p$ with arbitrary $p>0$ and establish\nsome facts about the function $A(p)$. \n\n\\medskip\n{\\em Keywords}: AM-GM inequality, asymptotics, dynamic programming, enveloping curve, recurrence relations.\n\n\\medskip\nMSC: \n26D15,\n26D20 \n\\end{abstract}\n\n\n\\section{Statement of results}\n\\label{sec:intro}\n\nThe main result of this work concerns the asymptotic behaviour as $x\\to+\\infty$ of the function\n$$\n F(x)=\\inf_{n\\in\\mathbb{N}} F_n(x),\n$$\nwhere\\footnote{\nThe function $F(x)$ in another guise (see Proposition~\\ref{prop:altoptfn}) appeared in the study of a certain cyclic inequality\n\\cite{Sadov_2022_maxcyc}, which motivated this paper.} \n\\begin{align}\n&F_n(x)=\\inf_{t_1,\\dots,t_{n-1}\\geq 0} S(t_1,\\dots,t_{n-1},x),\n\\label{op_main}\n\\\\[1ex]\n& S(t_1,\\dots,t_n)= t_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{t_n}{t_{n-1}+1}\n\\label{S1}\n\\end{align}\nIn other words, $C=F(x)$ is the best constant, independent of $n$ and $\\{t_j\\}$, in the inequality\n$$\n t_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\\geq C.\n$$ \n\nWe will use the notation\n$$\n \\ab{x}=\\mathrm{dist}\\,(x,\\mathbb{Z}).\n$$\nThe function $x\\mapsto\\ab{x}$ is a $1$-periodic piecewise-linear, continuous function oscillating between $0$ and $1\/2$.\n\n\\begin{theorem}\n\\label{thm:main}\nLet\n$\n u=\\log x.\n$\nThere exist numerical constants\n$$\n\\ba{l}\n A\\approx 1.7046560372,\n\\\\\n b\\approx 0.6973885601, \n\\end{array}\n$$\nsuch that \nthe function $F(x)$\nhas the asymptotics \n\\begin{equation}\n\\label{asf}\nF(x)=eu-A+\\frac{e}{2}\\,\\frac{{\\ab{u+b}}^2}{u}\n+O\\left(\\frac{1}{u^2}\\right)\n\\end{equation}\nas $x\\to\\infty$.\n\\end{theorem}\n\nTheorem~\\ref{thm:main} will be better understood \nin the context of two \ntheorems stated below: \nTheorem~\\ref{thm:AMGM}, which is a simpler result of the same kind, and Theorem~\\ref{thm:paramcurves} of a technical nature, which\nshows that a periodic (up to a small error)\nremainder term appears in a class of optimization problems\nconcerning the lower envelope of parametric curves. \nThe asymptotic formula \\eqref{asf} will be eventually obtained through Theorem~\\ref{thm:paramcurves}.\n\n\\medskip\nLet us consider a simpler analogue of the functions\n$F_n(x)$: \n\\begin{equation}\n\\label{ep_AM-GM}\nF^{(0)}_n(x)=\\inf_{t_1,\\dots,t_{n-1}>0} \\left(\n t_1+\\frac{t_2}{t_1}+\\dots+\\frac{t_{n-1}}{t_{n-2}}+\\frac{x}{t_{n-1}}\n \\right).\n\\end{equation}\n\nThe definition of $F^{(0)}_n(x)$ can be written as\n$$\n F^{(0)}_n(x)=\\inf_{a_1,\\dots,a_{n}>0}\n(a_1+\\dots+a_n)\n\\quad\\text{subject to $\\;a_1\\cdot\\dots\\cdot a_n=x$}.\n$$\n\nOne recognizes the constrained optimization problem associated with the inequality between the arithmetic and geometric means, hence\n$$\n F^{(0)}_n(x)=n x^{1\/n}.\n$$ \n\n\\begin{theorem}\n\\label{thm:AMGM}\nLet $u=\\log x$. The function\n$$\n F^{(0)}(x)=\\inf_{n\\in\\mathbb{N}} F^{(0)}_n(x)\n$$\nhas the asymptotics\n\\begin{equation}\n\\label{asf0}\n F^{(0)}(x)=eu+\\frac{e}{2}\\,\\frac{\\ab{u}^2}{u}+O\\left(\\frac{1}{u^2}\\right)\n\\end{equation}\nas $x\\to\\infty$.\n\\end{theorem}\n\n\\begin{figure}\n\\begin{picture}(260,175)\n\\put(0,0){\\includegraphics[scale=0.5]{f0_approximation}}\n\\put(265,16){$u$}\n\\end{picture}\n\\caption{Illustration of Theorem~\\ref{thm:AMGM}. Blue line: the function $u\\mapsto F^{(0)}(x)-e u$, $u=\\log x$. Gray line:\nthe correction term $(e\/2){\\ab{u}}^2\/u$.}\n\\label{fig:f0corr}\n\\end{figure}\n\nThe asymptotic approximation \\eqref{asf0} is illustrated in Fig.~\\ref{fig:f0corr}.\n\n\n\\medskip\nTheorems~\\ref{thm:main} and \\ref{thm:paramcurves} both\ndeal with a function defined as infimum over a family.\nFor instance, the graph of the function $F^{(0)}(x)$ is the lower envelope of the curves $y=nx^{1\/n}$, which can be written parametrically as $\\log x=nt$, $y=ne^t$. From this point of view Theorem~\\ref{thm:AMGM} is an easy consequence of our next theorem. To keep things simple where possible, we will prove Theorem~\\ref{thm:AMGM} directly; however, Theorem~\\ref{thm:paramcurves} will be fully relevant for the proof of Theorem~\\ref{thm:main}.\nThe relevance of the general asymptotic pattern\n\\eqref{asfabs} is apparent already.  \n\n\n\\begin{theorem}\n\\label{thm:paramcurves}\nConsider a family of parametric curves\n$u=\\xi_n(t)$, $v=\\eta_n(t)$, $t\\in I$, where $I$ is a segment of the real line. \nSuppose the functions $\\xi_n(t)$, $\\eta_n(t)$ have the asymptotic behaviour%\n$$\n\\ba{l}\n\\displaystyle\n \\xi_n(t)=p_0(t) n+ q_0(t)+ r_0(t)n^{-1}+ O(n^{-2}),\n\\displaystyle\n\\\\\n \\eta_n(t)=p_1(t) n+ q_1(t)+ r_1(t)n^{-1}+ O(n^{-2})\n\\end{array}\n$$\nas $n\\to\\infty$, uniformly in $t\\in I$.\n\nLet $v=f(u)$ be the lower envelope of the curves so defined. That is, given $u$, we determine the set%\n\\footnote{This set is nonempty if $u$ is sufficiently large.}\nof those $n$ for which the equation $\\xi_n(t)=u$ has a solution and define\n$$\n f(u)=\\inf_{(n,t):\\, \\xi_n(t)=u} \\eta_n(t).\n$$\n\nDenote\n$$\n \\beta(t)=\\frac{p_1(t)}{p_0(t)}, \n\\qquad \n\\delta(t)=p_0(t)r_1(t)-p_1(t)r_0(t).\n$$\nWe make the following assumptions.\n\n\\smallskip\n{\\rm(i)} The \ncoefficients $p_i(\\cdot)$, $q_i(\\cdot)$, $r_i(\\cdot)$ are of class $C^2(I)$. \n\n\\smallskip\n{\\rm(ii)}  $p_0(t)>0$, $p_1(t)>0$, $p_0(t)'>0$ on $I$.\n\n\\smallskip\n{\\rm(iii)} The function $\\beta(t)$ has the unique point of minimum $t=t_0$ on $I$ and\n$$\n\\beta(t)=b_0+\\frac{b_2}{2}(t-t_0)^2+o(t-t_0)^2\n\\quad \\text{near $t_0$}.  \n$$\n\nThen\n\\beq{asfabs}\n f(u)=a_0 u+a_1+\\frac{\\Phi(u)}{u}+O\\left(\\frac{1}{u^2}\\right)\n\\quad\\text{as $u\\to\\infty$},\n\\end{equation}\nwhere\n\\beq{coefa0a1}\n a_0=b_0,\n\\qquad\n a_1=q_1(t_0)-b_0 q_0(t_0),\n\\end{equation}\nand\n\\begin{align}\n& \\Phi(u)=a_2+a_3\\ab{\\frac{u-q_0(t_0)}{p_0(t_0)}}^2,\n\\label{asgenPhi}\n\\\\[2ex]\\displaystyle\n& a_2\n=-\\frac{(q_1'(t_0)-b_0 q_0'(t_0))^2}{2b_2}+\\delta(t_0),\n\\qquad\na_3=\\frac{b_2}{2}\\,\\left(\\frac{(p_0(t_0))^2}{p_0'(t_0)}\\right)^2.\n\\label{coefa2a3}\n\\end{align}\n\\end{theorem}\n\nProof of Theorem~\\ref{thm:paramcurves} is given in Appendix~\\ref{app:paramcurves}.\n\n\\medskip\nOur final theorem concerns an interpolation between\nTheorems~\\ref{thm:main} and~\\ref{thm:AMGM}.\n\nIndeed, it is natural to treat the functions $F^{(0)}(x)$\nand $F(x)=F^{(1)}(x)$ as members of a one-parameter family\nof functions\n$$\n F^{(p)}(x)=\\inf_{n\\in\\mathbb{N}} F^{(p)}_n(x)\n$$\nwith any $p\\geq 0$,\nwhere\n$$\n F^{(p)}_n(x)=\\inf_{t_1,\\dots,t_{n-1}\\geq 0,\\;t_n=x} \nS^{(p)}_n\n(t_1,\\dots,t_n),\n$$\n\\begin{equation}\n\\label{Snpt}\n S^{(p)}_n(t_1,\\dots,t_n)=t_1+\\sum_{j=2}^{n}\\frac{t_j}{t_{j-1}+p}.\n\\end{equation}\nIn particular,\n$$\n F^{(p)}_1(x)=x,\\qquad F^{(p)}_2(x,p)=\\inf_{t\\geq 0}\\left(t+\\frac{x}{t+p}\\right)=\\begin{cases}\n2\\sqrt{x}-p, \\quad x\\geq p^2,\\\\\nx\/p, \\qquad x\\leq p^2.\n\\end{cases}\n$$\n\nWe will not chase the asymptotics of $F^{(p)}(x)$ as precisely as we did in Theorems~\\ref{thm:main}--\\ref{thm:AMGM} but rather\nfocus on the constant term in the asymptotics. In particular, we determine its behaviour as $p\\to +0$. We also reveal that the parameter value $p=1$ separates the regions with different analytic form of $F^{(p)}$: it is much simpler for $p>1$. This latter fact gives an additional support to our attention to the function $F(x)=F^{(1)}(x)$ as the main subject of this paper. \n\n\\begin{theorem}\n\\label{thm:genpar}\nLet $u=\\log x$. \n\n\\smallskip\n{\\rm(a)} For any $p>0$ \n\\begin{equation}\n\\label{Fpasym}\n F^{(p)}(x)=eu-A(p)+O(u^{-1}) \\quad\\text{as $x\\to\\infty$}.\n\\end{equation}\nThe function $A(\\cdot)$ is increasing.\n\n\\smallskip\n{\\rm(b)} If $p>1$, then\n\\begin{equation}\n\\label{Fpgt1}\n F^{(p)}(x)=F\\left(\\frac{x}{p}\\right),\n\\end{equation}\nso $A(p)=A+e\\log p$.\n\n\\smallskip\n{\\rm(c)} If $0<p<1$, then\nthe function $A(p)$ satisfies the functional equation\n\\begin{equation}\n\\label{feq-Ap}\n A(p)=\\max_{1\\leq u\\leq e} \\left(A\\left(\\frac{p}{u}\\right)-u+e\\log u\\right)+p.\n\\end{equation}\nThe asymptotics of $A(p)$ as $p\\to +0$ is\n$$\nA(p)=\\frac{p}{1-e^{-1}}+o(p).\n$$\n\\end{theorem}\n\n\n\\medskip\nThe plan of the paper is as follows.\n\n\\smallskip\nIn Section~\\ref{sec:variants} we describe a few guises in which the optimization problem leading to the functions $F_n(x)$ or $F^{(p)}_n(x)$ may arise. In particular,\nwe discuss a ``least action formulation'' and introduce a recurrence that determines the ``extremal trajectory'' for each $n$.\n\n\\smallskip\nIn Section~\\ref{sec:simpleproofs} we prove what is easy to prove: Theorem~\\ref{thm:AMGM} and the crude asymptotics\n$F(x)=e\\log n+O(1)$. \n\n\\smallskip\nIn Section~\\ref{sec:minimizers} we begin the analysis\nof candidate minimizers (extremal trajectories) for\nthe main problem. We identify some special points\nfor every $n$ that play specific roles in the analysis.\nWe illustrate the introduced concept with graphs\nand point out some pitfalls.\n\n\\smallskip\nIn Section~\\ref{sec:proofmain} the facts observed in\nSection~\\ref{sec:minimizers} are fully formalized and proved and the proof of Theorem~\\ref{thm:main} is completed by reducing it to a form suitable for application of Theorem~\\ref{thm:paramcurves}.\n \n\\smallskip\nIn Section~\\ref{sec:Ap} we prove Theorem~\\ref{thm:genpar}\nand describe an efficient algorithm, based on a recurrence,\nto tabulate the function $A(p)$ for $0<p<1$.\n\n\\smallskip\nThere are two appendices containing proofs technically unrelated to the main part of the paper.\n\n\\smallskip\nIn Appendix~\\ref{app:paramcurves} Theorem~\\ref{thm:paramcurves} is proved. \n\n\\smallskip\nIn Appendix~\\ref{app:alpha} we rigorously prove a visually obvious property (Proposition~\\ref{prop:alpha})\nof a function $\\alpha_\\infty(x)$, which is important in the main proof.\n\nThe proof uses contour integration for estimating derivatives of a real function.\nSome elements of the proof refer to numerical evaluations but always in a ``provable'' way. \n\n\n\\iffalse\nSolving an optimization problem $S(\\mathbf{t})\\mathop{\\to}\\min$ is equivalent to establishing the inequality $S(\\mathbf{t})\\geq f$\nwhere the value $f=S_{min}$ is best possible.\nThroughout the paper the letters $S$ and $f$, possibly with sub\/superscripts, will designate, respectively,\nthe objective functions and their optimal values.\nThe optimization variables --- the components of \nthe vector $\\mathbf{t}$ --- will be assumed nonnegative (in denominators --- positive).\n\nIn the said sense, the classical inequality between the arithmetic and geometric means (AM-GM) can be considered as a closed form solution of the constrained optimization problem\n\\beq{cons0}\nS(\\mathbf{t}) \\stackrel{\\text{\\scriptsize\\rm def}}{=} \nt_1+\\dots+t_n\\to \\min \\quad\\text{\\rm subject to}\\;\\; t_1\\cdot\\dots\\cdot t_n=x;\n\\end{equation}\nthe minimum value is\n$$\nf_n^{(0)}(x)=nx^{1\/n}.\n$$\n\n\nIn this paper we study a recurrence that has a similar look to \\eqref{recurf0}:\n\\beq{recurf}\n f_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y+1}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith initial condition identical to that for $f_1^{(0)}$:\n\\beq{recurf1}\n f_1(x)=x.\n\\end{equation}\n\nFor example, \n\\beq{f2}\n f_2(x)=\\min_{y\\geq 0}\\left(y+\\frac{x}{y+1}\\right)\n=\n\\begin{cases}\nx,\\; \\text{if $x\\leq 1$},\\\\\n2\\sqrt{x}-1,\\; \\text{if $x> 1$}.\n\\end{cases}\n\\end{equation}\nA longer calculation shows that\n$$\n f_3(x)=\n\\begin{cases}\nf_2(x),\\; \\text{if $x\\leq 4$},\\\\\n-1+2u+\\sqrt{\\frac{x}{u}}=3x^{1\/3}+O(1),\\; \\text{if $x> 4$},\n\\end{cases}\n$$ \nwhere $u=u(x)$ is the larger (of the two) positive root of the equation $(u^2+1)^2=xu$.\n\nA legitimate quibble concerning the use of the symbol `$\\min$' as opposed to `$\\inf$' in \\eqref{recurf} will be addressed in Sec.~\\ref{sec:prelim}.\n\nOne simple fact about the sequence $(f_n(x))$ is that it is nonincreasing (Proposition~\\ref{prop:monotfn}). It allows one to define the function \n\\beq{deflimf}\n f(x)=\\lim_{n\\to\\infty} f_n(x)=\\inf_{n\\geq 1} f_n(x).\n\\end{equation}\nMoreover, we show (Proposition~\\ref{prop:fnstab}) that for any fixed $x>0$ the sequence $(f_n(x))$ stabilizes: \nthere exists a non-decreasing function $x\\mapsto \\nu(x)$ taking values in positive integers such that\n$$\n f(x)= f_n(x)\n, \\quad \\forall n\\geq \\nu(x).\n$$\n\nOur main result \nconcerns the asymptotics of $f(x)$ as $x\\to\\infty$.\n\nTo compare, the sequence $f_n^{(0)}(x)=nx^{1\/n}$\nis not monotone. It has a minimum for every fixed $x>0$; the function\n\\beq{defminf0}\n f^{(0)}(x)=\\min_{n\\geq 1} f_n^{(0)}(x),\n\\end{equation}\nhas the asymptotics $f^{(0)}(x)=e\\log x+O(1\/\\log x)$,\nsame as in \\eqref{asf1} with constant term $A=0$; also likewise in \\eqref{asf1}\nthe remaider is not $o(1\/\\log x)$. This is a simple fact, see Proposition~\\ref{prop:asf0} in Sec.~\\ref{sec:prelim}.\nA somewhat more elaborate yet also rather quick result\nis\nthe crude\nasymptotic formula \n$f(x)=e\\log x+O(1)$ (Proposition~\\ref{prop:crudeas} in Sec.~\\ref{ssec:crudeas}).\nA proof of the two-term asymptotics \\eqref{asf1} requires much more effort.\n\nThe described analogy is meant to justify the attention to the function $f(x)$ and the sequence $(f_n(x))$ in the eyes of a reader unconcerned about author's contemplation.\n\nIn truth, my motives to\n(a) study the function $f(x)$, and (b) work out its asymptotics in the precise form, were different. \n\nFor (a), the function $f(x)$ --- and the need to analyse its asymptotic behavious --- appeared in my study \\cite{Sadov_2022maxcyc} of a certain cyclic inequality (or, better, an extremal problem for a cyclic sum) that, unlike a number of previously published variations, allows for ``uncycling''. \nAs a matter of fact, the definition of the function $f(x)$ in \\cite{Sadov_2022maxcyc} is not quite the same as above but equivalent, see \\eqref{fe1} in Sec.~\\ref{sec:prelim}. The equivalence is proved in\nProposition~\\ref{prop:funeqf}.\n\nAs concerns (b), it is a purely aesthetical or sporting\ndrive. A result of a similar nature is proved in my recent paper \\cite{Sadov_2021}.\nIn spite of some semblance of the optimization problems --- see Remark in Sec.~\\ref{ssec:leastaction} --- technical parallels between that case and the present one don't go far\nand similarity is more in spirit. The eventual proof of the asymptotic formula in both cases depends on the analysis of trajectories of discrete dynamical systems determining the critical points of the objective functions. That analysis, in essense, consists of numerous steps, some plain and some delicate,\njustifying visually observed properties or relevant functions and sequences, such as monotonicity and convexity.\n\\fi\n\n\\iffalse \n--------------\n\n\nPrevious version \n\n\nLet $f(x)=x$ for $0<x\\leq 1$ and define $f(x)$\nin the intervals $(1,2], (1,3], \\dots$ recursively by means of  \nthe functional equation\n\\beq{fe1}\n f(x)=\\min_{0< y<x-1}\\left(f(y)+\\frac{x}{y+1}\\right).\n\\end{equation}\n\nFor example, if $1<x\\leq 2$, then\n$$\n f(x)=\\min_{0< y<x-1}\\left(y+\\frac{x}{y+1}\\right)\n=-1+\\min_{1<\\tilde y<x}\\left(\\tilde y+\\frac{x}{\\tilde y}\\right)\n=2\\sqrt{x}-1.\n$$\n\nThe main result \nconcerns the asymptotics of $f(x)$ as $x\\to\\infty$.\n\n\\begin{theorem}\n\\label{thm:mainasym}\nThere exists a constant \n$$\n A\\approx 0.70465603\n$$\nsuch that\n\\beq{asf1}\n f(x)=e\\log x-A+ O\\left(\\frac{1}{\\log x}\\right)\n\\end{equation}\nas $x\\to\\infty$.\nThe symbol $O$ in the remainder term\ncannot be replaced by $o$.\n\\end{theorem}\n\nAlong with function $f(x)$ we consider the sequence of functions $f_n(x)$, $n=1,2,\\dots$, defined recurrently for $x>0$\nby \n\\beq{recurf}\n f_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y+1}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith initial condition\n$$\n f_1(x)=x.\n$$\nWe will show that there exists a non-decreasing function $x\\mapsto \\nu(x)$ taking values in positive integers such that\n$$\n f(x)=\\lim_{n\\to\\infty} f_n(x)=\\min_{n\\geq 1} f_n(x)\n=f_{k}(x), \\quad \\forall k\\geq \\nu(x).\n$$\n\nThe recurrence \\eqref{recurf} has a similar look to the recurrence\n\\beq{recurf0}\n f^{(0)}_n(x)=\\min_{y\\geq 0}\\left(f_{n-1}(y)+\\frac{x}{y}\\right),\n\\quad n\\geq 2,\n\\end{equation}\nwith the initial condition \n$f^{(0)}_1(x)=x$.\nOne recognizes the latter as a \ndynamical programming formulation of the constrained optimization problem\n\\beq{cons0}\nf^{(0)}_n(x)=\\min_{t_1\\cdot\\dots\\cdot t_n=x} (t_1+\\dots+t_n),\n\\end{equation}\nThe classical inequality between the arithmetic and geometric means (AM-GM) amounts to the fact that \n$$\nf_n^{(0)}(x)=nx^{1\/n}.\n$$\nIn \\cite[\\S~7]{BeckBel_1961} a ``dual'' dynamical programming formulation is considered, in which one maximizes the product of $n$ undeterminates subject to the prescribed value of their sum.\n\nThe function \n$$\n f^{(0)}(x)=\\min_{n\\geq 1} f_n^{(0)}(x)\n$$\nhas the asymptotics $f^{(0)}(x)\\sim e\\ln x$\nsimilar to \\eqref{asf1}. However, $f^{(0)}(x)\\neq \\lim_{n\\to\\infty}f_n^{(0)}(x)$. \nSome mark points in our analysis of the\nfunctions $f_n(\\cdot)$ are analogous to their much simpler counterparts pertaining to the functions $f_n^{(0)}(x)$. \n\nThe described analogy is meant to justify the attention to the function $f(x)$ and the sequence $(f_n(x))$ in the eyes of a general reader,\nand, from the author's perspective, to justify a\nseparate title for an item that can be of interest in its own right notwithstanding motives that brought it\ninto being.\n\nIn truth, I needed Theorem~\\ref{thm:mainasym} as part of  the study of a certain cyclic inequality (or, better, an extremal problem for a cyclic sum) that allows for ``uncycling'' \\cite{Sadov_2022maxcyc}. \n\nLastly, and to mention the author's other internal motive, let me point out the possibility to treat the function $f_n(x)$ in terms of\nthe ``least action principle'' with discrete time,\n\\beq{lapfn}\n f_n(x)=\\min_{\\{(t_0,\\dots,t_n)\\mid t_0=0,t_n=x\\}}\nS_n(t_0,\\dots,t_n).\n\\end{equation}\nThe ``action function'' $S_n(\\dots)$ is analogous to the action in classical mechanics (the integral of the Lagrangian along a virtual trajectory)\n\\beq{actionn}\n S_n(t_0,t_1,\\dots,t_n)=\\sum_{j=1}^n L(t_{j-1},t_j).\n\\end{equation}\nThe Lagrangian in the present case is\n\\beq{lagr1}\n L(p,q)=\\frac{p}{q+1}.\n\\end{equation}\nAn asymptotic problem of the similar nature, with the Lagrangian\n$$\n L(p,q)=q+\\frac{1+p}{q}\n$$\nhas been studied in the author's recent paper \\cite{Sadov_2021}. The corresponding asymptotics of the constrained minimum defined as in \\eqref{lapfn} is\n$\\min S_n=3n-C+O(\\rho^{-n})$ with specific constants $C$ and (best possible) $\\rho<1$.\nThere, as well as here, we do not attempt to generailze (introducing more parameters, say), but rather focus on the precise estimate of the remainder term in the asymptotics.\nIn both cases the analysis of optimal trajectories plays the crusial role and the proof, in essence, consists of steps justifying experimentally observed properties or relevant functions and sequences (monotonicity, convexity). However, in spite of some visual resemblance of the two Lagrangians, the details are very different.\n\n\n\n\nThe plan is as follows.\nWe begin with some simple observations concerning the functions $f(x)$, $f_n(x)$, and their relations with\n$f^{(0)}(x)$, $f_n^{(0)}(x)$. Also in Sec.~\\ref{sec:prelim} we prove the crude asymptotics\n$f(x)=e\\log x+O(1)$, which is much simpler result than\nTheorem~\\ref{thm:mainasym}.\nThen in Sec.~\\ref{sec:proofmain} we prove Theorem~\\ref{thm:mainasym} modulo an auxiliary theorem concerning the properties of a function that determines the ``asymptotic shape'' of the optimal trajectories for the problem \\eqref{lapfn}.\nThat auxiliary theorem is proved in Sec.~\\ref{sec:proofaux}. Some elements of the (admittedly boring) proof rely on numerical evaluations; however, we never recourse to an ``experimental evidence'' as the last argument. \n\n\\fi\n\n\\section{Alternative forms of the extremal problem}\n\\label{sec:variants}\n\n\\subsection{Prototype: formulations for the AM-GM}\n\nThe AM-GM optimization problem as stated in \\eqref{ep_AM-GM} involves\nonly ``soft'' constraints $t_j>0$.\nThe objective function can be written more symmetrically by introducing two extra indeterminates $t_0$ and $t_n$ and subjecting them to the artificial, boundary-value constraints:  \n\\begin{align}\n& F^{(0)}_n(x)=\\min_{\\mathbf{t}>0,\\; t_0=1,t_n=x} \\hat S_n^{(0)}(t_0,t_1,\\dots,t_n),\n\\label{optfn0}\n\\\\[3ex]\n& \\hat S_n^{(0)}(t_0,t_1,\\dots,t_n)\\stackrel{\\text{\\scriptsize\\rm def}}{=}\n\\frac{t_1}{t_0}+\\frac{t_2}{t_1}+\\dots+\\frac{t_{n-1}}{t_{n-2}}+\\frac{t_n}{t_{n-1}}\n\\label{L0n}\n.\n\\end{align}\n\nA proof of the AM-GM inequality by R.~Bellmann's dynamic programming approach%\n\\footnote{\\cite[\\S~7]{BeckBel_1961} presents a dual dynamic programming formulation of the AM-GM inequality, in which one maximizes the product of $n$ indeterminates subject to the prescribed value of their sum.}\n amounts to replacing the multivariate optimization problem by\na sequence of univariate optimization problems, \nwhere the functions $F_n^{(0)}$ are defined recurrently by\nputting $F^{(0)}_1(x)\\stackrel{\\text{\\scriptsize\\rm def}}{=} x$ and \n\\begin{equation}\n\\label{recurf0}\n F^{(0)}_n(x)=\\min_{y> 0}\\left(F_{n-1}^{(0)}(y)+\\frac{x}{y}\\right),\n\\quad n\\geq 2.\n\\end{equation}\n\n\\smallskip\n\n\\subsection{Formulation with boundary constraints}\n\\label{ssec:bconst}\n\nThe constrained optimization problem that parallels\n\\eqref{optfn0} is\n\\begin{align}\n& F_n(x)=\\inf_{\\mathbf{t}\\geq 0,\\; t_0=0,t_n=x}\n\\hat S_n(t_0,t_1,\\dots,t_n), \n\\label{optfn}\n\\\\[3ex]\n& \\hat S_n(t_0,t_1,\\dots,t_n)\\stackrel{\\text{\\scriptsize\\rm def}}{=}\n\\frac{t_1}{t_0+1}+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{t_n}{t_{n-1}+1}\n\\label{Ln}\n.\n\\end{align}\n\nIn Sec.~\\ref{ssec:leastaction} we will elaborate on this\npresentation of the problem.\n\n\\subsection{Formulation with additive constraint}\n\n\\begin{prop}\n\\label{prop:altoptfn}\nFor every $n\\in\\mathbb{N}$, the function $\\bar F_n(x)$ defined by\n\\begin{equation}\n\\label{op_additive}\n \\bar F_n(x)=\\inf_{\\{u_1,\\dots,u_n\\mid \\sum u_j=1\\}} \\left(\\frac{u_1}{u_2}+\\dots+\\frac{u_{n-1}}{u_n}+xu_n\\right)\n\\end{equation}\nis identical to $F_n(x)$. \n\\end{prop}\n\nThe objective function in \\eqref{op_additive}\ncan be written as $\\hat S^{(0)}_{n}(1\/u_1,\\dots,1\/u_n, x)$,\nusing the notation \\eqref{L0n}. Thus \\eqref{op_additive}\ncan also be seen as a result of the replacement of\nthe boundary condition $t_0=1$ in \\eqref{optfn0} by the nonlocal constraint $1\/t_0+\\dots+1\/t_{n-1}=1$.\n\n\nThe functions $F_n(x)$ in the guise \\eqref{op_additive} appeared in my work \\cite{Sadov_2022_maxcyc}.\n\n\\begin{proof}\n\\iffalse\nClearly, $\\tilde f_1(x)=x$. We make the inductive assumption that\n$$\n \\tilde f_{n-1}(y)=\\min_{\\{p'_1,\\dots,p'_{n-1}\\mid \\sum p'_j=1\\}} \\left(\\frac{p'_1}{p'_2}+\\dots+\\frac{p'_{n-2}}{p'_{n-1}}+yp'_{n-1}\\right).\n$$\nNow we put $p_n=(1+y)^{-1}$ and \n$$\n p_j=\\frac{p'_j}{y+1},\\quad j=1,\\dots,n-1.\n$$\n\\fi\n\\iffalse\nConsider the case $n=2$. Here\n$$\n f_2(x)=\\min_{t_1>0}\\left(t_1+\\frac{x}{t_1+1}\\right),\n$$\nwhile\n$$\n \\tilde f_2(x)=\\min_{p_1+p_2=1}\\left(\\frac{p_1}{p_2}+xp_2\\right).\n$$\nPut\n$$\n p_2=\\frac{1}{t_1+1},\\qquad p_1=t_1 p_2=\\frac{t_1}{t_1+1}.\n$$\nClearly, $p_1+p_2=1$. Conversely, given $p_1\\in(0,1)$\nand $p_2=1-p_1$, put\n$t_1=p_1\/p_2$.\n\\fi\nDenote the objective function in \\eqref{op_additive}\nby $\\bar S_n(\\dots)$.\nWe will exhibit a one-to-one correspondence \nbetween the points $(u_1,\\dots,u_n)$ of the simplex $\\sum u_j=1$ and the points\n$(t_1,\\dots,t_{n-1})$ of  $\\mathbb{R}_+^{n-1}$ preserving the\nobjective function: $S_n(t_1,\\dots,t_{n-1},x)=\\bar S_n(u_1,\\dots,u_n,x)$. \n\nDenote\n$$\n s_j=u_1+\\dots+u_j \\quad (j=1,\\dots,n).\n$$\n\nTo a vector $\\mathbf{u}$ with $\\sum u_j=1$ (that is,\n$s_n=1$) we\nput in correspondence the vector $\\mathbf{t}=(t_{1},\\dots,t_{n-1})$ with\ncoordinates\n$$\n t_{j}=\\frac{s_j}{u_{j+1}}\\quad (j=1,\\dots,n-1).\n$$\nThen \n$$\n t_j+1=\\frac{s_j+u_{j+1}}{u_{j+1}}=\\frac{s_{j+1}}{u_{j+1}},\n$$\nso\n$$\nS_n(t_1,\\dots,t_{n-1},x)=\\frac{s_1}{u_2}+\\sum_{j=2}^{n-1}\n\\frac{u_j}{u_{j-1}}+\\frac{xu_n}{s_n}=\n\\bar S_n(u_1,\\dots,u_n,x),\n$$\nsince $s_1=u_1$ and $s_n=1$.\n\n\\iffalse\n$$\n\\frac{t_j}{t_{j-1}+1}=\\frac{p_j}{p_{j+1}}\n\\quad (2\\leq j\\leq n-1)\n$$\nand\n$$\n t_1=\\frac{s_1}{p_2}=\\frac{p_1}{p_2}.\n$$\nFinally,\n$$\n \\frac{x}{t_{n-1}+1}=\\frac{xp_n}{s_n}=xp_n.\n$$\nWe see that the objective function from\nthe definition of $\\tilde f_n(x)$  is transformed term-wise to the objective function from \\eqref{op_main}.\n\\fi\n\n\nThe inverse transformation $\\mathbf{t}\\mapsto \\mathbf{u}$ is defined by\nthe formulas\n$$\n u_n=\\frac{1}{t_{n-1}+1},\n$$\nthen recurrently for $j=n-1,\\dots,2$\n$$\nu_j=\\frac{t_j u_{j+1}}{t_{j-1}+1}\n=\\frac{1}{t_{j-1}+1}\\prod_{i=j}^{n-1}\\frac{t_i}{t_i+1},\n$$\nand lastly,\n$$\n u_1=t_1 u_2.\n$$\nWe prove by induction (on ascending $j$) that \n$s_j=t_j u_{j+1}$, $1\\leq j\\leq n-1$.\nIt is true for $j=1$, since $s_1=u_1$. The induction step ($2\\leq j\\leq n-1)$ goes: $s_j=s_{j-1}+u_j=(t_{j-1}+1)u_j=t_j u_{j+1}$.\nFinally,\n$s_n=(t_{n-1}+1)u_n=1$, so that \nthe $\\mathbf{u}$ satisfies the required constraint.\n\\end{proof}\n\n\\subsection{Existence of a minimizer}\n\\label{ssec:existence}\nWe have written `$\\min$' in Eq.~\\eqref{optfn0}, since the\nminimum of the objective function in is attained at critical point where $t_j=x^{j\/n}$ ($j=0,1,\\dots,n)$. It is also true, thouhg not immediately obvious, that the greatest lower bound of the objective function in \\eqref{optfn} is attained. We prove this now along with first elementary properties of the functions $F_n(\\cdot)$.\n\n\\iffalse\nWe can, of course, introduce an interpolatory family of optimization problems:\n$$\n\\ba{l}\\displaystyle\n f_n(x|s)=\n\\min_{t_1,\\dots,t_{n-1}\\geq 0} \\left(t_1+\\frac{t_2}{t_1+s}+\\dots+\\frac{t_{n-1}}{t_{n-2}+s}+\\frac{x}{t_{n-1}+s}\n\\right),\n\\\\[2ex]\nf_0(x|s)=x.\n\\end{array}\n$$\nEquivalently (upon substitution $t_j\\mapsto st_j$): $f_n(x|s)=g_n(x\/s\\mid s)$, where\n$$\n\\ba{l}\\displaystyle\n g_n(x|s)=\n\\min_{t_1,\\dots,t_{n-1}\\geq 0} \\left(st_1+\\frac{t_2}{t_1+1}+\\dots+\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\n\\right),\n\\\\[2ex]\ng_1(x|s)=sx.\n\\end{array}\n$$\nThere is the recursive definition, the same as one for $f_n$:\n$$\n g_n(x|s)=\\min_{y>0}\\left(g_{n-1}(y)+\\frac{x}{y+1}\\right),\\quad n\\geq 2.\n$$\nOnly the initial condition (case $n=1$) depends on $s$.\n\\fi\n\n\\begin{prop}\n\\label{prop:monotfn}\n{\\rm(a)} For every $n$, the function $F_n(\\cdot)$ is nondecreasing.\n\n{\\rm(b)} For every fixed $x$, the sequence $(F_n(x))$ is nonincreasing, hence the function $F(x)$ can be defined as\na monotone limit\n$$\nF(x)=\\downarrow\\lim_{n\\to\\infty} F_n(x).\n$$\n\n{\\rm(c)} \n\\iffalse\nFor every $n$ the inequality \n\\begin{equation}\n\\label{compf0fn}\n\\frac{1}{2}F^{(0)}(x)\\leq F_n(x)\\leq F_n^{(0)}(x)\n\\end{equation}\nholds. Consequently,\n\\begin{equation}\n\\label{compf0f}\n\\frac{1}{2}F^{(0)}(x)\\leq F(x)\\leq F^{(0)}(x).\n\\end{equation}\n\n\n{\\rm(d)} \n\\fi\nThe objective function $(t_1,\\dots,t_{n-1})\\mapsto L_n(0,t_1,\\dots,t_{n-1},x)$ attains its minimum value at some nonnegative\n$(n-1)$-tuple.\nHence the symbol `$\\inf$' in  \\eqref{op_main}, \\eqref{optfn} and \\eqref{op_additive} can be replaced by `$\\min$'. \n\\end{prop}\n\n\\begin{proof}\n(a) Trivial: the function $x\\mapsto S_n(t_0,\\dots,t_{n-1},x)$ is increasing for every\n$n$-tuple \n$(t_0,\\dots,t_{n-1})$.\n\n\\smallskip\n(b) The inequality $F_{n}(x)\\leq F_{n-1}(x)$ \nis due to the fact that imposing the additional constraint $t_1=0$ in \\eqref{optfn} yields $F_{n-1}(x)$. \nThat is,\n$$\nL_{n-1}(0,t_2,\\dots,t_n)=L_n(0,0,t_2,\\dots,t_n).\n$$\n\n\\iffalse\n(c) \nThe right inequality in \\eqref{compf0fn} is obvious: $L_n^{(0)}(\\mathbf{t})>L_n(\\mathbf{t})$ for any\n$\\mathbf{t}>0$.\n\nThe left inequality is equivalent to the claim:\nfor any $n$, $x$ and $t_1,\\dots,t_{n-1}$ there exists $k$ such that\n$$\n L_n(0,t_1,\\dots,t_{n-1},x)\\geq \\frac{1}{2} F_k^{(0)}(x).\n$$\nWe prove this by induction on $n$.\nThe base case $n=1$ is obvious, since \n$L_1(0,x)=x$ and one can take $k=1$.\n\nAssuming that the claim is true with $n-1$ in place of $n$ and some $k'$ in place of $k$, we write\n$$\n L_n(0,t_1,\\dots,t_{n-1},x)=\nL_{n-1}(0,t_1,\\dots,t_{n-1})+\\frac{x}{t_{n-1}+1}.\n$$\nIf $t_{n-1}\\leq 1$, then $x\/(t_{n-1}+1)\\geq x\/2$,\nso the claim is true with $k=1$.\nIf $t_{n-1}> 1$, then $x\/(t_{n-1}+1)> x\/(2t_{n-1})$.\nUsing the recursive definition \\eqref{recurf0}, we infer\n$$\n S_n(0,t_1,\\dots,t_{n-1},x)\n\\geq\n\\frac{1}{2}\\left(f_{k'}^{(0)}(x)+\\frac{x}{t_{n-1}}\\right)\n\\geq \\frac{1}{2}f_{k}^{(0)}(x),\n$$\nwhere $k=k'+1$. \n\nThe left-hand side of \\eqref{compf0fn} does not depend on $n$, hence  \\eqref{compf0fn} implies \\eqref{compf0f}.\n\n\\smallskip\n(d) \n\\fi\n\n(c) \nIt suffices to show that the optimization region\nin \\eqref{op_main} can be reduced to a compact.\nLet us fix $x$ and put $C=F_n(x)$. \nIf $t_1>C+1$, then $S(t_1,\\dots,t_{n-1},x)>C+1$,\nso the optimization region can be reduced to\n$\\{0\\leq t_1\\leq C+1,\\;t_2,\\dots,t_{n-1}\\geq 0\\}$.\n\nSuppose that $t_1\\leq C_1=C+1$ and define the constants\n$C_k$ recurrently: $C_k=(C_{k-1}+1)(C+1)$.\nIf $k\\in\\{2,\\dots,n-1\\}$ is the least index such that $t_k>C_k$ (assuming such an index exists),\nwe have $t_k\/(t_{k-1}+1)>C+1$. Hence $S_n(t_1,\\dots,t_{n-1},x)>C+1$ in the region \n$\\{t_1>C_1\\}\\cup\\{t_2>C_2\\}\\cup\\dots\\{t_{n-1}>C_{n-1}\\}$.\nTherefore the optimization region is reduced\nto the parallelotop $\\{0\\leq t_k\\leq C_k,\\;\\;k=1,\\dots,n-1\\}$.\n\\end{proof}\n\n\\begin{definition}\nAn $(n-1)$-tuple $(t_1,\\dots,t_{n-1})$ is called a {\\em minimizer}\\ for the problem \\eqref{op_main} if\n$S_{n-1}(t_1,\\dots,t_{n-1},x)=F_{n-1}(x)$.\nWe will also say ``a minimizer for $F_n(x)$''. \n\nWith reference to the problem \\eqref{optfn} we will call\nthe minimizing $(n+1)$-tuple $(0,t_1,\\dots,t_{n-1},x)$\na minimizer. \n\\end{definition}\n\nIn preparation to the proof of Proposition~\\ref{prop:funeqf} let us prove the first result\non minimizers. \n\n\\begin{lemma}\n\\label{lem:minimizer0}\nIf $x\\leq 1$, then the unique minimizer for $F_n(x)$\nis the zero tuple.\n\\end{lemma}\n\n\\begin{proof}\nWe need to prove that if not all $t_j$ equal zero, then\n$S_{n}(t_1,\\dots,t_{n-1},x)>S_{n}(0,\\dots,0,x)=x$.\nEquivalently, the inequality to prove is: if $\\mathbf{t}\\neq\\mathbf{0}$, then (in the worst case, when $x=1$) \n$$\n S_{n-1}(t_1,\\dots,t_{n-1})>\\frac{t_{n-1}}{t_{n-1}+1}.\n$$\n\nIt is obvious, if $n-1=1$. By induction we proceed from the estimate for $S_{n-1}$ to the one for $S_n$.\n\nIf $t_1=\\dots=t_{n-1}=0$ and $t_{n}>0$, then\nwe are in the same situation as in the case $n-1=1$.\nIf not all $t_j$ with $j\\leq n-1$ are equal to 0, then\nby the induction hypothesis we have\n$$\n S_{n}(t_1,\\dots,t_{n})=S_{n-1}(t_1,\\dots,t_{n-1})+\\frac{t_n}{t_{n-1}+1}\n>\\frac{t_{n-1}+t_n}{t_{n-1}+1}.\n$$\nSince\n$$\n \\frac{t_{n-1}+t_n}{t_{n-1}+1}-\\frac{t_{n}}{t_{n}+1}\n=\\frac{t_n^2+t_{n-1}}{(t_{n-1}+1)(t_{n}+1)}\\geq 0,\n$$\nthe induction step is complete.\n\\end{proof}\n\n\\subsection{Dynamic programming formulation}\n\\label{ssec:dynprog}\n\\smallskip\nWe give recurrent equations for\n$F_n(\\cdot)$ analogous to \\eqref{recurf0}.\n\nAlso it appears possible to define the function $F(x)$ is expressed without explicit reference to the $F_n$'s\n--- by means of\nthe functional equation \\eqref{fe1}.\n\n\\begin{prop}\n\\label{prop:funeqf}\nLet $\\tilde F_n(x)$, $n=1,2,\\dots$, be a sequence of functions\ndefined for $x>0$ as follows.\n\n\\smallskip\n{\\rm (i)} For $0<x\\leq 1$ and all $n\\;\\;$ $\\tilde F_n(x)= x$.\n\n\\smallskip\n{\\rm (ii)}  \nFor $x>1$ and $n\\geq 2$\n\\begin{equation} \n\\label{recurf}\n \\tilde F_n(x)= \\min_{0< y<x-1}\\left(\\tilde F_{n-1}(y)+\\frac{x}{y+1}\\right).\n\\end{equation} \n\nDefine also the function $\\tilde F(x)$, $x>0$, as follows.\n\n\\smallskip\n{\\rm (i$'$)} If $0<x\\leq 1$, then $\\tilde F(x)= x$.\n\n\\smallskip\n{\\rm (ii$'$)} In the intervals $(1,2], (1,3], \\dots$ the values $\\tilde F(x)$ are defined recursively by means of the functional equation\n\\begin{equation} \n\\label{fe1}\n \\tilde F(x)=\\min_{0< y<x-1}\\left(\\tilde F(y)+\\frac{x}{y+1}\\right).\n\\end{equation} \n(The right-hand side refers to values of $\\tilde F(\\cdot)$ defined earlier.)\n\nThen \\\\\n{\\rm (a)} \n$\\tilde F(x)=\\tilde F_n(x)$ for $0<x\\leq n$;\n\\\\\n{\\rm (b)} \n$\\tilde F_n(x)=F_n(x)$ for all $n$ and $x$, and\n\\\\\n{\\rm (c)} \n$\\tilde F(x)=F(x)$ for all $x$. \n\\end{prop}\n\n\\begin{proof}\n(a) $\\tilde F_1(x)=\\tilde F(x)=1$ for $0<x\\leq 1$ by definition.\nThe claimed identity for any $n\\geq 2$ follows by induction, comparing the recursive definitions of $\\tilde F_n(x)$ and $\\tilde F(x)$.\n\n\\smallskip\n(b)\nWe will prove by induction, simultaneously, that (I) $\\tilde F_n(x)=F_n(x)$,\n$0<x\\leq n$; and (II) $0<t_{n-1}^*<x-1$, so that $t_{n-1}^*$ can be identified with the point of minimum\n$y_*$ in the right-hand side of \\eqref{recurf}.\n\nFor $n=1$, the claim (I) is trivial and the claim (II) is vacuous.\n\nSuppose that the objective function in \\eqref{op_main} attains its minimum at $\\mathbf{t}^*=(t_1^*,\\dots,t_n^*)$,\nso that\n$$\n F_n(x)=S_{n-1}(t^*_1,\\dots,t^*_{n-1})+\\frac{x}{t^*_{n-1}+1}.\n$$\nThe existence of such $\\mathbf{t}^*$ is ensured by Proposition~\\ref{prop:monotfn}.\n\nWe have\n$$\n\\frac{\\partial S_n(\\mathbf{t},x)}{\\partial t_{n-1}}\n=\\frac{\\partial}{\\partial t_{n-1}}\n\\left(\\frac{t_{n-1}}{t_{n-2}+1}+\\frac{x}{t_{n-1}+1}\\right)=\\frac{1}{t_{n-2}+1}-\\frac{x}{(t_{n-1}+1)^2}.\n$$\n(For $n=2$, we formaly take $t^*_{0}=0$.) \n\nSince $x>1$, there can be no minimum at $t_{n-1}=0$. \nHence $t^*_{n-1}>0$. \n\nBy the choice of $\\mathbf{t}^*$ we have $S_{n-1}(t^*_1,\\dots,t^*_{n-2},t^*_{n-1})\n=F_{n-1}(t_{n-1}^*)$. In order to complete the induction step\nit remains to prove that $t^*_{n-1}<x-1$.\n\nThe necessary condition of extremum yields\n$$\n (t^*_{n-1}+1)^2=x(t^*_{n-2}+1).\n$$\nSuppose that $t^*_{n-1}+1\\geq x$. Then $t^*_{n-1}\\leq t^*_{n-2}$. If $n=2$, we get $1<x\\leq t^*_1+1\\leq t^*_0+1=1$, a contradiction. \n\nIf $n>2$ and $t^*_{n-1}>1$, then by part (II) of the induction hypothesis the last component $t^*_{n-2}$ in the minimizer for $F_{n-1}(t^*_{n-1})$ satisfies the inequality $t^*_{n-2}<t^*_{n-1}-1$.\n\nFinally, if $n>2$ and $t^*_{n-1}\\leq 1$, then by Lemma~\\ref{lem:minimizer0} $t^*_{n-2}=0$.\n\nIn both cases, we arrive to a contradiction with assumption\n$t^*_{n-1}\\leq t^*_{n-2}$. \n\nThe induction step is complete.\n\n\\smallskip\n(c) By parts (a) and (b) we get $F_n(x)=\\tilde F(x)$\nfor any integer $n\\geq x$. Since the sequence $F_n(x)$\nis monotone (Proposition~\\ref{prop:monotfn}(b)), we conclude that $F(x)=\\tilde F(x)$.\n\\end{proof}\n\n\\subsection%\n{General look: least action formulation}\n\\label{ssec:leastaction}\n\n\n\\iffalse\nConsider the optimization problem in the form~\\eqref{op_main}. A {\\em minimizer} is .\nThe existence of a minimizer \nhas been proved in Proposition~\\ref{prop:altoptfn}.\nSince there are no constraints in the form of equalities here, there are two possibilities for a minimizer:\n(i) a minimizer\nlies in the interior of the admissible domain;\nthen it is a critical point of the \nobjective function;\n(ii) a minimizer is a boundary point, that is, at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nCase (ii) yields a minimization problem with fewer variables and its critical points are to be analysed.\n\nFor a particular value of $n$ there may be more than one candidate point for a minimizer, and subsequent minimization over $n$ (to find a minimizer for $f(x)$)\npotentially involves another act of selection. \n\\fi\n\nThe boundary value formulation in Sec.~\\ref{ssec:bconst} and the dynamic programming formulation in Sec.~\\ref{ssec:dynprog} do not by themselved offer\nmuch of a progress in finding the asymptotics of $F(x)$.\nWe will need to study minimizers. For that purpose it\nis useful to look at the optimization problem from a more general point of view.\n\nConsider a minimization problem with objective function of the form\n\\beq{Sn-general}\n\\mathcal{S}_n(\\mathbf{t})=\\sum_{j=1}^n L(t_{j-1},t_j).\n\\end{equation} \nBy analogy with classical mechanics, we call the function $L(\\cdot,\\cdot)$ the {\\em Lagrangian}.  \nA {\\em virtual trajectory}\\ $\\mathbf{t}=(t_0,\\dots,t_n)$\nis subject to the constraints $t_0=x_0$, $t_n=x$;\nthe boundary values $x_0$ and $x$ are assumed given.\nThe function $\\mathcal{S}_n(\\mathbf{t})$ (``integral of the Lagrangian along a virtual trajectory'') is the analog of {\\em action} in mechanics. The object of our attention is the extremal value\n\\beq{fn-general}\n g_n(x_0,x)=\\min_{(t_0,\\dots,t_n)\\mid\\, t_0=x_0,\\,t_n=x}\n\\mathcal{S}_n(t_0,\\dots,t_n).\n\\end{equation}\nWe recognize in this setting ``the least action principle'' with discrete time.\n\nIn mechanics, one usually pays little attention to the minimum value of the action as such; the goal is to determine the extremal trajectory, which describes the actual evolution of a mechanical system. \nHere, we are originally interested in extremal values $g_n$, but the focus will eventually shift to extremal trajectories.\n\nWe assume that the Lagrangian is differentiable in its domain and denote $L_1(u,v)=\\partial L(u,v)\/\\partial u$\nand $L_2(u,v)=\\partial L(u,v)\/\\partial v$.\n\nLet $\\mathbf{t}^*$ be an extremal (more precisely, minimizing) trajectory\nfor the problem \\eqref{fn-general}. Suppose that $(t^*_1,\\dots,t^*_{n-1})$ is an interior point of the domain of the function $\\mathcal{S}_n|_{t_0=x_0,\\;t_n=x}$.\nThe necessary conditions of extremum $\\partial \\mathcal{S}_n\/\\partial t_j|_{\\mathbf t=\\mathbf t^*}=0$ ($1\\leq j\\leq n-1$), known in mechanics as the Euler-Lagrange equations, in the expanded form read\n\\beq{ELeq-general}\n L_2(t_{j-1},t_j)+\n L_1(t_{j},t_{j+1})=0,\\quad j=1,\\dots,n-1.\n\\end{equation}\n\nSuppose that the equation $L_1(u,v)+p=0$ with given $u$ and $p$ is uniquely solvable for $v$ in all occurrences and denote the solution as $v=V(u,p)$.\nThen the system \\eqref{ELeq-general} can be written in the form of the second order recurrence equations\n\\beq{ELeq-recur}\n t_{j+1}=V(t_j,L_2(t_{j-1},t_j)),\\quad j=1,\\dots,n-1.\n\\end{equation}\nMore precisely, we have the boundary value problem comprising the equations \\eqref{ELeq-recur} and the\nboundary conditions\n$$\n t_0=x_0,\\qquad t_n=x.\n$$\nIntroduce a free parameter $\\tau$,\nset $T_0(\\tau)=x_0$, $T_1(\\tau)=\\tau$, and define  further functions $T_j(\\tau)$ by \nmaking the substitutions \n$t_i\\mapsto T_i(\\tau)$ in the recurrence \\eqref{ELeq-recur}:\n\\beq{defTj}\nT_{j+1}(\\tau)=V(T_j(\\tau),L_2(T_{j-1}(\\tau),T_j(\\tau))), \\quad j=1,\\dots,n-1.\n\\end{equation}\nThe boundary value problem can be in principle solved by the shooting method: \nthe equation \n$\nT_n(\\tau)=x,\n$\ndetermines the value of $\\tau$ and hence the whole extremal trajectory. The existence and uniqueness of solution are two immediate concerns. We will attend to them in our concrete case --- first, in Section~\\ref{sec:minimizers}, by presenting numerical results and revealing fine points, and then, analytically.\n\nThe dynamic programming approach to the minimization\nproblem \\eqref{fn-general} leads to the recurrence\n\\beq{recg}\n g_{j+1}(x)=\\min_y (g_{j}(y)+L(y,x)),\\quad j\\geq 1,\n\\end{equation}\nwith initial condition  \n$$\n g_1(x)=L(x_0,x).\n$$\nAn extremal trajectory defined by the recurrence \\eqref{defTj} induces the recurrence\n\\beq{Gn-rec}\n G_{j+1}(\\tau)=G_j(\\tau)+L(T_j(\\tau),T_{j+1}(\\tau)),\n\\quad j\\geq 0,\n\\end{equation}\nwith initial condition  \n$$\n G_0(\\tau)=0.\n$$\nIn a simple scenario, we expect that $G_n(\\tau)$ is the minimum value $g_n(x_0,x)$ sought in \\eqref{fn-general}. However, if there are several \nsolutions of the equation $T_n(\\tau)=x$ and\nseveral extremal trajectories, one needs to pick the minimum among the several corresponding\nvalues $G_n(\\tau)$.  Also one should not ignore the possibility that the minimum value may be attained at the boundary of the parameter domain.\n\n\\begin{remark}\nUpon the change of variables $(u,v)\\mapsto (u,p)$, $p=L_2(u,v)$ applied to all pairs $(u,v)=(t_{j},t_{j+1})$, $j=0,\\dots,n-1$,\n the recurrence \\eqref{ELeq-recur} can be recast in the form \n\\beq{EL-H}\n p_{j+1}=L_2(t_j,V(t_j,p_j)),\n\\qquad t_{j+1}=V(t_j,p_j),\n\\qquad j=1,\\dots,n-1.\n\\end{equation}\nThis can be viewed as a ``Hamiltonian system with discrete time'' referring to the fact that\nthe map $(t_j,p_j)\\mapsto(t_{j+1},p_{j+1})$ is symplectic, that is, its Jacobian is equal to $1$. \nCf.~\\cite[end of Sec.~9.1]{McDuff-Salomon_1998}.\n\nIn the present paper, the Lagrangian is\n$$\n L(u,v)=\\frac{v}{u+1}.\n$$\n\nIncidentally, another asymptotic problem solved recently by the author \\cite{Sadov_2021} involves the Lagrangian \n$$\n L(u,v)=u+\\frac{1+v}{u}\n$$\nand the boundary conditions $p_0=t_n=0$ for the system in the Hamiltonian form \\eqref{EL-H}.%\n\\footnote{The comparison applies \nto the trajectory $(t_0,\\dots,t_n)$ \nin \\cite{Sadov_2021}\nre-indexed backwards.}\n(Unlike in the present case, there is no variable $x$; the integer $n$ is the only parameter.)\nCrusial to the asymptotic analysis in \\cite{Sadov_2021} is the presence of a fixed point  \nof the map $(u_j,p_j)\\mapsto(u_{j+1},p_{j+1})$, which is not the case here. \n\\end{remark}\n\n\\section{Asymptotics that allow for simple proofs}\n\\label{sec:simpleproofs}\n\nHere we prove two results that do not rely on the analysis of extremal trajectories. The material of this section is not used in the sequel.\n\n\\subsection{Proof of Theorem~\\ref{thm:AMGM}}\n\\label{ssec:asf0}\n\nWe write $u=\\log x$, as in the formulation of the Theorem.\n\nLet $u_n=n(n+1)\\log(1+n^{-1})=n+1\/2+O(1\/n)$. We have:\n$F^{(0)}_{n+1}(x)>F^{(0)}_n(x)$ if and only if $u>u_n$. \n\nSuppose $n>1$, $u\\in I_n=[u_{n-1},u_n]$.\nThen $F^{(0)}(x)=F^{(0)}_{n}(x)=u\\varphi(u\/n)$,\nwhere $\\varphi(t)=t^{-1}e^t$. The function $\\varphi(t)$ has the minimum at $t=1$\nand $\\varphi(1+\\varepsilon)=e(1+\\varepsilon^2\/2)+O(\\varepsilon^3)$ as $\\varepsilon\\to 0$. \n\nPut $u=n+s$, $|s|\\leq 1\/2+O(1\/n)$. We have $\\varphi(u\/n)=e+(e\/2)(s\/n)^2+O(n^{-3})$. \n\nNote that $|s|=|\\left\\langle  u\\right\\rangle|+O(1\/n)$;\nindeed, $s\\neq \\left\\langle  u\\right\\rangle$ can happen\nonly near half-integer values of $n$, where $s$ is close to $\\pm 1\/2$.\n\nSince $n^{-1}=u^{-1}+O(u^{-2})$, we \nobtain the asymptotics \\eqref{asf0}. \n\\qed\n\n\\begin{remark}\nThe left and right derivative numbers of $F^{(0)}(x)$\nat the endpoints of the intervals $I_n$ are different.\nIn Section~\\ref{ssec:experiment} we will see that the situation with function $F(x)$ is more interesting in this respect.\n\\end{remark}\n\n\\iffalse\nWe have $f^{(0)}(x)=\\log x\\cdot \\min_{\\mathbb{Z}\\ni n\\geq 1}\\varphi(\\log x\/n)$,\nwhere $\\varphi(t)=t^{-1}e^t$. \nThe function $\\varphi(t)$ has the minimum at $t=1$\nand $\\varphi(1)=e$. \n\nPut $u=\\log x$.\nThe value $n_*=n_*(x)$ for which\n$f^{(0)}(x)=u\\varphi(u\/n_*)$ is always one of the two integers nearest to $u$: either $n_*=\\lfloor u\\rfloor$ or $n_*=\\lceil u\\rceil$. Hence\n$u\/n_*=1+\\epsilon$, where $\\epsilon=O(1\/u)$ but $\\epsilon\\neq o(1\/u)$.\n\nSince $\\varphi(1+\\epsilon)=e+c\\epsilon^2+ o(\\epsilon^2)$ with $c>0$, we conclude that $\\varphi(u\/n_*)=e+O(1\/u^2)$ and the \nremainder estimate is optimal. The claimed asymptotics \nof $f^{(0)}(x)$ follows.\n\\fi\n\n\\subsection{Crude asymptotics of the function \\texorpdfstring{$F(x)$}{F(x)}}\n\\label{ssec:crudeas}\n\n\\begin{prop}\n\\label{prop:crudeas}\n\\iffalse\nLet $f(x)$ be a real-valued function defined for $x> 0$ recursively as follows:\n$$\n f(x)=x, \\qquad 0< x\\leq 1,\n$$\nand\n\\begin{equation}\n\\label{recg}\n f(x)=\\inf_{0<y\\leq x-1}\\left(f(y)+\\frac{x}{y+1}\\right),\n \\qquad x>1.\n\\end{equation}\n\\fi\nThe function $F(x)$ has the asymptotic behavior\n$F(x)=e\\log x+O(1)$ as $x\\to\\infty$. Specifically,\nif $x\\geq 1$, then\n$F(x)$ satisfies the inequalities\n\\begin{equation}\n\\label{rbndf}\n-a_1+\\frac{b_1}{x+1}\\leq\nF(x)-e\\log(x+1)\\leq -a_2+\\frac{b_2}{x+1},\n\\end{equation}\nwhere \n$b_1\\approx 1.77$\nis the (smaller of the two) root of the equation \n\\begin{equation}\n\\label{eqb1}\n2\\log\\frac{b+1}{2}=\\frac{b}{e};\n\\end{equation}\nthe constant $a_1$\nis defined by\n$$\na_1=\\max_{0\\leq x\\leq 1}\\left(e\\log(x+1)+\\frac{b_1}{x+1}-x\\right)\n\\approx 1.78,\n$$\nand $a_2=b_2=e\/(e-1)\\approx 1.58$.\n\\end{prop}\n\n\\begin{proof}\nWe prove that \\eqref{rbndf} is true for $0<x\\leq n$ by induction on $n$.  For $n=1$ the left inequality is fullfilled by the definition of $a_1$, while the left inequality is due to the fact that \nthe function $y\\mapsto  e\\log(x+1)+b_2\/(x+1)-x$ attains it minimum on $[0,1]$ at $x=0$.\n\nFor the induction step consider the function\n$t\\mapsto e\\log t+u\/t$ with given $u>0$. The critical\npoint is $t^*=u\/e$, hence\n\\beq{minf_tmp}\n \\min_{t>0}\\left(e\\log t+\\frac{u}{t}\\right)=\ne\\log t^*+\\frac{u}{t^*}=\ne\\log u.\n\\end{equation}\n\nLet us first prove the left inequality\nin \\eqref{rbndf} for $x\\leq n+1$.\nUsing the inductive assumption and the formula \\eqref{fe1} from Proposition~\\ref{prop:funeqf}, we get \n$$\n F(x)\\geq e\\log(y+1)-a_1+\\frac{b_1}{y+1}+\\frac{x}{y+1}\n$$\nfor any $y\\in(0,x-1]$. By \\eqref{minf_tmp} with $u=b_1+x$, we have\n$$\n F(x)\\geq e \\log(b_1+x)-a_1.\n$$\nIt remains to check the inequality\n$$\ne\\log(b_1+x)\\geq e\\log(x+1)+\\frac{b_1}{x+1};\n$$\nequivalently,\n$$\n\\frac{\\log(1+(b_1-1)s)}{s}\\geq \\frac{b_1}{ e},\n$$\nwhere\n$s=(x+1)^{-1}$. \n\nThe left-hand side is a decreasing function of $s$.  We may assume that $x>1$, hence $s<1\/2$.\nTherefore\n$$\n\\frac{\\log(1+(b_1-1)s)}{s}> 2\\log(1+(b_1-1)\/2).\n$$\nBy  definition of $b_1$, the right-hand side equals $b_1\/e$. The proof of the lower estimate for $F(x)$ is complete.\n\nWe prove the right inequality\nin \\eqref{rbndf} for $x\\leq n+1$ similarly.\nAgain, using the inductive assumption and the formula \\eqref{fe1}, we get \n$$\n F(x)\\leq \\min_{y<x-1} \\left(e\\log(y+1)-a_2+\\frac{b_2}{y+1}+\\frac{x}{y+1}\\right).\n$$\nPut $u=b_2+x$ in \\eqref{minf_tmp}. The critical point\n$t^*=(b_2+x)\/e$ corresponds to $y^*=t^*+1$. \nNow, $y^*$ lies in the admissible interval \n$0<y<x-1$ provided that\n$t_*< x$, that is, $x> b_2\/(e-1)\\approx 0.92$.\nThis condition is fulfilled, since we assume $x\\geq 1$.\nSo\n$$\n F(x)\\leq e\\log(b_2+x)-a_2.\n$$\nIt remains to prove that\n$$\ne\\log(b_2+x)\\leq e\\log(x+1)+\\frac{b_2}{x+1}.\n$$\nSubtracting $e\\log(x+1)$ from both sides, we estimate:\n$$\n e\\log\\frac{x+b_2}{x+1}=e\\log\\left(1+\\frac{b_2-1}{x+1}\\right)< e\\frac{b_2-1}{x+1}.\n$$\nSince $e(b_2-1)=b_2$, the proof is complete.\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:ill-mainasym}\n The double-sided estimate\n\\eqref{rbndf} agrees with\nasymptotic formula \\eqref{asf}, since $a_2<A<a_1$, while the rational\nterms $\\mathrm{const}\\cdot(x+1)^{-1}$, as well as the difference between $\\log x$ and $\\log(x+1)$, are absorbed by the remainder.\nA freedom to choose a particular form of asymptotically negligible terms can be used to \ndevise an approximation to $F(x)$ that would be practical not only for large values of $x$ but also for small ones.\n\nThe approximation with an empirically chosen \ncorrection $b_2\/(x+1)$\n\\begin{equation}\n\\label{empir_approx}\nF(x)\\approx\nF_{\\rm approx}(x)\\stackrel{\\text{\\scriptsize\\rm def}}{=} e\\ln(x+1)-A+\\frac{b_2}{x+1},\n\\end{equation}\nis illustrated in Figure~\\ref{fig:f1corr}(a).\nNote that if we had plotted only the initial part of the graph, up to approximately $\\log(x+1)=8$ (shown in red), then it could\nlead to a misleading assumption that already the second term of the asymptotics, that of order $O(1)$, is periodic.\nThe right part of the graph shows that the error of approximation in fact decays at a rate $O((\\log x)^{-1})$,\nas the theory predicts. The equation of the dashed majorizing curve is $y=(e\/8)(\\log(x+1)+7)^{-1}$. (Here $e\/8=e\/2\\cdot\\max\\ab{\\cdot}^2$. The ``magic'' number $7$ makes the enveloping curve closer to the graph without affecting the asymptotics up to terms of order $O((\\log x)^{-1})$.)\n\nThe right part of the same graph is shown in larger scale on Fig.~\\ref{fig:f1corr}(b) in comparison with third term of asymptotics from Theorem~\\ref{thm:main}.\n\\end{remark}\n\n\\iffalse\n\\begin{figure}\n\\begin{picture}(260,190)\n\\put(0,0){\\includegraphics[scale=0.5]\nempirical_approximation_error-logx}\n}\n\\put(315,25){\\small $\\log (x+1)$}\n\\end{picture}\n\\caption{Difference $f(x)-f_{\\rm approx}(x)$ vs $\\log(x+1)$} \n\\label{fig:emperror}\n\\end{figure}\n\\fi\n\n\\begin{figure}\n\\begin{picture}(311,520)\n\\put(50,260)%\n{\\includegraphics[scale=0.5]{f1_approximation_error_logx}}\n\\put(315,274)%\n{\\small $\\log (x+1)$}\n\n\\put(50,0){\\includegraphics[trim={0 0 0 50pt},clip,scale=0.5]{f1_approximation_34-40}}\n\\put(315,16){$\\log(x+1)$}\n\\put(5,380){(a)}\n\\put(5,120){(b)}\n\\end{picture}\n\\caption\n(a) The error of empirical approximation \\eqref{empir_approx}. (b) Blue line: the same error; gray line: the correction term $\\Phi(u)\/u$ from Theorem~\\ref{thm:main}.}\n\\label{fig:f1corr}\n\\end{figure}\n\n\n\n\\section{Extremal trajectories and minimizers}\n\\label{sec:minimizers}\n\nHere we make initial steps towards the reduction of Theorem~\\ref{thm:main} to Theorem~\\ref{thm:paramcurves}.\n\nTaking Eq.~\\eqref{optfn} as the definition of $F_n(x)$, the minimizer candidates are critical points of $\\hat S_n(\\mathbf{t})|_{t_0=0,t_n=x}$ inside the admissible domain $\\{t_i>0,\\,i=1,\\dots,n-1\\}$ and on its boundary, where $t_i=0$ for at least one $i\\geq 1$.\n\n\n\nOn the other hand, from Proposition~\\ref{prop:funeqf}(c) we know that $F(x)=F_n(x)$ for all sufficiently large $n$\n(e.g. $n\\geq x$), and we will observe the stabilization of\nminimizers in a precise componentwise sense. \n\nIn this section we will use the termiology of the general least action problem with discrete time as described in Section~\\ref{ssec:leastaction}.\n\n\n\\iffalse\nConsider the optimization problem in the form~\\eqref{op_main}. A {\\em minimizer} is .\nThe existence of a minimizer \nhas been proved in Proposition~\\ref{prop:altoptfn}.\nSince there are no constraints in the form of equalities here, there are two possibilities for a minimizer:\n(i) a minimizer\nlies in the interior of the admissible domain;\nthen it is a critical point of the \nobjective function;\n(ii) a minimizer is a boundary point, that is, at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nCase (ii) yields a minimization problem with fewer variables and its critical points are to be analysed.\n\nFor a particular value of $n$ there may be more than one candidate point for a minimizer, and subsequent minimization over $n$ (to find a minimizer for $f(x)$)\npotentially involves another act of selection. \n\\fi\n\n\\iffalse\n\\subsection{Critical points, extremal trajectories}\n\\label{ssec:leastaction}\n\nFrom the formulation \\eqref{op_main} of the optimization problem defining $f_n(x)$ \nit is clear that the point of minimum (which exists, as we have proved in Proposition~\\ref{prop:altoptfn})\nis either a critical point, where all partial derivatives of the objective function turn to $0$, or a boundary point, where at least one of $t_j$ ($1\\leq j\\leq n-1$) is zero. \n\nUsing analogy with classical mechanics, we call the objective function $S_n(\\mathbf{t})$ in the optimization problem \\eqref{optfn}\n {\\em the action}\\ and\nthe vector $\\mathbf{t}=(t_0,\\dots,t_n)$ a {\\em virtual trajectory}. It must satisfy the boundary conditions $t_0=0$, $t_n=x$.\n\nThe action is ``the integral of the Lagrangian along\na virtual trajectory''; here this description corresponds to the expression\n\\begin{equation}\n\\label{action}\nS_n(\\mathbf{t})=\\sum_{j=1}^n L(t_{j-1},t_j),\n\\end{equation}\nwhere the {\\em Lagrangian} is\n$$\n L(u,v)=\\frac{u}{v+1}.\n$$\n\nThe problem of minimization of the function $S_n(\\mathbf{t})$ under the given constraints\ncan be thought of as \nthe ``least action principle'' with discrete time.\nIn mechanics, one usually pays little attention to the minimum value of the action:\nIn mechanics, \nthe focus is on describing the actual (extremal) trajectory and so it will be in this section.\n\n\nSuppose $\\mathbf{t}$  is the minimizing vector \nfor $S_n(\\cdot)$.\nIf \n$t_j\\neq 0$ for some $j$ (between $1$ and $n-1$), then the necessary condition of extremum, $\\partial S_n\/\\partial t_j=0$,\nmust be fulfilled. This equation can be written as  \n\\beq{ELj}\n\\partial_v L(t_{j-1},t_j)+\\partial_u L(t_{j},t_{j+1})=0.\n\\end{equation}\nExtending analogy with mechanics, we call it the Euler-Lagrange equation.\nIt determines $t_{j+1}$ in terms of $t_{j-1}$ and $t_j$ (provided the solution\nis unique).\n\nThe equation \\eqref{ELj} does not need to hold if $t_j=0$. \n\nIn the asymptotic problem from \\cite{Sadov_2021} mentioned in the Introduction the Lagrangian is\n$$\n L(u,v)=v+\\frac{1+u}{v}.\n$$\nNo parameter $x$ is present there and the boundary condition is one-sided, $t_0=0$.\nIn both cases the Lagrangian $L(u,v)$ is linear in $u$, Eq.~\\eqref{ELj} is uniquely solvable for $t_{j+1}$; however, technical parallels do not go much further. \n\nFrom now on we engage with explicit details of \nproblem \\eqref{optfn}.\n\\fi\n\n\\subsection{Extremal trajectories: basic properties}\n\\label{ssec:basicrelations}\n\nThe Euler-Lagrange equations \\eqref{ELeq-recur}\nfor the concrete problem \\eqref{optfn}\nbecome\n$$\n\\frac{1}{t_{j-1}+1}-\\frac{t_{j+1}}{(t_{j}+1)^2}=0.\n$$ \nThe recurrence relations\n\\eqref{ELeq-recur} and \\eqref{Gn-rec} take the form\n$$\n T_{j+1}(\\tau)=\\frac{(T_{j}(\\tau)+1)^2}{T_{j-1}(\\tau)+1},\n\\qquad\nG_{j+1}(\\tau)=G_j(\\tau)+\\frac{T_{j+1}(\\tau)}{T_j(\\tau)+1}.\n$$\n\nTo make the recurrence relations more compact, we introduce the functions where the parameter and the values are shifted by $1$: \n$$\n \\xi_j(t)=T_j(t-1)+1,\n\\qquad \n \\eta_j(t)=G_j(t-1)+1, \\quad t\\geq 1.\n$$ \nThen\n\\begin{equation}\n\\label{recxi}\n\\xi_{j+1}(t)=\\frac{\\xi_{j}^2(t)}{\\xi_{j-1}(t)}+1.\n\\end{equation}\nIntroduce also the auxiliary functions\n\\begin{equation}\n\\label{defalpha}\n \\alpha_j(t)=\\frac{\\xi_j(t)}{\\xi_{j-1}(t)}.\n\\end{equation}\nIn places where $t$ does not change, we will often simply write $\\xi_j$, $\\eta_j$, $\\alpha_j$. \n\nThe governing system of recurrence relations becomes\n\\begin{equation}\n\\label{recrelalxi}\n\\begin{array}{l}\n\\displaystyle\n\\alpha_{j+1}=\\alpha_j+\\frac{1}{\\xi_j},\n\\\\[2ex]\n\\xi_{j+1}=\\alpha_{j+1} \\xi_{j}=\\alpha_{j}\\xi_j+1.\n\\end{array}\n\\end{equation}\nIt is complemented by the subordinate recurrence\n\\begin{equation}\n\\label{recreleta}\n\\eta_{j+1}\n=\\eta_j+\\alpha_{j}.\n\\end{equation}\n\nClearly, all the introduced functions are rational functions of $t$.\nHere are the first few, including the initial values ($n=0,1$) set by definition:\n\n\\bigskip\\noindent\n\\begin{tabular}{c|c|c|c}\n$n$ & $\\alpha_n$ & $\\xi_n$ & $\\eta_n$ \\\\\n\\hline\n$0$ & $t-1$    & $1$      & $1$        \\\\[1ex]\n$1$ & $t$ & $t$ & $t$ \\\\[2ex]\n$2$ & $t+t^{-1}$ & $t^2+1$ & $2t$\\\\[2ex]\n$3$ & $\\displaystyle\\frac{t^2+1}{t}+\\frac{1}{t^2+1}$ & $t^3+2t+1+t^{-1}$ & $3t+t^{-1}$\\\\[2ex]\n$4$ & $\\cdots$ & $\\displaystyle\\frac{(t^2+1)^{3}}{t^2}+\\frac{2(t^2+1)}{t}+\\frac{1}{t^2+1}+1$ & $\\displaystyle\\frac{4t^4+6t^2+t+2}{t(t^2+1)}$\\\\\n\\end{tabular}\n\n\\bigskip \n\nSome basice consequences of the recurrence relations\n\\eqref{recxi}--\\eqref{recreleta} are collected in the following proposition.\n\n\\begin{prop}\n\\label{prop:xieta-basic}\n{\\rm(a)} The introduced rational functions behave at inifinity as follows (for any fixed $n$): $\\alpha_n(t)\\sim t$, $\\xi_n(t)\\sim t^n$,\n$\\eta_n(t)\\sim nt$.\n\n\\smallskip\n{\\rm(b)} For any $t$, the sequences $(\\alpha_n(t))$, \n$(\\xi_n(t))$ and $(\\eta_n(t))$ are increasing.\nThe inequalities\n$\n \\alpha_n(t)\\geq 2 \n$ $(n\\geq 1)$\nand \n$\n \\xi_n(t)\\geq 2^{n-1}\n$\n$(n\\geq 2)$\nhold true. Also $\\alpha_n(t)<3$ for all $n$.\nConsequently, there exists the finite limit\n$$\n \\alpha_{\\infty}(t)=\\lim_{n\\to\\infty}\\alpha_n(t).\n$$\n\n\\smallskip\n{\\rm(c)} For $n\\geq 1$, define  $X^{0}_n=\\min_{t\\geq 1}\\xi_n(t)$.\nThen $\\xi_n(\\cdot)$ maps $[1,\\infty)$ to $[X^{0}_n,\\infty)$. The sequence $(X^{0}_n)$ is increasing and $X^{0}_n\\geq 2^{n-1}$. \n\n\\smallskip\n{\\rm(d)} For any $n\\geq 1$ the following relations hold true:\n\\begin{equation}\n\\label{ic12}\n\\xi_n(1)=\\xi_{n-1}(2), \n\\qquad\n\\eta_n(1)=\\eta_{n-1}(2),\n\\qquad\n\\alpha_n(1)=\\alpha_{n-1}(2),\n\\end{equation}\nand\n\\begin{equation}\n\\label{dxieta}\n\\eta'_n(t)=\\frac{\\xi'_n(t)}{\\xi_{n-1}(t)}.\n\\end{equation}\n\\end{prop}\n\n\\begin{remark}\n1. In (c), it is not true generally that $\\xi_n(1)=X_n^0$. See Sec.~\\ref{ssec:experiment}. \n\n\\smallskip\n2. To elucidate the formula \\eqref{dxieta}, let us look at the dynamic programming formulation \\eqref{recg}.\nWe have the \nobjective function $\\Lambda_j(x,y)=g_j(y)+L(y,x)$. Let \n$y^*=y^*(x)$ be some critical point. \nThen $\\frac{\\partial}{\\partial y}\\Lambda_j(x,y^*)=0$.\nHence\n$g_{j+1}'(x)=\n\\frac{\\partial}{\\partial x}\\Lambda_j(x,y^*)=L_2(y^*,x)\n$. With our Lagrangian, and putting $j=n-1$, we get  $L_2(y^*,x)=L_2(T_{n-1}(\\tau),T_{n}(\\tau))=(T_{n-1}(\\tau)+1)^{-1}$.\nThe result is the relation $d\\eta_{n}\/d\\xi_{n}=\\xi_{n-1}^{-1}$, equivalent to \\eqref{dxieta}.\n\\end{remark}\n\n\\begin{proof}\n(a) Immediate by induction.\n\n(b) Monotonicity of $(\\alpha_n)$ and $(\\eta_n)$\nis obvious from \\eqref{recrelalxi} and \\eqref{recreleta}. Since $\\min\\alpha_2=2$, the inequalities $\\alpha_n> 2$ and $\\xi_{n+1}>2\\xi_n$ for $n>2$ follow. \n\nTherefore, for $n>2$ we have $\\xi_n>2^{n-2}\\xi_2\\geq 2^{n-1}$. By \\eqref{recrelalxi}, \n$\\alpha_n-\\alpha_2<\\sum_{j=3}^{n-1} 2^{1-j}<1$.\nThus the sequence $(\\alpha_n(t))$ is bounded and the\n$\\alpha_{\\infty}(t)<\\infty$. \n\n\\smallskip\n(c) We know from (b) that $\\alpha_n\\geq 2$ for $n\\geq 2$. Now \\eqref{recrelalxi} implies $X^{0}_{n+1}\\geq 2X^{0}_n+1$, hence the strict monotonicity of $(X^{0}_n)$ and the estimate $X^{0}_n\\geq 2^{n-1}$. \n\n\\smallskip\n(d)\nDue to the identical initial conditions $(\\alpha_0(2),\\xi_0(2),\\eta_0(2))=(1,1,1)=\n(\\alpha_1(1),\\xi_1(1),\\eta_1(1))$, \nthe sequences $(\\alpha_n(2))$ etc.\\ are identical to \n$(\\alpha_{n+1}(1))$ etc.\n\nThe relations \\eqref{dxieta} follow by induction:\nfor $n=1$ we have $\\eta'_1(t)=1=\\xi'_1(t)\/\\xi_0(t)$; \nthe induction step goes:\n$$\n \\eta_{n+1}'=\\eta_n'+\\left(\\frac{\\xi_{n+1}-1}{\\xi_n}\\right)'=\\frac{\\xi'_n}{\\xi_{n-1}}+\n\\frac{\\xi'_{n+1}}{\\xi_n}-\\frac{(\\xi_{n+1}-1)\\xi'_n}{\\xi_n^2}=\\frac{\\xi'_{n+1}}{\\xi_n},\n$$\ndue to \\eqref{recxi}.\n\\iffalse\n\\smallskip\n(e) Put $\\lambda_n(t)=\\xi'_n(t)\/\\xi_n(t)$. From \\eqref{recxi} we have the recurrence relation \n\\begin{equation}\n\\label{logdifxirec}\n \\lambda_{n+1}=(1-\\xi_{n+1}^{-1})(2\\lambda_n-\\lambda_{n-1}).\n\\end{equation}\nHence%\n\\footnote{This is only true if $\\lambda_j\\geq 0$ up to $j=n+1$.}\n the sequence $(\\lambda_n)$ is convex down for any $t\\geq 1$. The same is true for the sequence\n$\\tilde\\lambda_{n}=\\lambda_{n}-n\\lambda_1$. \nWe have $\\tilde\\lambda_1=0$ and $\\tilde\\lambda_2(t)=2t\/(1+t^2)-2\/t<0$, hence by induction\nfor any $n\\geq 2$\n$$\n \\frac{n-1}{n}\\tilde\\lambda_{n+1}\\leq\\tilde\\lambda_n-\n\\frac{1}{n}\\tilde\\lambda_1<0.\n\\eqno\\qedhere\n$$\n\\fi\n\\end{proof}\n\n\\subsection{Experimental observations}\n\\label{ssec:experiment}\n\nFor $n=1,2,\\dots$, denote by  $\\gamma_n$, resp., $\\gamma^T_n$, the parametric curve defined in the coordinate $(x,y)$-plane by the equations\n$$\n x=\\xi_n(t), \\quad y=\\eta_n(t), \\quad t\\geq 1,\n$$\nresp.,\n$$\n x=T_n(\\tau), \\quad y=G_n(\\tau), \\quad \\tau\\geq 0.\n$$\nThe curve $\\gamma_n$ is obtained from $\\gamma^T_n$\nby the shift $(x,y)\\mapsto(x+1,y+1)$.\n\nLet $\\gamma_n[1,2]$ be the part of the curve $\\gamma_n$\ncorresponding to the parameter values $1\\leq t\\leq 2$.\nProposition~\\ref{prop:xieta-basic}(d) asserts that \n$\\gamma_{n-1}[1,2]$ and $\\gamma_n[1,2]$ are geometrically adjacent to each other and have a common tangent at the adjacency point. \n\nSimilarly defined partial curves $\\gamma^T_n[0,1]$\nwill be used with reference to the curves $\\gamma^T_n$.\n\n\nThe lower envelope of the the graphs of the functions $F_n(x)$\nmakes, by definition, the graph of the function $F(x)$.\n\nOne anticipates a relation between \nthe graph of $F_n(x)$ \nand the curve\n$\\gamma^T_n$ \nbased on the premise that the abscissa $x=T_n(\\tau)$ corresponds to the value of $\\tau$ that determines the extremal trajectory yielding the extremal value $y=G_n(\\tau)$ of the action. We take this view as a ``first approximation''\nand discuss necessary corrections below.\n\nThe curve $\\gamma^T_n$ lies in the half-plane\n$x\\geqX^{0}_n-1$ (so that the equation $T_n(\\tau)=x$ has a solution).\nConsequently, $\\gamma^T_n$ cannot represent the whole\nof the graph of $F_n(x)$\n(which is defined for all $x>0$). \nParts of the graph will be represented by segments\nof curves $\\gamma^T_k$, $k<n$; we elaborate on this \nin \\S~\\ref{ssec:bndmin}.\n\n\\begin{figure}[htb]\n\\begin{picture}(260,266)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta1-7-logxy}\n}\n\\put(201,116){\\small $x$}\n\\put(44,65){\\small $y$}\n\\put(42,171){\\small $\\tilde\\gamma_1$}\n\\put(66,76){\\small $\\tilde\\gamma_2$}\n\\put(96,31){\\small $\\tilde\\gamma_3$}\n\\put(126,16){\\small $\\tilde\\gamma_4$}\n\\put(161,10){\\small $\\tilde\\gamma_5$}\n\\put(196,8){\\small $\\tilde\\gamma_6$}\n\\put(232,7.4){\\small $\\tilde\\gamma_7$}\n\\put(141,246){\\framebox{\\parbox{3.6cm}{\\small {Solid lines: $1\\leq t\\leq 2$}\\\\\n{Dotted lines: $t> 2$}}}}\n\\put(211,3){\\circle{5}}\n\\end{picture}\n\\caption{Curves $\\tilde\\gamma_n$, $n=1,\\dots,7$,\nwith parametric equations $x=\\log\\xi_n(t)$, $y=\\eta_n(t)-e x$. Inside the little circle is a region displayed in Fig.~\\ref{fig:xieta6-7}.} \n\\label{fig:xieta1-6}\n\\end{figure}\n\nThe mutual position of the curves $\\gamma_n$ (or $\\gamma^T_n$) with $1\\leq n\\leq 7$ is illustrated in Figure \\ref{fig:xieta1-6}.  \nAs a matter of fact, shown are the curves $\\tilde\\gamma_n$ obtained from $\\gamma_n$ by the transformation $(x,y)\\mapsto (\\log x,y-e\\log x)$ (scaling conveniently and removing the asymptotic \ndrift).\nThe\nunion of the solid parts forms a part of the graph of the function $F(e^{x}-1)+1-ex$.\n\nFigure~\\ref{fig:xieta1-6} seems to support the view that: (a) the parametric curves $\\gamma^T_n$ are the graphs of the functions\n$F_n(x)$\nrestricted to $x\\geq X^{0}_n-1$; \n(b) moreover, $X^{0}_n=\\xi_n(1)$; and (c) the lower envelope of the curves \n$\\gamma_n$\ncoincides with union of the segments $\\gamma_n[1,2]$. \n\n\\begin{figure}[ht]\n\\begin{picture}(260,255)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta30-32-logxy}\n}\n\\put(250,235){\\small $x$}\n\\put(30,-4){\\small $y$}\n\\put(53,120){\\small $\\tilde\\gamma_{30}$}\n\\put(123,10){\\small $\\tilde\\gamma_{31}$}\n\\put(249,120){\\small $\\tilde\\gamma_{32}$}\n\\put(111,180){\\small $t=1$}\n\\put(207,180){\\small $t=2$}\n\\put(83.2,68.7){\\circle*{4}}\n\\put(78,74){\\small $t=t^{0}_{31}$}\n\\put(101,28){\\circle*{4}}\n\\put(64,27){\\small $t=t^{\\ell}_{31}$}\n\\put(156.9,27.41){\\circle*{4}}\n\\put(166,27){\\small $t=t^{r}_{31}$}\n\\end{picture}\n\\caption{Parametric curves $\\tilde\\gamma_n$, $n=30,31,32$.\nThe coordinates and legend are the same as in Fig.~\\ref{fig:xieta1-6}. The $t$-value marks pertain to $\\tilde\\gamma_{31}$.} \n\\label{fig:xieta30-32}\n\\end{figure}\n\nThese impressions \nare refuted by observing the curves with greater values of $n$.\nFigure~\\ref{fig:xieta30-32} shows that:\n(a) the curve $\\tilde\\gamma_n$, and hence $\\gamma_n$ or $\\gamma^T_n$, in general is not a graph of a single-valued function;\n(b) the value $X^{0}_n$ is the abscissa of the cusp\nand corresponds to some $t^{0}_n\\in (1,2)$;\n(c) the lower envelope of the\ncurves $\\gamma_n$ \nis a proper subset of the\nunion of the segments $\\gamma_n[1,2]$. \n\n\n\\begin{figure}[thb]\n\\begin{picture}(257,255)\n\\put(0,0){\\includegraphics[scale=0.5]{xieta6-7b-logxy}\n}\n\\put(251,16){\\small $x$}\n\\put(40,244){\\small $y$}\n\\put(64,39){\\small $\\tilde\\gamma_6$}\n\\put(156,129){\\small $\\tilde\\gamma_7$}\n\\put(96,18){\\small $\\logX^{\\!\\times}_7$}\n\\put(191,18){\\small $\\logX^{0}_7$}\n\\end{picture}\n\\caption\nA magnified\nview of the curves\n$\\tilde\\gamma_6$ in parameter region $t\\to 2^-$ and $\\tilde\\gamma_7$\nin parameter region $t\\to 1^+$. Solid lines form a part of the graph of\n$F(x)$.\n} \n\\label{fig:xieta6-7}\n\\end{figure}\n\nLet us describe what {\\em is}\\ true and will be proved in the sequel (Sec.~\\ref{ssec:special-points}).\n\n\\smallskip\n(i) Case $n\\leq 6$. The observations based on Fig.~\\ref{fig:xieta1-6} are mostly adequate (with subtle exception described in (iii) below), the crucial fact being that the functions $\\xi_n(t)$ and $\\eta_n(t)$ are monotone increasing in $t\\in[1,\\infty)$.\nAt $x=T_n(0)=T_{n-1}(1)=X^{0}_n-1$, we have $F(x)=G_n(0)=G_{n-1}(1)=\\eta_n(1)-1$; \nthe function $F(\\cdot)$ has a continuous derivative at that point. \n\n\\smallskip\n(ii) Case $n\\geq 7$. There exist three special values of the parameter: $1<t^{0}_n<t^{\\ell}_n<t^{r}_n<2$.\nThe functions $\\xi_n(t)$ and $\\eta_n(t)$ are convex in $t\\in[1,\\infty)$;\nthey decrease in $[1,t^{0}_n]$ and increase in $[t^{0}_n,\\infty)$.\nThe value $X^{0}_n=\\min_t \\xi_n(t)$ equals $\\xi_n(t^{0}_n)$.\nThe curve $\\gamma_n$ has a cusp at the point\n$(X^{0}_n,\\eta_n(t^{0}))$.\nThe lower branches of the curves $\\gamma_n$ and\n$\\gamma_{n-1}$ meet at the point with coordinates\n$(\\xi_n(t^{\\ell}_n),\\eta_n(t^{\\ell}_n))=\n(\\xi_{n-1}(t^{r}_{n-1}),\\eta_{n-1}(t^{r}_{n-1}))$.\nSet by definition $X^{\\!\\times}_n=\\xi_n(t^{\\ell}_n)$.\nSo $X^{\\!\\times}_{n+1}=\\xi_n(t^{r}_n)$.\n\nThe branch of $\\gamma_n$ corresponding to parameter values $t\\geqt^{\\ell}_n$ is the graph of the function $F_n(x-1)+1$ restricted to $x\\geq X^{\\!\\times}_n$.\n(As to why $X^{\\!\\times}_n$ and not $X^{0}_n$ here -- see the end of \\S~\\ref{ssec:bndmin}.)\nThe point $(T_n(0),G_n(0))$ does not belong to the lower branch of the curve $\\gamma_n$.\nThe equality\n$F(x)=F_n(x)$ is valid for $x\\in[X^{\\!\\times}_n,X^{\\!\\times}_{n+1}]$.\n The left and right derivative numbers of $f(x)$\nat $x=X^{\\!\\times}_n$ are not equal to each other. \n\n\\smallskip\n(iii) The exception mentioned in (a) applies to \nthe case $n=6$.\nLooking at Fig.~\\ref{fig:xieta1-6},\none is led to mistakenly believe that the equality\n$F(x)=F_6(x)$ holds true for $\\xi_6(1)\\leq x\\leq \\xi_6(2)$,\nwhile in fact it is valid in the shorter interval\n$\\xi_6(1)\\leq x\\leqX^{\\!\\times}_7$, see Figure~\\ref{fig:xieta6-7}. Let us clarify that\nthe value $n=7$ is the first for which $X^{\\!\\times}_n$ is defined,\nsince $\\gamma_7$ is the first curve having a cusp.\nThe value $t^{\\ell}_6$ is not defined, but $t^{r}_6\\approx 1.9975$\nis defined and \n$X^{\\!\\times}_7=\\xi_7(t^{\\ell}_7)=\\xi_6(t^{r}_6)$. \n\n\n\\subsection{Case of a minimizer at the boundary}\n\\label{ssec:bndmin}\n\nWe will show that if the minimum of the objective function\n$\\hat S_n(\\mathbf{t})$ is attained at the boundary of the admissible region, then the minimizer corresponds to \nan extremal trajectory of a shorter length.\n\n\n\\begin{lemma}\n\\label{lem:min-at-bnd}\n(a) Suppose $\\mathbf{t^*}=(0,t^*_1,\\dots,t^*_{n-1},x)$ is a minimizer for the problem \\eqref{optfn}.\nIf $t^*_j=0$ for some $j\\in\\{1,\\dots,n-1\\}$, then\n$t^*_i=0$ for any $i\\in\\{1,\\dots,j-1\\}$.\nIn this case $F_n(x)=F_{n-j}(x)$.\n\n(b) If,\nin addition, \n$t^*_{j+1}\\neq 0$,\nthen the truncated vector\n$$\\hat{\\mathbf{t}}^*=(0=t_{j}^*,t_{j+1}^*,\\dots,t_{n-1}^*,x)\n$$\nis an extremal trajectory for the problem \\eqref{optfn}\nwith $n$ replaced by $n-j$. In this case $T_{n-j}(t^*_{j+1})=x$ and $G_{n-j}(t^*_{j+1})=F_n(x)$.\n\\end{lemma}\n\n\\iffalse\n\\begin{prop}\n\\label{prop:fnstab}\nFor every $n\\geq 1$ there exists $X_n$ such that\n$f_{n+1}(x)=f_{n}(x)$ for $0\\leq x\\leq X_n$.\nFor instance, $X_1=1$.\n\\end{prop}\n\n\\begin{proof}\nSuppose the $n$-tuple $(t_1,\\dots,t_{n}$ \nminimizes $S_{n-1}(t_1,\\dots,t_n)$ under constraint\n$t_n=x$. If \n$f_{n+1}(x)<f_{n}(x)$, then \nthere exists an $(n+1)$-tuple $t'_0,\\dots,t'_{n-1},t_n$ with $t'_n=x$ such that\n$$\nS_n(t'_0,\\dots,t'_{n-1},x)<f_{n-1}(x)=S_{n-1}(t_1,\\dots,t_n).\n$$\n\n\nWe need to show that for some $X_n$\nand any $t_1,\\dots,t_n$ with $t_n\\leq X_n$ the inequality\n$$\n S_n(t_0,\\dots,t_n)\\leq S_{n-1}(t_1,\\dots,t_n)\n$$\nimplies $t_0=0$.\n\\end{proof}\n\\fi\n\n\\begin{proof}\n(a) Put $i=j-1$. If $i>0$ and $t^*_i>0$, then the necessary condition of\nextremum $\\partial \\hat S_n\/\\partial t_i|_{\\mathbf{t}=\\mathbf{t}^*}=0$\nyields\n$$\n\\frac{1}{t_{i-1}^*+1}-\\frac{t_{j}^*}{(t_{i}^*+1)^2}=0,\n$$ \nwhich contradicts the assumption $t_j^*=0$. Hence $t_{j-1}^*=0$. By downward induction on $i$ we get $t_i=0$ for any $i<j$. \n\nSince\n$$\n \\hat S_n(0,\\dots,0,t_{j+1},\\dots,x)=S_{n-j}(0,t_{j+1},\\dots,x),\n$$\nwe have $F_n(x)=\\hat S_{n-j}(\\hat{\\mathbf{t}}^*)\\geq F_{n-j}(x)$. It follows that $F_n(x)=F_{n-j}(x)$.\n\n\\smallskip\n(b) For the same reason, $\\hat{\\mathbf{t}}^*$ is the minimizer for\n\\eqref{optfn}\nwith $n$ replaced by $n-j$. Moreover,\n$t^*_i>0$ for $i=j+1,\\dots,n$. Hence $\\hat{\\mathbf{t}}^*$ is an extremal trajectory of length $n+1-j$ and $T_{n-j}(t_{j+1}^*)=x$.\nFinally, $\\hat S_{n-j}(\\hat{\\mathbf{t}}^*)=G_{n-j}(t^*_{j+1})$,\nhence $F_n(x)=G_{n-j}(t^*_{j+1})$.\n\\end{proof}\n\nWe see that a minimizer that lies on the boundary of\nthe admissible domain is represented by an extremal trajectory of length $k\\in\\{2,\\dots,n\\}$. \nConsequently, the graph of $F_n(x)$ \nis the lower envelope of the parametric curves \n$\\gamma^T_k$ with $1\\leq k\\leq n-1$.\n\nFor $n\\leq 6$ and $1\\leq k\\leq n-1$, the graph of the restriction $F_n(x)|_{[T_k(0),T_k(1)]}$ coinsides with\n$\\gamma^T_k[1,2]$.\n\nFor $n\\geq 7$, the part of the curve $\\gamma^T_n$ corresponding to parameter values $t^{0}_n<t<t^{\\ell}_n$\nlies above the curve \n$\\gamma^T_{n+1}$ and hence does not belong to the\ngraph of $F_n(x)$. \n\n\n\\subsection{Monotonicity of the critical index}\n\\label{ssec:crind}\n\n\\begin{definition}\n\\label{def:crind}\nFor the given $x>0$, the {\\em critical index} $\\nu(x)$ is the integer equal to minimum value of $n$ such that $F(x)=F_{n}(x)$.\n\\end{definition}\n\n\\begin{prop}\n\\label{prop:crind}\nThe function $x\\mapsto \\nu(x)$ is nondecreasing.\n\\end{prop}\n\n\\begin{proof}\nBy the definition of extremal trajectories and in view ofLemma~\\ref{lem:min-at-bnd}, the equality $n=\\nu(x)$ holds if and only if both of the following\nare true:\n\n\\smallskip\n(i) there exists $\\tau>0$ \n such that $T_n(\\tau)=x$ and $G_n(\\tau)=F(x)$;\n equivalently, the vector $\\mathbf{t}\\in\\mathbb{R}^{n+1}$ with \n $t_0=0$ and $t_{j}=T_j(\\tau)$\n($j=1,\\dots,n)$ is a minimizer for the problem \\eqref{optfn};\n\n\\smallskip\n(ii) for any $k<n$ either the equation $T_{k}(\\tau)=x$ does not have a solution or, if \n$\\tau$ is a solution, then $G_{k}(\\tau)>F(x)$.  \n\n\\smallskip\nConsequently, if $\\mathbf{t}$ is an extremal trajectory specified in (i), then $\\nu(t_j)=j$ ($1\\leq j\\leq n$).  \n\nUsing this observation, we will show  by induction on $n$ that the inequality $n=\\nu(x)>\\nu(x')$ implies $x>x'$.\n\nLet us assume that the said implication is true with $n-1$ instead of $n$. (The special case $n-1=1$ is included.) \n\nSuppose that the induction step fails.\nIt means that there exist $x$ and $x'$ such that $x<x'$ and $n=\\nu(x)>\\nu(x')=k$.\n\nLet $\\mathbf{t}=(0,t_1,\\dots,t_{n-1},x)$ and\n$\\mathbf{t'}=(0,t'_1,\\dots,t'_{k-1},x')$\nbe the extremal trajectories with $G_n(t_1)=F(x)$\nand $G_k(t'_1)=F(x')$.\nThen $\\nu(t_{n-1})=n-1$ and $\\nu(t'_{k-1})=k-1$. By the inductive assumption, $t'_{k-1}<t_{n-1}$.\n(If $n=2$, then this is a tautology: $0=t'_0<t_1$.)\nWe have\n$$\n F(x')=S_k(\\mathbf{t'})=\\hat S_k(0,t'_1,\\dots,t'_{k-1},x)\n+\\frac{x'-x}{1+t'_{k-1}}.\n$$\nBy the assumption of {\\em ad absurdum}\\ argument, $k<n$, so\n$$\n\\hat S_k(0,t'_1,\\dots,t'_{k-1},x)\\geq F_k(x)>F(x)\n=\\hat S_n(0,t_1,\\dots,t_{n-1},x).\n$$\nDue to the inequality $t'_{k-1}<t_{n-1}$, \n$$\n \\frac{x'-x}{1+t'_{k-1}}>\\frac{x'-x}{1+t_{n-1}}.\n$$\nTherefore\n$$\n F(x')> \\hat S_n(0,t_1,\\dots,t_{n-1},x)\n+\\frac{x'-x}{1+t_{n-1}}=\\hat S_n(0,t_1,\\dots,t_{n-1},x'),\n$$\nwhich contradicts the assumption that $\\mathbf{t}$\nis a minimizer for $F(x')$.\n\\end{proof}\n\n\\section{Reduction of Theorem~\\ref{thm:main} to  Theorem~\\ref{thm:paramcurves}}\n\\label{sec:proofmain}\n\nIn the course of the proof, which involves many small technical steps, we will prove that Fig.~\\ref{fig:xieta30-32} adequately illustrates the relevant features of the curves $\\gamma_n$ \n(defined in Sec.~\\ref{sec:minimizers})\nwith large enough $n$.\n\nWe will explore in great detail a parametrization of the curves $\\gamma_n$. As a result, we will be able to describe\na piece-wise parametrization of their lower envelope\nand to derive the asymptotics of the function $f(x)$.\n\n\n\\subsection{Only the partial curves \\texorpdfstring{$\\gamma_n[1,2]$}{gamma[1,2]} are relevant}\n\\label{ssec:gamma12}\n\nObserving the dotted lines in Figs.~\\ref{fig:xieta1-6} and \\ref{fig:xieta30-32},\none is led to conjecture that the parts of the curves $\\gamma_n$\ncorresponding to the parameter values $t>2$\ndo not contribute to the lower envelope of the\ncurves $\\gamma_n$. Equivalently, \nthe parts of the curves $\\gamma^T_n$\ncorresponding to the parameter values $\\tau>1$\ndo not contribute to the graph of the function $F(x)$.\nWe will prove this conjecture now.\n\nAs shown in \\S~\\ref{ssec:bndmin}, the coordinates of any point of the graph of $F(x)$ can be written as\n$x=T_n(\\tau)$, $F(x)=G_n(\\tau)$ with some $n\\geq 1$\nand $\\tau>0$. \n\n\\begin{prop}\n\\label{prop:tau01}\nGiven $x>0$, suppose that $n$ and $\\tau$ are such that $x=T_n(\\tau)$ and $F(x)=F_n(x)=G_n(\\tau)$. Then $\\tau\\leq 1$.\n\\end{prop}\n\n\\begin{proof}\nWe have\n$$\n F_n(x)=\\hat S_n(0,t^*_1,\\dots,t^*_n),\n$$\nwhere $t^*_1=T_1(\\tau)=\\tau$ and $t^*_n=T_n(\\tau)=x$.\n\nSuppose, contrary to what is claimed, that $\\tau>1$.\nDefine a vector $\\mathbf{t}=(0,t_1,\\dots,t_{n+1})\n\\in\\mathbb{R}_+^{n+2}$ as follows:\n$$\n t_1=s, \\quad\\text{and}\\quad\n t_{j+1}=t^*_j \\;\\; (j=1,\\dots,n),\n$$\nwhere $s$ is an arbitrary number such that $0<s<\\tau-1$.\nThen\n$$\n \\hat S_n(\\mathbf{t^*})-\\hat S_{n+1}(\\mathbf{t})\n=t^*_1-\\left(t_1+\\frac{t_2}{t_1+1}\\right)=\n\\tau-\\left(s+\\frac{\\tau}{s+1}\\right)=\\frac{s(\\tau-1-s)}{s+1}>0,\n$$\nso $F(x)\\leq F_{n+1}(x)\\leq \\hat S_{n+1}(\\mathbf{t})<\\hat S_n(\\mathbf{t^*})=F_n(x)$, a contradiction. \n\\end{proof}\n\n\n\\subsection{Convexity of the functions \\texorpdfstring{$\\alpha_n(t)$ and $\\alpha_\\infty(t)$,\n$t\\in[1,2]$}{alpha[n] on [1,2]}}\n\\label{ssec:alpha}\n\nRecall that the functions $\\alpha_j(t)$ is defined as\n$\\alpha_j(t)=\\xi_j(t)\/\\xi_{j-1}(t)$.\nThey are a part of the recurrent scheme \\eqref{recrelalxi} defining extremal trajectories.\nThe function $\\alpha_\\infty(t)=\\uparrow\\lim_{j\\to\\infty}\\alpha_j(t)$ determines the eventual rate of the exponential \ngrowth of the components of an extremal trajectory\nstarting at $\\tau=t-1$. The graph of the function\n$\\alpha_\\infty(t)$ on the interval $[1,2]$ is shown in\nFig.~\\ref{fig:alphaplot}. Crusial for the derivation of the asymptotic formulas is the existence of the solution\nof the equation $\\alpha_\\infty(t)=e$ in $[1,2]$.\nThe roots are marked $t_a$ and $t_b$ on the figure.\n\n\\begin{prop}\n\\label{prop:alpha}\nThe function $\\alpha_\\infty(t)$ is real-analytic in $[1,2]$, convex, and has one point of minimum at $t_o\\in(1,2)$.\nThe inequality\n$$\n\\alpha_\\infty(t_o)<e<\\alpha_\\infty(1)=\\alpha_\\infty(2)\n$$\nholds. \nConsequently, there are uniquely determined $t_a$, $t_b$ with $1<t_a<t_o<t_b<2$ and $\\alpha_\\infty(t_a)=\\alpha_\\infty(t_b)=e$.\n\nAlso the functions $\\alpha_n(t)$ are convex.\n\\end{prop}\n\n\\begin{figure}[htb]\n\\begin{picture}(260,255)\n\\put(0,0){\\includegraphics[scale=0.5]{alphaplot}\n}\n\\put(260,14){\\small $t$}\n\\put(60,16){\\small $t_a$}\n\\put(132,16){\\small $t_o$}\n\\put(199,16){\\small $t_b$}\n\\end{picture}\n\\caption{Function $\\alpha_\\infty(t)$ on the interval $[1,2]$} \n\\label{fig:alphaplot}\n\\end{figure}\n\nIn \\eqref{num-alpha} the numerical values are given for reference. As a matter of fact, only\n$t_b$ will be relevant in the proof of Theorem~\\ref{thm:main}.\n\n\\begin{equation}\n\\label{num-alpha}\n\\ba{l}\n\\alpha_\\infty(1)=\\alpha_\\infty(2)\\approx 2.815572650,\n\\\\[0.7ex]\nt_o\\approx 1.447847, \n\\quad \n\\alpha_\\infty(t_o)\\approx 2.673953412\n\\\\[0.7ex] \nt_a\\approx 1.185591828,\n \\quad\nt_b\\approx 1.742084284.\n\\end{array}\n\\end{equation}\n\nThe behaviour of $\\alpha_\\infty$ is simple, but to prove it analytically is not an easy task.\nOur proof is rather long and technical. It is given in Appendix~\\ref{app:alpha}.\n\n\\smallskip\nThe next lemma, which is a simple corollary\nof Proposition~\\ref{prop:alpha}, will be important in\nthe proof of Proposition~\\ref{prop:Xmin}(b) below.\n\n\\begin{lemma}\n\\label{lem:conv-xi}\nThe functions $\\xi_n(t)$, $n=1,2,\\dots$, are convex\nin $t\\in[1,2]$.\n\\end{lemma}\n\n\\begin{proof}\nThe claim is true for $n=1$, since $\\xi_1(t)=t$.\nThe general case follows by induction due to\nthe identity $\\xi_{n+1}(t)=\\alpha_n(t)\\xi_n(t)+1$\nand the convexity of $\\alpha_n(t)$. \n\\end{proof}\n\n\\subsection{Special points on the curves \\texorpdfstring{$\\gamma_n[1,2]$}{gamma[n]}}\n\\label{ssec:special-points}\n\nRecall that the special values $X^{0}_n=\\min_{t\\geq 1}\\xi_n(t)$ \nhave been defined in Proposition~\\ref{prop:xieta-basic}(c).\nThus $X^{0}_n$ is such a value that the equation\n\\beq{Tnx}\nT_n(\\tau)=x\n\\end{equation}\nhas a solution if and only if $x\\geqX^{0}_n-1$.\n\nIn Sec.~\\ref{ssec:experiment} we have identified and illustrated (Fig.~\\ref{fig:xieta30-32}) special values of parameter $t$: $t^{0}_n$, $t^{\\ell}_n$, and $t^{r}_n$.\nHere we define them analytically. \n($t^{\\ell}_n$ is defined for $n\\geq 7$.) For the curves $\\gamma_n^T$, the corresponding values of $\\tau=t-1$  will be denoted similarly: $\\tau^{0}_n$, $\\tau^{\\ell}_n$, $\\tau^{r}_n$.\n\n\\begin{prop}\n\\label{prop:Xmin}\n{\\rm(a)} $\\xi_n'(t)>0$ for all $t\\geq 2$ and $n\\geq 1$.\nConsequently, the solution $t=t^{0}_n$ of the equation  $\\xi_n(t)=X^{0}_n$, belongs to $[1,2)$.\nEquivalently, the solution $\\tau_n^0$ of the equation \n\\eqref{Tnx} with $x=X^{0}_n-1$ lies in $[0,1)$.\n\n\\smallskip\n{\\rm(b)} For every $n\\geq 1$ and every $x\\geq X^{0}_n-1$\nthe equation~\\eqref{Tnx} has at most two solutions. \n\n\\smallskip\n{\\rm(c)} The sequences $(t_n^0)$ (hence $(\\tau_n^0)$) and $(X^{0}_n)$ are nondecreasing. More precisely, \n$t_n^0=1$ for $n\\leq 6$ and $t_n^0>t_{n-1}^0$ for $n\\geq 7$. \n\\end{prop}\n\n\\begin{definition}\n\\label{def:t_pm}\nIf the equation~\\eqref{Tnx} has two distinct solutions,\nthe smaller will be denoted $\\tau_n^-(x)$ and the larger $\\tau_n^+(x)$. \n\nIf there is a unique solution, it will be denoted $\\tau_n^+(x)$. In this case we leave $\\tau_n^-(x)$ undefined, except when $x=X^{0}_n-1$. We put $\\tau_n^-(X^{0}_n-1)=\\tau_n^+(X^{0}_n-1)=\\tau_n^0$.  \n\nSimilar notation $t_n^{\\pm}(x)$ will be used in reference to the equation $\\xi_n(t)=x$.\n\\end{definition}\n\n\\begin{proof}\n(a) Since $\\xi_n=\\alpha_n\\alpha_{n-1}\\dots\\alpha_1$,\nit suffices to prove that $\\alpha'_n>0$ for\n$t\\geq 2$ and all $n\\geq 1$. We take this inequality\nas the induction hypothesis. It is true for $n=1$.\n\nWe have $\\alpha_n=\\alpha_1+\\sum_{j=1}^{n-1}\\xi_j^{-1}$\nand $\\alpha_1=t$.\nWe need to prove that\n$$\n \\sum_{j=1}^{n-1}\\frac{\\xi'_j}{\\xi_j^2}< 1=\\alpha_1'.\n$$\nAs a consequence of the induction hypothesis, \n$\\xi'_k>0$ for $1\\leq k\\leq n-1$ and $t\\geq 2$.\nHence \n$\\alpha'_j<\\alpha'_1$, so $0<\\xi'_j\/\\xi_j=\\sum_{k=1}^j\n\\alpha'_k\/\\alpha_k<j\/t\\leq j\/2$ for $j=1,\\dots,n-1$.\nAlso, $\\xi_j\\geq\\alpha_1^j\\geq 2^j$. The required estimate follows:\n$$\n \\sum_{j=1}^{n-1}\\frac{\\xi'_j}{\\xi_j^2}\\leq\n\\sum_{j=1}^{n-1}\\frac{j}{2^{j+1}}<\\sum_1^\\infty\\frac{j}{2^{j+1}}=1.\n$$\n\n\\smallskip\n(b) The functions $\\xi_n(t)$, are convex in $1\\leq t\\leq 2$ by Lemma~\\ref{lem:conv-xi}; more precisely, for $n\\geq 2$ they are strictly convex. They are increasing in $[2,\\infty)$ by part (a).\nTherefore the equation $\\xi_n(t)=x$ has at most two solutions.\n\n\\smallskip\n(c) Due to the convexity of $\\xi_n(\\cdot)$, there are two possibilities: either (i) $t^0_n=1$ and $X^{0}_n=\\xi_n(1)$ or (ii) $t^0_n>1$, then $\\xi'(t^0_n)=0$ and the function $\\xi(\\cdot)$ decreases from $\\xi_n(1)$ to $X^{0}_n$ as $t$ changes from $1$ to $t^0_n$. (Cf.\\ the backtracking segment of the curve $\\tilde\\gamma_7$ in Fig.~\\ref{fig:xieta6-7}.)\n\nIn both cases, $\\xi_n'(t)>0$ for $t>t_n^0$.\n\nSuppose that the sequence $(t_n^0)$ is not monotone as claimed. Let $n$ be the least index for which $t^0_n>t^0_{n+1}$. Then $\\xi'_{n+1}(t^0_n)<0$ and $\\xi'_n(t^0_n)=0$. Also\n $t^0_{n-1}\\leq t^0_n$, so $\\xi'_{n-1}(t^0_n)\\leq 0$. \nThe combination of signs $\\xi'_{n+1}<0$, $\\xi'_n=0$,\n$\\xi'_{n-1}\\leq 0$ contradicts the recurrence relation\n$$\n \\frac{\\xi'_{n+1}}{\\xi_{n+1}-1}=\\frac{2\\xi'_n}{\\xi_n}-\\frac{\\xi'_{n-1}}{\\xi_{n-1}},\n$$ \nwhich follows from \\eqref{recxi} by logarithmic differentiation.\n\nBy the recurrence relations \\eqref{recxi} and the one above we find that $\\xi'_n(1)>0$ for $1\\leq n\\leq 6$,\nwhile $\\xi'_7(1)=-\\frac{19661554943536}{328636389375}<0$. \nTherefore $t^0_n>1$ for $n\\geq 7$.\n\n\\smallskip\nFor any $t$, by definition of $X^{0}_n$, we have\n$X^{0}_n\\leq \\xi_n(t)$. Since $\\xi_n(t)<\\xi_{n+1}(t)$,\nit follows that\n$X^{0}_n\\leq\\inf_{t\\geq 1}\\xi_{n+1}(t)=X^{0}_{n+1}$.\nIt is easy to see that the inequality is strict when \n$t^0_n<t^0_{n+1}$, i.e. for $n\\geq 6$.\n\\end{proof}\n\nNext we will analytically prove that the branches of the curve $\\gamma_n$ are always situated as shown in  Fig.~\\ref{fig:xieta30-32} or Fig.~\\ref{fig:xieta6-7}:\nthe ``backward'' branch (where $t<t^0_n$)\nlies above the ``forward'' branch. \n\n\\begin{prop}\n\\label{prop:pmbranches}\nIf the equation $\\xi_n(t)=x$ has two distinct solutions, then $\\eta_n(t_n^+(x))<\\eta_n(t_n^-(x))$.\nEquivalently,\n$G_n(\\tau_n^+(x))<G_n(\\tau_n^-(x))$.\n\\end{prop}\n\n\\begin{proof}\nThe equation $\\xi_n(t)=y$ has two distinct solutions for\nall $y\\in(X^{0}_n,x]$. For $y=X^{0}_n$ the value\n$t_n^-(y)=t^0_n=t_n^+(y)$ is also defined. Put\n$$\n \\kappa(y)=\\xi_{n-1}(t_n^{+}(y))-\\xi_{n-1}(t_n^{-}(y)).\n$$\n\nClearly, $\\kappa(X^{0}_n)=0$.\nWe will prove that: \n(i) $\\kappa(X^{0}_n+\\varepsilon)>0$\nfor sufficiently small $\\varepsilon>0$, \nand \n(ii) $\n \\kappa(y)\\neq 0 \n$\nfor any $y\\in(X^{0}_n,x]$.\nBy continuity of $\\kappa(\\cdot)$ it follows then that\n$\\kappa(x)>0$. \n\n\\smallskip\nProof of (i): Due to the inequality $t_n^0>t_{n-1}^0$ (Proposition~\\ref{prop:Xmin}(c)), the function \n$\\xi_{n-1}(t)$ increases in the neighborhood of $t^0_n$.\n\n\n\\smallskip\nProof of (ii): Suppose, by way of contradiction, that \n$\\kappa(y_*)=0$ for some $y_*>X^{0}_n$. \nLet $t_*^\\pm=t_n^\\pm(y_*)$.\nThen \n$y=\\xi_{n}(t_*^-)=\\xi_{n}(t_*^+)$\nand $\\xi_{n-1}(t_*^-)=\\xi_{n-1}(t_*^+)$.\nBy the inverse recurrence relation we conclude that\n$\\xi_{j}(t_*^-)=\\xi_{j}(t_*^+)$ for $j$ from $n$ down to $1$. For $j=1$ it results in $t_*^-=t_*^+$,\na contradiction.\n\nNow, by \\eqref{dxieta}\n$$\n \\frac{d\\eta_n(t^{\\pm}(y))}{d y}=\\frac{1}{\\xi_{n-1}(t^{\\pm}(y))}.\n$$\nThe inequality $\\kappa(y)>0$ implies\n$$\n \\frac{d\\eta_n(t^{+}(y))}{d y}<\n\\frac{d\\eta_n(t^{-}(y))}{d y}.\n$$\nIntegrating from $y=X^{0}_n$ to $x$ we get $\\eta_n(t^{+}(x))<\\eta_n(t^{-}(x))$.\n\\end{proof}\n\n\\begin{definition}\n\\label{def:xcross}\nFor $n\\in\\mathbb{N}$, let $X^{\\!\\times}_n$ be defined by%\n\\footnote{The subtraction of $1$ \n makes this definition consistent with notation $X^{\\!\\times}_n$ in Sec.~\\ref{ssec:experiment}.}\n$X^{\\!\\times}_n-1=\\inf\\{x\\mid\\nu(x)=n\\}\n=\\max\\{x\\mid\\nu(x)=n-1\\}$, where $\\nu(\\cdot)$\nis the critical index (Definition~\\ref{def:crind}).\n\\end{definition}\n\n\\begin{prop}\n\\label{prop:argmin_intervals}\n{\\rm (a)} $F(x)=F(x_{n-1})$ for $0<x\\leq X^{\\!\\times}_n$\nand  $F(x)<F(x_{n-1})$ for $x>X^{\\!\\times}_n-1$.\nThe point $(X^{\\!\\times}_n-1,\\,F(X^{\\!\\times}_n-1))$ is common to the\ncurves $\\gamma_{n}^T$ and $\\gamma_{n+1}^T$.\nEquivalently, the point $(X^{\\!\\times}_n,\\,1+F(X^{\\!\\times}_n-1))$ is common to the\ncurves $\\gamma_{n}$ and $\\gamma_{n+1}$.\n\n\\smallskip\n{\\rm (b)} $X^{\\!\\times}_n=X^{0}_{n}=\\xi_n(1)$ for $1\\leq n\\leq 6$\nand $X^{\\!\\times}_n<\\xi_n(1)$ for $n\\geq 7$.\n\\end{prop}\n\n\\begin{proof}\n(a) If $x\\leqX^{\\!\\times}_n-1$, then by the monotonicity of\n$\\nu(\\cdot)$ (Proposition~\\ref{prop:crind}) $\\nu(x)\\leq n-1$, hence $F(x)=F_{\\nu(x)}=F_{n-1}(x)$. \n\nIf $x>X^{\\!\\times}_n-1$, then $\\nu(x)>n-1$, hence by Definition~\\ref{def:crind} (of $\\nu(x)$) $F_{n-1}(x)>F(x)$.\n\nThe point $(X^{\\!\\times}_n-1,F(X^{\\!\\times}_n-1))$ lies on the graph of the function $F(\\cdot)$. In the left neighbourhood of\nthis point the graph of $F$ coincides with graph of $F_{n-1}$ and in the right neighbourhood -- with graph of $F_n$. Hence the curves $\\gamma_{n-1}$ and $\\gamma_n$ \nmeet at this point.\n\n\\smallskip\n(b) For $1\\leq n\\leq 6$ the intervals $[\\xi_n(1),\\xi_n(2))$\ndo not overlap, hence $\\nu(x-1)=n-1$ for $x\\leq \\xi_n(1)$. \n\nIn general, there are no points on the curve $\\gamma_{n-1}[1,2]$ with abscissas greater than $\\xi_{n-1}(2)$, since $\\xi_{n-1}'(t)>0$ for $t\\geq 2$ (Proposition~\\ref{prop:Xmin}(a)). \n\nFor $n\\geq 7$, $X^{0}_n<\\xi_n(1)=\\xi_{n-1}(2)$.\nThe equality $X^{\\!\\times}_n=\\xi_{n}(1)$ is impossible, since\nthe backward branch of the curve $\\gamma_n$ starting\nat $(\\xi_n(1),\\eta_n(1))$ lies above the branch with\nparameter values $t>X^{0}_n$ by Proposition~\\ref{prop:pmbranches}.\n\\end{proof}\n\nThe graph of the function $F(x)$ is the union of the segments of the curves $\\gamma_n^T$ corresponding to the parameter values $[\\tau^\\ell_n, \\tau^r_n]\\subset(\\tau^0_n,1)$. We have\n$$\n T_n(\\tau^\\ell_n)=T_{n-1}(\\tau^r_{n-1})=X^{\\!\\times}_n\n$$ \nand \n$$\n G_n(\\tau^\\ell_n)=G_{n-1}(\\tau^r_{n-1}).\n$$\nWith reference to the lower envelope of the curves\n$\\gamma_n$, the parameter values $t_n^\\ell=\\tau_n^\\ell+1$\nand $t_n^r=\\tau_n^r+1$ play the same role. \n\n\\iffalse\n\\section{More general problem}\n\n* AM-GM:\n$$\n \\frac{x_n}{x_{n-1}}+ \\frac{x_{n-1}}{x_{n-2}}+\\dots+\n \\frac{x_1}{x_0}+ \\frac{x_0}{p}\\to\\min \n\\quad\\text{\\rm over $x$ and $n$}.\n$$\n\n$$\n RHS\\geq n\\left(\\frac{x_n}{p}\\right)^{1\/n}.\n$$\nThere must be $x_n\/p\\geq 1$. Since $1\/p=N$ (by interpretation in Part I), we get $n\\sim\\ln N$.\n\n* Bellmann, Lee \\cite{BelLee1978} consider\n$$\n f(p)=\\max_q H(p,q,f(T(p,q))\n$$\nIn our case, $p$ becomes $\\xi$, $q$ becomes $t$ ($0<q< 1$); $T(p,q)=q(=t)$; $H(p,q,z)=ph(q)+z$.\n\nOr (?) $q=1\/(1+t)$; $t=q^{-1}-1$; $T(p,q)=T(q)=q^{-1}-1$.\n\n* Our case (one-parameter extension of the dynamic programming method as applied to the AM-GM problem)\n$$\n\\ba{l}\n g_{n+1}(\\xi)=\\inf\\left(g_n(t)+\\xi h(t)\\right)=\\inf_t G_n(t,\\xi)=g_n(t_n)+\\xi h(t_n),\n\\\\[2ex]\n g_1(\\xi)=s\\xi\n\\end{array}\n$$\nThe critical point $t_n$ is a root of the equation $\\partial_t G_n(t,\\xi)=0$, i.e.\n$$\n \\xi h'(t_n)=-g_n'(t_n).\n$$\nOn the other hand,\n$$\n g'_{n+1}(\\xi)=\\partial_t G_n(t,\\xi)|_{t=t_n}\\cdot t_n'(\\xi)+\n\\partial_\\xi G_n(t,\\xi)=0+\\partial_\\xi G_n(t,\\xi)=\nh(t_n).\n$$\nShifting the index, \nwe get $g_{n}(t_{n-1})=-\\xi h'_{t_{n-1}}$.\nHence\n$$\n \\xi h'(t_{n})=-h(t_{n-1}),\n$$\nthe implicit recurrence equation for determining $t_n(\\xi)$.\n\nIt begins with $s+\\xi h'(t_1)=0$, that is,\n$$\n h'(t_1)=-\\frac{s}{\\xi}.\n$$\n\nLet us assume that for any initial value $\\tau\\geq 0$ the sequence \n$(t_1=\\tau,t_2,\\dots)$ is determined uniquely.\nWe will write\n$$\n t_j=T_j(\\tau) \\quad (j=1,2,\\dots).\n$$\n\nOur minimization problem can be stated in the form\n$$\n g_n(\\xi)=\\min S_n(t_0,\\dots,t_n),\n$$\nwhere \n$$\n S_n(t_0,t_1,\\dots,t_n)=\\sum_{k=1}^n L(t_{k-1},t_k),\n$$\n$$\n L(p,q)=q h(p),\n$$\nwith boundary conditions $t_0=t_*$ (given) and $t_n=\\xi$.\n\n\n* Put\n$$\n G_n(t_0,\\dots,t_n\\mid c)=t_0+\\frac{t_1}{t_0+c}+\n\\frac{t_2}{t_1+c}+\\dots+\\frac{t_n}{t_{n-1}+c}.\n$$\nIf $t_j=ct'_j$, then\n$$\nG_n(0,t_1,\\dots,t_n\\mid c)=G_{n-1}(t'_1,\\dots,t'_n\\mid 1).\n$$\n\nDefine\n$$\n g_n(\\xi)=\\min_{(t_0,\\dots,t_{n-1})} G_n(t_0,\\dots,t_n\\mid c)|_{t_n=\\xi}\n=\\min_{t(=t_{n-1})}\\left(g_{n-1}(t)+\\frac{\\xi}{t+c}\n\\right).\n$$\n\n\n* Given a function $h$, define\n\\beq{recurgh}\n g(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+ h(\\xi,\\rho)\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n* Given a function $h$, define\n\\beq{recurgh}\n g(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+\\xi h(\\rho)\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n\n* For $s>0$:\n\\beq{recurgs}\n g_s(\\xi)=\\inf_{0<\\rho\\leq\\xi-1}\\left(g(\\rho)+\\frac{\\xi}{\\rho+s}\\right)\n\\end{equation}\nwith initial condition\n$$\n g(\\xi)=\\xi, \\qquad 0<\\xi\\leq 1.\n$$\n\n\\begin{prop}\n$g_0(\\xi)=g_1(\\xi-1).$\n\\end{prop}\n\n\n\n\\section{Analysis of the sequence $F^*(p)$}\n\nThe proofs of qualitative results rely on computations only insofar as the required accuracy is rather low and there is no doubt about stability etc.\nWe also provide graphs and more precise values of various numerical constants but not as part of rigorous arguments.  \n\nWhen referring to decimal numerals, we indicate rounding down or up by superscripts $+$ and $-$; precise decimal results are indicated\nby the $\\doteq$ sign as in the following example: \n$$\n\\frac{45}{32}=1.4^+,\\qquad\n\\frac{45}{32}=1.407^-,\n\\qquad \n\\frac{45}{32}\\doteq 1.40625.\n$$\n\nWe won't be excessively pedantic. The choice of parameters is not critical and no attempt is made to optimize them in numerical evaluations. The deviation from \"absolute rigor\" is no more than the customary allowances of using the same name for technically non-identical objects such as, say, group as a 4-tuple $(G,e\\in G, \\mathrm{mult}:G\\times G\\to G, \\mathrm{inv}:G\\to G)$ and the underlying set $G$.\n\nIn this section we prove that there exists constant \n$$\nB\\approx-0.70465603\n$$\nsuch that the solution of the recursion\n$$\n g(\\xi)=\\inf_{0<\\rho\\leq \\xi-1} \\left( g(\\rho)+\\frac{\\xi}{\\rho+1}\\right)\n$$\nwith initial condition\n$$\n g(\\xi)=\\xi,\\quad 0<\\xi\\leq 1,\n$$\nhas asymptotics\n$$\n g(\\xi)=e\\log\\xi+B+O(1\/\\log\\xi)\n$$\nas $\\xi\\to\\infty$.\n\nThis fact should not be an isolated peculiarity of the particular equation, but we do not pursue its investigation as a structural property of a class of similar equation, but rather take advantage of the specifics of our system explicitly --- notably the Eq.~\\eqref{gprime_rec}. \n\\fi\n\n\\iffalse\n---------------\n\n... We supply some additional facts that are not strictly necessary for our main line, but add to understanding of the fine features of the \nrecurrent sequences...\n\n-----------------------\n\nPut\n$$\n \\chi_n(t)=\\eta_n(t)-e\\log\\xi_n(t).\n$$\n\nProperties of the two roots of $\\alpha_\\infty(t)=e$:\n\n\\begin{lemma}\n(a) There exists $n_0$ such that for any $n\\geq n_0$\nthere are $t_1^{(n)}$ and $t_2^{(n)}$ in $(1,2)$ uniquely determined by the conditions:\n\\beq{deft1n}\n\\chi_n(t_1^{(n)})=0 \n\\end{equation}\nand\n\\beq{deft2n}\nt_2^{(n)} \\;\\text{\\rm is the point of minimum of $\\chi_n(t)$ in $[1,2]$}.\n\\end{equation}\n\n(b) $t_1=\\lim_{n\\to\\infty} t_1^{(n)}$ and\n$t_2=\\lim_{n\\to\\infty} t_2^{(n)}$.\n\\end{lemma}\n\\fi\n\n\\subsection{Asymptotics of\n\\texorpdfstring{$\\xi_n$}{xi[n]} and \n\\texorpdfstring{$\\eta_n$ as $n\\to\\infty$}{eta[n]}}\n\\label{ssec:asxieta}\n\nPut \n\\begin{equation}\n\\label{deltan}\n\\delta_{n}(t)=\\sum_{j=n}^\\infty \\frac{1}{\\xi_j(t)}=\\alpha_\\infty(t)-\\alpha_{n}(t).\n\\end{equation}\nand define the functions $\\phi(t)$ and $\\psi(t)$ for $t\\in[1,2]$ by\n\\begin{equation}\n\\label{phi}\n\\phi(t)=\\sum_{j=1}^\\infty\\log\\left(1-\\frac{\\delta_j(t)}{\\alpha_\\infty(t)}\\right),\n\\end{equation}\n\\begin{equation}\n\\label{psi}\n\\psi(t)=t-\\alpha_\\infty(t)-\\sum_{j=1}^\\infty \\delta_j(t).\n\\end{equation}\n\n\n\\begin{prop}\n\\label{prop:asphipsi}\nThe functions $\\phi(t)$ and $\\psi(t)$ are real-analytic and\nthe following asymptotic formulas hold:\n\\beq{asxin}\n\\log\\xi_n(t)=n\\log\\alpha_\\infty(t)+ \\phi(t)+\nO\\left(2^{-n}\\right),\n\\end{equation}\n\\beq{asetan}\n\\eta_n(t)=n\\alpha_\\infty(t)+ \\psi(t)+\nO\\left(2^{-n}\\right).\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} \nThe recurrence relations \\eqref{recrelalxi} imply\n$$\n \\log\\xi_n(t)=\\log\\xi_0(t)+\\sum_{j=1}^n\\log\\alpha_j(t)\n=\\log t+\\sum_{j=1}^n\\log\\frac{\\alpha_j(t)}{\\alpha_\\infty(t)}\n+n\\log\\alpha_\\infty(t).\n.\n$$\nBy definition, \n$\\alpha_j(t)=\\alpha_\\infty(t)-\\delta_j(t)$, so \n$$\n\\log\\xi_n(t)-\\phi(t)-n\\log\\alpha_\\infty(t)=-\n\\sum_{j=n+1}^\\infty\\log\\left(1-\\frac{\\delta_j(t)}{\\alpha_\\infty(t)}\\right).\n$$\nSince the sequence $\\xi_j(t)$ grows exponentially, taking into account the uniform estimate $\\alpha_n(t)\\geq 2$ from Proposition~\\ref{prop:xieta-basic}(b), \nfor sufficiently large $n_0$ the conditions of Lemma~\n\\ref{lem:est-al-xi} are met with $a=2$ and it follows by \\eqref{estdeltan} that $\\delta_j(t)=O(2^{-j})$, $j>n_0$. This estimate implies \\eqref{asxin}.\n\nSimilarly, by the recurrence relations \\eqref{recreleta}\n$$\n\\ba{rcl}\n\\displaystyle\n \\eta_n(t)=\\eta_0(t)+\\sum_{j=0}^{n-1}\\alpha_j(t)\n&=&\n\\displaystyle\n1+n\\alpha_\\infty(t)-\\sum_{j=0}^{n-1}\\delta_j(t)\n\\\\[2ex]\n&=&\n\\displaystyle\n\\psi(t)+n\\alpha_\\infty(t)+\\sum_{j=n}^{\\infty}\\delta_j(t).\n\\end{array}\n$$\nThe remainder term is estimated as above and we come to \\eqref{asetan}.\n\\end{proof}\n\n\\subsection{Narrowing down the parameter domain}\n\\label{ssec:narrowing}\n\nFrom the crude asymptotic result --- Proposition~\\ref{prop:crudeas} --- we know that \n$$\\frac{\\eta_n(t)}{\\log\\xi_n(t)}=e+o(1)$$ \non the segment of the\ncurve $\\gamma_n$ that belongs to the lower envelope.\nHence $\\log\\alpha_\\infty(t)\/\\alpha_\\infty(t)=e+o(1)$\nfor the corresponding values of the parameter and\nwe conclude that $\\alpha_\\infty(t)$ must be close to $e$.\nTherefore $t$ must be close to one of the two roots,\n$t_a$ or $t_b$, of the equation $\\alpha_\\infty(t)=e$.\n\nBy Proposition~\\ref{prop:pmbranches}, \namong the two values $t_1<t_2$ corresponding\nto the same value of $\\xi_n$, the inequality\n$\\eta_n(t_2)<\\eta_n(t_1)$ takes place. Therefore\nonly the neighbourhood of $t_b$ is of interest for\ndetermining the lower envelope of the curves $\\gamma_n$.\n\nSummarizing, we state\n\n\\begin{prop}\n\\label{prop:minint}\nThe values $t_n^\\ell$ and $t_n^r$ defined at the end of\nSec.~\\ref{ssec:special-points} tend to $t_b$ as $n\\to\\infty$. \n\\end{prop} \n\n\\subsection{End of proof of Theorem~\\ref{thm:main}}\n\\label{ssec:eop-main}\nDenote\n$$\n \\beta(t)=\\frac{\\alpha_\\infty(t)}{\\log\\alpha_\\infty(t)}\n$$\nand\n$$\n\\zeta(t)=\\psi(t)-\\beta(t)\\,\\phi(t).\n$$\nDue to Proposition~\\ref{prop:minint}, in the asymptotic calculation we may choose some compact neighborhood $I\\subset(t_o,2)$ \nof the point $t_b$\nas the parameter interval. \n\nSince $\\beta'(t)=(\\log\\alpha_\\infty(t)-1)\\alpha_\\infty'(t)\/(\\log\\alpha_\\infty(t))^2$, the \nfunction $\\beta(t)$ has a unique critical point (the minimum) on $I$, namely, $t=t_b$. By Proposition~\\ref{prop:alpha}, $\\alpha'(t_b)>0$.\n\nThe family of parametric curves $x=\\log\\xi_n(t)$, $y=\\eta_n(t)$ is shown to satisfy all conditions of\nTheorem~\\ref{thm:paramcurves}. \n\nUsing the asymptotic formulas \\eqref{asxin} and \\eqref{asetan}\nLet us find the coefficients in the final asymptotic formula \\eqref{asfabs} in our case.\n\n\nThe point $t_0$ in Theorem~\\ref{thm:paramcurves}\ncorresponds to $t_b$ in this context.\nThe terms $r_0(t)$ and $r_1(t)$ of the general formulation\nare not present in Eqs.~\\eqref{asxin}--\\eqref{asetan},\nso $\\delta(t)\\equiv 0$. We have \n$a_0=\\beta(t_b)=e$,\n$a_1=\\zeta(t_b)\\approx -0.704656$. The constant $A$ in Theorem~\\ref{thm:main} equals $1-\\zeta(t_b)$ (remember that $G_n(t-1)=\\eta_n(t)-1$ on the original extremal trajectory).\n\nIn the formula \\eqref{asgenPhi}, $p_0(t_0)=\\log\\alpha_\\infty(t_b)=1$\nand $q_0(t_0)=\\phi(t_b)\\approx 0.6974$;\nthis is the numeric constant $b$ in Theorem~\\ref{thm:main}.\n\nWe will derive the coefficients \\eqref{coefa2a3} analytically in the concluding two lemmas, showing that $a_3=e\/2$ (Lemma~\\ref{lem:a3})\nand $\\zeta'(t_b)=0$ (Lemma~\\ref{lem:Dzeta}) hence $a_2=0$.\n\nThus the proof of Theorem~\\ref{thm:main} is complete.\n\\qed\n\n\\begin{lemma}\n\\label{lem:a3}\nThere holds the identity\n$$\n \\beta''(t_b)\\left(\\frac{(p_0(t_b))^2}{p_0'(t_b)}\n\\right)^2=e,\n$$\nwhere $p_0(t)=\\log\\alpha_\\infty(t)$.\n\\end{lemma}\n\n\\begin{proof} Since $\\beta=e^{p_0}\/p_0$,\nwe have $\\beta'=p_0'\\cdot e^{p_0}\/p_0^2\\cdot (p_0-1)$.\nTo evaluate $\\beta''(t_b)$, it suffices to differentiate\nthe last factor (which vanishes at $t_b$):\n$$\n\\beta''(t_b)=\\left.\\frac{p_0'(t)\\cdot e^{p_0(t)}}{p_0^2(t)}\\cdot p_0'(t)\\right|_{t=t_b}.\n$$\nTherefore \n$$\n\\beta''(t_b)\\left(\\frac{(p_0(t_b))^2}{p_0'(t_b)}\n\\right)^2=e^{p_0(t_b)}p_0^2(t_b)=e\\cdot 1^2=e.\n\\eqno\\qedhere\n$$\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:Dzeta}\nThere holds the identity\n$\\zeta'(t_b)=0$, that is, $\\psi'(t_b)-e\\phi'(t_b)=0$.\n\\end{lemma}\n\n\\begin{proof}\nWriting $\\psi'=-\\sum_{j=0}^\\infty (\\alpha'_\\infty-\\alpha'_j)$ and $\\phi'=\\sum_{j=1}^\\infty\n(\\alpha'_j\/\\alpha_j-\\alpha'_\\infty\/\\alpha_\\infty)$,\nwe see that\n$$\n\\left[\\psi'-e\\phi'\\right]_{t_b}\n=\n1-\\alpha'_\\infty(t_b)-\\sum_{j=1}^\\infty\\left[\n\\alpha'_\\infty\\left(1-\\frac{e}{\\alpha_\\infty}\n\\right)-\\alpha'_j\\left(1-\\frac{e}{\\alpha_j}\n\\right)\\right]_{t_b}\n$$\nIn view of the identities $\\alpha_\\infty(t_b)-e=0$\nand $\\alpha_j(t_b)-e=-\\delta_j(t_b)$, the right-hand side\nsimplifies to \n$$\n 1-\\alpha'_\\infty(t_b)-\\sum_{j=1}^\\infty\\delta_j(t_b)\\,\\frac{\\alpha'_j(t_b)}{\\alpha_j(t_b)}.\n$$\nWe will prove that in general\n\\begin{equation}\n\\label{alpha_identity}\n \\sum_{j=1}^\\infty \\delta_j(t) \\frac{\\alpha_j'(t)}{\\alpha_j(t)}=1-\\alpha_\\infty'(t).\n\\end{equation}\n\nBy the definition \\eqref{defalpha} of $\\alpha_j$,\n$$\n \\frac{\\alpha_j'}{\\alpha_j}=\n\\frac{\\xi_j'}{\\xi_{j}}-\\frac{\\xi_{j-1}}{\\xi_{j-1}}.\n$$\nApplying Abel's summation-by-parts formula to the partial sum in the l.h.s. of \\eqref{alpha_identity}, we get\n$$\n \\sum_{j=1}^N \\delta_j \\frac{\\alpha_j'}{\\alpha_j}=\n\\delta_N\\frac{\\xi_N'}{\\xi_N}-\\delta_1\\frac{\\xi_0'}{\\xi_0}\n+\\sum_{j=1}^{N-1} (\\delta_j-\\delta_{j+1}) \\frac{\\xi_j'}{\\xi_{j}}.\n$$\n\nWe have $\\xi_0=1$ and $\\xi'_0=0$; $\\xi'_N\/\\xi_N=O(1)$\nand $\\delta_N=o(1)$ as $N\\to\\infty$, so the boundary terms are $o(1)$.\nNow, \n$\\delta_{j+1}-\\delta_j=\\alpha_j-\\alpha_{j+1}=-\\xi_j^{-1}$. Therefore\n$$\n\\sum_{j=1}^N \\delta_j \\frac{\\alpha_j'}{\\alpha_j}\n=\\sum_{j=1}^{N-1}(\\xi_j^{-1})'+o(1)\n=\\alpha'_1-\\alpha'_N+o(1),\n$$\nand the limit is $\\alpha'_1-\\alpha'_\\infty=1-\\alpha'_\\infty$.\n\\end{proof}\n\n\\iffalse\n\\begin{corollary}\nThe critical index $\\nu(x)$ satisfies the asymptotic estimate\n$$\n e=\\min_{1\\leq t\\leq 2}\\log\\alpha_\\infty(t)\\leq\\liminf_{x\\to\\infty}\\frac{\\log x}{\\nu(x)}\n\\leq\\limsup_{x\\to\\infty}\\frac{\\log x}{\\nu(x)} \n\\leq \\max_{1\\leq t\\leq 2}\\log\\alpha_\\infty(t).\n$$\n\\end{corollary}\n\n\\begin{proof}\nWe combine Proposition~\\ref{prop:tau01} with the asymptotics of $\\log\\xi_n(t)$ from Lemma~\\ref{lem:asphipsi}.\n\n[Need details]\n\n(f) [Not quite; it's too early here. Put where discuss the asymptotics of the minimizing critical point.] \n\nA refinement of the functional equation \n\\eqref{fe1} reads\n\\beq{fe2}\n f(x)=\\min_{0<y\\leqX^{**}_{\\nu(x)-1}}\\left(f(y)+\\frac{x}{y+1}\\right).\n\\end{equation}\n\\end{proof}\n\\fi\n\n\n\n\\input{sec_Ap}\n\n\\input{app_proof-paramcurves}\n\n\\input{app_alpha}\n\n\\iffalse\n\\appndx{Stability analysis of the recurrence \\eqref{recrelalxi}}{app:stability}\n\n\\begin{prop}\nLet $(\\tilde\\alpha_n)_{n\\geq 0}$, $(\\tilde\\xi_n)_{n\\geq 0}$,\n$(\\varepsilon_n)_{n\\geq 0}$ and $(\\mu_n)_{n\\geq 0}$ be four sequences of positive numbers. \nSuppose that\n$$\n a=\\sum_{n=1}^\\infty\\frac{1}{\\tilde\\alpha_n}<\\infty\n$$\nand \n$$\n b=\\sum_{n=0}^\\infty\\frac{1}{\\tilde\\xi_n}<\\infty.\n$$\nGiven some parameter $c>1$, \nsuppose further that\n$$\n k=1-c^2 ab>0.\n$$\nFinally, suppose that $(\\varepsilon_n)$ and $(\\mu_n)$ \nsatisfy the relations\n$$\n \\ba{l}\n\\displaystyle\n \\varepsilon_{n+1}<\\varepsilon_n+c\\frac{\\mu_n}{\\tilde\\xi_n},\n\\\\[2ex]\n \\displaystyle\n \\mu_{n+1}<\\mu_n+c\\frac{\\varepsilon_{n+1}}{\\tilde\\alpha_{n+1}}.\n\\end{array}\n$$\nDenote\n$$\n\\ba{l}\n\\displaystyle\n \\bar{\\varepsilon}=\\frac{\\varepsilon_0+cb\\mu_0}{k},\n\\\\[2ex]\\displaystyle\n \\bar{\\mu}=\\frac{\\mu_0+ca\\varepsilon_0}{k}.\n\\end{array}\n$$\nThen for any $n\\geq 0$ the inequalities\n$\n \\varepsilon_{n}<\\bar\\varepsilon\n$\nand\n$\n \\mu_{n}<\\bar\\mu\n$\nhold true.\n\\end{prop}\n\n\\begin{proof}\nThe claim is obviously true for $n=0$.\nAssuming it is true up to a certain $n$, we have\n$$\n \\varepsilon_{n+1}<\\varepsilon_0+c\\bar\\mu\\sum_{j=0}^n\\frac{1}{\\tilde\\xi_j}\n<\\varepsilon_0+\\frac{cb}{k}(\\mu_0+ca\\varepsilon_0)=\\bar\\varepsilon\n$$ \nand\n$$\n \\mu_{n+1}<\\mu_0+c\\bar\\varepsilon\\sum_{j=1}^{n+1}\\frac{1}{\\tilde\\alpha_j}\n<\\mu_0+\\frac{ca}{k}(\\varepsilon_0+cb\\mu_0)=\\bar\\mu.\n$$\nThe validity of our claim for all $n$ follows by induction.\n\\end{proof}\n\n\n\\appndx{Monotonicity of critical points\nand the mutual position of branches of the curve\n$\\eta_n(\\xi_n)$}{app:critpts}\n\n\\begin{prop}\nIf $\\tau_{on}$ is the critical point of the function\n$\\alpha_n(\\tau)$ (which exists for $n\\geq 5$), then\n$t_{on}<t_{o{n+1}}$ ($n\\geq 5$).\n\\end{prop}\n\n\\begin{proof}\nSince $\\alpha_n''(\\tau)>0$ (item 3 in proof of Proposition~\\ref{prop:alpha}) and $\\xi_1(\\tau)''\\equiv 0$, we conclude by induction that the functions\n$\\xi_n(\\tau)$ are convex for $1\\leq \\tau\\leq 2$\nand any $n$.\n\nTherefore the equation $\\xi_n'(\\tau)=0$ can have at most one root in $[1,2]$ and hence in $[1,\\infty)$. (Recall that for $\\tau>2$ the inequality $\\xi_n'(\\tau)>0$ is asserted by Corollary of Lemma~\\ref{lem:estUn}.)\n\nOur inductive assumption is: $\\xi'_{n-1}(\\tau)>0$ for \nall $\\tau\\geq \\tau_{on}$. \n\nDifferentiating the recurrence relation, we get\n$$\n \\xi'_{n+1}(\\tau)=\\frac{2(\\xi_n(\\tau)+1)}{\\xi_{n-1}(\\tau)+1}\\xi'_{n}(\\tau)-\\left(\\frac{\\xi_n(\\tau)+1}{\\xi_{n-1}(\\tau)+1}\\right)^2\\xi'_{n-1}(\\tau).\n$$\nAt $\\tau=\\tau_{on}$, $\\xi'_n(\\tau)=0$. Hence\n$\\xi'_{n+1}(\\tau_{on})$ and $\\xi'_{n-1}(\\tau_{on})$ have different signs. \n\nThe inequality $\\xi'_{n-1}(\\tau_{on})<0$ contradicts the inductive assumption. Hence $\\xi'_{n+1}(\\tau_{on})<0$. Therefore\n$\\xi'_{n+1}(\\cdot)$ has a root in $(\\tau_{on},1]$.\nThat root is unique and equals $\\tau_{o,n+1}$.\nWe conclude that $\\tau_{o,n+1}>\\tau_{on}$ and\n$\\xi_n'(\\tau)>0$ for $\\tau\\geq\\tau_{o,n+1}$,\ncompleting the induction step.\n\\end{proof}\n\n\n\\begin{prop}\nIf $\\tau_{1n}<\\tau_{2n}$ are the roots of the function $\\alpha_n(\\tau)=e$, then\n$\\eta_{1n}>\\eta_{2n}$. In general, \nif $\\xi_n(\\tau')=\\xi_n(\\tau'')$ with $\\tau'<\\tau''$,\nthen $\\eta_n(\\tau')>\\eta_n(\\tau'')$.\n\\end{prop}\n\n\\begin{proof}\n\\end{proof}\n\\fi\n\n\n\\section{Constants \\texorpdfstring{$A(p)$}{A(p)}: existence and computation}\n\\label{sec:Ap}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:genpar}} \n\n(a) For any $n$-tuple $\\mathbf{t}$, the function\n$p\\mapsto F^{(p)}(\\mathbf{t})$ is decreasing. Hence,\nif the asymptotics \\eqref{Fpasym} takes place, the function\n$A(p)$ is at least nondecreasing.\n\nWe will consider the cases $p>1$ and $p<1$ separately \nto justify the formula \\eqref{Fpasym} and to deduce that $A(p)$ is strictly increasing. That way, a proof of (a) will be finished.\n\nLet us note an identity that will be useful in both cases\n$p>1$ and $p<1$.\n\nFor any $q>0$, putting $\\tilde t_j=t_j\/q$, $j=1,\\dots,n$, we get\n$t_j\/(t_{j-1}+p)=\\tilde t_j\/(\\tilde t_{j-1}+p\/q)$.\nHence\n\\begin{align}\nS^{(p)}_n(t_1,\\dots,t_n)\n&=t_1+\\frac{\\tilde{t}_2}{\\tilde t_1+p\/q}+\\dots+\\frac{\\tilde t_n}{\\tilde{t}_{n-1}+p\/q}\n\\label{Sp-transform1}\n\\\\[1ex]\n&=t_1-\\tilde{t}_1+S^{(p\/q)}_n(\\tilde{t}_1,\\dots,\\tilde {t}_n).\n\\label{Sp-transform}\n\\end{align}\n\n\\medskip\n(b) Case $p>1$. Let us take $q=p$ in \\eqref{Sp-transform}.\nWe get\n$$\nS^{(p)}_n(t_1,\\dots,t_n)\n=t_1\\left(1-\\frac{1}{p}\\right)\n+S_n(\\tilde t_1,\\dots,\\tilde t_n)\\geq \nS_n(\\tilde t_1,\\dots,\\tilde t_n).\n$$\nHence $F^{(p)}_n(x)\\geq F_n(x\/p)$ and $F^{(p)}(x)\\geq F(x\/p)$.\n\nOn the other hand, taking again $\\tilde t_j=t_j\/p$\nand setting $t_1=0$, we obtain  \n$$\n\\left. S^{(p)}_n(t_1,\\dots,t_n)\n\\right|_{t_1=0}=S_{n-1}(\\tilde t_1,\\dots,\\tilde t_{n-1}).\n$$\nHence $F^{(p)}_n(x)\\leq F_{n-1}(x\/p)$, so $F^{(p)}(x)\\leq F(x\/p)$.\n\nThe identity \\eqref{Fpgt1} is thus proved. The asymptotics and the expression for $A(p)$ follow\nfrom Theorem~\\ref{thm:main}.\n\n\\medskip\n(c) Case $0<p<1$. Here we give only a sketch of proof, using heuristics at some steps.\n\nThe asymptotics \\eqref{Fpasym} can be derived in the same way as we did it for $p=1$, by obtaining parametric\ndescription of the curves $y=F^{(p)}_n(x)$.\n\nTo derive the functional equation, take $q=t_1+p$ in Eq.~\\eqref{Sp-transform1}. Then $\\tilde t_1+p\/q=1$,\nhence\n$$\n S^{(p)}_n(t_1,\\dots,t_n)=t_1+S^{(p\/q)}_{n-1}(\\tilde t_2,\\dots,\\tilde t_n).\n$$\nIt follows that\n$$\n F^{(p)}_n(x)=\\inf_{q\\geq p} \\left(q-p+F^{(p\/q)}_{n-1}(x\/q)\\right).\n$$\nConsequently,\n$$\n F^{(p)}(x)=\\inf_{q\\geq p} \\left(q-p+F^{(p\/q)}(x\/q)\\right).\n$$\nFrom \\eqref{Sp-transform} we see that \n$F^{(p)}(x)\\leq F^{(p)}(x\/q)$ for $q\\leq 1$. As $q-p\\geq 0$, the minimization range can be reduced to $q\\geq 1$.\n\nThe next step needs a justification that we omit. Instead of the functions $F^{(\\dots)}(\\dots)$ we substitute their asymptotics\n(and change the letter $q$ into $u$).\nThe result is \n$$\n e\\log x-A(p)+o(1)=\\inf_{u\\geq 1} \\left(u-p+e\\log\\frac{x}{u}-A\\left(\\frac{p}{u}\\right)+o(1)\\right).\n$$\nAfter cancellation of the $e\\log x$ terms the remaining leading terms are constants (with respect to $x$). We obtain the equation \\eqref{feq-Ap} with a weaker constraint: $u\\geq 1$ rather that $1\\leq u\\leq e$.\nHowever, the values $u>e$ can be excluded from the maximization range since the function $A(p\/u)$ is nonincreasing w.r.t. $u$ and the function $e\\log u-u$ strictly decreases for $u>e$.\n\n\\smallskip\nFinally, let us explain the asymptotics of $A(p)$\nas $p\\to +0$. \nChoosing $u=e$ in the right-hand side of \\eqref{feq-Ap} we\nget $A(p)\\geq A(p\/e)+p$.\nTherefore\n$A(p)\\geq A(pe^{-k})+p\\sum_{j=0}^{k-1} e^{-j} $ for any $k\\in\\mathbb{N}$. By monotonicity of $A(\\cdot)$ we get\n$A(p)\\geq p(1-e^{-1})^{-1}$. Putting $k=\\liminf_{p\\to +0} A(p)\/p$, we see that $k\\geq k_0=(1-e^{-1})^{-1}$. \n\nSkipping a necessary justification, we assume that the $\\lim_{p\\to +0} A(p)\/p$ exists; its value is thus $k$.\nLet us show that $k=k_0$. \n\nTake small $\\varepsilon>0$. For sufficiently small $p$ \nand any $u\\in[1,e]$ we have\n$A(p\/u)<(k+\\varepsilon)p\/u$, so \n$$\n A(p)\\leq \\max_{1\\leq u\\leq e}\\left((k+\\varepsilon)\\frac{p}{u}+e\\log u-u+p\\right).\n$$\nDifferentiating, we see that the maximum is attained at the point $u=u^+$ which is the larger root (close to $e$) of the quadratic equation\n$$\n u^2-eu+(k+\\varepsilon)p=0.\n$$\nThe smaller root is $u_-=e-u_+=(k+\\varepsilon)p\/u_+\\sim (k+\\varepsilon)p\/e$. \nSince $|e\\log u-u|=O(|e-u|^2)$ as $u\\to e$, we have\n$$ \nA(p) \\leq (k+\\varepsilon)\\frac{p}{u_+}+O(p^2)+p=\\left(\\frac{k+\\varepsilon}{e}+1\\right)p+O(p^2).\n$$\nTherefore \n$$\n k=\\lim_{p\\to +0}\\frac{A(p)}{p}\\leq \\frac{k+\\varepsilon}{e}+1,\n$$\nso $k\\leq e\/(e-1-\\varepsilon)$. Making $\\varepsilon\\to 0$, we obtain\n$k\\leq k_0$, hence $k=k_0$.\n\n\\subsection{Tabulation of the function \\texorpdfstring{$A(p)$}{A(p)}}\nLet us rewrite the functional equation\n\\eqref{feq-Ap} in the form\n$$\n A(p)=\\max_{p\/e\\leq s\\leq p} \\left(A(s)+\\theta\\left(\\frac{p}{s}\\right)\\right)+p ,\n$$\nwhere\n$$\n\\theta(t)=e\\log t-t.\n$$\nAssuming that $A(\\cdot)$ is differentiable, we have the\ncondition of extremum:\n$$\n A'(s_*)-\\frac{p}{s_*^2}\\cdot\\theta'\\left(\\frac{p}{s_*}\\right)=0.\n$$\nEquivalently,\n$$\n  s_*^2\\,A'(s_*)-es_*+p=0.\n$$\n\nWe have\n$$\nA'(p)=1+\\frac{1}{s_*}\\theta' \\left(\\frac{p}{s_*}\\right)\n=1+\\frac{e}{p}-\\frac{1}{s_*}.\n$$\n\nConsider the triples $(s_*,A'(s_*),A(s_*))$\nand $(p,A'(p),A(p))$ as consequtive points of the iteration process:\n\\begin{align}\n(x_{n-1},y_{n-1},z_{n-1})&= (s_*,A'(s_*),A(s_*)),\n\\nonumber\n\\\\\n(x_n,y_n,z_n)&=(p,A'(p),A(p)).\n\\nonumber\n\\end{align}  \nThe equations defining the recurrence are\n$$\n \\ba{l}\n\\displaystyle\n x_n=ex_{n-1}-x_{n-1}^2 y_{n-1},\n\\\\[1ex]\n\\displaystyle \ny_n=1-\\frac{1}{x_{n-1}}+\\frac{e}{x_n},\n\\\\[2ex]\n\\displaystyle\nz_n=z_{n-1}+x_n+\\theta\\left(e-x_{n-1}y_{n-1}\\right).\n\\end{array}\n$$\nStarting with some arbitrary small $x_1$ and setting\n$y_1=k_0$, $z_1=k_0 x_1$, we can continue iterations\nuntil $x_n$ exceeds $1$ for the first time. This way we can tabulate the function $A(p)$, $0<p<1$.\n\nThe actual calculation of $A(1)$ performed by this method\ndemonstrated an\nagreement with the\nvalue stated in Theorem~\\ref{thm:main}, which was originally found using Theorem~\\ref{thm:paramcurves}\nand the functions \\eqref{phi}, \\eqref{psi}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany countries are ethnically diverse, regardless of their level of development.\\footnote{See \\cite{Alesina2003DevleeschauwerEasterlyKurlatWacziarg2003} and \\cite{Fearon2003}, who propose time-invariant  measures of ethnic diversity that are widely used in the literature, and report relatively high levels of ethnic fractionalization by country (as well as significant cross-country variation). For alternative --- time-variant --- measures, see \\cite{PatsiurkoCampbellHall2012}, who show that even the most historically homogenous countries (i.e. the OECD countries) are rapidly becoming more heterogeneous. In addition, see \\cite{VogtBormannRueggerCedermanHunzikerGirardin2015}, who propose the Ethnic Power Relations (EPR) Core Dataset, a very rich dataset that is extensively used in this paper.}  Despite the benefits of ethnic heterogeneity,\\footnote{One example is increased productivity, as ethnic diversity may be associated with more trade and innovation. However, ethnic diversity is also associated with negative outcomes, particularly lower levels of generalized trust \\citep{AlesinaLaFerrara2005} and an underprovision of public goods \\citep{AlesinaBaqirEasterly1999}. See \\cite{AlesinaLaFerrara2005} for a review of the possible costs and benefits of ethnic diversity, and \\cite{LaitinJeon2015} for a longer list of the potential benefits of ethnic diversity.} ethnic-based political inequality and discrimination are pervasive.\\footnote{From 1946 to 2017, at least 32\\% of ethnic groups were explicitly politically discriminated against at some point (see EPR Core Dataset). Group discrimination issues have also been on the agenda of the United Nations for many years (for example, see Resolution 47\/135 of Dec. 18, 1992) and, specifically, it has been argued that it is not sufficient for states to siimply ensure formal political participation for minority groups, but that this political participation must have a substantial influence on decisions that are taken (see OHCHR Forum on Minority Issues, 2009).}  Why is this?  This study suggests that part of the variation in ethnic-based political inequality depends of the relative size of ethnic groups within a country. \n\nUsing group-level data for 569 ethnic groups in 175 countries from 1946 to 2017, I find evidence of an inverted-U-shaped relationship between the relative size of an ethnic group and its access to power. This single-peaked relationship is robust to many alternative specifications, which include group fixed effects, time-varying group-specific controls, country-time fixed effects and group-specific linear trends. This relationship is also robust to using lagged relative group size as an instrument for its current value.  \n\nThough caution is necessary in interpreting the results as causal, they are consistent with relative group size affecting access to power. Taking this into consideration, I propose an explanation based on an initial high level of political inequality, and on the incentives that more powerful groups have,  \\emph{ex ante}, to continue limiting other groups' access to power. Through a very simple model, I argue that these incentives crucially depend on the relative size of politically marginalized groups, and follow an inverted-U-shaped pattern. First, when a politically marginalized group is very small, the group tends to be irrelevant, so more powerful groups have little incentive to cede any control to this small group. Second, when a politically marginalized group is relatively large, individuals in very powerful --- but also relatively small --- groups greatly value a government controlled by their own group, as it allows them to extract very high per capita rents. Therefore, individuals in small powerful groups do not have incentives to share power because it would limit their ability to keep rents for themselves. Only when a politically excluded group is neither very big nor very small can it expect to receive greater access to power; the excluded group is big enough to potentially overthrow the government if it isn't voluntarily given access to power, but small enough that it wouldn't detract too much from the incumbent group's ability to extract rents in a power-sharing arrangement. \n\nThe model I propose to explain the inverted-U relationship between group size and access to power leads to a new empirical prediction that identifies a crucial role of the persistence of political institutions: the single-peaked pattern should be weaker in countries where political institutions have historically been less open. Insofar as undemocratic institutions tend to persist, in countries with political institutions that have traditionally been less open, any increase in the \\emph{de jure} access to power of politically marginalized groups might be seen as less threatening by more powerful groups. Thus, insofar as the costs of maintaining the status quo increase with the size of marginalized groups, in countries where political institutions have historically been less open, the model predicts a monotonically increasing relation between group size and access to power. \n\nUsing PolityIV's measures of openness and competitiveness of executive recruitment as a proxy for the openness of political institutions, I find evidence consistent with this additional prediction: the inverted-U relationship between group size and access to power is specific to countries where political institutions have been more open in the past; in countries with less open political institutions, this relationship tends to be  monotonically increasing. \n\nThe rest of the paper is organized as follows. Section \\ref{background} discusses the existing literature on the relationship between political exclusion and group size. The data and empirical strategy are discussed in Section \\ref{dataandidentification}. The main results are presented in Section \\ref{baselineresults}.  Section \\ref{model} presents the model, and Section \\ref{additionalevidence} provides additional evidence related to the role of the persistence of political institutions. Section \\ref{conclusion} concludes.\n\n\n\\section{Related literature}\\label{background}\n\nThe sociology and economics literatures have developed distinct approaches to examining the relationship between group discrimination and group size. The classical sociological approach identifies the size of the groups as a crucial factor in explaining how individuals in dominant groups perceive potential threats posed by individuals in other groups. This literature has proposed two hypotheses about the specific relationship between this perception and the size of the subordinate groups. First, it may be that the larger the size of a subordinate group, the more individuals in a dominant group perceive the subordinate group as a threat to their economic and social privileges and, therefore, the higher their motivation to discriminate against the subordinate group (see for instance \\citealt{Blumer1958} or  \\citealt{Blalock1967}).\\footnote{This idea constitutes the core of what is known as the ``power-threat'' or ``group threat'' theory.} Second, it may be that the larger the relative size of a subordinate group, the greater the opportunity for positive intergroup contact, which reduces the motivation for individuals in the dominant group to discriminate against individuals in the subordinate group (see for instance \\citealt{Allport1954} or  \\citealt{Pettigrew1998}).\\footnote{This idea constitutes the core of what in known as the ``intergroup contact'' theory.} The empirical evidence for these two theories is inconclusive (see \\cite{SchlueterScheepers2010} for a critical review), with problems related to identification and external validity.\n\nThe economics literature relating discrimination and group size is less abundant, and focuses on discrimination in labor markets and  segregation. Departing from a standard model of statistical discrimination in labor markets, \\cite{MoroNorman2004}  propose a theoretical extension that examines how group size affects conflicting interests between groups. Specifically, they find that as the marginalized group grows, the per capita gains from discrimination for individuals in the dominant group also grow, as do the incentives of individuals in the subordinate group to invest in human capital. This makes it harder to sustain a discriminatory equilibrium, and suggests one rationale for apartheid or other discriminatory measures; however, \\cite{MoroNorman2004} do not model this explicitly. \n\n\\cite{Eeckhout2006} proposes a model of segregation in which segregation can make minority groups worse off while making the majority better off, so that any change in equilibrium that eliminates segregation will make the majority worse off. Even though it is not explicitly modeled, \\citeauthor{Eeckhout2006}'s main result is consistent with \\citeauthor{MoroNorman2004}'s rationale for the existence of institutional devices to ensure that the preferred equilibrium for the group with political control is realized.\n\nThis paper seeks to contribute to the sociological literature by providing new and relatively well-identified evidence about the relationship between group discrimination and group size, with a specific focus on the access to power of individuals in subordinate groups (rather than on out-group attitudes). In addition, this paper proposes a new and very stylized micro-funded theory that allows for the possibility of a non-monotonic relationship.  This paper seeks to contribute to the economics literature by providing evidence about the relationship between group discrimination and group size, and on the role of institutional mechanisms that affect this relationship.\n \n\n \n\\section{Empirical specification and identification}\\label{dataandidentification}\n\n\\subsection{Data}\n\nI study 569 ethnic groups in 175 countries from 1946 to 2017. The baseline specification is divided into 10-year sub-periods (starting in 1945) to account for potential lags in the relationship between group size and group access to state power.\\footnote{I also construct a five-year panel, which achieves similar results.}  To construct observations for each sub-period, I take an average of the annual data from that sub-period.\\footnote{I explore robustness to an alternative specification in which, instead of averaging the 10-year data, I take one year's observation from within each sub-period (i.e. one observation every tenth year). The motivation behind this robustness check is the additional serial correlation that averaging may introduce. As will be shown below, the results are robust to the use of this alternative specification.} \n\nThe analysis is based on the Ethnic Power Relations (EPR) Core Dataset 2018, which identifies all politically relevant ethnic groups and their access to state power in every country of the world from 1946 to 2017 \\citep{VogtBormannRueggerCedermanHunzikerGirardin2015}.\\footnote{This data can be found at \\url{https:\/\/icr.ethz.ch\/data\/epr\/core\/}.} It provides annual data on politically relevant ethnic groups,\\footnote{The EPR dataset defines ``ethnicity'' as a subjectively experienced sense of commonality based on a belief of common ancestry and a shared culture. It includes ethnolinguistic, ethnoreligious, and ethnosomatic (or ``racial'') groups.} their relative sizes as a share of the total\npopulation, and their access to executive  state power in all countries with a population of at least 250,000.\\footnote{For a more detailed description of this dataset, see \\cite{BesleyMueller2018}, who also use this dataset.} \n\n\nThe EPR dataset characterizes an ethnic group as politically relevant if at least one significant political actor claims to represent the interests of that group in the national political arena, or if group members are systematically and intentionally discriminated against in public politics. The EPR measures access to power with a roughly ordinal scale composed of three main categories (and several subcategories), depending on whether a group controls power alone, shares power with other ethnic groups, or is excluded from executive state power. Specifically,  the EPR measures power access according to the following categories and subcategories: \n\\begin{itemize}[noitemsep]\n  \\item[1] The group rules alone:\n  \\begin{itemize}\n  \\item[1.1] Monopoly -- Members hold monopoly power in the executive. \n  \\item[1.2]   Dominance -- Members hold dominant power in the executive.\n\\end{itemize}  \n  \\item[2]  The group shares central power:\n  \\begin{itemize}\n  \\item[2.1]   Senior partner -- Members participate as senior partners in a\nformal or informal power-sharing arrangement. \n  \\item[2.2]   Junior partner -- Members participate as junior partners in government.\n\\end{itemize}\n   \\item[3]  The group is excluded from central power:\n   \\begin{itemize}\n  \\item[3.1]   Powerless -- Members  do not have influence on decision-making at the national level of executive power, but are not explicitly discriminated against.\n  \\item[3.2]  Discrimination -- Members are subjected to active, intentional and targeted discrimination by the state, with the intent of excluding them from political power.\n  \\item[3.3]   Self-exclusion (or separatist autonomy) -- Members have excluded themselves from central state power.\n   \\item[3.4]   Irrelevance -- Members' ethnicity is not politicized (but the members comprise a sufficiently large group).\n\\end{itemize} \n\\end{itemize}\nAlthough these categories are qualitative, they have a clear hierarchy: ``monopoly'' means more access to power than ``dominance,'' which is better than sharing power. For groups that are excluded, being powerless plausibly implies having more access to power than being irrelevant or self-excluded, and the worst scenario is likely discrimination.\\footnote{The ERC dataset does not provide reasons why a group is irrelevant, so a group may become irrelevant after being heavily discriminated against. Thus, there is a case to be made that irrelevance is worse than discrimination. In addition, it is not clear whether self-exclusion is better (or worse) than irrelevance. I discuss these issues in more detail below.} I exploit this implicit order to construct a scale of access to power from 0 to 5.\\footnote{In this sample, the average index of access to central power is 1.545 (and the standard deviation is 1.216).}\n\n\\begin{table}[H]\n  \\centering \n  \\caption{ }\\label{indexaccesspower_tab}\n  \\begin{tabular}{|c|c|c|}\n\\hline\nEPR classification  &  Access to power score \\\\\n\\hline\n Monopoly  &  5  \\\\\n Dominance  &  4  \\\\\n Senior partner &  3  \\\\\n Junior partner &  2  \\\\\n Powerless\/Irrelevance\/Self-exclusion &  1  \\\\\nDiscrimination &  0  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe index in Table \\ref{indexaccesspower_tab} relies on two main assumptions. First, groups that are discriminated against have less access to central power than groups that are irrelevant, powerless or self-excluded. The justification for this assumption is that being explicitly discriminated against implies greater costs --- for accessing power --- than being  powerless, irrelevant or self-excluded.  Second, groups that are powerless, irrelevant or self-excluded have the same (low) level of access to central power. The justification for this assumption is based on the notion of irrelevance: if a group is self-excluded or powerless, dominant groups will likely expect that the other group's members will be unable to gain access to central power.\\footnote{However, since these two assumptions may be still seen as ad hoc, I explore robustness to the use of an alternative measure that relies on the possibility of being part of a government (e.g. as a junior partner). The main results are robust to the use of a dichotomous measure of access to central power, which is equal to 1 if the group is not excluded and 0 if it is excluded. In addition, the results are robust to dropping all the groups with an irrelevant level of access to power.} \n\nThe other main dataset used in the paper is the Polity IV dataset, which evaluates the openness and competitiveness of countries' political institutions. Specifically, I use  Polity IV's ``Openness of Executive Recruitment'' and ``Competitiveness of Executive Recruitment'' to measure how politically open and competitive a country's institutions are in a given year (or decade).\\footnote{See Section \\ref{additionalevidence} for a description of how I code these measures.} \n\n\n\n\\subsection{Empirical Strategy and Identification}\n\nAs previously mentioned, there is no consensus among scholars about the relationship between group size and access to power. My empirical analysis allows for the possibility of either a negative or positive relationship. \n\nIn the main specification, I model the outcome $y_{gct}$ (my index of access to central power based on the EPR power access characterization)\\footnote{In the baseline specification, the analysis is limited to groups that are not dominant, i.e. that don't have monopoly power or are not a senior partner (i.e. groups with an access to power score of 2 or less in Table \\ref{indexaccesspower_tab}).} of group $g$ in country $i$ and 10-year period $t$ as\n\\begin{equation}\n\\label{eqbaseline}\naccess\\_to\\_power_{gct}=\\alpha+\\beta_1 \\times size_{gct} +\\beta_2 \\times size^2_{gct}+ \\gamma_{ct}+\\eta_g+\\epsilon_{gct}\n\\end{equation}\nwhere $size_{gct}$ is the relative size of $g$  in country $i$ and period $t$, $size^2_{gct}$ is the square of this size, $\\gamma_{ct}$ are country\/period fixed effects,  $\\eta_g$ are group fixed effects and $\\epsilon_{gct}$ is the error term.   In some specifications, I also add group-specific linear trends, and control for the presence of each group in other countries (which varies at the group  $\\times$ 10-year level). The two coefficients of interest are $\\beta_1$ and $\\beta_2$, which are associated with the relative size of the groups; $\\beta_2$ is introduced to allow for a nonlinear relationship.\n\n\\medskip\n\nWith respect to identification, since the baseline specification (\\ref{eqbaseline}) includes country\/period fixed effects, it controls for any omitted variable varying at the country $\\times$ 10-year level.\\footnote{As previously mentioned, all the results are robust to using five-year panel, which includes country\/5-year period fixed effects.} In addition, the inclusion of group-specific linear trends helps control for any omitted linear trend that varies at the group level. Finally, note that the control for the presence  of each group in other countries accounts for a potentially omitted (and crucial) channel: a group may have less (or more) access to central power because its members were (or haven't been) forced to migrate to other countries, and this migration (or lack of it) may also explain the small (or large) group size. \n\nSimultaneity is also a concern in (\\ref{eqbaseline}) if, for instance, groups with less access to power have lower fertility rates. To address this, I first replace the explanatory variables (group size and the square of group size) with their lagged values. Second, I instrument group size (and the square of group size) with a lagged group size (and a lag of the squared group size). As long as the lagged values for group size do not themselves exert a direct effect on power access, this second alternative specification provides an effective estimation strategy.\n\nEven though I recognize that fixed effects regressions (such as (\\ref{eqbaseline})) cannot necessarily estimate the causal effect of group size on power access (e.g. there may be still omitted variables varying non-linearly at the group $\\times$ period level that are weakly correlated with a group's presence in other countries),\\footnote{One example may be the level of cohesiveness of the groups, or, more specifically, the level of trust of individuals within these groups. However, it can be argued that although this level changes over time, it does so only over long periods \\cite[in this respect see for instance][who show that inherited trust is strongly persistent]{AlganCahuc2010}.} the results are robust to many alternative specifications, which significantly alleviates these concerns.\n\n\n\n\\section{Baseline empirical results}\\label{baselineresults}\n\nFigure \\ref{integrVSsizeOLSav10yr_fig} shows, for each non-dominant minority group in each 10-year period, the correlation between the group's access to power score and the relative size of the group (i.e. the group's membership as a proportion of the country's population). It shows a clear inverted-U shape.\n\nTable \\ref{integrVSsizeOLSav10yr_tab} reports the estimated coefficients for variations of the model in Eq. (\\ref{eqbaseline}). The specification in column (1) does not include any controls or fixed effects, and corresponds to the specification used in Figure \\ref{integrVSsizeOLSav10yr_fig}. Column (2) includes country and 10-year-period fixed effects, and column (3) adds  country\/10-year period fixed effects. Columns (4) and (5) include group fixed effects and group-specific linear trends.  In all the specifications in Table \\ref{integrVSsizeOLSav10yr_tab}, the estimated coefficients for $\\beta_1$ are positive, the coefficients for $\\beta_2$ are negative, and all are individually and jointly significant. In addition, the peak is located at 0.47, which lies comfortably within the range of values for relative group size (i.e, between 0 and 1).\\footnote{More specifically, the value 0.47 falls between the 75th and 85th percentiles of relative size.} Thus, these results confirms the inverted-U shape in Figure \\ref{integrVSsizeOLSav10yr_fig}.\n\nI also provide results for several additional specifications to verify the robustness of the results in Table \\ref{integrVSsizeOLSav10yr_tab}. First, instead of averaging annual results over a 10-year period, I use one observation from within each 10-year period (e.g one observation every tenth year). These estimates are shown in Panel A of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. Second, instead of using 10-year periods, I use a five-year panel. These estimates are shown in Panel B of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. Third, instead of the continuous measure for access to state power, I use a dichotomous measure of access to power that is equal to 1 if a group is not excluded from central power and 0 if it is excluded. These estimates are shown in Panel C of Table \\ref{integrVSsizeOLSav10yrrobustness1_tab}. In all three additional specifications, the results from Table \\ref{integrVSsizeOLSav10yr_tab} hold, i.e. all specifications show a clear inverted-U-shaped relationship between the relative size of a group and its level of access to central power.\n\nIn interpreting causality, we must be cautious of reverse causality bias: a group's access to power may affect its relative size (e.g. groups that are discriminated against may migrate to other countries and\/or have lower fertility rates). In addition, groups that differ in relative size might also differ in other unobserved and non-linearly time-varying characteristics that affect their access to central power (e.g. members of large groups may receive support from exiles in other countries, and this may improve their chances to gaining access to central power). If this is the case, the previous results may be affected by omitted-variable bias. \n\nTable \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} addresses these potential issues.  Columns (1) to (3) of panel A consider specifications that use lagged values of group size as an independent variable instead of current group size. Columns (4) and (5) of panel A consider specifications that use the lagged values of relative group size as instruments for the contemporaneous values. Under certain conditions, these two strategies help to avoid the inconsistency problems associated with simultaneity \\cite[see][]{Reed2015}. All the estimates are virtually the same as those in Table \\ref{integrVSsizeOLSav10yr_tab}, which indicates that simultaneity is likely not a problem. \n\nOmitted-variable bias may be less worrisome given the different fixed effects included in the baseline specifications. In addition, the specifications in panel B of Table \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} add one crucial covariate: the presence of members of the same group in other countries. All the estimates remain the same whether we consider contemporaneous values for the relative size (columns (1) to (3)), lagged values (column (4)), or lagged values as instruments for the contemporaneous values (column (5)). \n \nWhile one must be cautious in claiming that the estimates in Tables  \\ref{integrVSsizeOLSav10yr_tab} to \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} are caused by purely exogenous shocks, the fact that the results in Table \\ref{integrVSsizeOLSav10yr_tab} are robust to all the specifications in Table \\ref{integrVSsizeOLSav10yrlagIVcontrol_tab} guards against the most obvious forms of endogeneity. \n\n\n\\section{Explaining the inverted U: A simple model}\\label{model}\n\nIn this section, I propose an explanation of the single-peaked relationship between access to central power and relative group size. I use a very simple stylized model in which one group, with an \\emph{ex ante} monopoly of political power, decides whether to continue limiting other groups' access to central power. \n\n\\subsection{Set-up}\n\nConsider a society composed of individuals who belong to \\emph{n} internally homogenous groups. Let $N$ be the total number of individuals in the society, with $N_i$ representing the number of individuals in group $i$, so that $\\sum_{i=1}^nN_i=N$. There are two time periods, each with a government that may be controlled by a single group or by several groups. Assume that in the first period, one group (or a cluster of groups) fully controls the government and decides whether or not to continue limiting other groups' access to power in the second period. Without loss of generality, we set $n=2$, and assume that group 1 has an \\emph{ex ante} monopoly over political power in period 1. Group 2 can be interpreted as a cluster of groups with limited access to power in period 1; \\emph{de jure}, this group is totally excluded from government decisions in period 1 (and potentially in period 2). If  group 2's access to power in period 2 is not restricted, both group 1 and group 2 will participate in the period-2 government. Crucially, I assume that if group 2's access to power in period 2 is not \\emph{de jure} limited, its level of control over the period-2 government will be proportional to its share of the population. Specifically, let $s_i$ denote group $i$'s level of control over the  period-2 government, and assume that\\footnote{\\label{footnotepi}This would correspond to an electoral system with perfectly proportional representation; in this scenario, $s_i$ could represent the number of seats in the national parliament for a political party that represents group $i$. Note that (\\ref{si}) assumes that when group 2 has access to power, its level of control over the government depends only on its relative size. This  is a strong assumption, as one might expect that group 1 could use its control of the government in period 1 to \\emph{de facto} limit group 2's level of control in period 2 (e.g. by passing constitutional measures that restrict the extent of political reforms that the government can pass in period 2). More generally, this assumption implies that a country can easily transition from having a very closed political system to having a very open system, which may not be realistic. In the next subsection, I relax this assumption and obtain new predictions.} \n\\begin{equation}\n\\label{si}\ns_i=\\frac{N_i}{\\sum_{j=1}^2N_j}.\n\\end{equation}\nAs previously mentioned, the analysis focuses on group 1's decision about whether or not to,  \\emph{de jure}, limit  group's 2 access to power in period 2. This decision depends on the benefits that individuals in group 1 obtain from a government fully or partially controlled by their own group. Specifically, I assume that the government has an exogenously determined budget that it divides and transfers to individuals using the following approach. First, each group that participates in the government receives an allocation of the budget proportional to that group's level of control of the government. Second, the budgetary allocation for that group is divided equally among the group's members. In other words, if group $i$ has monopoly control of the government, and if the \\emph{per capita} value of the government budget is normalized to unity, then each member of this group will receive $N\/N_i$.\\footnote{The model can be generalized to allow part of the budget to be allocated toward public goods that benefit all groups. In this alternative scenario, we could weight the two different categories of expenditures, and the share that is allocated to public goods could be interpreted as an indicator of the importance of the public good payoff relative to the ``monetary'' payoff.  All results are robust to this extension.}  \n\nIndividuals use their government transfer to purchase a private good in period 2, from which they derive utility that increases linearly in their consumption of the private good.\\footnote{Alternatively, if we include the possibility of public goods, and if group's 2 access to power is not limited, we could define $\\alpha_i$  as the utility that a member of group 1 derives from the public good if a single unit per capita of the optimal mix for group $i$ is produced (using the private good as numeraire). Then, the  \\emph{per capita} payoff to group 1 could be written as $\\alpha_1+(N\/N_1)$ if group 1 fully controls the government in period 2, and $\\alpha_2$ if group 2 fully controls the government in period 2. Note that it must be the case that $\\alpha_1\\geq \\alpha_2$. As previously mentioned, the results are robust to this extension.}  More precisely, we define the overall payoff to an individual in group 1 when group's 2 access to power is not restricted as\n\\begin{equation}\n\\label{payofflimited}\ns_1(N\/N_1).\n\\end{equation}\nWhen group 1 decides to restrict group's 2 access to power, I assume that group 2 attempts to overthrow group 1's government, which results in a costly contest between the two groups.  The group that wins this contest fully controls the government in period 2. The success of each group depends on its relative size and on its capacity to coordinate its actions. Specifically, I assume the following contest function:\n\\begin{equation}\n\\label{pi}\np_i=\\frac{N_i+a_i}{\\sum_{j=1}^2(N_j+a_j)}\n\\end{equation}\nwhere  $a_i\\geq 0$ for $i=1,2$ is a parameter that captures how sensitive a group's probability of success is to the group's relative population. This can be interpreted as the ability of each group to solve the collective-action problem.\\footnote{\\label{noises}Specifically, $a_i$ can be understood as the \\emph{de facto} capacity of individuals in group 2 to coordinate their actions to overthrow the existing regime, or the capacity of individuals in group 1 to repress group 2. This assumption helps to simply correct an undesirable feature of Tullock's success function \\cite[see][]{Hirshleifer1989, Blavatskyy2010, CorchonDahm2010, Jia2012}.}  \n\nI model the costs of the conflict as follows. I assume that a fraction of the government budget is destroyed due to violence, and only the reminder can be divided among the members of the winning group. The  fraction of the budget that is destroyed is assumed to be proportional to the relative size of group 2, since the bigger the group without political rights, the greater its capacity to cause turbulence and destruction. I therefore summarize the overall  payoff to an individual in group 1 when group's 2 access to power is limited as\n\\begin{equation}\n\\label{payoffnotlimited}\np_i(N\/N_1)(1-\\lambda(N_2\/N))\n\\end{equation}\nwhere $\\lambda(N_2\/N)$ is the fraction of  the government budget that is destroyed, and $\\lambda\\in(0,1)$ is a parameter that captures the marginal increase in the level of destruction as the relative size of group 2 increases.\\footnote{The main results hold if the proportion of the government budget that is destroyed also depends on $a_2$. However, to simplify the exposition, I assume that $\\lambda$ only depends on group size.}  \n\nFinally, I make two assumptions about the parameters $\\lambda$, $a_1$ and $a_2$. First, I assume that $a_1>a_2$, and normalize $a_2=0$, with which $a_1>0$. As discussed in footnote \\ref{noises}, this implies that the \\emph{de facto} power of group 1 in period 2 is greater than that of group 2. This assumption can be justified by the fact that group 1 had the monopoly of power in period 1, so it is plausible that this group,  in period 2, has more resources to invest in increasing its  \\emph{de facto} political power.\\footnote{Even though the model assumes \\emph{de facto} power is exogenous, this assumption is consistent with some literature that endogenizes \\emph{de facto} power \\cite[e.g. see][]{AcemogluRobinson2008AER}.}\n\nSecond, I assume that  \n \\begin{equation}\n\\label{assumption1}\n\\vfrac{\\lambda}{(1-\\lambda)}>a_1.\n\\end{equation}\nBy implying that the costs of conflict are sufficiently high and that the \\emph{de facto} power of group 1 is sufficiently low, this second assumption basically guarantees that limiting group 2's access to power is not a dominant strategy for group 1. \n\n\n\\subsection{Equilibrium}\n\nThe main objective of this section is to propose an explanation for the empirical relationship between access to power and relative size found in Section \\ref{baselineresults}. This section examines the conditions under which individuals in group 1 decide whether to limit group 2's access to power in period 2 and, specifically, how those conditions depend on the relative size of the two groups.\n\nFrom the previous analysis, it is clear that group 1 will not give group 2 access to power in period 2 when the expression in (\\ref{payoffnotlimited}) is greater than the expression in (\\ref{payofflimited}), i.e. when \n\\begin{equation}\n\\label{condnotlimiting}\np_i(N\/N_1)(1-\\lambda(N_2\/N))\\geq s_1(N\/N_1).\n\\end{equation}\nTo facilitate the exposition and interpretation of the results, I normalize the population to 1 (i.e. $N=1$), and define $\\delta$ as the relative size of group 2 (so $N_1=1-\\delta$ and  $N_2=\\delta$). Thus, replacing the expressions for $s_i$ and $p_i$ in (\\ref{condnotlimiting}) and rearranging, it is easy to see that (\\ref{condnotlimiting}) is equivalent to \n\\begin{equation}\n\\label{maincond}\n \\frac{(1-\\delta+a_1)}{(1+a_1)}\\frac{(1-\\lambda \\delta)}{(1-\\delta)}-1\\geq 0.\n\\end{equation}\nThe expression in (\\ref{maincond}) constitutes the main finding of this section. Importantly, it entails an  inverted-U relationship between group 2's relative size (i.e. $\\delta$) and its access to central power. This result is summarized in the following proposition: \n\n  \\begin{prop1} Consider the above-described game. In equilibrium, the relationship between access to central power for a group with an ex ante limited access and its relative size ($\\delta$) follows an inverted-U pattern:  this group's access to central power is expected to be lower for smaller and larger values of  $\\delta$, and higher for intermediate values of $\\delta$. \n \\end{prop1}\n \\begin{proof}\nSee the Appendix.\n\\end{proof} \nThe inverted-U shape can be explained as follows. First, when the relative size of group 2 is very small, individuals in group 1 have few incentives to increase group 2's access to power because the cost of maintaining the status quo, which benefits group 1, is small: the ensuing contest with group 2 is expected to be minimally destructive and group 1's chance of success is very high. As the relative size of group 2 increases, a contest becomes more destructive, so the period-2 budget associated with this scenario is reduced; this gives individuals in group 1 an incentive to increase group 2's access to power.\\footnote{Provided that the size of group 1 is not very small so that the budget in the contest scenario would be divided among a still large number of individuals, and that the effect of the bias in favor of group 1 in the contest success function is still small.} Finally, when the relative size of group 2 is very big, this means group 1 is relatively very small, which implies that individuals in group 1 greatly value a government fully controlled by their group (because the \\emph{per capita} transfers they receive are very large). In these circumstances, individuals in group 1 have few incentives to increase group 2's access to power, even though this implies that the government will have a significantly reduced budget in period 2 (which is mitigated somewhat by a contest function that is biased in favor of group 1). \n \nImportantly, note that the  U-shaped relationship described in Prop. 1 is consistent with the empirical evidence in Section 3. In particular, it rationalizes the results in Figure \\ref{integrVSsizeOLSav10yr_fig} and Columns (1) to (6) in Table \\ref{integrVSsizeOLSav10yr_tab}.  It is also important to note that even though the explanation I propose is new, it is consistent with the main predictions of the existing theoretical literature on group discrimination. Specifically, it is consistent with the idea that the smaller a minority, the more it suffers \\citep{Eeckhout2006}, and the larger a minority, the more likely the coercive measures against it \\citep{MoroNorman2004}, and the greater the antipathy felt towards it  \\citep{Blumer1958, Blalock1967, OliverWong2003}. One of the contributions of this paper is to propose a very simple mechanism that explains (and tests) both predictions. \n \n\n\n\\subsection{Persistence of political institutions}\\label{modelpesistence}\n\nIn the last subsection it was assumed that if group 2 is given access to power in period 2, its level of power will be proportional to its size. As mentioned in footnote \\ref{footnotepi}, this means that the institutions that affect group 2's access to political power can transition from being completely closed to being completely open. In this subsection, I remove this assumption to allow for the possibility that institutions in period 2 are biased in favor of group 1 even if group 2's access to power is not limited. The motivation behind this extension is based on the idea that institutions tend to persist for long periods \\citep{AcemogluJohnsonRobinson2001, AcemogluJohnsonRobinson2002, BanerjeeIyer2005}. \n  \nSpecifically, I assume that when group 2's access to power is not \\emph{de jure} limited,  the participation of group 2 in the period-2 government is  given by\n  \\begin{equation}\n\\label{q2}\nq_2=\\gamma s_2\n\\end{equation}\nwhere $\\gamma\\in[0,1]$ is a parameter that measures how unbiased political institutions are expected to be in period 2.  Since period-1 political institutions are assumed to be very biased in favor group 1, $\\gamma$ can be understood as the  inverse of how persistent institutions are.\n\nFrom (\\ref{q2}), note that since $q_1=1-q_2$, we have that $q_1=\\gamma s_1+1-\\gamma$. From these expressions, note that when $\\gamma=1$, $q_i=s_i$ for $i=1,2$, which corresponds to the scenario examined in the last subsection. Importantly, note that $q_1\\geq s_1$, $q_2\\leq s_2$, and that as $\\gamma$ decreases $q_1$ increases, meaning that group 1's level of control in the period-2 government can be disproportionally larger despite the fact that group 2's access to power is not \\emph{de jure} limited. \n\nReplacing $q_2$ in (\\ref{maincond}) and rearranging, it is easy to see that we have\n\\begin{equation}\n\\label{democcond}\n \\frac{(1-\\delta+a_1)}{(1+a_1)}\\frac{(1-\\lambda \\delta)}{(1-\\delta)}-\\frac{(1-\\gamma\\delta)}{(1-\\delta)} \\geq 0.\n\\end{equation}\nBy analyzing (\\ref{democcond}), the following proposition summarizes how the results in Prop. 1 change when institutions in period 2 are expected to be biased in favor of group 1, even though both groups have access to power. \n\n  \\begin{prop2} Consider the above-described game. In equilibrium, if political institutions are sufficiently persistent, then the relationship between a group's access to central power and its relative size does not follow an inverted-U pattern and, more specifically, the larger a group's relative size, the more likely it will have access to power. \n \\end{prop2}\n \\begin{proof}\nSee the Appendix\n\\end{proof} \n\nThe intuition behind Prop. 2 is straightforward: the more biased institutions are expected to be in favor of group 1, the less costly it is for group 1 to give group 2 access to power, particularly when group 2 is relatively large. In addition, and importantly, Prop. 2 establishes a new empirical prediction: the inverted-U relationship between relative size and access to power crucially depends on the historical quality of institutions; for historically closed political institutions, it is more likely that this relationship, if existent, is positive (rather than non-monotonic). In the next section I examine the empirical plausibility of this new prediction. \n\n\n \n \\section{Additional evidence}\\label{additionalevidence}\n \nThis section empirically examines whether the results in Section \\ref{baselineresults} depend on the historical level of openness of the political institutions, as predicted in Section \\ref{model}. To measure the openness of political institutions, I use two proxies from PolityIV. First, I use a variable called ``Openness of Executive Recruitment (\\emph{xropen}).\" This variable measures  ``the extent that the politically active population has an opportunity, in principle, to attain the position through a regularized process\" \\cite[see][]{PolityIV2018}. It ranges from 0 to 4, with 4 representing the most open institutions. Second, I use a variable called ``Competitiveness of Executive Recruitment (\\emph{xrcomp}),'' which measures  ``the extent that prevailing modes of advancement give subordinates equal opportunities to become superordinates.'' This variable ranges from 0 to 3, with 3 representing the most competitive institutions.  Since the hypothesis regarding how the level of political openness should affect each group's access to power is based on the persistence of institutions, I compute, for each country-year, the average for each variable since 1800. Then, I define a dummy variable equal to 1 if, for the case of the variable \\emph{xropen}, its historical average is equal to 4 (which is very close to the median, and represents a highly open system), and for the case of the variable (\\emph{xrcomp}), its historical average is above the median (which represents a highly competitive system). \n\nTable \\ref{integrVSsizeOLSav10yropenness_tab} shows estimates of the same specifications used in columns (4) and (5) of Table \\ref{integrVSsizeOLSav10yr_tab}, but distinguishes between countries with high and low historical levels of openness (Panel A), and between countries with high and low historical levels of competitiveness (Panel B).  These estimates show that the non-monotonic (inverted-U-shaped) effect of group size on access to power is stronger in countries with historically high levels of openness and competitiveness (columns (1) and (2)), and weaker (and almost nonexistent) in countries that are less open and competitive (columns (3) and (4)). In addition, for countries with historically low levels of openness and competitiveness, the relationship between group size on access to power appears to follow a linear and positive pattern (see column (5)).   Table \\ref{integrVSsizeOLSav10yropennesslagIV_tab} explores the robustness of the previous results to the use of lagged values for group size as i) independent variables (columns (1) to (3)) and as ii) instruments for the contemporaneous values (columns (4) to (6)).  The robustness results are all consistent with those in Table \\ref{integrVSsizeOLSav10yropenness_tab}. \n\nImportantly, the results in Tables \\ref{integrVSsizeOLSav10yropenness_tab}  and \\ref{integrVSsizeOLSav10yropennesslagIV_tab} are consistent with the prediction of the model proposed in the Section \\ref{model}. In particular, they show that as we move from high to low levels of openness, the relationship between group size on access to power passes from having a marked inverted-U shape to being better described as monotonically increasing. As mentioned in the last section, this can be explained by a decrease in the costs (to the group with access to power) associated with not limiting other groups' access to power. \n \n \n\n\n \\section{Conclusion}\\label{conclusion}\n \nThis study examines whether the relative size of ethnic groups within a country affects the extent to which governments limit these groups' access to central power. Using data at the group level in 175 countries from 1946 to 2017, I find evidence of an inverted-U-shaped relationship between the relative size of groups and their access to central power. This single-peaked relationship is robust to many alternative specifications, and several robustness checks suggest that relative size causes access to power. Through a very simple model, I propose an explanation based on an initial high level of political inequality, and on the incentives that more powerful groups have to continue limiting other groups' access to power. This explanation incorporates essential elements of existing theories about the relationship between group size and political and social exclusion, and leads to a new empirical prediction: the single-peaked pattern should be weaker in countries where political institutions have been less open in the past. This additional prediction is supported by the data.\n \nSeveral opportunities exist for future research. One could examine the effect of ethnic groups' relative size on other --- more de facto --- forms of political participation, such as the formation of political organizations and participation in protests. It would also be interesting to examine whether other kind of institutions matter (e.g. less formal political institutions and cultural institutions). Finally, there is a question of whether individuals in dominant groups actually agree that the institutions that their groups control limit access to the power of minority groups.\n \n \n\n\n\\hbox {}\n\\hbox {} \\newpage\n\\section*{Graphs and Tables}\n\n\n\\begin{figure}[H]\n\\begin{center}\n\\caption{Scatter plot of relative size and access to central power}\\label{integrVSsizeOLSav10yr_fig}\n\\resizebox{10cm}{8cm}{\\includegraphics[width=4in]{figure1.png}}\n\\caption*{\\footnotesize The figure plots a measure of the size of each ethnic group on the x-axis against each group's access to  power score --- as defined in Table \\ref{indexaccesspower_tab} --- on the y-axis. Each point represents a group over a 10-year period. The quadratic curve that is overlaid is reported in column (1) of Table \\ref{integrVSsizeOLSav10yr_tab}.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power}  \\label{integrVSsizeOLSav10yr_tab}\n\\begin{center}  \n\\vspace{-0.5cm}\\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\  \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n\\primitiveinput{tableII.tex}\n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]           \nCountry fixed effects & N & Y & Y & Y & Y   \\\\         \n10-year period fixed effects & N & Y & Y & Y & Y   \\\\   \n  Country\/period fixed effects & N & N & Y & Y & Y \\\\       \n Group fixed effects & N & N & N & Y & Y \\\\    \n  Group-specific linear trends & N & N & N & N & Y \\\\    \n      \\addlinespace[0.15cm] \\hline\\hline         \n\\multicolumn{6}{p{14cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period.  In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power (robustness to alternatives measures of access to central power)}    \\label{integrVSsizeOLSav10yrrobustness1_tab}\n\\begin{center}\n\\vspace{-0.6cm} \\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n &\\multicolumn{5}{c}{Panel A: Using first ob. within each 10-year period}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIa.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: Using averages over 5-year periods}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIb.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel C: Dep. var. is prob. of access to power}     \\\\\\cmidrule[0.2pt](l){2-6}   \\addlinespace[0.15cm]\n\\primitiveinput{tableIIIc.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]             \nCountry fixed effects & N & Y & Y & Y & Y   \\\\         \n Period fixed effects & N & Y & Y & Y & Y   \\\\   \n  Country\/period fixed effects & N & N & Y & Y & Y \\\\       \n Group fixed effects & N & N & N & Y & Y \\\\    \n  Group-specific linear trends & N & N & N & N & Y \\\\    \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{6}{p{14.8cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is based on ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less. Panel A uses 10-year panel, but rather than averaging the 10-year data, it takes one observation within each sub-period (e.g one every tenth year). In this sample, the average relative size is 0.121 (with s.d. 0.228) and the average level of access to power is 1.023 (with s.d. 0.609). Panel B uses a 5-year panel. In this sample, the average relative size is 0.118 (with s.d. 0.226) and the average level of access to power is 1.030 (with s.d. 0.586). Panel C uses 10-year panel and a dichotomous measure of access to power which is equal to one if the group is not excluded from central power and zero if it is excluded. In this sample, the average probability of not being excluded from central power is 0.216 (with s.d. 0.411). Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{4pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power (robustness to the use lagged size, IV and control for presence of each group in other countries)}  \\label{integrVSsizeOLSav10yrlagIVcontrol_tab}\n\\begin{center}  \n\\vspace{-0.5cm}\\begin{tabular}{lccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n    & (1)& (2) & (3)& (4)& (5) \\\\  \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n &\\multicolumn{5}{c}{Panel A:}     \\\\\\cmidrule[0.2pt](l){2-6} \n        & \\multicolumn{3}{c}{Lagged effect} &\\multicolumn{2}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-6}\n\\primitiveinput{tableIVa.tex} \n\\cmidrule[0.2pt](l){2-6} \n  Country and period fixed effects & Y & Y & Y & Y & Y \\\\       \n  Country\/period fixed effects & Y & Y & Y & N & Y \\\\       \n Group fixed effects & N & Y & Y & N & N \\\\    \n  Group-specific linear trends & N & N & Y & N & N \\\\    \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: }     \\\\\\cmidrule[0.2pt](l){2-6} \n         & \\multicolumn{3}{c}{Baseline specification} & \\multicolumn{1}{c}{Lagged} & \\multicolumn{1}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-6}\n \\primitiveinput{tableIVb.tex} \n \\cmidrule[0.2pt](l){2-6}        \n  Country\/period fixed effects & Y & Y & Y & Y & Y  \\\\       \n Group fixed effects & N & Y & Y& N  & N  \\\\        \n   Group-specific linear trends & N & N & Y & N & N  \\\\    \n      \\addlinespace[0.15cm] \\hline\\hline         \n\\multicolumn{6}{p{15.2cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}).  The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period. In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Robust standard errors  clustered by country are in parentheses. * denotes results are statistically significant at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\\begin{figure}[H]\n\\caption{Relative size and access to central power by level of political openness}\\label{intVSsizeCORRav10yrsindexropen_fig}\n\\begin{center}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{High degree of openness}\n\\includegraphics[width=3in,height=2.6in]{figure2a.png}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{Low degree of openness}\n\\includegraphics[width=3in,height=2.6in]{figure2b.png}\n\\end{subfigure}\n\\begin{minipage}{15cm} \\footnotesize The figures plot a measure of the size of each ethnic group on the x-axis against each group's access to power score on the y-axis, by historical level of political openness (i.e. PolityIV's measure of Openness of Executive Recruitment, computed since 1800).  Each point represents a group over a 10-year period. Figure (a) includes countries with an above-median level of historical political openness, and Figure (b) includes countries with a below-median level of historical political openness. \n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\caption{Relative size and access to central power  by level of political competitiveness}\\label{intVSsizeCORRav10yrsindexrcomp_fig}\n\\begin{center}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{High degree of competitiveness}\n\\includegraphics[width=3in,height=2.6in]{figure3a.png}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\caption{Low degree of competitiveness}\n\\includegraphics[width=3in,height=2.6in]{figure3b.png}\n\\end{subfigure}\n\\begin{minipage}{15cm}  \\footnotesize The figures plot a measure of the size of each ethnic group on the x-axis against each group's access to power score on the y-axis, by historical level of political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800).  Each point represents a group over a 10-year period. Figure (a) includes countries with an above-median level of historical political competitiveness, and Figure (b) includes countries with a below-median level of historical political competitiveness. \n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{5pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power by level of political openness and political competitiveness}    \\label{integrVSsizeOLSav10yropenness_tab}\n\\begin{center}\n \\begin{tabular}{lcccccc}\n\\hline\\hline \\addlinespace[0.15cm]\n    & \\multicolumn{5}{c}{Dep. variable: Level of access to central power} \\\\\\cmidrule[0.2pt](l){2-6}\n& (1)& (2) & (3)& (4)& (5) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm]     \n     &\\multicolumn{5}{c}{Panel A: By openness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-6}        \n            & \\multicolumn{2}{c}{High degree of openness} & \\multicolumn{3}{c}{Low degree of openness} \\\\\\\n        & \\multicolumn{2}{c}{(avg. xropen$=$4)} & \\multicolumn{3}{c}{(avg. xropen$<$4)} \\\\\\cmidrule[0.2pt](l){2-3}\\cmidrule[0.2pt](l){4-6}\n\\primitiveinput{tableVa.tex}\n\\addlinespace[0.15cm]\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]       \n &\\multicolumn{5}{c}{Panel B: By competitiveness of executive recruitment}     \\\\\\cmidrule[0.2pt](l){2-6}     \n             & \\multicolumn{2}{c}{High degree of competitiveness} & \\multicolumn{3}{c}{Low degree of competitiveness} \\\\\\\n        & \\multicolumn{2}{c}{(avg. xrcomp above median)} & \\multicolumn{3}{c}{(avg. xrcomp below median)} \\\\\\cmidrule[0.2pt](l){2-3}\\cmidrule[0.2pt](l){4-6}\n \\primitiveinput{tableVb.tex}\n \\addlinespace[0.15cm]\n \\hline \\addlinespace[0.15cm]      \nCountry fixed effects & Y & Y & Y & Y  & Y \\\\         \n Period fixed effects & Y & Y & Y & Y &  Y \\\\       \n  Country\/period fixed effects & Y & Y & Y & Y  & Y \\\\         \n Group fixed effects & N & Y & N & Y &  Y \\\\     \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{6}{p{17cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}).  The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period. In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).  Columns (1) and (2) in Panel A include countries with an above-median level of historical political openness, and Columns (3) to (5) in Panel A include countries with a below-median level of historical political openness (i.e. PolityIV's measure of  Openness of Executive Recruitment, computed since 1800).   Columns (1) and (2) in Panel B include countries with an above-median level of historical political competitiveness, and Columns (3) to (5) in Panel B include countries with a below-median level of historical political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800). Robust standard errors  clustered by country are in parentheses. $\\dagger$ denotes results are statistically significant at the 15\\% level, * at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\\begin{table}[H]\n{ \n\\renewcommand{\\arraystretch}{0.8} \n\\setlength{\\tabcolsep}{2pt}\n\\captionsetup{font={normalsize,bf}}\n\\caption {Relative size and access to central power by level of political openness and political competitiveness (lagged size and IV)}    \\label{integrVSsizeOLSav10yropennesslagIV_tab}\n\\begin{center}\n \\begin{tabular}{lcccccc}\n\\hline\\hline  \\addlinespace[0.15cm]\n    & \\multicolumn{6}{c}{Dep. variable: Level of access to state power} \\\\\\cmidrule[0.2pt](l){2-6}\n            & \\multicolumn{3}{c}{Lagged effect} & \\multicolumn{3}{c}{IV} \\\\\\cmidrule[0.2pt](l){2-4}\\cmidrule[0.2pt](l){5-7}\n            & (1)& (2) & (3)& (4)& (5)& (6) \\\\    \\addlinespace[0.15cm] \\hline \\addlinespace[0.15cm] \n                 &\\multicolumn{6}{c}{Panel A: By openness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-7}       \n        & \\multicolumn{1}{c}{High openness} & \\multicolumn{2}{c}{Low openness}& \\multicolumn{1}{c}{High openness} & \\multicolumn{2}{c}{Low openness} \\\\\\cmidrule[0.2pt](l){2-2}\\cmidrule[0.2pt](l){3-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-7}\n\\primitiveinput{tableVIa.tex} \n\\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]     \n                 &\\multicolumn{6}{c}{Panel B: By competitiveness of executive recruitment}      \\\\\\cmidrule[0.2pt](l){2-7}       \n        & \\multicolumn{1}{c}{High compet.} & \\multicolumn{2}{c}{Low compet.}& \\multicolumn{1}{c}{High compet.} & \\multicolumn{2}{c}{Low compet.} \\\\\\cmidrule[0.2pt](l){2-2}\\cmidrule[0.2pt](l){3-4}\\cmidrule[0.2pt](l){5-5}\\cmidrule[0.2pt](l){6-7}\n         \\primitiveinput{tableVIb.tex}\n         \\addlinespace[0.15cm]\\hline\\addlinespace[0.15cm]                \n  Country\/period fixed effects & Y & Y & Y & Y & N & Y \\\\       \n Group fixed effects & N & N & Y & Y & N & N \\\\    \n  Group-specific linear trends & N & N & N & Y & N & N \\\\    \n       \\addlinespace[0.15cm] \\hline\\hline                       \n\\multicolumn{7}{p{16.6cm}}{\\footnotesize{\\textbf{Notes:} \nAll columns report OLS estimates for estimates from Eq (\\ref{eqbaseline}). The dependent variable in all columns is each ethnic group's access to power score (as defined in Table \\ref{indexaccesspower_tab}). The sample is limited to groups with a score of 2 or less, and the index is averaged over a 10-year period.  In this subsample, the average relative size is 0.117 (with s.d. 0.224) and the average level of access to central power is 1.036 (with s.d. 0.575).   Columns (1) and (2) in Panel A include countries with an above-median level of historical political openness, and Columns (3) to (5) in Panel A include countries with a below-median level of historical political openness (i.e. PolityIV's measure of  Openness of Executive Recruitment, computed since 1800).   Columns (1) and (2) in Panel B include countries with an above-median level of historical political competitiveness, and Columns (3) to (5) in Panel B include countries with a below-median level of historical political competitiveness (i.e. PolityIV's measure of Competitiveness of Executive Recruitment, computed since 1800).   Robust standard errors  clustered by country are in parentheses. $\\dagger$ denotes results are statistically significant at the 15\\% level, * at the 10\\% level, ** at the 5\\% level, and *** at the 1\\% level.} } \\\\\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\n\n\n\n\\hbox {}\n\\hbox {} \\newpage\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}