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Measuring the concentration of random variables is a fundamental concept in probability and statistics. Here, we explore a type of concentration measure for continuous random variables with bounded support and use it to provide a notion of stochastic order by concentration. We give an application to the Beta family of distributions, and specifically to the one-parameter subfamily with constant mean. This leads to using U.S. household income data to fit generalized Beta distributions and offers a new measure of income concentration., Comment: 19 pages, 4 figures

A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such sets show up in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points p and q, then the set of points x satisfying d(x, p) = d(x, q) is a minimal separating set. In this paper we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graphs embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and their underlying graphs., Comment: 19 pages, 6 figures

Suppose that the surfaces K0 and Kr are the boundaries of two convex, complete, connected C^2 bodies in R^3. Assume further that the (Euclidean) distance between any point x in Kr and K0 is always r (r > 0). For x in Kr, let {\Pi}(x) denote the nearest point to x in K0. We show that the projection {\Pi} preserves geodesics in these surfaces if and only if both surfaces are concentric spheres or co-axial round cylinders. This is optimal in the sense that the main step to establish this result is false for C^{1,1} surfaces. Finally, we give a non-trivial example of a geodesic preserving projection of two C^2 non-constant distance surfaces. The question whether for any C^2 convex surface S0, there is a surface S whose projection to S0 preserves geodesics is open., Comment: 9 figures, 15 pages

We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.

We present evidence that the Bak-Sneppen model of evolution on $N$ vertices requires $N^3$ iterates to reach equilibrium. This is substantially more than previous authors suggested (on the order of $N^2$). Based on that estimate, we present a novel algorithm inspired by previous rank-driven analyses of the model allowing for direct simulation of the model with populations of up to $N = 25600$ for $2\cdot N^3$ iterations. These extensive simulations suggest a cutoff value of $x^* = 0.66692 \pm 0.00003$, a value slightly lower than previously estimated yet still distinctly above $2/3$. We also study how the cutoff values $x^*_N$ at finite $N$ approximate the conjectured value $x^*$ at $N=\infty$. Assuming $x^*_N-x^*_\infty \sim N^{-\nu}$, we find that $\nu=0.978\pm 0.025$, which is significantly lower than previous estimates ($\nu\approx 1.4$)., Comment: 18 figures, 12 pages

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to efficiently analyze the average behavior of certain common nonlinear dynamical systems, such as the angle-doubling map, $x \mapsto 2x$ modulo 1. In particular, one can use this information to analyze how the behavior of individual orbits deviates from the global average (called fluctuations). More generally, the formula is valid in $\mathbb{R}^m$, where expanding maps give rise to so-called number systems. To illustrate the usefulness in this setting, we compute the fluctuations of a certain map on the plane.

This paper analyzes the global dynamics of 1-dimensional agent arrays with nearest neighbor linear couplings. The equations of motion are second order linear ODEs with constant coeffcients. The novel part of this research is that the couplings are different for each distinct agent. We allow the forces to depend on the positions and velocity (damping terms) but the magnitudes of both the position and velocity couplings are different for each agent. We, also, do not assume that the forces are "Newtonian" (i.e. the force due to A on B equals the minus the force of B on A) as this assumption does not apply to certain situations, such as traffic modeling. For example, driver A reacting to driver B does not imply the opposite reaction in driver B. There are no known analytical means to solve these systems, even though they are linear, and so relatively little is known about them. This paper is a generalization of previous work that computed the global dynamics of 1-dimensional sequences of identical agents [3] assuming periodic boundary conditions. In this paper, we push that method further, similar to [2], and use an extended periodic boundary condition to to gain quantitative insights to the systems under consideration. We find that we can approximate the global dynamics of such a system by carefully analyzing the low-frequency behavior of the system with (generalized) periodic boundary conditions., Comment: 17 pages, 10 figures

The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances. In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment of the so-called deficiency zero systems of CRN theory. We show that much of the dynamics of these polynomial systems of differential equations can be understood by analyzing the directed graph Laplacian associated with the system. Beside the more concise mathematical treatment, this leads to considerably stronger results. In particular, (i) we show that our Laplacian deficiency zero theorem is markedly stronger than the traditional one and (ii) we derive simple equations for the locus of the equilibria in all (Laplacian) deficiency zero cases. This paper is written in a way to make the material easily accessible to a mathematical audience. In particular, no knowledge of chemistry or physics is assumed., Comment: 27 pages, 5 figures

We study statistical properties of the truncated flat spot map $f_t(x)$. In particular, we investigate whether for large $n$, the deviations $\sum_{i=0}^{n-1} \left(f_t^i(x_0)-\frac 12\right)$ upon rescaling satisfy a $Q$-Gaussian distribution if $x_0$ and $t$ are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if $f_t$ is the rotation by $t$, then it has been shown that in this case the rescaled deviations are distributed as a $Q$-Gaussian with $Q=2$ (a Cauchy distribution). This is the only case where a non-trivial (i.e. $Q\neq 1$) $Q$-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, $\lim_n S_n/n$ converges to a random variable with a curious distribution which is clearly not a $Q$-Gaussian or any other standard smooth distribution.

We consider the integrable (zero perturbation) two--dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for KAM tori., Comment: 6 pages, 2 figures

We analyze the asymptotic behavior of general first order Laplacian processes on digraphs. The most important ones of these are diffusion and consensus with both continuous and discrete time. We treat diffusion and consensus as dual processes. This is the first complete exposition of this material in a single work., Comment: 24 pages, 5 figures, summer-school/conference on mathematical modeling of complex systems in Pescara, 2019

We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point $\gamma(t)$ on a geodesic $\gamma$ and a compact set $K$ is a right-differentiable function of $t$. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles., Comment: 22 pages, 8 figures

We review some basic results of convex analysis and geometry in $\mathbb{R}^n$ in the context of formulating a differential equation to track the distance between an observer flying outside a convex set $K$ and $K$ itself., Comment: 14 pages, 12 figures

In this paper, we give necessary conditions for stability of coupled autonomous vehicles in R. We focus on linear arrays with decentralized vehicles, where each vehicle interacts with only a few of its neighbors. We obtain explicit expressions for necessary conditions for stability in the cases that a system consists of a periodic arrangement of two or three different types of vehicles, i.e. configurations as follows: ...2-1-2-1 or ...3-2-1-3-2-1. Previous literature indicated that the (necessary) condition for stability in the case of a single vehicle type (...1-1-1) held that the first moment of certain coefficients of the interactions between vehicles has to be zero. Here, we show that that does not generalize. Instead, the (necessary) condition in the cases considered is that the first moment plus a nonlinear correction term must be zero.

The study of the movement of flocks, whether biological or technological is motivated by the desire to understand the capability of coherent motion of a large number of agents that only receive very limited information. In a biological flock a large group of animals seek their course while moving in a more or less fixed formation. It seems reasonable that the immediate course is determined by leaders at the boundary of the flock. The others follow: what is their algorithm? The most popular technological application consists of cars on a one-lane road. The light turns green and the lead car accelerates. What is the efficient algorithm for the others to closely follow without accidents? In this position paper we present some general questions from a more fundamental point of view. We believe that the time is right to solve many of these questions: they are within our reach., Comment: 7 pages, 10 figures

Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\dot x=-{\cal L}x$ for consensus and $\dot p=-p{\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\{\bar \gamma_i\}_{i=1}^k$ of the left null-space of ${\cal L}$ and a basis of column vectors $\{\gamma_i\}_{i=1}^k$ of the right null-space of ${\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characterization of the asymptotic behavior of both diffusion and consensus --- discrete and continuous --- in terms of these eigenvectors. As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one. We further show that the teleporting feature usually included in the algorithm is not strictly necessary. This is a complete and self-contained treatment of the asymptotics of consensus and diffusion on digraphs. Many of the ideas presented here can be found scattered in the literature, though mostly outside mainstream mathematics and not always with complete proofs. This paper seeks to remedy this by providing a compact and accessible survey., Comment: 19 pages, Survey Article, 1 figure

It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute $(3/2)^n$ modulo $1$ to $n = 10^8$. We then statistically analyze its distribution. Our results strongly agree with the hypothesis that $(3/2)^n$ modulo 1 is uniformly distributed., Comment: 12 pages, 2 figures

We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal $n$ by $n$ matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. %By boundary conditions, we mean the first and last row of the matrix. For large $n$, we show there are up to $4$ eigenvalues, the so-called \emph{special eigenvalues}, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called \emph{regular eigenvalues} vary very little as function of the boundary conditions. For large $n$, we determine the regular eigenvalues up to ${\cal O}(n^{-2})$, and the special eigenvalues up to ${\cal O}(\kappa^n)$, for some $\kappa\in (0,1)$. The components of the eigenvectors are determined up to ${\cal O}(n^{-1})$. The matrices we study have important applications throughout the sciences. Among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling, and discretizations of second order linear partial differential equations. In both cases, we give examples where specific choices of boundary conditions substantially influence leading eigenvalues, and therefore the global dynamics of the system., Comment: 20 pages, 5 figures

Consider a surface $S$ and let $M\subset S$. If $S\setminus M$ is not connected, then we say $M$ \emph{separates} $S$, and we refer to $M$ as a \emph{separating set} of $S$. If $M$ separates $S$, and no proper subset of $M$ separates $S$, then we say $M$ is a \emph{minimal separating set} of $S$. In this paper we use methods of computational combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus $g=2$ and $g=3$. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology., Comment: 24 pages, 5 figures, 2 tables (11 pages)

Since the 1940's there has been an interest in the question why social networks often give rise to two antagonistic factions. Recently a dynamical model of how and why such a balance might occur was developed. This note provides an introduction to the notion of social balance and a new (and simplified) analysis of that model. This new analysis allows us to choose general initial conditions, as opposed to the symmetric ones previously considered. We show that for general initial conditions, four factions will evolve instead of two. We characterize the four factions, and we give an idea of their relative sizes., Comment: 9 pages, 5 figures

The faithful state transfer is an important requirement in the construction of classical and quantum computers. While the high-speed transfer is realized by optical-fibre interconnects, its implementation in integrated optical circuits is affected by cross-talk. The cross-talk between densely packed optical waveguides limits the transfer fidelity and distorts the signal in each channel, thus severely impeding the parallel transfer of states such as classical registers, multiple qubits and qudits. Here, we leverage on the suitably engineered cross-talk between waveguides to achieve the parallel transfer on optical chip. Waveguide coupling coefficients are designed to yield commensurate eigenvalues of the array and hence, periodic revivals of the input state. While, in general, polynomially complex, the inverse eigenvalue problem permits analytic solutions for small number of waveguides. We present exact solutions for arrays of up to nine waveguides and use them to design realistic buses for multi-(qu)bit and qudit transfer. Advantages and limitations of the proposed solution are discussed in the context of available fabrication techniques.

We consider large but finite systems of identical agents on the line with up to next nearest neighbor asymmetric coupling. Each agent is modelled by a linear second order differential equation, linearly coupled to up to four of its neighbors. The only restriction we impose is that the equations are decentralized. In this generality we give the conditions for stability of these systems. For stable systems, we find the response to a change of course by the leader. This response is at least linear in the size of the flock. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with at least next nearest neighbor interactions. Analytical predictions are tested in numerical simulations.

We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover, an optimization-based procedure is used to find the controller properties.

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property.

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves exhibiting at most countably many singularities, with finite total angular deficiency., Comment: 9 pages

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition., Comment: 13 pages, 4 figures

The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities (see [3]) in each of the two directions. As corollaries we show that symmetric interactions are far from optimal and that all these results independent of (reasonable) boundary conditions., Comment: 11 pages, 4 figures

We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles. The model considers asymmetric, linear, decentralized dynamics, where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. We first derive necessary and sufficient conditions for asymptotic stability, then derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocities $c_+>0$ and $c_-<0$ such that low frequency disturbances travel through the flock as $f(x-c_+t)$ in the direction of increasing agent numbers and $f(x-c_-t)$ in the other., Comment: 18 pages, 4 figures

We investigate a class of models related to the Bak-Sneppen model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of "complex behavior" such as self-organized criticality that is often observed in physical and biological systems. In this model, random fitnesses in $[0,1]$ are associated to agents located at the vertices of a graph $G$. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the worst fitness and some others \emph{with a priori given rank probabilities} are replaced by new agents with random fitnesses. We consider two cases: The \emph{exogenous case} where the new fitnesses are taken from an a priori fixed distribution, and the \emph{endogenous case} where the new fitnesses are taken from the current distribution as it evolves. We approximate the dynamics by making a simplifying independence assumption. We use Order Statistics and Dynamical Systems to define a \emph{rank-driven dynamical system} that approximates the evolution of the \emph{distribution} of the fitnesses in these rank-driven models, as well as in the Bak-Sneppen model. For this simplified model we can find the limiting marginal distribution as a function of the initial conditions. Agreement with experimental results of the BS model is excellent., Comment: 12 gigures, 20 pages

Consider a string of N+1 damped oscillators moving on the line of which the motion of the first (called the "leader") is independent of the others. Each of the followers `observes' the relative velocity and position of only its nearest neighbors. Inasmuch as these are different from 0, this information is then used to determine its own acceleration. Fix all parameters except the number N in such a way that the system is asymptotically stable. Now as N tends tends we consider the following problem. At t=0 the leader gets kicked and starts moving with unit velocity away from the flock. Due to asymptotic stability the followers will eventually fall in behind the leader and travel each at its own predetermined distance from the leader. In this note we conjecture that before equilibrium ensues, the perturbations to the orbit of the last oscillator grow exponentially in N except when there is a symmetry in the interactions and the growth is then linear in N. There are two cases. We prove the conjecture in one case, and give a strong heuristic argument in the other., Comment: 13 pages, 1 figure

We discuss some stability problems when each agent of a linear flock on the line interacts with its two nearest neighbors (one on either side)., Comment: 11 pages, 5 figures

The movement of a flock with a single leader (and a directed path from it to every agent) can be stabilized. Nonetheless for large flocks perturbations in the movement of the leader may grow to a considerable size as they propagate throughout the flock and before they die out over time. As an example we consider a string of N+1 oscillators moving in the line. Each one `observes' the relative velocity and position of only its nearest neighbors. This information is then used to determine its own acceleration. Now we fix all parameters except the number of oscillators. We then show (within a certain class of systems) that a perturbation in the leader's orbit is almost always amplified exponentially in N as it propagates towards the outlying members of the flock. The only exception is when there is a symmetry present in the interaction: in that case the growth of the perturbation is linear in N., Comment: 9 pages, 5 figures

A simple model for a granular ratchet corresponding to a single grain bouncing off a vertically vibrating sawtooth-like base is studied. Depending on the vibration strength, the sawtooth roughness and the restitution coefficient, horizontal transport in both the preferred and unfavoured directions is observed. A phase diagram indicating the regions in parameter space where each of the three possible regimes (no current, normal current, and current reversal) occurs is presented., Comment: 7 pages, 3 figures, submitted to Physica A

A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which the dynamics is described by a one-dimensional map. This map is studied in detail and it is shown to exhibit several dynamical regimes (steady state, chaotic behavior, and accelerated motion) as the model parameters vary. A phase diagram showing the corresponding domain of existence for these regimes is presented. The model is also found to be in good qualitative agreement with recent experiments on a ball moving on a rough inclined line., Comment: 6pages, 6 figures, to appear in Phys. Rev. E (May 1999)

Brillouin zones were introduced by Brillouin in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in $\R^n$. They play an important role in solid-state physics. It was shown by Bieberbach that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin Zones to apply to an arbitrary discrete set in a proper metric space, and show that analogs of Bieberbach's results hold in this context. We then use these ideas to discuss focusing of geodesics in orbifolds of constant curvature. In the particular case of the Riemann surfaces H^2/Gamma(k), (k=2,3, or 5), we explicitly count the number of geodesics of length t that connect the point i to itself., Comment: 21 pages, 18 PostScript figures

We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upper- and lower-bounds for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the moduli of the eigenvalues of the underlying affine contraction are all equal (this includes Jordan blocks). The tiles we discuss play an important role in the theory of wavelets. We calculate the dimension for a number of examples.

Local scaling of a set means that in a neighborhood of a point the structure of the set can be mapped into a finer scale structure of the set. These scaling transformations are compact sets of locally affine (that is: with uniformly $\alpha$-H\"older continuous derivatives) contractions. In this setting, without any assumption on the spacing of these contractions such as the open set condition, we show that the measure of the set is an upper semi-continuous of the scaling transformation in the $C^0$-topology. With a restriction on the 'non-conformality' (see below) the Hausdorff dimension is lower semi-continous function in the $C^{1}$-topology. We include some examples to show that neither of these notions is continuous.

Circle maps with a flat spot are studied which are differentiable, even on the boundary of the flat spot. Estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set are obtained. Also, a sharp transition is found from degenerate geometry similar to what was found earlier for non-differentiable maps with a flat spot to bounded geometry as in critical maps without a flat spot.

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$ When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $\alpha$ is different from one, the dynamics is no longer conformal. In particular, the dynamics is not completely determined by the orbit of the critical point. Nevertheless, for many values of the parameter c, the dynamics has strong similarities to that of the quadratic family. For other parameter values the dynamics is dominated by 2 dimensional behavior: saddles and the like. The objects of study are Julia sets, filled-in Julia sets and the connectedness locus. These are defined in analogy to the conformal case. The main drive in this study is to see to what extent the results in the conformal case generalize to that of maps which are topologically like quadratic maps (and when $\alpha$ is close to one, close to being quadratic).