on the convergence of gradient descent for finding the riemannian center of mass we study the problem of finding the global riemannian center of mass of a set of data points on a riemannian manifold specifically we investigate the convergence of constant step size gradient de

chaotic scattering theory of transport and reaction rate coefficients the chaotic scattering theory is here extended to obtain escape rate expressions for the transport coefficients appropriate for a simple classical fluid or for a chemically reacting system this theory allows various transport coefficients such as the coefficients of viscosity thermal conductivity etc to be expressed in terms of the positive lyapunov exponents and kolmogorov sinai entropy of a set of phase space trajectories that take place on an appropriate fractal repeller this work generalizes the previous results of gaspard and nicolis for the coefficient of diffusion of a particle moving in a fixed array of scatterers

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

intraband discrete breathers in disordered nonlinear systems ii localization abstract we find spatially localized time periodic solutions discrete breathers or dbs in disordered nonlinear systems with frequency inside the linear phonon spectrum under conditions that strictly prohibit their existence in their periodic counterparts for that purpose we develop a new in situ method for the accurate calculation of these solutions which does not make use of any continuation from an anticontinuous limit using this method we demonstrate that intraband localized modes intraband discrete breathers or idbs at a given site with frequencies inside the discrete linear spectrum do exist provided these frequencies do not belong to forbidden resonance gaps since there is a dense set of resonant frequencies we illustrate numerically in agreement with a theorem by albanese and frohlich that the localized dbs exist provided that their frequencies belong to fat cantor sets i e with finite measure such a set contains as accumulation points the linear frequency of the normal mode at the occupied site we check that many of these solutions are linearly stable and conjecture that their frequency belongs to another smaller fat cantor set our numerical methods provide a much wider set of exact solutions which are multisite breathers and suggest conjectures extending the existing theorems the physical implications of the existence of idbs and possible applications for glasses and the persistent spectral hole burning are discussed

riemannian consensus for manifolds with bounded curvature consensus algorithms are popular distributed algorithms for computing aggregate quantities such as averages in ad hoc wireless networks however existing algorithms mostly address the case where the measurements lie in euclidean space in this work we propose riemannian consensus a natural extension of existing averaging consensus algorithms to the case of riemannian manifolds unlike previous generalizations our algorithm is intrinsic and in principle can be applied to any complete riemannian manifold we give sufficient convergence conditions on riemannian manifolds with bounded curvature and we analyze the differences with respect to the euclidean case we test the proposed algorithms on synthetic data sampled from the space of rotations the sphere and the grassmann manifold

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

subdiffusion in the nonlinear schrodinger equation with disorder the nonlinear schrodinger equation in the presence of disorder is considered the dynamics of an initially localized wave packet is studied a subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk a probabilistic description of subdiffusion is suggested and a transport exponent of subdiffusion is obtained to be 2 5

algebraic decay in hierarchical graphs we study the algebraic decay of the survival probability in open hierarchical graphs we present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the frobenius perron operator using a perturbative scheme we derive the exponent of the classical algebraic decay in terms of two parameters of the model one parameter defines the geometrical relation between the length scales on the graph and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another the scattering resonances of the corresponding hierarchical quantum graphs are also studied the width distribution shows the scaling behavior p 1

chaotic scattering and diffusion in the lorentz gas a chaotic scattering theory of diffusion in the lorentz gas is presented the scattering process is considered on disk scatterers of increasing sizes in this way chaotic and fractal properties of the scattering process are related to diffusion a formula is obtained that gives the diffusion coefficient in terms of the lyapunov exponent and the hausdorff codimension of the fractal repeller of orbits trapped in the scatterer numerical results are presented that support our theoretical results

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

localization length of stationary states in the nonlinear schrodinger equation for the nonlinear schrodinger equation nlse in the presence of disorder exponentially localized stationary states are found we demonstrate analytically that the localization length is typically independent of the strength of the nonlinearity and is identical to the one found for the corresponding linear equation the analysis makes use of the correspondence between the stationary nlse and the langevin equation as well as of the resulting fokker planck equation the calculations are performed for the white noise random potential and an exact expression for the exponential growth of the eigenstates is obtained analytically it is argued that the main conclusions are robust

subdiffusion in classical and quantum nonlinear schrdinger equations with disorder the review is concerned with the nonlinear schrdinger equation nlse in the presence of disorder disorder leads to localization in the form of the localized anderson modes am while nonlinearity is responsible for the interaction between the ams and transport the dynamics of an initially localized wave packets are concerned in both classical and quantum cases in both cases there is a subdiffusive spreading which is explained in the framework of a continuous time random walk ctrw and it is shown that subdiffusion is due to the transitions between those ams which are strongly overlapped this overlapping being a common feature of both classical and quantum dynamics leads to the clustering with an effective trapping of the wave packet inside each cluster therefore the dynamics of the wave packet corresponds to the ctrw where the basic mechanism of subdiffusion is an entrapping of the wave packet with delay or waiting times distributed by the power law w t 1 t1 where is the transport exponent it is obtained that 1 3 for the classical nlse and 1 2 for the quantum nlse

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

classical dynamics on graphs we consider the classical evolution of a particle on a graph by using a time continuous frobenius perron operator that generalizes previous propositions in this way the relaxation rates as well as the chaotic properties can be defined for the time continuous classical dynamics on graphs these properties are given as the zeros of some periodic orbit zeta functions we consider in detail the case of infinite periodic graphs where the particle undergoes a diffusion process the infinite spatial extension is taken into account by fourier transforms that decompose the observables and probability densities into sectors corresponding to different values of the wave number the hydrodynamic modes of diffusion are studied by an eigenvalue problem of a frobenius perron operator corresponding to a given sector the diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the green kubo form moreover we study finite but large open graphs that converge to the infinite periodic graph when their size goes to infinity the lifetime of the particle on the open graph is shown to correspond to the lifetime of a system that undergoes a diffusion process before it escapes

dynamics of wave packets for the nonlinear schrodinger equation with a random potential the dynamics of an initially localized anderson mode is studied in the framework of the nonlinear schrodinger equation in the presence of disorder it is shown that the dynamics can be described in the framework of the liouville operator an analytical expression for a wave function of the initial time dynamics is found by a perturbation approach as follows from a perturbative solution the initially localized wave function remains localized at asymptotically large times the dynamics can be described qualitatively in the framework of a phenomenological probabilistic approach by means of a probability distribution function it is shown that the probability distribution function may be governed by the fractional fokker planck equation and corresponds to subdiffusion

experimental investigation of wigner s reaction matrix for irregular graphs with absorption we use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by t joints to make an experimental study of wigner s reaction k matrix for irregular graphs in the presence of absorption from measurements of the scattering matrix s for each realization of the microwave network we obtain distributions of the imaginary and real parts of k our experimental results are in good agreement with theoretical predictions

experimental and numerical investigation of the reflection coefficient and the distributions of wigner s reaction matrix for irregular graphs with absorption we present the results of an experimental and numerical study of the distribution of the reflection coefficient p r and the distributions of the imaginary p v and the real p u parts of the wigner reaction k matrix for irregular fully connected hexagon networks graphs in the presence of strong absorption in the experiment we used microwave networks which were built of coaxial cables and attenuators connected by joints in the numerical calculations experimental networks were described by quantum fully connected hexagon graphs the presence of absorption introduced by attenuators was modeled by optical potentials the distribution of the reflection coefficient p r and the distributions of the reaction k matrix were obtained from measurements and numerical calculations of the scattering matrix s of the networks and graphs respectively we show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory

hydrodynamic modes as singular eigenstates of the liouvillian dynamics deterministic diffusion hydrodynamic modes of diffusion and the corresponding nonequilibrium steady states are studied as an eigenvalue problem for the liouvillian dynamics of spatially extended suspension flows which are special continuous time dynamical systems including billiards defined on the basis of a mapping the infinite spatial extension is taken into account by spatial fourier transforms which decompose the observables and probability densities into sectors corresponding to the different values of the wave number the frobenius perron operator ruling the time evolution in each wave number sector is reduced to a frobenius perron operator associated with the mapping of the suspension flow in this theory the dispersion relation of diffusion is given as a pollicott ruelle resonance of the frobenius perron operator and the corresponding eigenstates are studied formulas are derived for the diffusion and the burnett coefficients in terms of the mapping of the suspension flow nonequilibrium steady states are constructed on the basis of the eigenstates and are given by mathematical distributions without density functions also referred to as singular measures the nonequilibrium steady states are shown to obey fick s law and to be related to zubarev s local integrals of motion the theory is applied to the regular lorentz gas with a finite horizon generalizations to the nonequilibrium steady states associated with the other transport processes are also obtained

dynamical chaos in the lorentz lattice gas this paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics in particular to a study of nonequilibrium phenomena in lorentz lattice gases with stochastic collision rules using simple arguments based upon discussions in the mathematical literature we show that such lattice gases belong to the category of dynamical systems with positive lyapunov exponents this is accomplished by showing how such systems can be expressed in terms of continuous phase space variables expressions for the lyapunov exponent of a one dimensional lorentz lattice gas with periodic boundaries are derived other quantities of interest for the theory of irreversible processes are discussed

stable polynomials and crystalline measures explicit examples of positive crystalline measures and fourier quasicrystals are constructed using pairs of stable polynomials answering several open questions in the area

contraction theory on riemannian manifolds abstract contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another in this work we present the fundamental results of contraction theory in an intrinsic coordinate free setting with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties we provide coordinate free proofs of the main results for autonomous vector fields and clarify the assumptions under which the results hold we state and prove several interesting corollaries to the main result study cascade and feedback interconnections of contracting systems study some simple examples and highlight how contraction theory has arisen independently in other scientific disciplines we conclude by illustrating the developed theory for the case of gradient dynamics

periodic orbit theory and the statistical analysis of scaling quantum graph spectra the explicit solution to the spectral problem of quantum graphs found recently by dabaghian and blumel phys rev e 68 055201 r 2003 70 046206 2004 jetp lett 77 530 2003 is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics including the distribution for the nearest neighbor separations sn kn kn 1 and the distribution of the spectral oscillations around the average deltakn kn kn

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

experimental simulation of quantum graphs by microwave networks we present the results of experimental and theoretical study of irregular tetrahedral microwave networks consisting of coaxial cables annular waveguides connected by t joints the spectra of the networks were measured in the frequency range 0 0001 16 ghz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity d3sld the comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry in particular we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs we also present experimental study of directional microwave networks consisting of coaxial cables and faraday isolators for which the time reversal symmetry is broken in this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of gaussian orthogonal ensembles in random matrix theory approaching especially for small eigenfrequency spacing s results for gaussian unitary ensembles experimental results are supported by the theoretical analysis of directional graphs

coordinated motion design on lie groups the present paper proposes a unified geometric framework for coordinated motion on lie groups it first gives a general problem formulation and analyzes ensuing conditions for coordinated motion then it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics it thereby shows that coordination can be studied in a systematic way once the lie group geometry of the configuration space is well characterized applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds a link with brockett s double bracket flows is also made the concepts are illustrated on so 3 se 2 and se 3

asymptotic localization of stationary states in the nonlinear schrodinger equation the mapping of the nonlinear schrodinger equation with a random potential on the fokker planck equation is used to calculate the localization length of its stationary states the asymptotic growth rates of the moments of the wave function and its derivative for the linear schrodinger equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear schrodinger equation and the localization length of the stationary states

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

parameter dependent spectral statistics of chaotic quantum graphs neumann versus circular orthogonal ensemble boundary conditions the parameter dependent spectral statistics of totally connected quantum graphs with n 4 30 vertices such as the parametric velocities correlation functions and the distribution of curvatures are studied the inverse participation ratio ipr an important measure of localization effects was also numerically investigated in the calculations we successfully used two different theoretical approaches the first approach was based on the graphs eigenenergies and wave functions calculations while the second one used the eigenphases and the eigenvectors of the bond scattering matrix s k we considered graphs with neumann and circular orthogonal ensemble coe boundary conditions we show that in contrast to large neumann graphs for which the departure of many parameter dependent spectral statistics from the random matrix theory rmt predictions is observed for large coe graphs the spectral statistics and ipr are in good agreement with the rmt predictions

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

synchronization of diffusively coupled systems on compact riemannian manifolds in the presence of drift abstract recently it has been shown that the synchronization manifold is an asymptotically stable invariant set of diffusively coupled systems on riemannian manifolds we regionally investigate the stability properties of the synchronization manifold when the systems are subject to drift when the drift vector field is quad i e satisfies a certain quadratic inequality and the underlying riemannian manifold is compact we prove that a sufficiently large algebraic connectivity of the underlying graph is sufficient for the synchronization manifold to remain asymptotically stable for drift vector fields which are quad or contracting we explicitly characterize the rate at which the solution converges to the synchronization manifold our main result is that the synchronization manifold is asymptotically stable even for drift vector fields which are only locally lipschitz continuous as long as the algebraic connectivity of the underlying graph is sufficiently large

markov tree model of transport in area preserving maps abstract transport in an area preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori a model retaining a discrete set of cantori approaching a boundary circle gives the markov chain description of hanson cary and meiss the inclusion of cantori surrounding island chains and islands about islands etc gives a markov tree model with a slower decay rate the survival probability distribution is shown to decay asymptotically as a power law the decay exponent agrees reasonably well with the computations of karney and of chirikov and shepelyanski

consensus optimization on manifolds the present paper considers distributed consensus algorithms that involve n agents evolving on a connected compact homogeneous manifold the agents track no external reference and communicate their relative state according to a communication graph the consensus problem is formulated in terms of the extrema of a cost function this leads to efficient gradient algorithms to synchronize i e maximizing the consensus or balance i e minimizing the consensus the agents a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time varying the cost function is linked to a specific centroid definition on manifolds introduced here as the induced arithmetic mean that is easily computable in closed form and may be of independent interest for a number of manifolds the special orthogonal group so n and the grassmann manifold text it grass p n are treated as original examples a link is also drawn with the many existing results on the circle

algebraic decay in self similar markov chains a continuous time markov chain is used to model motion in the neighborhood of a critical invariant circle for a hamiltonian map states in the infinite chain represent successive rational approximants to the frequency of the invariant circle for the case of a noble frequency the chain is self similar and the nonlinear integral equation for the first passage time distribution is solved exactly the asymptotic distribution is a power law times a function periodic in the logarithm of the time for parameters relevant to the critical noble circle the decay proceeds ast 4 05

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

synchronization and balancing on the n torus in this paper we study the behavior of a network of n agents each evolving on the circle we propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the possibly time varying and unidirectional communication graphs the global convergence analysis on the n torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the euclidean space

asymptotic stabilization of submanifolds embedded in riemannian manifolds we study control problems in which systems whose states spaces are riemannian manifolds must be steered towards embedded submanifolds of their state spaces in a stable fashion i e feedback laws rendering the given submanifold asymptotically stable are sought bringing the closed loop to the form of a gradient system with drift we find that the gradient part can be scaled in a fashion so as to guarantee asymptotic stability of the desired submanifold under this circumstance we show that solutions of the system can be brought to an arbitrarily small neighborhood of the submanifold in arbitrary time thereafter we derive an algebraic condition under which the control vector fields can be brought to the desired form lastly we recast our algebraic condition in terms of vertical and horizontal spaces for the case that the submanifold is an equivalence class

brief paper novel decentralized adaptive strategies for the synchronization of complex networks this paper is concerned with the analysis of the synchronization of networks of nonlinear oscillators through an innovative local adaptive approach in particular time varying feedback coupling gains are considered whose gradient is a function of the local synchronization error over each edge in the network it is shown that under appropriate conditions the strategy is indeed successful in guaranteeing the achievement of a common synchronous evolution for all oscillators in the network the theoretical derivation is complemented by its validation on a set of representative examples