measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

an efficient threshold dynamics method for wetting on rough surfaces the threshold dynamics method developed by merriman bence and osher mbo is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary direct generalization of mbo type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original mbo method the dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary in this paper we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces this method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase the method is simple stable with o n log n complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

phase transition with the line tension effect we make the connection between the geometric model for capillarity with line tension and the cahn hilliard model of two phase fluids to this aim we consider the energies f eps u eps int o du 2 1 over eps int o w u lambda int bd o v u where u is a scalar density function and w and v are double well potentials we show that the behaviour of f e in the limit e 0 and depends on the limit of e log if this limit is finite and strictly positive then the singular limit of the energies f e leads to a coupled problem of bulk and surface phase transitions and under certain assumptions agrees with the relaxation of the capillary energy with line tension these results were announced in abs1 and abs2

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

nonconvex variational problems with anisotropic perturbations we obtain the l 1 limit for a class of variational problems which arise as an anisotropic singular perturbation of a nonconvex functional any limit of minimizers of the perturbed problems must minimize the new variational problem which arises in the limit and the primary goal is to understand the dependence of this new variational problem on the type of perturbation chosen

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

z measures on partitions related to the infinite gelfand pair s 2 h abstract the paper deals with the z measures on partitions with the deformation jack parameters 2 or 1 2 we provide a detailed explanation of the representation theoretic origin of these measures and of their role in the harmonic analysis on the infinite symmetric group

the z measures on partitions pfaffian point processes and the matrix hypergeometric kernel abstract we consider a point process on one dimensional lattice originated from the harmonic analysis on the infinite symmetric group and defined by the z measures with the deformation jack parameter 2 we derive an exact pfaffian formula for the correlation function of this process namely we prove that the correlation function is given as a pfaffian with a 2 2 matrix kernel the kernel is given in terms of the gauss hypergeometric functions and can be considered as a matrix analogue of the hypergeometric kernel introduced by a borodin and g olshanski 2000 5 our result holds for all values of admissible complex parameters

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

derivation of the wenzel and cassie equations from a phase field model for two phase flow on rough surface in this paper the equilibrium behavior of an immiscible two phase fluid on a rough surface is studied from a phase field equation derived from minimizing the total free energy of the system when the size of the roughness becomes small we derive the effective boundary condition for the equation by the multiple scale expansion homogenization technique the wenzel and cassie equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition the homogenization results are proved rigorously by the convergence theory

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

complex network based time series analysis abstract recent works show that complex network theory may be a powerful tool in time series analysis we propose in this paper a reliable procedure for constructing complex networks from the correlation matrix of a time series an original stock time series the corresponding return series and its amplitude series are considered the degree distribution of the original series can be well fitted with a power law while that of the return series can be well fitted with a gaussian function the degree distribution of the amplitude series contains two asymmetric gaussian branches reconstruction of networks from time series is a common problem in diverse research the proposed strategy may be a reasonable solution to this problem

finding metastable states in real world time series with recurrence networks in the framework of time series analysis with recurrence networks we introduce a self adaptive method that determines the elusive recurrence threshold and identifies metastable states in complex real world time series as initial step we introduce a way to set the embedding parameters used to reconstruct the state space from the time series we set them as the ones giving the maximum shannon entropy of the diagonal line length distribution for the first simultaneous minima of recurrence rate and shannon entropy to identify metastable states as well as the transitions between them we use a soft partitioning algorithm for module finding which is specifically developed for the case in which a system shows metastability we illustrate our method with a complex time series example finally we show the robustness of our method for identifying metastable states our results suggest that our method is robust for identifying metastable states in complex time series even when introducing considerable levels of noise and missing data points

comparison of tests for embeddings it is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test the classical tests which depend on real numbers fractal dimensions lyapunov exponents averaged over an attractor are compared with a topological test that depends on integers the comparison can only be done for mappings into three dimensions we find that the classical tests fail to predict when a mapping is an embedding and when it is not we point out the reasons for this failure which are not restricted to three dimensions

random rectangular graphs a generalization of the random geometric graph rgg model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square 0 1 2 the topological properties of the random rectangular graphs rrgs generated by this model are then studied as a function of the rectangle sides lengths a and b 1 a and the radius r used to connect the nodes when a 1 we recover the rgg and when a the very elongated rectangle generated resembles a one dimensional rgg we obtain here analytical expressions for the average degree degree distribution connectivity average path length and clustering coefficient for rrg these results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle usually in a monotonic way the clustering coefficient however increases when the square is transformed into a slightly elongated rectangle and after this maximum it decays with the increase of the elongation of the rectangle we support all our findings by computational simulations that show the goodness of the theoretical models proposed for rrgs

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

potential theory and analytic properties of self similar fractal and multifractal distributions by the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential moments and fourier transform of exactly self similar fractal and multifractal charge distributions three specific examples are given the binomial distribution on the middle third cantor set which is a multifractal distribution the uniform distribution on the menger sponge which illustrates the added complication of higher dimensionality and the uniform distribution on the von koch snowflake which illustrates the effect of rotations in the defining transformations

extremes on trees this paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on x1 xnn where x1 x2 are i i d in 2 with distribution f and nn is independent of the xi and satisfies nn n p1 a new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which f has unbounded support the general results are applied to a number of special cases including elliptically contoured distributions distributions with independent weibull like margins and distributions with parallel level curves

spacing distributions for some gaussian ensembles of hermitian matrices the probability e n s that an interval of length s contains exactly n eigenvalues of a random matrix is expressed in terms of their correlation functions for the gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and antisymmetric parts the authors can thus write e n s as a convergent infinite product multiplied by an infinite sum

comparing fuzzy partitions a generalization of the rand index and related measures in this paper we introduce a fuzzy extension of a class of measures to compare clustering structures namely measures that are based on the number of concordant and the number of discordant pairs of data points this class includes the well known rand index but also commonly used alternatives such as the jaccard measure in contrast with previous proposals our extension exhibits desirable metrical properties apart from elaborating on formal properties of this kind we present an experimental study in which we compare different fuzzy extensions of the rand index and the jaccard measure

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

analyzing long term correlated stochastic processes by means of recurrence networks potentials and pitfalls long range correlated processes are ubiquitous ranging from climate variables to financial time series one paradigmatic example for such processes is fractional brownian motion fbm in this work we highlight the potentials and conceptual as well as practical limitations when applying the recently proposed recurrence network rn approach to fbm and related stochastic processes in particular we demonstrate that the results of a previous application of rn analysis to fbm liu et al phys rev e 89 032814 2014 are mainly due to an inappropriate treatment disregarding the intrinsic nonstationarity of such processes complementarily we analyze some rn properties of the closely related stationary fractional gaussian noise fgn processes and find that the resulting network properties are well defined and behave as one would expect from basic conceptual considerations our results demonstrate that rn analysis can indeed provide meaningful results for stationary stochastic processes given a proper selection of its intrinsic methodological parameters whereas it is prone to fail to uniquely retrieve rn properties for nonstationary stochastic processes like fbm

effective contact angle for rough boundary abstract we provide rigorous justification for the classical wenzel equation for the roughness enhanced effective contact angle the minimization of the total surface energy is reformulated into a variational problem as the size of the roughness becomes small we show convergence of the minimizer the limiting minimizer and effective contact angle are explicitly calculated to verify the wenzel equation

isolation and connectivity in random geometric graphs with self similar intensity measures random geometric graphs consist of randomly distributed nodes points with pairs of nodes within a given mutual distance linked in the usual model the distribution of nodes is uniform on a square and in the limit of infinitely many nodes and shrinking linking range the number of isolated nodes is poisson distributed and the probability of no isolated nodes is equal to the probability the whole graph is connected here we examine these properties for several self similar node distributions including smooth and fractal uniform and nonuniform and finitely ramified or otherwise we show that nonuniformity can break the poisson distribution property but it strengthens the link between isolation and connectivity it also stretches out the connectivity transition finite ramification is another mechanism for lack of connectivity the same considerations apply to fractal distributions as smooth with some technical differences in evaluation of the integrals and analytical arguments

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

asymptotic expansions in the poisson limit theorem asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the poisson distribution are derived together with explicit estimates of the truncation error expansions are also derived for the expectations of at most polynomially growing functions of such sums applications to the poisson binomial and poisson negative binomial approximations are considered the method used is an adaptation of the stein chen approach 1 introduction in chen 1975 stein s method of obtaining error bounds for normal approximations was introduced in the poisson context and was used to obtain rates of convergence in total variation to the poisson distribution for sums of stationary sequences of 0 1 random variables chen also observed that stein s method is in principle suited to developing asymptotic expansions as well as obtaining error estimates and used it for independent 0 1 summands to derive the second term in such an expansion together with an estimate of the remaining error kerstan 1964 had previously derived a similar result with a sharper error estimate by a quite different technique the stein chen method was refined in barbour and hall 1984 to yield the best known upper bounds for the discrepancy between the poisson distribution and that of a sum of independent 0 1 random variables as well as complementary lower bounds and new estimates for the error remaining after the second term in the asymptotic expansion were also established however although stein s method was potentially applicable also to higher order expansions it proved in practice too cumbersome to use in this paper it is shown how by means of a simple identity the stein chen method can be made to yield a full asymptotic expansion with a minimum of difficulty the error estimates obtained perform well in comparison with other estimates that are available for sums of independent 0 1 random variables they agree with those of barbour and hall when the series is truncated after one term or after two for the more general case of sums of independent nonnegative integer valued random variables the error estimates obtained after two terms in the expansion differ only slightly from those in barbour and hall comparison with kerstan s estimates for integral nonnegative summands of the error after one term of the expansion is more complicated by and large the techniques of this paper give better approximations except when the summands themselves are almost precisely poisson distributed in section 3 expansions are also obtained for the expectations of at most polynomially growing functions of the

random geometric graphs with general connection functions in the original 1961 gilbert model of random geometric graphs nodes are placed according to a poisson point process and links formed between those within a fixed range motivated by wireless ad hoc networks soft or probabilistic connection models have recently been introduced involving a connection function h r that gives the probability that two nodes at distance r are linked directly connect in many applications not only wireless networks it is desirable that the graph is connected that is every node is linked to every other node in a multihop fashion here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions it turns out that only a few quantities such as moments of the connection function appear good agreement is found with special cases from previous studies and with numerical simulations

computer simulations of phase field drops on super hydrophobic surfaces abstract we present a novel quasi newton continuation procedure that efficiently solves the system of nonlinear equations arising from the discretization of a phase field model for wetting phenomena we perform a comparative numerical analysis that shows the improved speed of convergence gained with respect to other numerical schemes moreover we discuss the conditions that on a theoretical level guarantee the convergence of this method at each iterative step a suitable continuation procedure develops and passes to the nonlinear solver an accurate initial guess discretization performs through cell centered finite differences the resulting system of equations is solved on a composite grid that uses dynamic mesh refinement and multi grid techniques the final code achieves three dimensional realistic computer experiments comparable to those produced in laboratory settings this code offers not only new insights into the phenomenology of super hydrophobicity but also serves as a reliable predictive tool for the study of hydrophobic surfaces

harmonic analysis on the infinite symmetric group the infinite symmetric group s whose elements are finite permutations of 1 2 3 is a model example of a big group by virtue of an old result of murray von neumann the one sided regular representation of s in the hilbert space l2 s generates a type ii1 von neumann factor while the two sided regular representation is irreducible this shows that the conventional scheme of harmonic analysis is not applicable to s for the former representation decomposition into irreducibles is highly non unique and for the latter representation there is no need of any decomposition at all we start with constructing a compactification mathfrak s supset s infty which we call the space of virtual permutations although mathfrak s is no longer a group it still admits a natural two sided action of s thus mathfrak s is a g space where g stands for the product of two copies of s on mathfrak s there exists a unique g invariant probability measure 1 which has to be viewed as a true haar measure for s more generally we include 1 into a family t t 0 of distinguished g quasiinvariant probability measures on virtual permutations by making use of these measures we construct a family tz z of unitary representations of g called generalized regular representations each representation tz with z 0 can be realized in the hilbert space l 2 mathfrak s mu t where t z 2 as z the generalized regular representations tz approach in a suitable sense the naive two sided regular representation of the group g in the space l2 s in contrast with the latter representation the generalized regular representations tz are highly reducible and have a rich structure we prove that any tz admits a unique decomposition into a multiplicity free continuous integral of irreducible representations of g for any two distinct and not conjugate complex numbers z1 z2 the spectral types of the representations t z 1 and t z 2 are shown to be disjoint in the case z a complete description of the spectral type is obtained further work on the case z reveals a remarkable link with stochastic point processes and random matrix theory

recurrence plot based measures of complexity and their application to heart rate variability data the knowledge of transitions between regular laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems while several linear approaches are often insufficient to describe such processes there are several nonlinear methods that however require rather long time observations to overcome these difficulties we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart rate variability data for the logistic map these measures enable us not only to detect transitions between chaotic and periodic states but also to identify laminar states i e chaos chaos transitions the traditional recurrence quantification analysis fails to detect the latter transitions applying our measures to the heart rate variability data we are able to detect and quantify the laminar phases before a life threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event our findings could be of importance for the therapy of malignant cardiac arrhythmias

estimating the shannon entropy recurrence plots versus symbolic dynamics recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics a few years later the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis among the different measures introduced a shannon entropy was found to be correlated with the inverse of the largest lyapunov exponent the discrepancy between this and the usual interpretation of a shannon entropy is solved here by using a new definition still based on the recurrence plots and it is verified that this new definition is correlated with the largest lyapunov exponent as expected from the pesin conjecture a comparison with a shannon entropy computed from symbolic dynamics is also provided

classical skew orthogonal polynomials and random matrices skew orthogonal polynomials arise in the calculation of the n point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry in particular the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials in the case that the eigenvalue probability density function involves a classical weight function explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials and the structure is used to give a closed form expression for the sum this theory treates all classical cases on an equal footing giving formulas applicable at once to the hermite laguerre and jacobi cases

spatially embedded random networks many real world networks analyzed in modern network theory have a natural spatial element e g the internet social networks neural networks etc yet aside from a comparatively small number of somewhat specialized and domain specific studies the spatial element is mostly ignored and in particular its relation to network structure disregarded in this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding specifically we model connectivity as dependent on the distance between network nodes our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real world networks but rather to gain intuition for the effects of spatial embedding on network structure nevertheless we are able to demonstrate in a quite general setting some constraints of spatial embedding on connectivity such as the effects of spatial symmetry conditions for scale free degree distributions and the existence of small world spatial networks we also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples

wetting of rough surfaces a homogenization approach the contact angle of a drop in equilibrium on a solid is strongly affected by the roughness of the surface on which it rests we study the roughness induced enhancement of the hydrophobic or hydrophilic properties of a solid surface through homogenization theory by relying on a variational formulation of the problem we show that the macroscopic contact angle is associated with the solution of two cell problems giving the minimal energy per unit macroscopic area for a transition layer between the rough solid surface and a liquid or vapor phase our results are valid for both chemically heterogeneous and homogeneous surfaces in the latter case a very transparent structure emerges from the variational approach the classical laws of wenzel and cassie baxter give bounds for the optimal energy and configurations of minimal energy are those leading to the smallest macroscopic contact angle in the hydrophobic case to the largest one in the hydrophilic case

correlations for superpositions and decimations of laguerre and jacobi orthogonal matrix ensembles with a parameter a superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue in the cases of the classical matrix ensembles with orthogonal symmetry it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry the basic identities expressing these facts can be extended to include a parameter which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles the parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues by the construction of new families of biorthogonal and skew orthogonal polynomials we are able to compute the corresponding correlation functions both in the finite system and in various scaled limits specializing back to the cases of orthogonal and symplectic symmetry we find that our results imply different functional forms to those known previously

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

classical discrete symplectic ensembles on the linear and exponential lattice skew orthogonal polynomials and correlation functions the eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice the corresponding correlation functions can be expressed in terms of certain discrete and q skew orthogonal polynomials respectively we give a theory of both of these classes of polynomials and the correlation kernels determining the correlation functions in the cases in which the weights for the corresponding discrete unitary ensembles are classical crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones and have a tridiagonal action on the corresponding discrete or q orthogonal polynomials

a free boundary problem with facets we study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous hamiltonian we find an explicit formula for the hamiltonian prove that the solutions are unique and prove that the limiting free boundary has a facets in every rational direction our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems the problem is motivated by physical experiments involving liquid drops on patterned solid surfaces

matrix kernels for measures on partitions we consider the problem of computation of the correlation functions for the z measures with the deformation jack parameters 2 or 1 2 such measures on partitions are originated from the representation theory of the infinite symmetric group and in many ways are similar to the ensembles of random matrix theory of 4 or 1 symmetry types for a certain class of such measures we show that correlation functions can be represented as pfaffians including 2 2 matrix valued kernels and compute these kernels explicitly we also give contour integral representations for correlation kernels of closely connected measures on partitions

pfaffian expressions for random matrix correlation functions it is well known that pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of pfaffians in this article we review the formulations and applications of pfaffian formulas for that purpose we first present the general pfaffian expressions in terms of the corresponding skew orthogonal polynomials then we clarify the relation to eynard and mehta s determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials applications of pfaffian formulas to random matrix theory and other fields are also mentioned

ambiguities in recurrence based complex network representations of time series recently different approaches have been proposed for studying basic properties of time series from a complex network perspective in this work the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail we discuss the main requirements that permit a feasible system theoretic interpretation of network topology in terms of dynamically invariant phase space properties possible artifacts induced by disregarding these requirements are pointed out and systematically studied finally a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed

spatial network surrogates for disentangling complex system structure from spatial embedding of nodes networks with nodes embedded in a metric space have gained increasing interest in recent years the effects of spatial embedding on the networks structural characteristics however are rarely taken into account when studying their macroscopic properties here we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes embedding in a metric space comparing the original network s and the resulting surrogates global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links we apply our framework to various real world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes spatial embedding depending on the actual performance of the proposed null models the networks are categorized into different classes since many real world complex networks are in fact spatial networks the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields ranging from social systems over infrastructure and neurophysiology to climatology

universality for orthogonal and symplectic laguerre type ensembles we give a proof of the universality conjecture for orthogonal 1 and symplectic 4 random matrix ensembles of laguerre type in the bulk of the spectrum as well as at the hard and soft spectral edges our results are stated precisely in the introduction theorems 1 1 1 4 1 6 and corollaries 1 2 1 5 1 7 they concern the appropriately rescaled kernels k n correlation and cluster functions gap probabilities and the distributions of the largest and smallest eigenvalues corresponding results for unitary 2 laguerre type ensembles have been proved by the fourth author in ref 23 the varying weight case at the hard spectral edge was analyzed in ref 13 for 2 in this paper we do not consider varying weights our proof follows closely the work of the first two authors who showed in refs 7 8 analogous results for hermite type ensembles as in refs 7 8 we use the version of the orthogonal polynomial method presented in refs 22 25 to analyze the local eigenvalue statistics the necessary asymptotic information on the laguerre type orthogonal polynomials is taken from ref 23

robust perron cluster analysis in conformation dynamics the key to molecular conformation dynamics is the direct identification of metastable conformations which are almost invariant sets of molecular dynamical systems once some reversible markov operator has been discretized a generalized symmetric stochastic matrix arises this matrix can be treated by perron cluster analysis a rather recent method involving a perron cluster eigenproblem the paper presents an improved perron cluster analysis algorithm which is more robust than earlier suggestions numerical examples are included

generalized recurrence plots for the analysis of images from spatially distributed systems abstract we propose a new method for the analysis of images showing patterns emerging from the evolution of spatially distributed systems the generalized recurrence plot grp and the generalized recurrence quantification analysis grqa are exploited for the investigation of such patterns we focus on snapshots of spatio temporal processes such as the formation of turing structures and traveling waves in the belousov zhabotinsky reaction satellite images of spatial chlorophyll distribution in seas and oceans similar to turbulent flows colonies of dyctiostelium discoideum fractals and noise the method is based on the grp and grqa and particularly on the measures determinism det and entropy ent providing a new criterion for the assessment and classification of images based on the simultaneous evaluation of their global and local structure the det ent diagram is introduced and compared with the classical image analysis entropy defined on the pixels values the method proposed provides appealing performances in the case of images showing complex spatial patterns

lower semicontinuity of surface energies using the theory of indicator measures a lower semicontinuity result for quasiconvex functions in w 1 1 and assuming only l 1 convergence is obtained

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

analytical framework for recurrence network analysis of time series recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems while there are already many successful applications ranging from medicine to paleoclimatology a solid theoretical foundation of the method has still been missing so far here we interpret an ensuremath varepsilon recurrence network as a discrete subnetwork of a continuous graph with uncountably many vertices and edges corresponding to the system s attractor this step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ensuremath varepsilon in particular we introduce local measures such as the ensuremath varepsilon clustering coefficient mesoscopic measures such as ensuremath varepsilon motif density path based measures such as ensuremath varepsilon betweennesses and global measures such as ensuremath varepsilon efficiency this new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ensuremath varepsilon via a percolation threshold and it shows that estimation can be improved by so called node splitting invariant versions of the measures we finally illustrate the framework for a number of archetypical chaotic attractors such as those of the bernoulli and logistic maps periodic and two dimensional quasiperiodic motions and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments more generally the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science

finite clusters in high density continuous percolation compression and sphericality a percolation process inrd is considered in which the sites are a poisson process with intensity and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other the distribution of the number of sites in the clusterc of the origin is examined and related to the geometry ofc it is shown that when andk are large there is a characteristic radius such that conditionally on c k the convex hull ofc closely approximates a ball of radius with high probability when the normal volumek thatk points would occupy is small the cluster is compressed in that the number of points per unit volume in this ball is much greater than the ambient density for larger normal volumes there is less compression this can be compared to bernoulli bond percolation on the square lattice in two dimensions where an analog of this compression is known not to occur

relaxation of quasiconvex functional in bv ω ℝp for integrands f x u u in this paper it is shown that if p x u is a quasiconvex function with linear growth then the relaxed functional in bv p of u to int limits omega f x u x nabla u x dx with respect to the l1 topology has an integral representation of the form begin gathered mathfrak f u int limits omega f x u x nabla u x dx int limits sigma u k x u x u x v x dh n 1 x hfill int limits omega f infty x u x dc u hfill end gathered where du u dx u u v dhn 1l u c u the proof relies on a blow up argument introduced by fonseca muller in the case where u w1 1 and on a recent result by alberti showing that the cantor part c u is rank one valued