extended double complex linear systems and new multiple infinite dimensional hidden symmetries for the general symplectic gravity models by using a so called extended double ed complex method the previously found doubleness symmetry for each member of the class of general symplectic gravity models is further exploited and extended a 2 n 1 2 n 1 matrix double complex h potential is constructed for any non negative integer n and the motion equations in two dimensions are written in a double complex formulation a double duality mapping is proposed and two pairs of ed complex hauser ernst type linear systems j math phys 21 1126 1980 are established based on these linear systems explicit formulations of new multiple hidden symmetry transformations for the studied theories are given for any fixed n these symmetry transformations are verified to constitute multiple infinite dimensional lie algebras each of which is a semidirect product of the kac moody sp 2 n 1 r and virasoro algebras without center charges these results demonstrate that the ed complex method is necessary and more effective and the general symplectic g

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

a simplified derivation of the geroch group in two dimensional reduced gravity by generalizing our previous treatment of hidden symmetry in two dimensional chiral models a simple and explicit approach is proposed to the geroch group for the vacuum einstein equations with the metric tensor depending only on two variables two dimensional reduced gravity an infinite number of infinitesimal transformations for the metric tensor preserving the equations of motion are summarized by an explicit parametric transformation and the commutators among them are calculated in a simple and straightforward way these transformations and their group structure are further identified to be those of the well known geroch group

bounds for eigenvalues of certain stochastic matrices abstract we consider the class of stochastic matrices m generated in the following way from graphs if g is an undirected connected graph on n vertices with adjacency matrix a we form m from a by dividing the entries in each row of a by their row sum being stochastic m has the eigenvalue 1 and possibly also an eigenvalue 1 we prove that the remaining eigenvalues of m lie in the disk 1 n 3 and show by examples that the order of magnitude of this estimate is best possible in these examples g has a bar bell structure in which n 3 of the vertices are arranged along a line with n 3 vertices fully interconnected at each end we also obtain better bounds when either the diameter of g or the maximal degree of a vertex is restricted

normalized graph laplacians for directed graphs abstract we consider the normalized laplace operator for directed graphs with positive and negative edge weights this generalization of the normalized laplace operator for undirected graphs is used to characterize directed acyclic graphs moreover we identify certain structural properties of the underlying graph with extremal eigenvalues of the normalized laplace operator we prove comparison theorems that establish a relationship between the eigenvalues of directed graphs and certain undirected graphs this relationship is used to derive eigenvalue estimates for directed graphs finally we introduce the concept of neighborhood graphs for directed graphs and use it to obtain further eigenvalue estimates

spectrum of the laplacian on a covering graph with pendant edges i the one dimensional lattice and beyond abstract in this paper we examine covering graphs that are obtained from the d dimensional integer lattice by adding pendant edges in the case of d 1 we show that the laplacian on the graph has a spectral gap and establish a necessary and sufficient condition under which the laplacian has no eigenvalues in the case of d 2 we show that there exists an arrangement of the pendant edges such that the laplacian has no spectral gap

a class of multi objective supply chain networks optimal model under random fuzzy environment and its application to the industry of chinese liquor an important issue when shipping cost and customers demand are random fuzzy variables in supply chain network scn design problem is to find the network strategy that can simultaneously achieve the objectives of minimization total cost comprised of fixed costs of plants and distribution centers dcs inbound and outbound distribution costs and maximization customer services that can be rendered to customers in terms of acceptable delivery time in this paper we propose a random fuzzy multi objective mixed integer non linear programming model for the scn design problem of luzhou co ltd which is representative in the industry of chinese liquor by the expected value operator and chance constraint operator the model has been transformed into a deterministic multi objective mixed integer non linear programming model then we use spanning tree based genetic algorithms st ga by the prufer number representation to find the scn to satisfy the demand imposed by customers with minimum total cost and maximum customer services for multi objective scn design problem of this company under condition of random fuzzy customers demand and transportation cost between facilities furthermore the efficacy and the efficiency of this method are demonstrated by the comparison between its numerical experiment results and those of tradition matrix based genetic algorithm

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

a non dominance based online stopping criterion for multi objective evolutionary algorithms a non dominance criterion based metric that tracks the growth of an archive of non dominated solutions over a few generations is proposed to generate a convergence curve for multi objective evolutionary algorithms moeas it was observed that similar to single objective optimization problems there were significant advances toward the pareto optimal front in the early phase of evolution while relatively smaller improvements were obtained as the population matured this convergence curve was used to terminate the moea search to obtain a good trade off between the computational cost and the quality of the solutions two analytical and two crashworthiness optimization problems were used to demonstrate the practical utility of the proposed metric

tight estimates for convergence of some non stationary consensus algorithms the present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete time multi agent systems in the systems considered here both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time under the hypothesis that some spanning tree structure is preserved along time and that some non zero minimal weight of the information transfer along this tree is guaranteed an estimate of the contraction rate is given the latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights

a markov chain analysis of genetic algorithms with power of 2 cardinality alphabets abstract in this paper we model the run time behavior of gas using higher cardinality representations as markov chains define the states of the markov chain and derive the transition probabilities of the corresponding transition matrix we analyze the behavior of this chain and obtain bounds on its convergence rate and bounds on the runtime complexity of the ga we further investigate the effects of using binary versus higher cardinality representation of a search space

extended double riemann hilbert problem and multiple representations of the infinite dimensional symmetric group for the stationary axisymmetric vacuum field equations the double complex function method suggested in previous papers is extended and the extended method is used to establish two mutually dual extended double hauser ernst equations and related riemann hilbert problems then quadruple representations of the semidirect product of the kac moody and virasoro groups are given these show that for the stationary axisymmetric vacuum space times there are more and richer symmetry structures than previously expected the multiple forms of some well known transformations can be obtained as some special cases

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems this paper introduces the concept of distribution distance for the measurement of errors in exact and approximate methods for stochastic simulation of chemically reacting systems two types of distance are discussed the kolmogorov distance and the histogram distance the self distance an important property of monte carlo methods that quantifies the accuracy limitation at a given resolution for a given number of realizations is defined and studied estimation formulas are established for the histogram and the kolmogorov self distance these formulas do not depend on the distribution of the samples and thus show a property of the monte carlo method itself numerical results demonstrate that the formulas are very accurate application of these results to two problems of current interest in the simulation of biochemical systems is discussed

convergence criteria for genetic algorithms in this paper we discuss convergence properties for genetic algorithms by looking at the effect of mutation on convergence we show that by running the genetic algorithm for a sufficiently long time we can guarantee convergence to a global optimum with any specified level of confidence we obtain an upper bound for the number of iterations necessary to ensure this which improves previous results our upper bound decreases as the population size increases we produce examples to show that in some cases this upper bound is asymptotically optimal for large population sizes the final section discusses implications of these results for optimal coding of genetic algorithms

new infinite dimensional quadruple symmetry groups for the general symplectic gravity model the symmetry structures of the general symplectic gravity model are further studied by using the so called extended double ed complex function method the usual riemann hilbert rh problem is extended to an ed complex formulation for any fixed non negative integer n two pairs of ed rh transformations are constructed and they are verified to give infinite dimensional quadruple symmetry groups of the general symplectic gravity model each of these symmetry groups has the structure of semidirect product of kac moody group sp 2 n 1 r and virasoro group moreover the infinitesimal forms of these rh transformations are calculated out and they are found to give exactly the same results as previous these demonstrate that the two pairs of ed rh transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper the finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generatio

the convergence of bird flocking we bound the time it takes for a group of birds to stabilize in a standard flocking model each bird averages its velocity with its neighbors lying within a fixed radius we resolve the worst case complexity of this natural algorithm by providing asymptotically tight bounds on the time to equilibrium we reduce the problem to two distinct questions in computational geometry and circuit complexity

spectral gaps and discrete magnetic laplacians abstract the aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete laplacian on periodic graphs for proving this we analyse the discrete magnetic laplacian dml on the finite quotient and interpret the vector potential as a floquet parameter we develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues written in ascending order and counting multiplicities specify the bracketing intervals where the spectrum of the laplacian is localised we prove higuchi shirai s conjecture for z periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence of spectral gaps

on the cover time and mixing time of random geometric graphs the cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated recently with the advent of ad hoc and sensor networks an interesting class of random graphs namely random geometric graphs has gained new relevance and its properties have been the subject of much study a random geometric graph g n r is obtained by placing n points uniformly at random on the unit square and connecting two points iff their euclidean distance is at most r the phase transition behavior with respect to the radius r of such graphs has been of special interest we show that there exists a critical radius r o p t such that for any r r o p tg n r has optimal cover time of q nlogn with high probability and importantly r o p email protected r c o n where r c o n denotes the critical radius guaranteeing asymptotic connectivity moreover since a disconnected graph has infinite cover time there is a phase transition and the corresponding threshold width is o r c o n on the other hand the radius required for rapid mixing r r a p i email protected r c o n and in particular r r a p i email protected 1 poly logn we are able to draw our results by giving a tight bound on the electrical resistance and conductance of g n r via certain constructed flows

convergence rates for markov chains this is an expository paper that presents various ideas related to nonasymptotic rates of convergence for markov chains such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science they also have applications to analysis of card shuffling and other areas in this paper we attempt to describe various mathematical techniques that have been used to bound such rates of convergence in particular we describe eigenvalue analysis random walks on groups coupling and minorization conditions connections are made to modern areas of research wherever possible elements of linear algebra probability theory group theory and measure theory are used but efforts are made to keep the presentation elementary and accessible

convergence speed of unsteady distributed consensus decay estimate along the settling spanning trees results for estimating the convergence rate of nonstationary distributed consensus algorithms are provided on the basis of qualitative mainly topological as well as basic quantitative information lower bounds on the matrix entries the results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples the main idea is to follow propagation of information along certain spanning trees which arise in the communication graph

laplacians on periodic graphs with guides abstract we consider laplace operators on periodic discrete graphs perturbed by guides i e graphs which are periodic in some directions and finite in other ones the spectrum of the laplacian on the unperturbed graph is a union of a finite number of non degenerate bands and eigenvalues of infinite multiplicity we show that the spectrum of the perturbed laplacian consists of the unperturbed one plus the additional so called guided spectrum which is a union of a finite number of bands we estimate the position of the guided bands and their length in terms of geometric parameters of the graph we also determine the asymptotics of the guided bands for guides with large multiplicity of edges moreover we show that the possible number of guided bands their length and position can be rather arbitrary for some specific periodic graphs with guides

accelerated dual descent for network flow optimization we present a fast distributed solution to the convex network flow optimization problem our approach uses a family of dual descent algorithms that approximate the newton direction to achieve faster convergence rates than existing distributed methods the approximate newton directions are obtained through matrix splitting techniques and sparse taylor approximations of the inverse hessian we couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute we show that similarly to conventional newton methods the proposed algorithm exhibits super linear convergence within a neighborhood of the optimal value numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods a connection with recent developments that use consensus to compute approximate newton directions is also presented

on synchronous robotic networks part ii time complexity of rendezvous and deployment algorithms this paper analyzes a number of basic coordination algorithms running on synchronous robotic networks we provide upper and lower bounds on the time complexity of the move toward average and circumcenter laws both achieving rendezvous and of the centroid law achieving deployment over a region of interest the results are derived via novel analysis methods including a set of results on the convergence rates of linear dynamical systems defined by tridiagonal toeplitz and circulant matrices

essential spectrum of the discrete laplacian on a perturbed periodic graph abstract we address the laplacian on a perturbed periodic graph which might not be a periodic graph we give a criterion for the essential spectrum of the laplacian on the perturbed graph to include that on the unperturbed graph this criterion is applicable to a wide class of graphs obtained by a non compact perturbation such as adding or removing infinitely many vertices and edges using this criterion we demonstrate how to determine the spectra of cone like graphs the upper half plane and graphs obtained from z 2 by randomly adding vertices

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

new stopping criterion for genetic algorithms abstract genetic algorithms have been successfully applied in a wide variety of problems although widely used there are few theoretical guidelines for determining when to terminate the search one result by aytug and koehler provides a loose bound on the number of ga generations needed to see all populations and hence an optimal solution with a specified probability in this paper we derive a tighter bound this new bound is on the number of iterations required to achieve a level of confidence to guarantee that a genetic algorithm has seen all strings and hence an optimal solution

convergence speed in distributed consensus and averaging we study the convergence speed of distributed iterative algorithms for the consensus and averaging problems with emphasis on the latter we first consider the case of a fixed communication topology we show that a simple adaptation of a consensus algorithm leads to an averaging algorithm we prove lower bounds on the worst case convergence time for various classes of linear time invariant distributed consensus methods and provide an algorithm that essentially matches those lower bounds we then consider the case of a time varying topology and provide a polynomial time averaging algorithm

genetic algorithm for constrained global optimization in continuous variables we present a stochastic global optimization algorithm referred to as a genetic algorithm ga for solving constrained optimization problems over a compact search domain it is a real coded ga that converges in probability to the optimal solution the constraints are treated through a repair operator a specific repair operator is included for linear inequality constraints

degree fluctuations and the convergence time of consensus algorithms we consider a consensus algorithm in which every node in a sequence of undirected b connected graphs assigns equal weight to each of its neighbors under the assumption that the degree of each node is fixed except for times when the node has no connections to other nodes we show that consensus is achieved within a given accuracy e on n nodes in time b 4n3 bln 2n e because there is a direct relation between consensus algorithms in time varying environments and in homogeneous random walks our result also translates into a general statement on such random walks moreover we give a simple proof of a result of cao spielman and morse that the worst case convergence time becomes exponentially large in the number of nodes n under slight relaxation of the degree constancy assumption

degree of population diversity a perspective on premature convergence in genetic algorithms and its markov chain analysis in this paper a concept of degree of population diversity is introduced to quantitatively characterize and theoretically analyze the problem of premature convergence in genetic algorithms gas within the framework of markov chain under the assumption that the mutation probability is zero the search ability of ga is discussed it is proved that the degree of population diversity converges to zero with probability one so that the search ability of a ga decreases and premature convergence occurs moreover an explicit formula for the conditional probability of allele loss at a certain bit position is established to show the relationships between premature convergence and the ga parameters such as population size mutation probability and some population statistics the formula also partly answers the questions of to where a ga most likely converges the theoretical results are all supported by the simulation experiments

a general steady state distribution based stopping criteria for finite length genetic algorithms we propose two general stopping criteria for finite length simple genetic algorithms based on steady state distributions and empirically investigate the impact of mutation rate string length crossover rate and population size on their convergence our first stopping criterion is based on the second largest eigenvalue of the genetic algorithm transition matrix and the second stopping criterion is based on minorization conditions

the effect of multiple optima on the simple ga run time complexity genetic algorithms are stochastic search algorithms that have been applied to optimization problems in this paper we analyze the run time complexity of a genetic algorithm when we are interested in one of a set of distinguished solutions one such case occurs when multiple optima exist we define the worst case scenario and derive a probabilistic worst case bound on the number of iterations required to find one of these multiple solutions of interest

consensus algorithms with state dependent weights we provide an analysis of a distributed consensus type algorithm with weights dependent only on the received data our approach does not require any global knowledge of the network which distinguishes our approach from most other contributions in general the weights may be further modified by any function this in turn may model various network behaviors e g link failures and time varying networks we provide general convergence conditions prove several important properties derive steady states for such locally derived weights as well as analyze the convergence rate of the algorithm applications of such weight model include distributed algorithms which do not require exact average and have no global information about the network topology

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples

convergence analysis of canonical genetic algorithms this paper analyzes the convergence properties of the canonical genetic algorithm cga with mutation crossover and proportional reproduction applied to static optimization problems it is proved by means of homogeneous finite markov chain analysis that a cga will never converge to the global optimum regardless of the initialization crossover operator and objective function but variants of cga s that always maintain the best solution in the population either before or after selection are shown to converge to the global optimum due to the irreducibility property of the underlying original nonconvergent cga these results are discussed with respect to the schema theorem