a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

numerical solutions for systems of fractional differential equations by the decomposition method in this paper we use adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi term fractional differential equation by reducing it to a system of fractional equations each of order at most unity we begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method moreover we show how the method can be applied to a general linear multi term equation and solve several applied problems

lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay abstract in this paper lie symmetry method is employed to solve the fractional neutral ordinary differential equations the symmetry analysis was carried out and infinite dimension symmetry algebras are obtained an application of lie symmetry techniques is used to obtain the exact solutions

on the one leg θ methods for solving nonlinear neutral functional differential equations abstract this paper discusses the one leg methods for the numerical solution of nonlinear neutral functional differential equations with pantograph delay y t f t y t y pt y pt t 0 0 p 1 y 0 y in recent years the numerical techniques for the solution to this kind of equations have been developed by numerous authors who have mainly considered the linear system instead in the present paper we study the nonlinear case the first step integration of above systems on 0 t 0 by one leg methods is analyzed it is proved that the numerical solution preserves the contractivity property for some t0 on the quasi geometric mesh the numerical stability of one leg methods for above systems is investigated some sufficient conditions for global stability and asymptotic stability are established two numerical examples are also included

numerical solution of fractional partial differential equations with variable coefficients using generalized fractional order legendre functions abstract in this paper a general formulation for the generalized fractional order legendre functions gflfs is constructed to obtain the numerical solution of fractional partial differential equations with variable coefficients the special feature of the proposed approach is that we define generalized fractional order legendre functions over 0 h based on fractional order legendre functions we use these functions to approximate the unknown function on the interval 0 h 0 l in addition the gflfs fractional differential operational and product matrices are driven these matrices combine with tau method to transform the problem to solve systems of linear algebraic equations by solving the linear algebraic equations we can obtain the numerical solution the error analysis shows that the algorithm is convergent the method is tested on examples the results show that the gflfs yields better results

quantum query complexity of entropy estimation estimation of shannon and renyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing in this paper we give the first quantum algorithms for estimating alpha renyi entropies shannon entropy being 1 renyi entropy in particular we demonstrate a quadratic quantum speedup for shannon entropy estimation and a generic quantum speedup for alpha renyi entropy estimation for all alpha geq 0 values including tight bounds for the shannon entropy the hartley entropy alpha 0 and the collision entropy alpha 2 we also provide quantum upper bounds for estimating min entropy alpha infty as well as the kullback leibler divergence we complement our results with quantum lower bounds on alpha renyi entropy estimation for all alpha geq 0 values our approach is inspired by the pioneering work of bravyi harrow and hassidim bhh however with many new technical ingredients 1 we improve the error dependence of the bhh framework by a fine tuned error analysis together with montanaro s approach to estimating the expected output of quantum subroutines for alpha 0 1 2 we develop a procedure similar to cooling schedules in simulated annealing for general alpha geq 0 and 3 in the cases of integer alpha geq 2 and alpha infty we reduce the entropy estimation problem to the alpha distinctness and lceil log n rceil distinctness problems respectively

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

structure of real lie algebras abstract restrictions are presented on the radical structure of an indecomposable lie algebra these restrictions are consequences of both the levi theorem and the representation theory of semisimple lie algebras as an application all real lie algebras of dimension 9 that admit a nontrivial levi decomposition are found these algebras are expressed in terms of known solvable and simple algebras of smaller dimensions and in terms of representations of simple algebras

functional separable solutions of nonlinear reaction diffusion equations with variable coefficients abstract the paper presents a number of new functional separable solutions to nonlinear reaction diffusion equations of the form c x u t a x u x x b x u x p x f u where f u is an arbitrary function it is shown that any three of the four variable coefficients a x b x c x p x of such equations can be chosen arbitrarily and the remaining coefficient can be expressed through the others examples of specific equations and their exact solutions are given the results obtained are generalized to more complex multidimensional nonlinear reaction diffusion equations with variable coefficients also some functional separable solutions to nonlinear reaction diffusion equations with delay u t u x x a x f u w w u x t where 0 is the delay time and f u w is an arbitrary function of two arguments are obtained it is important to note that the exact solutions of nonlinear pdes and delay pdes that contain arbitrary functions and therefore have sufficient generality are of the greatest practical interest for testing and evaluating the accuracy of various numerical and approximate analytical methods for solving corresponding initial boundary value problems

group analysis of the one dimensional wave equation with delay abstract in this paper we establish a lie type invariance condition for second order delay partial differential equations the determining equations are obtained using taylor s theorem for a function of several variables the symmetries of the wave equation with delay its kernel and extensions of the kernel have been found we make a complete group classification of the wave equation containing an arbitrary differentiable functional with delay for which there is no existing literature further the complete set of invariant solutions led by this classification have been found

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

estimating multiple pathways of object growth using nonlongitudinal image data this article presents an infinite bayesian mixture of monotonic regression models for analyzing multiple growth pathways of star shaped objects growing over time by using nonlongitudinal data a motivating example is the analysis of nanocrystal growth processes a radius function representation used for the outlines of star shaped objects allows us to represent an object growth i e expansion of outlines by an increasing sequence of random radius functions we propose a monotonic regression model to fit the increasing sequence to nonlongitudinal data ensuring that the fitted radius functions always monotonically increase in the sequence to model the multiple pathways of object growth we propose the dirichlet infinite location mixture of multiple monotonic regression models and use a block gibbs sampler as a numerical solver we apply the proposed model to nonlongitudinal sets of microscopic image data for inferring multiple growth pathways of nanocrystals comparing the inference result with real nan

adaptive simulated annealing a near optimal connection between sampling and counting we present a near optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set an important application of our work is to approximating the partition function z of a discrete system such as the ising model matchings or colorings of a graph the typical approach to estimating the partition function z at some desired inverse temperature is to define a sequence which we call a cooling schedule 0 e 0 for well studied problems such as estimating the partition function of the ising model or approximating the number of colorings or matchings of a graph our cooling schedule is of length o s n which implies an overall savings of o n in the running time of the approximate counting algorithm since roughly e samples are needed to estimate each ratio a similar improvement in the length of the cooling schedule was recently obtained by lovasz and vempala in the context of estimating the volume of convex bodies while our reduction is inspired by theirs the discrete analogue of their result turns out to be significantly more difficult whereas a fixed schedule suffices in their setting we prove that in the discrete setting we need an adaptive schedule that is the schedule depends on z more precisely we prove any nonadaptive cooling schedule has length at least o ln a and we present an algorithm to find an adaptive schedule of length o s ln a

accelerating simulated annealing for the permanent and combinatorial counting problems we present an improved cooling schedule for simulated annealing algorithms for combinatorial counting problems under our new schedule the rate of cooling accelerates as the temperature decreases thus fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature easy region to the low temperature difficult region we present applications of our technique to colorings and the permanent perfect matchings of bipartite graphs moreover for the permanent we improve the analysis of the markov chain underlying the simulated annealing algorithm this improved analysis combined with the faster cooling schedule results in an o n 7 log 4 n time algorithm for approximating the permanent of a 0 1 matrix

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

solving systems of fractional differential equations using differential transform method this paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method the fractional derivatives are described in the caputo sense the application of differential transform method developed for differential equations of integer order is extended to derive approximate analytical solutions of systems of fractional differential equations the solutions of our model equations are calculated in the form of convergent series with easily computable components some examples are solved as illustrations using symbolic computation the numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations the method introduces a promising tool for solving many linear and nonlinear fractional differential equations

numerical simulation for coupled systems of nonlinear fractional order integro differential equations via wavelets method abstract in this paper a new method for solving coupled systems of nonlinear fractional order integro differential equations is proposed the idea is to use bernoulli wavelets and operational matrix the main purpose of the technique is to transform the studied systems of fractional order integro differential equations into systems of algebraic equations which can be solved easily illustrative examples and comparisons with haar wavelets and legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis

numerical bifurcation analysis of immunological models with time delays in recent years a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences to analyze the models dynamics numerical methods are necessary since analytical studies can only give limited results in turn the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling which may lead to more realistic models we outline recently developed numerical methods for bifurcation analysis of ddes and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis b virus infection

chebyshev polynomials approach for numerically solving system of two dimensional fractional pdes and convergence analysis abstract in this paper an efficient numerical technique based on the chebsyhev orthogonal polynomials is established to obtain the approximate solutions of system of two dimensional fractional order pdes with initial conditions we construct the corresponding differential operational matrix of fractional order and then transform the problem into a system of linear algebra equations compared with other analytical or semi analytical methods ours can achieve better convergence accuracy only small terms are expanded moreover the proposed algorithm is simple in theoretical derivation and numerical simulation in our study the convergence analysis of the system is emphatically investigated than other numerical approaches lastly three numerical examples are applied to test the algorithm and that the obtained numerical results show that our approach is effective and robust

approximate solution of nonlinear fractional integro differential equations using fractional alternative legendre functions abstract in this paper a new set of functions called fractional alternative legendre is defined for solving nonlinear fractional integro differential equations the concept of the fractional derivative in this problem is in the caputo sense to solve the problem first we obtain the operational matrix of the riemann liouville fractional integration of these functions for the first time then this matrix and collocation method are utilized to reduce the solution of the nonlinear fractional integro differential equations to a system of algebraic equations also the convergence analysis of the proposed method is investigated finally some examples are included to demonstrate the validity and applicability of the approach

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

optimization problem for a portfolio with an illiquid asset lie group analysis abstract management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance one of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio cannot be sold until a certain finite infinite or random moment of time this approach arises a certain amount of models that are actively studied at the moment working in the merton s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid a risky and a risk free asset our goal in this paper is to carry out a complete lie group analysis of pdes describing value function and investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random moment of time with a prescribed liquidation time distribution the problem of such type leads to three dimensional nonlinear hamilton jacobi bellman hjb equations such equations are not only tedious for analytical methods but are also quite challenging form a numeric point of view to reduce the three dimensional problem to a two dimensional one or even to an ode one usually uses some substitutions yet the methods used to find such substitutions are rarely discussed by the authors we find the admitted lie algebra for a broad class of liquidation time distributions in cases of hara and log utility functions and formulate corresponding theorems for all these cases we use found lie algebras to obtain reductions of the studied equations several of similar substitutions were used in other papers before whereas others are new to our knowledge this method gives us the possibility to provide a complete set of non equivalent substitutions and reduced equations

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

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

lie symmetries of finite difference equations discretizations of the helmholtz heat and wave equations on uniform lattices are considered in various space time dimensions the symmetry properties of these finite difference equations are determined and it is found that they retain the same lie symmetry algebras as their continuum limits solutions with definite transformation properties are obtained identities and formulas for these functions are then derived using the symmetry algebra

fractional order legendre laguerre functions and their applications in fractional partial differential equations abstract in this paper we consider a new fractional function based on legendre and laguerre polynomials for solving a class of linear and nonlinear time space fractional partial differential equations with variable coefficients the concept of the fractional derivative is utilized in the caputo sense the idea of solving these problems is based on operational and pseudo operational matrices of integer and fractional order integration with collocation method we convert the problem to a system of algebraic equations by applying the operational matrices pseudo operational matrices and collocation method also we calculate the upper bound for the error of integral operational matrix of the fractional order we illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph we also describe the physical application of some examples

simulated annealing for convex optimization we apply the method known as simulated annealing to the following problem in convex optimization minimize a linear function over an arbitrary convex set where the convex set is specified only by a membership oracle using distributions from the boltzmann gibbs family leads to an algorithm that needs only o n phases for instances in rn this gives an optimization algorithm that makes o n4 5 calls to the membership oracle in the worst case compared to the previous best guarantee of o n5 the benefits of using annealing here are surprising because such problems have no local minima that are not also global minima hence we conclude that one of the advantages of simulated annealing in addition to avoiding poor local minima is that in these problems it converges faster to the minima that it finds we also give a proof that under certain general conditions the boltzmann gibbs distributions are optimal for annealing on these convex problems

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

application of legendre wavelets for solving fractional differential equations in this paper we develop a framework to obtain approximate numerical solutions to ordinary differential equations odes involving fractional order derivatives using legendre wavelet approximations the properties of legendre wavelets are first presented these properties are then utilized to reduce the fractional ordinary differential equations fodes to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

quantum speedup of monte carlo methods monte carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically one important example is the use in statistical physics of rapidly mixing markov chains to approximately compute partition functions in this work we describe a quantum algorithm which can accelerate monte carlo methods in a very general setting the algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance achieving a near quadratic speedup over the best possible classical algorithm combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions which use multiple stage markov chain monte carlo techniques the quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently

numerical solution of differential equations by using chebyshev wavelet operational matrix of integration in this paper we present operational matrix of integration omi of chebyshev wavelets basis and the product operation matrix pom of it some comparative examples are included to demonstrate the superiority of operational matrix of chebyshev wavelets to those of legendre wavelets

a deterministic polynomial time approximation scheme for counting knapsack solutions given n elements with nonnegative integer weights w 1 ldots w n and an integer capacity c we consider the counting version of the classic knapsack problem find the number of distinct subsets whose weights add up to at most the given capacity we give a deterministic algorithm that estimates the number of solutions to within relative error 1 pm varepsilon in time polynomial in n and 1 varepsilon fully polynomial approximation scheme more precisely our algorithm takes time o n 3 varepsilon 1 log n varepsilon our algorithm is based on dynamic programming previously randomized polynomial time approximation schemes were known first by morris and sinclair via markov chain monte carlo techniques and subsequently by dyer via dynamic programming and rejection sampling

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

estimating average growth trajectories in shape space using kernel smoothing in this paper we show how a dense surface point distribution model of the human face can be computed and demonstrate the usefulness of the high dimensional shape space for expressing the shape changes associated with growth and aging we show how average growth trajectories for the human face can be computed in the absence of longitudinal data by using kernel smoothing across a population a training set of three dimensional surface scans of 199 male and 201 female subjects of between 0 and 50 years of age is used to build the model

on the complete group classification of the reaction diffusion equation with a delay abstract the reaction diffusion delay differential equation u t x t u x x x t g x u x t u x t arises in many applications in the sciences group analysis is applied in the study of this equation a new definition of an equivalence lie group for delay differential equations is given as for the lie group theory of differential equations the determining equations for the equivalence and admitted lie groups are constructed the general solutions of the determining equations are obtained the complete group classification of the reaction diffusion equation with delay is presented in the manuscript and the invariant solutions of this equation are constructed

discrete hit and run for sampling points from arbitrary distributions over subsets of integer hyperrectangles we consider the problem of sampling a point from an arbitrary distribution over an arbitrary subset s of an integer hyperrectangle neither the distribution nor the support set s are assumed to be available as explicit mathematical equations but may only be defined through oracles and in particular computer programs this problem commonly occurs in black box discrete optimization as well as counting and estimation problems the generality of this setting and high dimensionality of s precludes the application of conventional random variable generation methods as a result we turn to markov chain monte carlo mcmc sampling where we execute an ergodic markov chain that converges to so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to unfortunately classical markov chains such as the nearest neighbor random walk or the coordinate direction random walk fail to converge to because they can get trapped in isolated regions of the support set to surmount this difficulty we propose discrete hit and run dhr a markov chain motivated by the hit and run algorithm known to be the most efficient method for sampling from log concave distributions over convex bodies in rn we prove that the limiting distribution of dhr is as desired thus enabling us to sample approximately from by delivering the last iterate of a sufficiently large number of iterations of dhr in addition to this asymptotic analysis we investigate finite time behavior of dhr and present a variety of examples where dhr exhibits polynomial performance

the variational iteration method for solving a neutral functional differential equation with proportional delays in this paper the variational iteration method is applied to neutral functional differential equations with proportional delays illustrative examples are given to show the efficiency of the method we also compare the performance of the method with that of a particular runge kutta method and a one leg q method

an efficient algorithm for solving volterra integro differential equations based on alpert s multi wavelets galerkin method abstract this paper is concerned with the application of the multi wavelets galerkin method to linear and nonlinear second order volterra integro differential equations using the operational matrices of integration and wavelet transform matrix we reduce the problem to a set of algebraic equations for linear type we use thresholding to reduce the computational effort we demonstrate that our solutions may be efficiently represented in a multi wavelets basis due to flexible vanishing moments property of this type of multi wavelets the l 2 convergence of the scheme for the proposed equation has been investigated a series of numerical tests is provided to demonstrate the validity and applicability of the technique

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

computational approaches to parameter estimation and model selection in immunology one of the significant challenges in biomathematics and other areas of science is to formulate meaningful mathematical models our problem is to decide on a parametrized model which is in some sense most likely to represent the information in a set of observed data in this paper we illustrate the computational implementation of an information theoretic approach associated with a maximum likelihood treatment to modelling in immunology the approach is illustrated by modelling lcmv infection using a family of models based on systems of ordinary differential and delay differential equations the models which use parameters that have a scientific interpretation are chosen to fit data arising from experimental studies of virus cytotoxic t lymphocyte kinetics the parametrized models that result are arranged in a hierarchy by the computation of akaike indices the practical illustration is used to convey more general insight because the mathematical equations that comprise the models are solved numerically the accuracy in the computation has a bearing on the outcome and we address this and other practical details in our discussion

on fractional calculus and fractional multipoles in electromagnetism using the concept and tools of fractional calculus we introduce a definition for fractional order multipoles of electric charge densities and we show that as far as their scalar potential distributions are concerned such fractional order multipoles effectively behave as intermediate sources bridging the gap between the cases of integer order point multipoles such as point monopoles point dipoles point quadrupoles etc this technique which involves fractional differentiation or integration of the dirac delta function provides a tool for formulating an electric source distribution whose potential functions can be obtained by using fractional differentiation or integration of potentials of integer order point multipoles of lower or higher orders as illustrative examples the cases of three dimensional point source and two dimensional line source problems in electrostatics are treated in detail and an extension to the time harmonic case is also addressed in the three dimensional electrostatic example we suggest an electric charge distribution which can be regarded as an intermediate case between cases of the electric point monopole point charge and the electric point dipole point dipole and we present its electrostatic potential which behaves as r sup 1 spl alpha p sub spl alpha cos spl theta where 0 spl alpha 1 and p sub spl alpha spl middot is the legendre function of noninteger degree spl alpha thus denoting this charge distribution as a fractional 2 sup spl alpha pole at the two limiting cases of spl alpha 0 and spl alpha 1 this fractional 2 sup spl alpha pole becomes the standard point monopole and point dipole respectively a corresponding intermediate fractional order multipole is also given for the two dimensional electrostatic case potential applications of this treatment to the image method in electrostatic problems are briefly mentioned physical insights and interpretation for such fractional order 2 sup spl alpha poles are also given

improved mixing condition on the grid for counting and sampling independent sets the hard core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics a multicast model of calls in communication networks and as a weighted independent set problem in combinatorics probability and theoretical computer science in this model each independent set i in a graph g is weighted proportionally to i for a positive real parameter for large computing the partition function namely the normalizing constant which makes the weighting a probability distribution on a finite graph on graphs of maximum degree 3 is a well known computationally challenging problem more concretely let lambda c mathbb t delta denote the critical value for the so called uniqueness threshold of the hard core model on the infinite regular tree recent breakthrough results of weitz proceedings of the 38th annual acm symposium on theory of computing stoc pp 140 149 2006 and sly proceedings of the 51st annual ieee symposium on foundations of computer science focs pp 287 296 2010 have identified lambda c mathbb t delta as a threshold where the hardness of estimating the above partition function undergoes a computational transition we focus on the well studied particular case of the square lattice mathbb z 2 and provide a new lower bound for the uniqueness threshold in particular taking it well above lambda c mathbb t 4 our technique refines and builds on the tree of self avoiding walks approach of weitz resulting in a new technical sufficient criterion of wider applicability for establishing strong spatial mixing and hence uniqueness for the hard core model our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure improving upon what can be achieved by just using the maximum degree applying our technique to mathbb z 2 we prove that strong spatial mixing holds for all 2 3882 improving upon the work of weitz that held for 27 16 1 6875 our results imply a fully polynomial deterministic approximation algorithm for estimating the partition function as well as rapid mixing of the associated glauber dynamics to sample from the hard core distribution