level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

the asymptotics of a bessel kernel determinant which arises in random matrix theory in random matrix theory the local correlations of the laguerre and jacobi unitary ensemble in the hard edge scaling limit can be described in terms of the bessel kernel b x y xyj x yj y j y xj x x2 y2 x y 0 1 in particular the so called hard edge gap probabilities p r can be expressed as the fredholm determinants of the corresponding integral operator b restricted to the finite interval 0 r using operator theoretic methods we are going to compute their asymptotics as r i e we show that p r det i b l2 0 r exp r24 r 22logr g 1 2 2 where g stands for the barnes g function in fact this asymptotic formula will be proved for all complex parameters satisfying re 1

universal parametric correlations of eigenvalues of random matrix ensembles eigenvalue correlations of random matrix ensembles as a function of an external parametric perturbation are investigated via the dyson brownian motion model in the situation where the level density has a hard edge singularity by solving a linearized hydrodynamical equation a universal dependence of the density density correlator on the external field is found as an application we obtain a formula for the variance of linear statistics with the parametric dependence exhibited as a laplace transform

a possibilistic multiple objective pricing and lot sizing model with multiple demand classes we address an inventory marketing system to determine the production lot size marketing expenditure and selling prices where a firm faces demand from two or more market segments in which the firm can set different prices considering pricing marketing and lot sizing decisions simultaneously the model maximizes the profit and return on inventory investment under multiple time varying demand classes the model is formulated as a fuzzy non linear multi objective one where some parameters are ill known and modeled by fuzzy numbers a hybrid possibilistic flexible programming approach is proposed to handle imprecise data and soft constraints concurrently after transforming the original model into an equivalent multi objective crisp model it is then converted to a classical mono objective one by a fuzzy goal programming method an efficient solution procedure using particle swarm optimization pso is also provided to solve the resulting non linear problem

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

a note on the lax pairs for painleveequations for the classical painleveequations besides the method of similarity reduction of lax pairs for integrable partial differential equations two ways are known for lax pair generation the first is based on the confluence procedure in fuchs linear ode with four regular singularities isomonodromy deformation which is governed by the sixth painleveequation the second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of odes in whose non triangular extensions give rise to the lax pairs for the painleveequations the theory of integrable integral operators suggests a new way of lax pair generation for the classical painleveequations this method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions some of the lax pairs we introduce are known others are new the question of gauge equivalence of different lax pairs for the painleveequations is considered as well

order allocation and procurement transport planning in apparel supply chain a utility based possibilistic flexible programming approach abstract the purpose of this paper is to study the integrated decisions of the physical supply channel on the global scale including the supplier selection order allocation and transportation planning under uncertainty in detail this study addresses the supply decision making problem in a multi source multi product and multi mode transportation in accordance with the price discounts to this end a multi objective mixed integer linear programming model is developed to jointly determine the best suppliers and purchasing quantities along with the number and type of transportation modes three important objective functions i e total logistics costs total late deliveries and total value of purchasing are optimized in our proposed mathematical model then a fuzzy multi choice goal programming model with the utility function is presented to solve the multi objective formulation a novel flexible possibilistic programming approach is proposed to cope with the epistemic uncertainty and the flexibility of constraints based on the expected value and chance constraint programming concept finally a numerical study in the apparel supply chain is carried out to show the model usefulness and the solution algorithm effectiveness

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

solving a multiobjective possibilistic problem through compromise programming abstract real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way it is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory here we propose a method for solving these kinds of models through a fuzzy compromise programming approach to formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced this concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their cuts the accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

asymptotics of level spacing distributions for random matrices asymptotic formulas for the probability of finding exactly n eigenvalues in an interval of length s for large s and fixed n are given for random matrices taken from the gaussian ensembles 1 2 4 these exact results are compared with the predictions of a continuum coulomb gas model due to dyson

approximation based fuzzy multi objective models with expected objectives and chance constraints application to earth rock work allocation in this paper we introduce a measure me which is more suitable for real life decision making problems under fuzzy environments then we establish a general fuzzy multi objective model with expected objectives and chance constraints ecm based on the me to handle the model and obtain solutions we propose two approximation models by dividing the fuzzy feasible region using rough set theory in particular we focus on a linear ecm and several properties and two equivalent models are presented finally as an illustration we apply the proposed models and approaches to an earth rock work allocation problem at the longfeng hydropower construction project

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

optimality conditions for fuzzy constrained programming problems abstract this paper solves optimization problems where both the objective and constraints are given by fuzzy functions in order to get it we first prove that these problems are equivalent to optimization problems where the constraints functions are non fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far this new stationary point concept is based on the gh differentiability and has many computational advantages that we describe it is well known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity and we are in that case due to the fact that the fuzzy numbers intervals space is a nonlinear one in this direction the gh derivative for fuzzy functions is a concept that is more general than hukuhara and level wise derivatives that are usually used in fuzzy optimization so far in the sense that they can be applied to a wider number of fuzzy function classes than above concepts with this new differentiability concept we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far moreover due to the fact that we do not have a linear space for fuzzy numbers the convex concepts and generalized convex fuzzy function notion are very restrictive also this implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful

multi choice goal programming with utility functions abstract goal programming gp has been and still is the most widely used technique for solving multiple criteria decision problems and multiple objective decision problems by finding a set of satisfying solutions however the major limitation of goal programming is that can only use aspiration levels with scalar value for solving multiple objective problems in order to solve this problem multi choice goal programming mcgp was proposed by chang 2007a following the idea of mcgp this study proposes a new concept of level achieving in the utility functions to replace the aspiration level with scalar value in classical gp and mcgp for multiple objective problems according to this idea it is possible to use the skill of mcgp with utility functions to solve multi objective problems the major contribution of using the utility functions of mcgp is that they can be used as measuring instruments to help decision makers make the best appropriate policy corresponding to their goals with the highest level of utility achieved in addition the above properties can improve the practical utility of mcgp in solving more real world decision management problems

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

dyson s constants in the asymptotics of the determinants of wiener hopf hankel operators with the sine kernel let k alpha pm stand for the integral operators with the sine kernels frac sin x y pi x y pm frac sin x y pi x y acting on l2 0 dyson conjectured that the asymptotics of the fredholm determinants of i k alpha pm are given by log det i k alpha pm frac alpha 2 4 mp frac alpha 2 frac log alpha 8 frac log 2 24 pm frac log 2 4 frac 3 2 zeta 1 o 1 as in this paper we are going to give a proof of these two asymptotic formulas

bayesian nonparametric estimation of the probability of discovering new species summary we consider the problem of evaluating the probability of discovering a certain number of new species in a new sample of population units conditional on the number of species recorded in a basic sample we use a bayesian nonparametric approach the different species proportions are assumed to be random and the observations from the population exchangeable we provide a bayesian estimator under quadratic loss for the probability of discovering new species which can be compared with well known frequentist estimators the results we obtain are illustrated through a numerical example and an application to a genomic dataset concerning the discovery of new genes by sequencing additional single read sequences of cdna fragments

hankel operators that commute with second order differential operators abstract suppose that is a continuous and self adjoint hankel operator on l 2 0 with kernel x y and that l f d d x a x d f d x b x f x with a 0 0 if a and b are both quadratic hyperbolic or trigonometric functions and satisfies a suitable form of gauss s hypergeometric differential equation or the confluent hypergeometric equation then l l the paper catalogues the commuting pairs and l including important cases in random matrix theory there are also results proving rapid decay of the singular numbers of hankel integral operators with kernels that are analytic and of exponential decay in the right half plane

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

integrable operators and the squares of hankel operators integrable operators arise in random matrix theory where they describe the asymptotic eigenvalue distribution of large self adjoint random matrices from the generalized unitary ensembles this paper gives sufficient conditions for an integrable operator to be the square of a hankel operator and applies the condition to the airy associated laguerre modified bessel and whittaker functions