moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

block orthogonal polynomials i definitions and properties constrained orthogonal polynomials have been recently introduced in the study of the hohenberg kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density more generally we define block orthogonal bo polynomials which are orthogonal with respect to a first euclidean inner product to a given dimensional subspace e i of polynomials associated with the constraints in addition they are mutually orthogonal with respect to a second euclidean inner product we recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an euclidean vector space endowed with distinct inner products an explicit two step gram schmidt orthogonalization g so process to determine these bases is given by definition the standard block orthogonal sbo polynomials are associated with a choice of e i equal to the subspace of polynomials of degree less than i we investigate their properties emphasizing similarities to and differences from the standard orthogonal polynomials applications to classical orthogonal polynomials will be given in forthcoming papers

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

the operator moment problem vector continued fractions and an explicit form of the favard theorem for vector orthogonal polynomials abstract we study the operator moment problem for a p 2 diagonal nonsymmetric matrix operator and give an explicit form of the favard theorem for vector orthogonal polynomials as a main tool we use the development of a system of resolvent functions of the operator in a vector continued fraction by a modified jacobi perron algorithm

partial quasi metrics in this article we introduce and investigate the concept of a partial quasi metric and some of its applications we show that many important constructions studied in matthews s theory of partial metrics can still be used successfully in this more general setting in particular we consider the bicompletion of the quasi metric space that is associated with a partial quasi metric space and study its applications in groups and bck algebras

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

some corollaries of the correspondence between partial metrics and multivalued equalities multivalued equalities originally arose in the context of the theory of sheaves while partial metrics originally arose in the context of the theory of domains for denotational semantics it turns out that multivalued equalities and partial metrics are closely related one can either consider them equivalent up to the choice of dual notation or one can formally capture the duality between logical and metric viewpoints by explicitly requiring logical values and distances to be represented by dual structures not only does this allow the transfer of the ideas and results between these two fields but the most interesting situations are those of interplay between logical and metric considerations the computational understanding of partial metrics as upper bounds and of multivalued equalities as lower bounds is discussed together with the lower bound counterparts to partial metrics and the corresponding upper bound counterparts to multivalued equalities it is shown that separated pre sheaves of sets and functions over complete heyting algebras can be understood as the pre sheaves of ultrametrics valued in the corresponding brouwerian algebras and non expansive maps of those ultrametrics it is proposed that the natural logical counterparts for partial metrics valued in non negative reals are multivalued equalities valued in the quantale of non positive reals the issue of canonical partial metrics on higher order and reflexive scott domains is revisited the intuition behind strong triangularity vickers form the computational version of strong triangularity and weighted quasi metrics are discussed in the context of quantaloid enrichment

a representation theorem for fuzzy pseudometrics in this paper we show that there exists a one to one correspondence between a certain class of fuzzy pseudometrics in the sense of kramosil and michalek and 0 1 indexed families of ordinary pseudometrics satisfying a property of lower semicontinuity the aforementioned bijection is proved to be independent of the t norm and it provides a representation theorem for a large class of fuzzy pseudometric spaces further the relations between the uniformities and topologies both generated by the fuzzy pseudometric and by the corresponding family of ordinary pseudometrics are also investigated

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

constrained orthogonal polynomials we define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average these are of interest for example for the study of the hohenberg kohn functional for electronic or nucleonic densities and for the study of density fluctuations in centrifuges we give explicit properties of such polynomial sets generalizing laguerre and legendre polynomials the nature of the dimension 1 subspace completing such sets is described a numerical example illustrates the use of such polynomials

lattice valued f transforms and similarity relations abstract the relationships between lattice valued f transforms and similarity relations are investigated several similarity relations are naturally associated with f transforms three types of such similarity relations related to lattice valued f transforms are investigated namely similarity relationships that are related to spaces with fuzzy partitions as ground structures of the f transform relationships between extensional hulls with respect to similarity relations and f transforms and relationships between similarities of pairs of functions on the one hand and similarities of the f transforms of these functions on the other some examples of possible applications are also presented

convex optimization and the epi distance topology let 17 x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology t of uniform convergence of distance functions on bounded sets a function f in 17 x is called well posed provided it has a unique minimizer and each minimizing sequence converges to this minimizer we show that well posedness of f e r x is the minimal condition that guarantees strong convergence of approximate minima of t approximating functions to the minimum of f moreover we show that most functions in 17 x taw are well posed and that this fails if 17 x is topologized by the weaker topology of mosco convergence whenever x is infinite dimensional applications to metric projections are also given including a fundamental characterization of approximative compactness

an extensional treatment of lazy data flow deadlock in an extensional treatment of dataflow deadlock wa81 wadge introduced an elegant non operational test for proving that many of kahn s data flow message passing networks ka74 must be free of deadlock a test that should extend to a much wider context in the study of program correctness such a context has now been provided with the introduction of partial metric spaces ma92 these spaces can be used to describe semantic domains such as those used in lazy data flow languages w a85 this paper develops wadge s ideas on establishing an extensional theory of program correctness by using partial metric spaces to give a non operational treatment of lazy data flow deadlock

continuity of the fenchel correspondence and continuity of polarities abstract the polar cone of the limit inferior of a sequence of cones of a banach space is shown to be the cesari s limit superior of the sequence of polar cones in the bounded weak topology in a similar way the lower semicontinuity of a not necessarily countable family of closed convex cones is characterized in terms of castaing s notion of pseudo upper semicontinuity more general results are given within the framework of convergence spaces and the pseudo upper semicontinuity of a closed valued multifunction is characterized as closedness with respect to a new convergence applications to the young fenchel correspondence are pointed out

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

continuity spaces reconciling domains and metric spaces abstract we use continuity spaces a common refinement of posets and metric spaces to develop a general theory of semantic domains which includes metric spaces and domains of cpo s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming our proposal for a general notion of semantic domain is a symmetrically compact continuity space where is a value quantale we show that the category of symmetrically compact continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures in a natural way the traditional examples in general the category will not be cartesian closed however powerdomains do exist and by adapting a construction of suenderhauf to continuity spaces we show that they define a computational monad in the sense of moggi

multiple orthogonal polynomials for classical weights a new set of special functions which has a wide range of applications from number theory to integrability of nonlinear dynamical systems is described we study multiple orthogonal polynomials with respect to p 1 weights satisfying pearson s equation in particular we give a classification of multiple orthogonal polynomials with respect to classical weights which is based on properties of the corresponding rodrigues operators we show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p 1 we also obtain explicit formulas and recurrence relations for these polynomials

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

calculation of some determinants using the s shifted factorial several determinants with gamma functions as elements are evaluated these kinds of determinants are encountered for example in the computation of the probability density of the determinant of random matrices the s shifted factorial is defined as a generalization for non negative integers of the power function the rising factorial or pochammer s symbol and the falling factorial it is a special case of a polynomial sequence of the binomial type studied in combinatorics theory in terms of the gamma function an extension is defined for negative integers and even complex values properties mainly composition laws and binomial formulae are given they are used to evaluate families of generalized vandermonde determinants with s shifted factorials as elements instead of power functions

an enriched category approach to many valued topology in this paper we discuss a basic question in fuzzy set theory what is a many valued topology on a set x an answer is presented by interpreting the axioms two logical formulas for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of x as preparation a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory

axiomatic of lattice valued f transform abstract axioms for upper and lower fuzzy transformation systems are introduced and it is proved that a transformation function satisfies axioms for upper or lower respectively transformation systems if and only if it is an upper or lower respectively f transform categories of upper lower respectively transformation systems are introduced and it is proved that both categories are isomorphic to the category of spaces with fuzzy partitions we prove that lower and upper f transforms preserve extensional fuzzy sets associated with spaces with fuzzy partitions

the order theoretic duality and relations between partial metrics and local equalities for a gl monoid l hohle introduced l valued sets to formalize the mathematical theory of identity and existence in monoidal logic for the purpose of analyzing deadlock situations in data flow networks the notion of partial metric space was proposed by matthews later on a lattice theoretic generalization of pseudo partial metric spaces extending the range of pseudo partial metrics to a value lattice v were studied under the name of v pseudo pmetric spaces referring to a gl monoid l and a dual gl monoid v we show that l valued sets are order theoretically dual to v pseudopmetric spaces and apply this duality to the representation of v pseudopmetric spaces and the determination of their various categorical properties in addition to this the present paper provides not only some new results about the representation of l valued sets and their categories but also some non trivial categorical connections between l valued sets and v pseudopmetric spaces

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

block orthogonal polynomials ii hermite and laguerre standard block orthogonal polynomials the standard block orthogonal sbo polynomials pi n x 0 i n are real polynomials of degree n which are orthogonal with respect to a first euclidean scalar product to polynomials of degree less than i in addition they are mutually orthogonal with respect to a second euclidean scalar product applying the general results obtained in a previous paper we determine and investigate these polynomials when the first scalar product corresponds to hermite resp laguerre polynomials these new sets of polynomials which we call hermite resp laguerre sbo polynomials provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an euclidean orthogonality relation

the epi distance topology continuity and stability results with applications to convex optimization problems let i x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets we show subject to some standard constraint qualifications that the operations of addition and restriction are continuous these results are applied to convex well posed optimization problems as well as to convergence of approximated solutions in infinite dimensional convex programming to linear functionals exposing convex sets and to metric projections also for any given function in i x we obtain results regarding the convergence of its inf convolution with smoothing kernels

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory