on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

matrices coupled in a chain ii spacing functions for the eigenvalues of p complex hermitian matrices coupled in a chain we give a method of calculating the spacing functions this is a generalization of the one matrix case which has been known for a long time

on some results of analysis for fuzzy metric spaces abstract a necessary and sufficient condition for a fuzzy metric space to be complete is given we prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable uniform limit theorem is generalized to fuzzy metric spaces

s topologies and bounded convergences abstract recently lechicki levi and spakowski have studied set convergence of the attouch wets type expressed in terms of truncations by members of a prescribed family of sets when this family is either the family of metrically bounded sets or the family of all nonempty subsets it is well known that convergence means uniform convergence of distance functions on members of the family in this paper we give necessary and sufficient conditions for coincidence to occur when the family is a general bornology of subsets

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

bulk asymptotics of skew orthogonal polynomials for quartic double well potential and universality in the matrix model we derive bulk asymptotics of skew orthogonal polynomials m 1 4 defined with respect to the weight exp 2nv x v x gx4 4 tx2 2 g 0 and t 0 such that m n cr where cr is the critical value that separates skew orthogonal polynomials with two cuts from those with one cut simultaneously we derive asymptotics for the recursive coefficients of skew orthogonal polynomials the proof is based on obtaining a finite term recursion relation between skew orthogonal polynomials and orthogonal polynomials and using asymptotic results of orthogonal polynomials derived by bleher and its semiclassical asymptotics of orthogonal polynomials riemann hilbert problems and universality in the matrix model ann math 150 185 1999 finally we apply these asymptotic results of skew orthogonal polynomials and their recursion coefficients in the generalized christoffel darboux formula ghosh s generalized christoffel darboux formula for skew orthogona

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

coefficient assignability and a block decomposition for systems over rings abstract we characterize the class of commutative rings r such that for any linear system a b with coefficients in r one can extract the reachable part of the system in a way similar to the classical kalman controllability decomposition for systems over fields the notion of strong ca ring is introduced as the class of commutative rings over which any system verifies a strong form of the coefficient assignability property it is shown that the class of strong ca rings lies strictly between the classes of rings with strong versions of the known pole assignability pa and feedback cyclization fc properties defined only for reachable systems we prove that for ucu rings for example polynomials with coefficients in a field the usual pa ca and fc properties are equivalent to the corresponding strong forms of these properties in particular c y is a strong ca ring

scalar convergence of convex sets abstract we study the scalar convergence of sequences of convex sets defined by lim sup a n a for all in dual space new properties are given relationship between scalar convergence and other known convergences is examined two natural distinct uniformities on nonvoid closed convex sets define the scalar convergence the associated topology is the weakest such that a d a h is continuous for each hyperplane h

pole assignability in polynomial rings power series rings and prufer domains the topics treated in this paper have their origins in the area of algebraic systems theory however the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text still it is appropriate to give a brief paragraph of motivation if a physical system is governed by a pair f g of matrices then the stability of the system can be determined by examining the eigenvalues of the matrix f if the system is unstable a feedback matrix k can sometimes be employed in such a way that the eigenvalues of the matrix f gk measure the stability of the modified system f gk g in this manner an unstable system can be rendered stable the pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices k the paper is divided into four sections section 1 is given over almost entirely to defining the properties in which we shall be interested it concludes with a theorem about residuating and lifting the properties section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation section 3 is concerned with feedback cyclization a strong form of pole assignability section 4 is concerned with pole assignability over priifer domains a complete elaboration of our results must await the introduction of the necessary terminology for now we mention the following results in somewhat vague language if r is a zero dimensional ring then the pole assignability problem is solvable in r x and in r xl from a ringtheoretic standpoint almost any class of commutative rings contains members for which the feedback cyclization problem is solvable on the other hand if r is a ring with 1 in its stable range then the feedback cyclization problem is solvable in r if and only if a certain nice matricial property

zeros of some bi orthogonal polynomials ercolani and mclaughlin have recently shown that the zeros of the bi orthogonal polynomials with the weight w x y exp v1 x v2 y 2cxy 2 relevant to a model of two coupled hermitian matrices are real and simple we show that their argument applies to the more general case of the weight w1 w2 wj x y a convolution of several weights of the same form this general case is relevant to a model of several hermitian matrices coupled in a chain their argument also works for more general weights such as w x y e x y x y 0 x y and for a convolution of several such weights

canonical forms for single input linear systems in this paper we obtain a canonical form for single input n dimensional linear systems over z and in general over a bezout domain we also give an algorithm to obtain that canonical form

weak topologies for the closed subsets of a metrizable space the purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space it can be shown that all of the standard hyperspace topologies including the hausdorff metric topology the vietoris topology the attouch wets topology the fell topology the locally finite topology and the topology of mosco convergence arise as weak topologies generated by families of geometric functionals defined on closed sets a key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of hausdorff excess functionals

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

set convergences an attempt of classification we endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics such structure is completely deter mined by a class which is a family of closed sets and a type which is a semi metric two types are sufficient to define and classify almost all convergences known till now these two types offer the possibility of defining other set con vergences

quasi uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions abstract we introduce several quasi uniformities on the hyperspace of a topological space in order to study convergence of nets of semicontinuous multifunctions the interplay between conjugate quasi uniformities yields results on both the topological structure of hyperspaces and the convergence of semicontinuous multifunctions

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

convergence of dynamic programming models weak conditions are presented for approximating dynamic programming models for a sequence of these models continuous convergence of the sequence of associated optimal value functions is obtained under the condition that state and action space converge in the sense of kuratowski and that the mappings of admissible actions as well as the transition law the discount factors and the reward functions converge continuously further a relation for the associated sets of optimal actions is given the analysis is based on results about convergence preserving properties of supremum value functions and integrals the approximation results are extended to so called upper semi continuous convergent sequences and are related to discretization procedures by using projections

a new approach for fuzzy topology ii abstract we point out a new approach for fuzzy topology with fuzzy logic and discuss the neighborhood structure of a point and the convergence of nets and filters in this new framework

the effective calculation of all assignable polynomials to a single input system over a bezout domain the set of all feedback assignable polynomials to a given single input system over a bezout domain is obtained the general case is reduced to the case of weakly reachable systems for these systems a canonical form for the algebraic equivalence is obtained algorithms and examples over a euclidean domain and in particular over the ring z of rational integers are also given the problem of deciding and feedback assignment of a given polynomial to a given non reachable n dimensional linear system is solved effectively up to a cost of o n4 arithmetic operations

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

l valued bornologies on powersets in m abel and a ostak 2011 1 the concept of an l fuzzy bornology was introduced actually an l fuzzy bornology on a set x is a certain ideal in the family lx of l fuzzy subsets of a set x here we propose an alternative approach to fuzzification of the concept of bornology we define an l valued bornology on a set x as an l fuzzy subset b of the powerset 2x satisfying l valued analogues of the axioms of a bornology basic properties of l valued bornological spaces are studied our special interest concerns l valued bornologies induced by fuzzy metrics and relative compactness type l valued bornologies in changgoguen l topological spaces

strong uniform continuity abstract let b be an ideal of subsets of a metric space x d this paper considers a strengthening of the notion of uniform continuity of a function restricted to members of b which reduces to ordinary continuity when b consists of the finite subsets of x and agrees with uniform continuity on members of b when b is either the power set of x or the family of compact subsets of x the paper also presents new function space topologies that are well suited to this strengthening as a consequence of the general theory we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

convergence of partial maps abstract given metric spaces x d and y a partial map between x and y is a pair d u where d is a closed subset of x and u d y is a function we introduce a general convergence notion for nets of such partial functions while our initial description is variational in nature we show that this description amounts to bornological convergence of the associated net of graphs as defined by lechicki levi and spakowski 26 with respect to a natural bornology on x y and which places the work on continuous partial functions of brandi ceppitelli and hola 12 13 20 21 in a general framework

strong feedback cyclization for systems over rings abstract for nonnecessarily reachable systems over a commutative ring r we define a strong form of feedback cyclization fc with this natural generalization of the fc property we obtain a feedback reduced form for systems over strong fc rings i e rings for which every system verifies the fc property in the particular case of reachable single input systems this gives the usual control canonical form of bumby et al remarks on the pole shifting problem over rings j pure appl algebra 20 1981 113 127 also it is proved that a commutative ring with 1 in its stable range has the strong fc property if and only if it has the ucu property which is the natural parallel form of the result given by brewer et al pole assignability in polynomial rings power series rings and prufer domains j algebra 106 1987 265 286 for the reachable case many classes of rings which were known to be fc rings are in fact strong fc rings but there are fc rings which are not strong fc rings

weak topology and browder kirk s theorem on hyperspace abstract let k be a weakly compact convex subset of a banach space x with normal structure browder kirk s theorem states that every non expansive mapping t which maps k into k has a fixed point in k suppose now that wcc x is the collection of all non empty weakly compact convex subsets of x we shall define a certain weak topology t w on wcc x and have the above mentioned result extended to the hyperspace wcc x t w

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

pole shifting for linear systems over commutative rings abstract this paper deals with the pole shifting problem for non necessarily reachable linear systems the notion of ps ring is introduced in the same way as the notion of pole assignable ring is given for reachable systems we prove that a bcs ring is a ps ring and that over a prufer domain these properties are equivalent finally we study when the converse of the pole shifting theorem is verified

wijsman convergence of convex sets under renorming let x p be a metric space a natural way to define convergence for a sequence of nonempty closed subsets a of x to a nonempty closed subset a is to insist that we have pointwise convergence of the associated sequence of distance functionals for the sets a to the distance functional for the set a for each x e x c x a lim d x a convergence in this sense is just convergence with respect to the weak topology on the nonempty closed subsets cl x of x determined by the family of distance functionals d x e x e x the so called wijsman topology z l 8 determined by the metric p here of course each functional of the form d x 0 is a function of a closed set variable unfortunately different metrics may give rise to different wijsman topologies even if the metrics are uniformly equivalent see e g 4 section 41 it turns out 6 theorem 3 11 that if we vary our metrics over the class c x of metrics that give rise to the topology of x as well as varying x over x the resulting family

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

arzela s theorem and strong uniform convergence on bornologies abstract in 1883 arzela 1983 1984 2 gave a necessary and sufficient condition via quasi uniform convergence for the pointwise limit of a sequence of real valued continuous functions on a compact interval to be continuous arzela s work paved the way for several outstanding papers a milestone was the p s alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space not necessarily compact to a metric space in 2009 in the realm of metric spaces beer and levi 2009 10 found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets we offer a direct proof of the equivalence of arzela alexandroff and beer levi conditions the proof reveals the internal gear of these important convergences and sheds more light on the problem we also study the main properties of the topology of strong uniform convergence of functions on bornologies initiated in beer and levi 2009 10

character expansion method for the first order asymptotics of a matrix integral the estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics geometry and free probability questions on a rigorous ground only integrals of one matrix or of several matrices with simple quadratic interaction called ab interaction could be evaluated so far see e g 19 17 or 9 in this article we follow an idea widely developed in the physics literature which is based on character expansion to study more complex interaction in this context we derive a large deviation principle for the empirical measure of young tableaux we then use it to study a matrix model defined in the spirit of the dually weighted graph model introduced in 13 but with a cutoff function such that the matrix integral and its character expansion converge we prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit

pointwise convergence of partial functions the gerlits nagy problem abstract for a set x r let b x r x denote the space of borel real valued functions on x with the topology inherited from the tychonoff product r x assume that for each countable a b x each f in the closure of a is in the closure of a under pointwise limits of sequences of partial functions we show that in this case b x is countably frechet urysohn that is each point in the closure of a countable set is a limit of a sequence of elements of that set this solves a problem of arnold miller the continuous version of this problem is equivalent to a notorious open problem of gerlits and nagy answering a question of salvador hernandez we show that the same result holds for the space of all baire class 1 functions on x we conjecture that in the general context the answer to the continuous version of this problem is negative but we identify a nontrivial context where the problem has a positive solution the proofs establish new local to global correspondences and use methods of infinite combinatorial topology including a new fusion result of francis jordan

nature inspired framework for measuring visual image resemblance a near rough set approach the problem considered in this paper is how to determine the degree of nearness between complex visual objects the proposed solution to this problem stems from a natural computing approach to solving the visual acuity problem in terms of a granular representation of visual information that is quantifiable as well as understandable for humans this is accomplished via a near rough set framework in the approximation of a pair of disjoint sets and measurement of distances between sets using various fuzzy pseudometrics pseudometrics in general and fuzzy pseudometrics in particular are useful in measuring the distance between pairs of objects such as sets such distances are indicators of the nearness of resemblance between visual objects these observations lead to a number of practical applications such as object recognition and object retrieval in digital image analysis one such application is reported in this article the contribution of this article is threefold introduction of a nature inspired framework for measurement of visual object resemblance four different incarnations of the standard fuzzy metric and application of fuzzy metrics in content based image retrieval experiments

phase structure of the three matrix chain model coupled to the magnetic field the author investigates the critical properties of the three matrix chain model corresponding to the blume emery griffiths model in a homogeneous magnetic field on a fluctuating lattice the third order critical line and the tricritical point disappear after the magnetic field is switched on it is however found that another critical line exists as a result of the naive perturbative expansion and the string susceptibility exponent is 1 3 on the line it is also shown that the critical exponent of the magnetic moment beta is 1 6 which coincides with the kpz solution

cyclizable matrix pairs over c x and a conjecture on toplitz pencils abstract the 22 year old conjecture is addressed which claims that any reachable matrix pair a b from c x n n m is cyclizable i e allows for f u over c x s t a bf k bu k 0 n 1 is a basis for c y n it is shown that for a whole class of pairs the correctness of the conjecture is a consequence of the correctness of a conjecture on certain toplitz pencils an algebraic computable test for the validity of the latter is given based on these results the validity of the conjecture on cyclizability can be extended up to dimension 5

pole assignment for systems over rings a very simple proof of the pole assignment theorem for systems over a principal ideal domain and other rings is given furthermore an algorithm is presented extensions are also indicated

a method of integration over matrix variables ii the integral over twon n hermitan matricesz g c dadbexp tr a2 b2 2cab g n a4 b4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory