the determinants of matrices with recursive entries abstract we prove a result concerning the evaluation of determinant of a matrix whose entries are given by certain linear inhomogeneous recurrence relations this result generalizes previous determinants evaluations due to krattenthaler neuwirth and the authors in c krattenthaler evaluations of some determinants of matrices related to the pascal triangle sem lothar combin 47 2002 19 article b47g e neuwirth treeway galton arrays and generalized pascal like determinants technical report university of vienna 2004 pp 1 16 a r moghaddamfar s m h pooya s navid salehy s nima salehy evaluating and generalizing certain determinants submitted for publication the main tool for proving our result is lu factorization of the recursive arrays introduced in c krattenthaler advanced determinant calculus a complement linear algebra appl 411 2005 68 166

a liapunov type inequality for sugeno integrals abstract the classical liapunov inequality shows an interesting upper bound for the lebesgue integral of the product of two functions this paper proposes a liapunov type inequality for sugeno integrals that is we show that h s t r s 0 1 f x s d r t s 0 1 f x t d r s s 0 1 f x r d s t holds for some constant h s t r where 0 t s r f 0 1 0 is a non increasing concave function and is the lebesgue measure on r we also present two interesting classes of functions for which the classical liapunov type inequality for sugeno integrals with h s t r 1 holds some examples are provided to illustrate the validity of the proposed inequality

notes on the stability of dynamic economic systems the lyapunov function method is the focus of this paper in proving marginal stability asymptotical or globally asymptotical stability of discrete dynamic systems we show that the slightly relaxed versions of the well known sufficient conditions are also necessary

generalization of portmanteau theorem for a sequence of interval valued pseudo probability measures abstract the main result of this paper is a generalization of portmanteau theorem for a sequence of interval valued pseudo probability measures the classical lebesgue integral from probability theory had been substituted with the pseudo integral of a real valued function with respect to an interval valued measure

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

on the automaticity of the hankel determinants of a family of automatic sequences abstract hankel determinants and automatic sequences are two classical subjects widely studied in mathematics and theoretical computer science however these two topics were considered totally independently until in 1998 when allouche peyriere wen and wen proved that all the hankel determinants of the thue morse sequence are nonzero this property allowed bugeaud to prove that the irrationality exponents of the thue morse mahler numbers are exactly 2 since then the hankel determinants of several other automatic sequences in particular the paperfolding sequence the stern sequence the period doubling sequence are studied by coons vrbik guo wu wen bugeaud fu han fokkink kraaikamp and shallit on the other hand it is known that the hankel determinants of a rational power series are ultimately zero and the hankel determinants of a quadratic power series over finite fields are ultimately periodic it is therefore natural to ask if we can obtain similar results about the hankel determinants of algebraic series in the present paper we provide a partial answer to this question by establishing the automaticity of the reduced hankel determinants modulo 2 of a family of automatic sequences as an application of our result we give upper bounds for the irrationality exponent of a family of automatic numbers

elliptic hypergeometric series on root systems abstract we derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems a n c n and d n in the special cases of classical and q series our approach leads to new elementary proofs of the corresponding identities

a generalized ordered bell polynomial abstract in this paper we consider a generalized ordered bell polynomial p n q defined by the following exponential generating function n 0 p n q n t n e t q q e t 1 using the method of exponential riordan arrays and orthogonal polynomials we give the jacobi continued fraction expansion of n 0 p n q t n then we get the q stieltjes moment property strong q log convexity 3 q log convexity and hankel determinants of p n q n 0

hankel transform of a series reversion of a certain rational function abstract starting from a sequence v n n n 0 defined by the series reversion of a certain rational function b x we use the method based on orthogonal polynomials to give closed form evaluation of the hankel transform of sequence v n n n 0 and corresponding shifted sequences that gives a proof the conjecture by barry 4 concerning the ratio equalities connecting those terms moreover in a special case we compute lu decomposition of the hankel matrix of a shifted sequence v n n n 0 which reproves the expression for the hankel transform of that sequence

riordan arrays generalized narayana triangles and series reversion abstract using elements of the group of riordan arrays we define a family of generalized narayana triangles and their associated generalized catalan numbers and study their links to series reversion in particular we use lagrange inversion techniques to determine the generating functions for these generalized catalan numbers

excursion processes associated with elliptic combinatorics researching elliptic analogues for equalities and formulas is a new trend in enumerative combinatorics which has followed the previous trend of studying q analogues recently schlosser proposed a lattice path model in the square lattice with a family of totally elliptic weight functions including several complex parameters and discussed an elliptic extension of the binomial theorem in the present paper we introduce a family of discrete time excursion processes on mathbb z starting from the origin and returning to the origin in a given time duration 2t associated with schlosser s elliptic combinatorics the processes are inhomogeneous both in space and time and hence expected to provide new models in non equilibrium statistical mechanics by numerical calculation we show that the maximum likelihood trajectories on the spatio temporal plane of the elliptic excursion processes and of their reduced trigonometric versions are not straight lines in general but are nontrivially curved depending on parameters we analyze asymptotic probability laws in the long term limit t rightarrow infty for a simplified trigonometric version of excursion process emergence of nontrivial curves of trajectories in a large scale of space and time from the elementary elliptic weight functions exhibits a new aspect of elliptic combinatorics

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

holder and minkowski type inequalities for pseudo integral abstract there are proven generalizations of the holder s and minkowski s inequalities for the pseudo integral there are considered two cases of the real semiring with pseudo operations one when pseudo operations are defined by monotone and continuous function g the second semiring a b sup where is generated and the third semiring where both pseudo operations are idempotent i e sup and inf

generalized pseudo probability measure abstract a new approach to non additive set functions is presented the notion of non additive measure is extended using non commutative and non associative operations with two parameters and the cumulative distribution function of a random variable one new non additive measure generalized pseudo probability measure is defined and some of its basic properties are proven

decomposition of some hankel matrices generated by the generalized rencontres polynomials abstract using the algebraic techniques of formal series we obtain a combinatorial decomposition of some matrices generated by the generalized rencontres polynomials specifically for the complete matrix and the hankel matrix we obtain an ldu decomposition r d s t where d is a diagonal matrix and r and s are two sheffer matrices then for the shifted matrices we obtain an ltu decomposition r t s t where t is a tridiagonal matrix and r and s are two sheffer matrices furthermore we give an algebraic characterization of the generalized rencontres numbers and polynomials in terms of hankel determinants finally we determine a relation between the complete matrices and the hankel matrices

sobolev orthogonal polynomials in computing of hankel determinants abstract in this paper we study closed form evaluation for some special hankel determinants arising in combinatorial analysis especially for the bidirectional number sequences we show that such problems are directly connected with the theory of quasi definite discrete sobolev orthogonal polynomials it opens a lot of procedural dilemmas which we will try to exceed a few examples deal with fibonacci numbers and power sequences will illustrate our considerations we believe that our usage of sobolev orthogonal polynomials in hankel determinant computation is quite new

permutations moments measures which combinatorial sequences correspond to moments of probability measures on the real line we present a generating function in the form of a continued fraction for a fourteenparameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups special cases include several classical and noncommutative probability laws along with a substantial subset of the orthogonalizing measures in the q askey scheme now given a new combinatorial interpretation in terms of elementary permutation statistics this framework further captures a variety of interesting combinatorial sequences including notably the moment sequences associated to distributions of the numbers of occurrences of classical and vincular permutation patterns of length three this connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries which generalize and unify results previously appearing in the literature while opening up new lines of inquiry the fourteen combinatorial statistics further generalize to signed and colored permutations and as an infinite family of statistics to the k arrangements permutations with k colored fixed points introduced here along with several related results and conjectures

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

approximations of upper and lower probabilities by measurable selections a random set can be regarded as the result of the imprecise observation of a random variable following this interpretation we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable we link this problem to the existence of selectors of a multi valued mapping and with the inner approximations of the upper probability and prove that under fairly general conditions although not in all cases the upper and lower probabilities are an adequate tool for modelling the available information in doing this we generalise a number of results from the literature finally we study the particular case of consonant random sets and we also derive a relationship between aumann and choquet integrals

sums of triangular numbers from the frobenius determinant abstract we show that the denominator formula for the strange series of affine superalgebras conjectured by kac and wakimoto and proved by zagier follows from a classical determinant evaluation of frobenius as a limit case we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4 m 2 d triangles whenever d 2 m and 4 m m 1 d triangles when d 2 m or d 2 m 2 this extends recent results of getz and mahlburg milne and zagier

power series expansion of a hankel determinant abstract in this article we find a power series expansion for the hankel determinant whose entries are the moments of a linear functional related to discrete semiclassical orthogonal polynomials we provide explicit formulas that allow the computation of coefficients to arbitrary order and give examples for the first few terms in the series we reinterpret these results in the context of the theory of young tableaux and schur functions and find closed form expressions for the cases corresponding to the charlier and meixner polynomials we also discuss further research directions

a direct method for evaluating some nice hankel determinants and proofs of several conjectures abstract in this paper we propose a direct method to evaluate hankel determinants for some generating functions satisfying a certain type of quadratic equations which cover generating functions of catalan numbers motzkin numbers and schroder numbers additionally four recent conjectures proposed by cigler 2011 3 are proved

generalization of the jensen inequality for pseudo integral generalizations of the jensen integral inequality for the pseudo integral are proven there are considered two cases of the real semiring with pseudo operations one when pseudo operations are defined by monotone and continuous function g then the pseudo integral reduces on g integral and the second semiring a b max where the pseudo multiplication is generated

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

macdonald denominators for affine root systems orthogonal theta functions and elliptic determinantal point processes rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t 0 t we give new expressions for the macdonald denominators using the karlin mcgregor lindstrom gessel viennot determinants for noncolliding brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding brownian brides with a time duration t which are specified by the pinned configurations at time t 0 and t t rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t

elliptic determinantal process of type a we introduce an elliptic extension of dyson s brownian motion model which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle using elliptic determinant evaluations related to the reduced affine root system of type a we give determinantal martingale representation dmr for the process when it is started at the configuration with equidistant spacing on the circle dmr proves that the process is determinantal and the spatio temporal correlation kernel is determined by taking temporally homogeneous limits of the present elliptic determinantal process trigonometric and hyperbolic versions of noncolliding diffusion processes are studied

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

general minkowski type inequalities for sugeno integrals minkowski type inequalities for the sugeno integral on abstract spaces are studied in a rather general form thus closing the series of papers on the topic dealing with special cases restricted to the pseudo additive operation

inequalities of holder and minkowski type for pseudo integrals with respect to interval valued measures abstract in the present paper the holder and minkowski type of inequality for the pseudo integral of a real valued function with respect to the interval valued pseudo additive measure is proven two cases of semirings are considered in the first case pseudo operations pseudo addition and pseudo multiplication are set by a strictly monotone continuous function in the second case the pseudo addition is the idempotent operation sup and pseudo multiplication is specified by a strictly monotone continuous function as in the first case trivial and nontrivial examples of interval valued pseudo additive measures are provided as well as holder and minkowski type of inequalities with respect to those measures

on the matrices related to the m arithmetic triangle abstract in this paper we first introduce the m arithmetic triangle which is a generalization of pascal s triangle then we put forward some computational results on the evaluations of determinants of matrices related to it we also try to find the inverse matrix and some factorizations of this type of matrices

hankel determinants of sums of consecutive motzkin numbers abstract we use combinatorial methods to evaluate hankel determinants for the sequence of sums of consecutive t motzkin numbers more specifically we consider the following determinant det m i j r t m i j r 1 t 0 i j n 1 where t is a real number and m k t is the total weight of all paths from 0 0 to k 0 that stay above the x axis and use up and down steps of weight one and level steps of weight t

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

idempotent integral as limit of g integrals we show that sup and inf decomposable measures can be obtained as limits of families of pseudo additive measures with respect to generated pseudo additions the corresponding integrals with respect to sup or inf decomposable measures can be obtained as limits of families of g integrals

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

some composition determinants abstract we compute several parametric determinants in which rows and columns are indexed by compositions where the entries are either products of binomial coefficients or products of powers these results generalize previous determinant evaluations due to the first and fourth author siam j matrix anal appl 23 2001 459 471 and a polynomial generalization of the power compositions determinant linear multilinear algebra in press and they prove two conjectures of the second author advanced determinant calculus a complement preliminary version

symmetry classes of alternating sign matrices under one roof in a previous article math co 9712207 we derived the alternating sign matrix asm theorem from the izergin korepin determinant for a partition function for square ice with domain wall boundary here we show that the same argument enumerates three other symmetry classes of alternating sign matrices vsasms vertically symmetric asms even htsasms half turn symmetric asms and even qtsasms quarter turn symmetric asms the vsasm enumeration was conjectured by mills the others by robbins math co 0008045 we introduce several new types of asms uasms asms with a u turn side uuasms two u turn sides osasms off diagonally symmetric asms oosasms off diagonally off antidiagonally symmetric and uosasms off diagonally symmetric with u turn sides uasms generalize vsasms while uuasms generalize vhsasms vertically and horizontally symmetric asms and another new class vhpasms vertically and horizontally perverse osasms oosasms and uosasms are related to the remaining symmetry classes of asms namely dsasms diagonally symmetric dasasms diagonally anti diagonally symmetric and tsasms totally symmetric asms we enumerate several of these new classes and we provide several 2 enumerations and 3 enumerations our main technical tool is a set of multi parameter determinant and pfaffian formulas generalizing the izergin korepin determinant for asms and the tsuchiya determinant for uasms solv int 9804010 we evaluate specializations of the determinants and pfaffians using the factor exhaustion method

addition theorems via continued fractions we show connections between a special type of addition formulas and a theorem of stieltjes and rogers we use different techniques to derive the desirable addition formulas we apply our approach to derive special addition theorems for bessel functions and confluent hypergeometric functions we also derive several additions theorems for basic hypergeometric functions applications to the evaluation of hankel determinants are also given

a jensen type inequality for fuzzy integrals in this paper we show a jensen type inequality for the sugeno integral we also discuss some conditions assuring the satisfaction of opposite inequality reverse jensen inequality

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

a unified approach for the hankel determinants of classical combinatorial numbers abstract we give a general formula for the determinants of a class of hankel matrices which arise in combinatorics theory we revisit and extend existent results on hankel determinants involving the sum of consecutive catalan motzkin and schroder numbers and we prove a conjecture of 10 about the recurrence relations satisfied by the hankel transform of linear combinations of catalans numbers

on directed compact percolation near a damp wall a percolation probability for directed compact percolation near a damp wall which interpolates between the previously examined cases is derived exactly we find that the critical exponent 2 in common with the dry wall rather than the value previously found in the wet wall and bulk cases the solution is found via a mapping to a particular model of directed walks we evaluate the exact generating function for this walk model which is also related to the asep model of traffic flow we compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models

hankel determinants of sums of consecutive weighted schroder numbers abstract we consider weighted large and small schroder paths with up steps 1 1 down steps 1 1 assigned the weight of 1 and with level steps 2 0 assigned the weight of t where t is a real number the weight of a path is the product of the weights of all its steps let r l t and s l t be the total weight of all weighted large and small schroder paths from 0 0 to 2 l 0 respectively for constants we derive the generating functions and the explicit formulae for the determinants of the hankel matrices r i j 2 t r i j 1 t i j 1 n r i j 1 t r i j t i j 1 n s i j 2 t s i j 1 t i j 1 n and s i j 1 t s i j t i j 1 n combinatorially via suitable lattice path models

berwald type inequality for sugeno integral abstract nonadditive measure is a generalization of additive probability measure sugeno integral is a useful tool in several theoretical and applied statistics which has been built on non additive measure integral inequalities play important roles in classical probability and measure theory the classical berwald integral inequality is one of the famous inequalities this inequality turns out to have interesting applications in information theory in this paper berwald type inequality for the sugeno integral based on a concave function is studied several examples are given to illustrate the validity of this inequality finally a conclusion is drawn and a problem for further investigations is given

chebyshev type inequalities for pseudo integrals abstract chebyshev type inequalities for two classes of pseudo integrals are shown one of them concerning the pseudo integrals based on a function reduces on the g integral where pseudo operations are defined by a monotone and continuous function g another one concerns the pseudo integrals based on a semiring a b max where is generated moreover a strengthened version of chebyshev s inequality for pseudo integrals is proved

some extensions of brioschi s double alternant abstract we evaluate some determinants related to brioschi s extension of the classical double alternant of cauchy and scott s extension of brioschi s

on set valued fuzzy measures so far there have been many discussions on sugeno s fuzzy measures and fuzzy integrals but most of them are concentrated on single valued functions motivated by aumann s set valued integral j math anal appl 12 1965 1 we have introduced the fuzzy integral of set valued functions fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341 ph d dissertation harbin institute of technology 1998 it is a well behaved extension of fuzzy integrals of single valued functions it is well known that arstein s set valued measure trans am math soc 165 1972 103 is an important branch in set valued analysis set valued analysis birkhauser berlin 1990 or theory of correspondences theory of correspondences wiley new york 1984 compared to it the present paper will try to establish the basic idea of set valued fuzzy measures which are monotone set valued set functions it is also a natural generalization of single valued fuzzy measures as well as an extension of set valued measures in the case of one dimension these works include the concept of set valued fuzzy measures set valued fuzzy measures defined by set valued fuzzy integrals and set valued pseudo additive measures it can be viewed as a continuation of previous work fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341

evaluation of minors associated to weighing matrices in the present paper we focus our research on calculating minors of weighing matrices of order n and weight n k denoted by w n n k we provide analytical determinant computations counting techniques for specifying the existence of certain submatrices inside a w n n k and an algorithm for computing the n j n j minors of a w n n k which is realized with the notion of symbolic manipulation these results are valid of general n the ideas presented in this work can be used as the fundamental basis on which the calculation of minors of other weighing matrices and in general of orthogonal matrices can be developed an application of the derived formulas to an interesting problem of numerical analysis the growth problem is also presented

an elementary result in the stability theory of time invariant nonlinear discrete dynamical systems the stability of the equilibria of time invariant nonlinear dynamical systems with discrete time scale is investigated we present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function all eigenvalues of the jacobian computed at the equilibrium must be inside or on the unit circle we also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle then the equilibrium maybe unstable or marginally stable or even asymptotically stable which show that the necessary condition cannot be further restricted in general in addition the necessary condition is given in terms of spectral radius and matrix norms

on characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions in this paper we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space through introducing two new concepts the lip constant of a nonlinear operator and strongly equivalent metrics of a norm we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics or if and only if the lip constant of the nonlinear operator is less than one in the latter case we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the lip constant based on the obtained results we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the lip constant and strongly equivalent metrics the obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems

confluent q extensions of some classical determinants abstract we evaluate two determinants the first is a q h extension of the classical confluent extension of the vandermonde determinant the second is a similar extension of cauchy s double alternant

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

matrix methods for determinants of pascal like matrices abstract using the ldu like factorization method the evaluations of the determinants of some pascal like matrices are derived the entries of these matrices satisfy the recurrence relation a i j a i 1 j a i j 1 a i 1 j 1 i j 1 and the values of the first row a 0 j and the column a i 0 have eight cases part of the results is due to c krattenthaler 2002 3

hankel continued fraction and its applications abstract the hankel determinants of a given power series f can be evaluated by using the jacobi continued fraction expansion of f however the existence of the jacobi continued fraction needs that all hankel determinants of f are nonzero we introduce hankel continued fraction whose existence and uniqueness are guaranteed without any condition for the power series f the hankel determinants can also be evaluated by using the hankel continued fraction it is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic we prove a similar result for power series if a power series f over a finite field satisfies a quadratic equation then the hankel continued fraction is ultimately periodic as an application we derive the hankel determinants of several automatic sequences in particular the regular paperfolding sequence thus we provide an automatic proof of a result obtained by guo wu and wen which was conjectured by coons vrbik

invariant algebras and major indices for classical weyl groups given a classical weyl group w that is a weyl group of type a b or d one can associate with it a polynomial with integral coefficients z w given by the ratio of the hilbert series of the invariant algebras of the natural action of w and w t on the ring of polynomials bf c x 1 ldots x n otimes t we introduce and study several statistics on the classical weyl groups of type b and d and show that they can be used to give an explicit formula for z d n more precisely we define two mahonian statistics that is statistics having the same distribution as the length function dmaj and ned on d n the statistic dmaj defined in a combinatorial way has an analogous algebraic meaning to the major index for the symmetric group and the flag major index of adin and roichman for b n namely it allows us to find an explicit formula for z d n our proof is based on the theory of t partite partitions introduced by gordon and further studied by garsia and gessel using similar ideas we define the mahonian statistic ned also on b n and we find a new and simpler proof of the adin roichman formula for z b n finally we define a new descent number ddes on d n so that the pair ddes dmaj gives a generalization to d n of the carlitz identity on the eulerian mahonian distribution of descent number and major index on the symmetric group

descent representations and multivariate statistics combinatorial identities on weyl groups of types a and b are derived from special bases of the corresponding coinvariant algebras using the garsia stanton descent basis of the coinvariant algebra of type a we give a new construction of the solomon descent representations an extension of the descent basis to type b using new multivariate statistics on the group yields a refinement of the descent representations these constructions are then applied to refine well known decomposition rules of the coinvariant algebra and to generalize various identities

quantitative studies on asymptotic growth behaviors of trajectories of nonlinear discrete dynamical systems this technical note studies quantitatively asymptotic growth behaviors of trajectories agbt of nonlinear autonomous discrete dynamical system that has unbounded domain non lipschitz continuous nonlinear operator and stable or unstable equilibrium point we explain how trajectory motion speed is quantitatively determined in the system and study exact computation and sharp estimation of the smallest exponential bound of trajectories we characterize exponential stability and asymptotic stability of the system from a new point of view and provide a simple condition to distinguish them from each other these results extend existing results that were obtained in some special cases of the system and are helpful for quantitative analysis and understanding of agbt of the system