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

differential equations for multiple orthogonal polynomials with respect to classical weights raising and lowering operators we obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to r n classical weights these multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials combining the lowering operator with the raising operators which have been obtained earlier in the literature we then also obtain a linear differential equation of order r 1

ladder operators and differential equations for multiple orthogonal polynomials in this paper we obtain the ladder operators and associated compatibility conditions for types i and ii multiple orthogonal polynomials these ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials our approach is based on riemann hilbert problems and the christoffel darboux formula for multiple orthogonal polynomials and the nearest neighbor recurrence relations as an illustration we give several explicit examples involving multiple hermite and laguerre polynomials and multiple orthogonal polynomials with exponential weights and cubic potentials

on some properties of q hahn multiple orthogonal polynomials this contribution deals with multiple orthogonal polynomials of type ii with respect to q discrete measures q hahn measures in addition we show that this family of multiple orthogonal polynomials has a lowering operator and raising operators as well as a rodrigues type formula the combination of lowering and raising operators leads to a third order q difference equation when two orthogonality conditions are considered an explicit expression of this q difference equation will be given indeed this q difference equation relates polynomials with a given degree evaluated at four consecutive non uniformed distributed points which makes these polynomials interesting from the point of view of bispectral problems

applications of riordan matrix functions to bernoulli and euler polynomials abstract we define riordan matrix functions associated with riordan arrays and study their algebraic properties we also give their applications in the construction of new classes of bernoulli and euler polynomials and bernoulli and euler numbers referred to as the duals and conjugate bernoulli and euler polynomials and dual and conjugate bernoulli and euler numbers respectively

row polynomial matrices of riordan arrays abstract let r r n k n k 0 be a riordan array define the row polynomials r n q k 0 n r n k q k and the row polynomial matrix r q r n k q n k 0 by r n k q j k n r n j q j k then r q is also a riordan array with the r n q located on the leftmost column of r q in this paper we investigate combinatorial properties of the matrix r q and the sequence r n q n 0 including their characterizations the q total positivity of r q and the q log convexity of r n q n 0

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

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

on askey scheme and d orthogonality i a characterization theorem this is the first in a series of papers dealing with generalized hypergeometric d orthogonal polynomials extending the polynomial families in the askey scheme in this paper we give a characterization theorem to introduce new examples of generalized hypergeometricd orthogonal polynomials to be studied in the forthcoming works for d 1 we obtain an unification of some known characterization theorems in the orthogonal polynomials theory

gaussian quadrature for multiple orthogonal polynomials we study multiple orthogonal polynomials of type i and type ii which have orthogonality conditions with respect to r measures these polynomials are connected by their recurrence relation of order r 1 first we show a relation with the eigenvalue problem of a banded lower hessenberg matrix l n containing the recurrence coefficients as a consequence we easily find that the multiple orthogonal polynomials of type i and type ii satisfy a generalized christoffel darboux identity furthermore we explain the notion of multiple gaussian quadrature for proper multi indices which is an extension of the theory of gaussian quadrature for orthogonal polynomials and was introduced by borges in particular we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of l n

the eight tetrahedron equations in this paper we derive from arguments of string scattering a set of eight tetrahedron equations with different index orderings it is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalizing the yang baxter equation under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form most known solutions fall under this class but it is by no means necessary comparison is made with the work on braided monoidal 2 categories also leading to eight tetrahedron equations

the elements of finite order in the riordan group over the complex field abstract an element of finite order in the riordan group over the real field must have order 1 or 2 if we extend all the entries to be complex numbers then it may have any finite order in the present paper we investigate the elements of finite order of the riordan group over the complex field this notion leads us to define an element of pseudo order k 2 and k pseudo involution respectively it turns out that the inverse of a k pseudo involution only differs from it in signs we clarify some relationship between the elements of pseudo order k and k pseudo involutions in particular k pseudo involutions for k 0 mod 4 are characterized by a single sequence the subgroups of the riordan group formed by the elements having pseudo order of a prime power p r are also introduced

asymptotics of multiple orthogonal polynomials associated with the modified bessel functions of the first kind we consider polynomials which are orthogonal with respect to weight functions which are defined in terms of the modified bessel functions i and i 1 these multiple orthogonal polynomials pn c of type ii were introduced earlier by the authors the aim of this paper is to study some asymptotics of these polynomials such as plancherel rotach asymptotics as a consequence of these results we obtain the accumulation set of the scaled zeros of pn c we also give a mehler heine type formula for these polynomials which is valid on compact subsets of c hence we obtain a more precise result about the asymptotic behavior of the small zeros of pn c

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

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

d orthogonality of little q laguerre type polynomials abstract in this paper we solve a characterization problem in the context of the d orthogonality that allows us on one hand to provide a q analog for the d orthogonal polynomials of laguerre type introduced by the first author and douak and on the other hand to derive new l q classical d orthogonal polynomials generalizing the little q laguerre polynomials various properties of the resulting basic hypergeometric polynomials are singled out for d 1 we obtain a characterization theorem involving as far as we know new l q classical orthogonal polynomials for which we give the recurrence relation and the difference equation

multiple little q jacobi polynomials we introduce two kinds of multiple little q jacobi polynomials pn with multi index nrarr n1 n2 nr and degree n n1 n2 nr by imposing orthogonality conditions with respect to r discrete little q jacobi measures on the exponential lattice qk k 0 1 2 3 where 0 q 1 we show that these multiple little q jacobi polynomials have useful q difference properties such as a rodrigues formula consisting of a product of r difference operators some properties of the zeros of these polynomials and some asymptotic properties will be given as well

double parameter recurrences for polynomials in bi infinite riordan matrices and some derived identities abstract we present some double parameter recurrence relations for families of polynomials in bi infinite riordan matrices from this point of view we treat some riordan families of generalized fibonacci and lucas polynomials we use these to obtain some identities related mainly with binomial coefficients and fibonacci numbers

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

the vertex formulation of the bazhanov baxter model in this paper we formulate an integrable model on the simple cubic lattice then valued spin variables of the model belong to edges of the lattice the boltzmann weights of the model obey the vertex type tetrahedron equation in the thermodynamic limit our model is equivalent to the bazhanov baxter model in the case whenn 2 we reproduce korepanov s and hietarinta s solutions of the tetrahedron equation as special cases

riordan arrays and d orthogonality abstract using the theory of the riordan group and d orthogonal polynomials we shall show that sequences of d orthogonal polynomials can be also generated by the riordan group we interpret some families of d orthogonal polynomials within the framework of the riordan group

matrix characterizations of riordan arrays abstract here we discuss two matrix characterizations of riordan arrays p matrix characterization and a matrix characterization p matrix is an extension of the stieltjes matrix defined in 28 and the production matrix defined in 8 by modifying the marked succession rule introduced in 21 a combinatorial interpretation of the p matrix is given the p matrix characterizations of some subgroups of riordan group are presented which are used to find some algebraic structures of the subgroups we also give the p matrix characterizations of the inverse of a riordan array and the product of two riordan arrays a matrix characterization is defined in 20 and it is proved to be a useful tool for a riordan array while on the other side the a sequence characterization is very complex sometimes by using the fundamental theorem of riordan arrays a method of construction of a matrix characterizations from riordan arrays is given the converse process is also discussed several examples and applications of two matrix characterizations are presented

simple proofs of open problems about the structure of involutions in the riordan group abstract we prove that if d g x f x is an element of order 2 in the riordan group then g x exp x xf x for some antisymmetric function x z also we prove that every element of order 2 in the riordan group can be written as bmb 1 for some element b and m 1 1 in the riordan group these proofs provide solutions to two open problems presented by l shapiro l w shapiro some open questions about random walks involutions limiting distributions and generating functions adv appl math 27 2001 585 596

on a system of classical polynomials of simultaneous orthogonality abstract we introduce a system of classical polynomials of simultaneous orthogonality study the differential equation of third order recurrence relation and precise the ratio asymptotic and zeros distribution of polynomials

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

representing polynomials as characteristic polynomials via the stieltjes transform abstract the stieltjes transform s a of an infinite lower triangular matrix a with nonzero diagonal entries is defined by s a a 1 a where a is the matrix obtained from a by deleting its initial row in this paper we express a sequence of polynomials as the characteristic polynomials of the stieltjes transforms using a highly structured infinite lower triangular matrix called a riordan matrix as a result computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues or to eigenvalue location theorems such as the gersgorin theorem we also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions as an application the recurrence relations for several polynomial sequences are obtained using the stieltjes transform

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

automorphisms of the heisenberg weyl algebra and d orthogonal polynomials we show that the d orthogonal charlier and hermite polynomials appear naturally as matrix elements of nonunitary transformations corresponding to automorphisms of the heisenberg weyl oscillator algebra basic properties duality recurrence and difference equations of the d orthogonal charlier polynomials can be derived directly from representations of the heisenberg weyl algebra

asymptotics for multiple meixner polynomials abstract we study the asymptotic behavior of multiple meixner polynomials of first and second kind respectively 6 we use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution moreover analyzing the limiting behavior of the coefficients of the recurrence relations for multiple meixner polynomials we obtain the main term of their asymptotics

some properties of generalized multiple hermite polynomials the purpose of this paper is to introduce and discuss a more general class of multiple hermite polynomials in this work the explicit forms operational formulas and a recurrence relation are obtained furthermore we derive several families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple hermite polynomials

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

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

some discrete multiple orthogonal polynomials in this paper we extend the theory of discrete orthogonal polynomials on a linear lattice to polynomials satisfying orthogonality conditions with respect to r positive discrete measures first we recall the known results of the classical orthogonal polynomials of charlier meixner kravchuk and hahn t s chihara an introduction to orthogonal polynomials gordon and breach new york 1978 r koekoek and r f swarttouw reports of the faculty of technical mathematics and informatics no 98 17 delft 1998 a f nikiforov et al classical orthogonal polynomials of a discrete variable springer berlin 1991 these polynomials have a lowering and raising operator which give rise to a rodrigues formula a second order difference equation and an explicit expression from which the coefficients of the three term recurrence relation can be obtained then we consider r positive discrete measures and define two types of multiple orthogonal polynomials the continuous case jacobi laguerre hermite etc was studied by van assche and coussement j comput appl math 127 2001 317 347 and aptekarev et al multiple orthogonal polynomials for classical weights manuscript the families of multiple orthogonal polynomials of type ii that we will study have a raising operator and hence a rodrigues formula this will give us an explicit formula for the polynomials finally there also exists a recurrence relation of order r 1 for these multiple orthogonal polynomials of type ii we compute the coefficients of the recurrence relation explicitly when r 2

d orthogonal polynomials and su 2 two families of d orthogonal polynomials related to su 2 are identified and studied the algebraic setting allows for their full characterization explicit expressions recurrence relations difference equations generating functions etc in the limit where su 2 contracts to the heisenberg weyl algebra h1 the polynomials tend to the standard meixner polynomials and d charlier polynomials respectively

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

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

mertens equimodular matrices of redheffer type abstract asymptotic behavior of the mertens function m n k 0 n k which is equal to the determinant of the n n redheffer matrix is known to be closely related to the riemann hypothesis an infinite matrix whose nth leading principal minor is equal to m n for all n 1 is called a mertens equimodular matrix we use riordan matrices to find a large class of mertens equimodular matrices each element of which is called by us a riordan redheffer matrix briefly an r r matrix we also give the generating function for the characteristic polynomials of r r matrices as a result we introduce several examples of r r matrices that reveal interesting spectral properties further we pose two conjectures on the eigenvalues of those r r matrices finally we find a sufficient condition for the riemann hypothesis using the smallest singular value of a r r matrix

a remark on a stability criterion for the radial cavitating map in nonlinear elasticity we study the integral functional i w adj w w w qdx on suitable maps w b r r and where 2q 2 3 the inequality i w i i which we establish on a subclass of the admissible maps was first proposed in 13 as one of two possible necessary conditions for the stability i e local minimality of the radial cavitating map in nonlinear elasticity here i is the identity map admissible maps w either do not vanish and in this case possess a single discontinuity x in b which produces a cavity about the origin or vanish at exactly one point x in b in which case w is a diffeomorphism in a neighbourhood of x we show that i dot operator behaves like a polyconvex functional and associate with it another functional k dot operator satisfying i w i i q k w k i we give conditions under which k w k i and from these infer i w i i it is also shown that i k is strictly decreasing along paths of admissible functions that move x away from the origin and ii k w exhibits some quite pathological behaviour when w is sufficiently close to i

shift operators defined in the riordan group and their applications abstract in this paper we discuss a linear operator t defined in riordan group r by using the upper shift matrix u and lower shift matrix u t namely for each r r t r u r u t some isomorphic properties of the operator t and the structures of its range sets for different domains are studied by using the operator t and the properties of bell subgroup of r the riordan type chu vandermonde identities and the riordan equivalent identities of format last theorem and beal conjecture are shown the applications of the shift operators to the complementary riordan arrays and to the riordan involutions and riordan pseudo involutions are also presented

parametric catalan numbers and catalan triangles abstract here presented a generalization of catalan numbers and catalan triangles associated with two parameters based on the sequence characterization of bell type riordan arrays among the generalized catalan numbers a class of large generalized catalan numbers and a class of small generalized catalan numbers are defined which can be considered as an extension of large schroder numbers and small schroder numbers respectively using the characterization sequences of bell type riordan arrays some properties and expressions including the taylor expansions of generalized catalan numbers are given a few combinatorial interpretations of the generalized catalan numbers are also provided finally a generalized motzkin numbers and motzkin triangles are defined similarly an interrelationship among parametrical catalan triangle pascal triangle and motzkin triangle is presented based on the sequence characterization of bell type riordan arrays

riordan arrays and related polynomial sequences abstract in this paper we generalize the results due to luzon and moron and present a characterization of the generalized riordan arrays using this characterization we establish a recurrence for the generalized sheffer sequences and study some special types of polynomial sequences including the generalized lucas u sequence and v sequence and the generalized humbert polynomial sequence in particular by appealing to the factorizations of riordan arrays the umbral composition of polynomial sequences and the hadamard product of series we establish various relations among these polynomial sequences thus we give a unified approach to some well known polynomial sequences appeared in combinatorics and the theory of special functions

some generalized hypergeometric d orthogonal polynomial sets in this paper we characterize the d orthogonal polynomial sets given by their explicit expressions in a specific basis as application we consider the generalized hypergeometric case to characterize d orthogonal polynomial sets of laguerre type meixner type meixner pollaczek type krawtchouk type continuous dual hahn type and dual hahn type for d 1 we obtain a unification of some characterization theorems in the orthogonal polynomials theory

labelling schemes for tetrahedron equations and dualities between them zamolodchikov s tetrahedron equations which were derived by considering the scattering of straight strings can be written in three different labelling schemes one can use as labels the states of the vacua between the strings the states of the string segments or the states of the particles at the intersections of the strings we give a detailed derivation of the three corresponding tetrahedron equations and also show how the frenkel moore equations fits in as a non local string labelling we then discuss how an analogue of the wu kadanoff duality can be defined between each pair of the above three labelling schemes it turns out that there are two cases for which one can simultaneously construct a duality between all three pairs of labellings

an algebraic structure for faber polynomials abstract let be the set of functions convergent for all z 1 with a laurent series of the form f z z n 0 a n z n in this paper we prove that the set of faber polynomial sequences over and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the bell k subgroup of the riordan group respectively further a relationship between such faber polynomial sequences and lucas and sheffer polynomial sequences is derived

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

large n limit of gaussian random matrices with external source part iii double scaling limit we consider the double scaling limit in the random matrix ensemble with an external source 1 over z n e n hbox tr 1 over 2 m 2 am dm defined on n n hermitian matrices where a is a diagonal matrix with two eigenvalues a of equal multiplicities the value a 1 is critical since the eigenvalues of m accumulate as n on two intervals for a 1 and on one interval for 0 a 1 these two cases were treated in parts i and ii where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles for the critical case a 1 new limiting behavior occurs which is described in terms of pearcey integrals as shown by brezin and hikami and tracy and widom we establish this result by applying the deift zhou steepest descent method to a 3 3 matrix valued riemann hilbert problem which involves the construction of a local parametrix out of pearcey integrals we resolve the main technical issue of matching the local pearcey parametrix with a global outside parametrix by modifying an underlying riemann surface

asymptotic zero distribution for a class of multiple orthogonal polynomials we establish the asymptotic zero distribution for polynomials generated by a four term recurrence relation with varying recurrence coefficients having a particular limiting behavior the proof is based on ratio asymptotics for these polynomials we can apply this result to three examples of multiple orthogonal polynomials in particular jacobi pineiro laguerre i and the example associated with modified bessel functions we also discuss an application to toeplitz matrices

vector orthogonal relations vector qd algorithm vector pade approximants to a function f f 1 f d from to d have been defined uniquely without any auxiliary choice than the degrees of the numerator and the denominator the same for all the components f i as in the scalar case 1 5 the denominators are associated to polynomials p r s which are given by vector orthogonal properties r and which satisfy for each s recurrence relations of order d 1 i e with d 2 terms called relations d we study here consequences of r and d first we prove an algorithm similar to the generalized mna al gorithm then we define a vector qd algorithm which links two diagonals p r s r and p r s 1 r conversely if a family p r s r 0 verifying d is given it is possible to build p r s r 0 s 0 and d linear functionals c a a 1 d such that p r 0 p r and p r s verify the orthogonal relations r with respect to the c a

error analysis of a dual parametric bi quadratic fem in cavitation computation in elasticity the orientation preservation conditions and approximation errors of a dual parametric bi quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed the analytical results allow us to establish based on an error equidistribution principle an optimal meshing strategy for the method in cavitation computation numerical results are in good agreement with the analytical results

on a class of gauss like quadrature rules we consider a problem that arises in the evaluation of computer graphics illumination models in particular there is a need to find a finite set of wavelengths at which the illumination model should be evaluated the result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene these values are then used to determine the rgb coordinates of the light by evaluating three definite integrals each with a common integrand the spd and interval of integration but with distinct weight functions we develop a method for selecting the sample wavelengths in an optimal manner more abstractly we examine the problem of numerically evaluating a set of m definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration it is shown that when m geq 3 it is not efficient to use a set of m gauss rules because valuable information is wasted we go on to extend the notions used in gaussian quadrature to find an optimal set of shared abcissas that maximize precision in a well defined sense the classical gauss rules come out as the special case m 1 and some analysis is given concerning the existence of these rules when m 1 in particular we give conditions on the weight functions that are sufficient to guarantee that the shared abcissas are real distinct and lie in the interval of integration finally we examine some computational strategies for constructing these rules

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

random matrices with equispaced external source we study hermitian random matrix models with an external source matrix which has equispaced eigenvalues and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity we obtain strong asymptotics for the multiple orthogonal polynomials associated to these models and as a consequence for the average characteristic polynomials one feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree nevertheless we are able to characterize them in terms of a pair of 2 1 vector valued riemann hilbert problems and to perform an asymptotic analysis of the riemann hilbert problems

multiple orthogonal polynomials on the semicircle and applications in this paper two types of multiple orthogonal polynomials on the semicircle with respect to a set of r different weight functions are defined such polynomials are generalizations of polynomials orthogonal on the semicircle with respect to a complex valued inner product f g email protected 0 pfe i qge i qwe i email protected the existence and uniqueness of introduced multiple orthogonal polynomials for certain classes of weight functions are proved some properties of multiple orthogonal polynomials on the semicircle including certain recurrence relations of order r 1 are presented finally an application in numerical integration is given

riordan arrays associated with laurent series and generalized sheffer type groups a relationship between a pair of laurent series and riordan arrays is formulated in addition a type of generalized sheffer groups is defined by using riordan arrays with respect to power series with non zero coefficients the isomorphism between a generalized sheffer group and the group of the riordan arrays associated with laurent series is established furthermore appell associated bell and hitting time subgroups of the groups are defined and discussed a relationship between the generalized sheffer groups with respect to different type of power series is presented the equivalence of the defined riordan array pairs and generalized stirling number pairs is given a type of inverse relations of various series is constructed by using pairs of riordan arrays finally several applications involving various arrays polynomial sequences special formulas and identities are also presented as illustrative examples

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

tetrahedral zamolodchikov algebras corresponding to baxter s l operators tetrahedral zamolodchikov algebras are structures that occupy an intermediate place between the solutions of the yang baxter equation and its generalization onto 3 dimensional mathematical physics the tetrahedron equation these algebras produce solutions to the tetrahedron equation and besides specific two layer solutions to the yang baxter equation here the tetrahedral zamolodchikov algebras are studied that arise froml operators of the free fermion case of baxter s eight vertex model

1 invariant sequences and truncated fibonacci sequences abstract let p i j i j 0 1 2 and d diag 1 0 1 1 1 2 as a linear transformation of the infinite dimensional real vector space r x0 x1 x2 t xi r for all i pd has only two eigenvalues 1 1 in this paper we find some matrices associated with p whose columns form bases for the eigenspaces for pd we also introduce truncated fibonacci sequences and truncated lucas sequences and show that these sequences span the eigenspaces of pd

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above

d orthogonality via generating functions in this paper starting from a suitable generating function of a polynomial set we show how to decide whether the considered polynomial set is d orthogonal and if it is so how to determine the corresponding d dimensional functional vector then we apply the obtained results to some known and new d orthogonal polynomial sets for the known ones we give new proofs for some already obtained results

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds