on a fractional integral equation of periodic functions involving weyl riesz operator in banach algebras abstract in this paper we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving weyl riesz fractional integral operator under the mixed generalized lipschitz caratheodory and monotonicity conditions the fixed point theorems due to dhage are the main tool in carrying out our proofs

exact solution and interfacial tension of the six vertex model with anti periodic boundary conditions we consider the six vertex model with anti periodic boundary conditions across a finite strip the row to row transfer matrix is diagonalized by the commuting transfer matrices method from the exact solution we obtain an independent derivation of the interfacial tension of the six vertex model in the anti ferroelectric phase the nature of the corresponding integrable boundary condition on the xxz spin chain is also discussed

toda type differential equations for the recurrence coefficients of orthogonal polynomials and freud transformation transformations of the measure of orthogonality for orthogonal polynomials namely freud transformations are considered jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations dynamics of the coefficients are described by generalized toda equations the classical toda lattice equations are the simplest special case of dynamics of the coefficients under the freud transformation of the measure of orthogonality also dynamics of hankel determinants its minors and other notions corresponding to the orthogonal polynomials are studied

three dimensional integrable models and associated tangle invariants in this paper we show that the boltzmann weights of the three dimensional baxter bazhanov model give representations of the braid group if some suitable spectral limits are taken in the trigonometric case we classify all possible spectral limits which produce braid group representations furthermore we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones

interlacing of zeros of gegenbauer polynomials of non consecutive degree from different sequences a theorem due to stieltjes states that if p n n 0 infty is any orthogonal sequence then between any two consecutive zeros of p k there is at least one zero of p n whenever k frac 12 if 0 t k 1 and also to the zeros of c n 1 lambda and c n 2 lambda k if k in 1 2 3 more generally we prove that stieltjes interlacing holds between the zeros of the kth derivative of c n lambda and the zeros of c n 1 lambda k in 1 2 dots n 1 and we derive associated polynomials that play an analogous role to the de boor saff polynomials in completing the interlacing process of the zeros

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

simplex equations and their solutions we investigate ann simplex generalization of the classical and quantum yang baxter equation for the case ofsl 2 we find the most general solution of the classicaln simplex equation for alln these classical solutions can be quantized in the sense of quantum group theory forn 2 3 and we exhibit a quantum solution to the tetrahedron equations n 3 the classical nondegenerate solutions cannot be quantized forn 4

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

on holomorphic solutions for nonlinear singular fractional differential equations the paper concerns with a nonlinear singular fractional differential equation t a au t z t a f t z u u z 0 a 1 where t j 0 1 and z u z c z 1 the existence and the uniqueness of holomorphic solution are established

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

existence and uniqueness of holomorphic solutions for fractional cauchy problem by employing majorant functions the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations the cauchy problems are introduced furthermore the analytic continuation of solutions is studied

persymmetric jacobi matrices isospectral deformations and orthogonal polynomials abstract persymmetric jacobi matrices are invariant under reflection with respect to the anti diagonal the associated orthogonal polynomials have distinctive properties that are discussed they are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice this is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric jacobi matrices from spectral points isospectral deformations of such matrices are also considered expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

remarks on the star triangle relation in the baxter bazhanov model we show that the restricted star triangle relation introduced by bazhanov and baxter can be obtained either from the star triangle relation of the chiral potts model or from the star square relation proposed by kashaev mangazeev and stroganov and give a response to a guess of bazhanov and baxter

solutions of n simplex equation from solutions of braid group representation it is shown that a kind of solution of the n simplex equation can be obtained from representations of the braid group the symmetries in its solution space are also discussed

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

a q generalization of the para racah polynomials abstract new bispectral orthogonal polynomials are obtained from an unconventional truncation of the askey wilson polynomials in the limit q 1 they reduce to the para racah polynomials which are orthogonal with respect to a quadratic bi lattice the three term recurrence relation and q difference equation are obtained through limits of those of the askey wilson polynomials an explicit expression in terms of hypergeometric series and the orthogonality relation are provided a q generalization of the para krawtchouk polynomials is obtained as a special case connections with the q racah and dual hahn polynomials are also presented

on some applications of the briot bouquet differential subordination recently srivastava et al j dziok h m srivastava certain subclasses of analytic functions associated with the generalized hypergeometric function integral transforms spec funct 14 2003 7 18 j dziok h m srivastava classes of analytic functions associated with the generalized hypergeometric function appl math comput 103 1999 1 13 y c kim h m srivastava fractional integral and other linear operators associated with the gaussian hypergeometric function complex var theory appl 34 1997 293 312 introduced and studied a class of analytic functions associated with the generalized hypergeometric function in the present paper by using the briot bouquet differential subordination new results in this class are obtained

baxter equations and the zamolodchikov model abstract a summary is given of the significance of the star triangle or yang baxter equations in the solution of two dimensional statistical mechanical models the extension to the three dimensional tetrahedron relations is quoted and it is shown how this can be used to obtain the partition function of the zamolodchikov model for an n lattice the n result differs from that recently obtained by bazhanov and stroganov

a characterization of classical and semiclassical orthogonal polynomials from their dual polynomials we study properties of the dual orthogonal polynomials op introduced by de boor and saff and present a new characterization of the classical and semiclassical op

subordination and superordination for univalent solutions for fractional differential equations in this article we establish the existence and uniqueness of univalent solution for fractional differential equation moreover we illustrate some properties of this solution containing differential and integral subordination properties

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

construction of acyclic matrices from spectral data abstract given real numbers 1 n 1 n 1 with 1 2 2 n 1 n an integer i with 1 i n and a tree with n vertices we prove the existence of a hermitian acyclic matrix a x rs with eigenvalues 1 n and x rs 0 whenever r s and r s is not an edge of and such that the principal submatrix of a obtained from a by deleting the i th row and column has eigenvalues 1 n 1

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

finite sequences of orthogonal polynomials connected by a jacobi matrix abstract if p0 pn is an orthogonal sequence with pj a monic polynomial of exact degree j all j then the sequence q0 qn of monic polynomials defined by the requirement that pj cjqn j 1pn 1 at the zeros of pn with cj suitable constants is also orthogonal with respect to a possibly differently weighted scalar product these two sequences can be shown to be the upper left respectively lower right principal minors of the matrix t j with j a suitable tridiagonal matrix the sequence q occurs in the characterization of the discrete least squares approximation to f from span p0 pj in terms of the n j 1 order divided differences of f p n 1 at the nodes i e the zeros of pn a complete characterization of the pair u v of zero sets of pj 1 and qn j is given and an application is made to the problem of recovering a jacobi matrix from such data

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

variations around classical orthogonal polynomials connected problems abstract we survey and prove the principal characterizations of the classical orthogonal polynomials with the method of the dual sequence two new characterizations are given connected problems are discussed

some exact solutions to stefan problems with fractional differential equations some exact solutions to the first second and extended stefan problems with fractional time derivative described in the caputo sense are given by means of fractional green s function and wright function in this paper by the aid of simple calculations many results of differential equations of integer order can be obtained as special cases of the results given by this paper

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

zeros of linear combinations of laguerre polynomials from different sequences we study interlacing properties of the zeros of two types of linear combinations of laguerre polynomials with different parameters namely r n l n a al n a and s n l n a bl n 1 a proofs and numerical counterexamples are given in situations where the zeros of r n and s n respectively interlace or do not in general with the zeros of l k a l k a k n or n 1 the results we prove hold for continuous as well as integral shifts of the parameter a

the uniqueness of an orthogonality measure given two finite sets of real points z n j j 1 n and z n 1 j j 1 n 1 satisfying the interlacing property z n 1 j

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

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

duality of orthogonal polynomials on a finite set we prove a certain duality relation for orthogonal polynomials defined on a finite set the result is used in a direct proof of the equivalence of two different ways using particles or holes of computing the correlation functions of a discrete orthogonal polynomial ensemble

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

a solvable lattice model and related rogers ramanujan type identities an exactly solvable lattice model is presented which is a natural generalization of baxter s hard hexagon model to three spin states the free energy and spin densities are calculated related rogers ramanujan type identities are also studied

is the zamolodchikov model critical evidence is presented in favour of the hypothesis that the zamolodchikov model an exactly solvable three dimensional lattice model without any temperature like parameters is critical the evidence is obtained by generalising the zamolodchikov model to include a temperature like variable the magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier also the authors show the two layer zamolodchikov model corresponds to a critical free fermion model

on toda lattices and orthogonal polynomials first we derive a simple connection between toda and langmuir lattices and give a characterization of toda lattices with the help of stieltjes functions then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic toda lattices the particles of the toda lattice are not even restricted as usual to move on the real line they may also move in the complex plane with the help of this result for special cases explicit solutions are obtained in terms of elliptic functions

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

on neighbouring matrices with quadratic elementary divisors this paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixa and relates them to those given by kakan it shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

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

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

extended lagrange interpolation in weighted uniform norm the author studies the uniform convergence of extended lagrange interpolation processes based on the zeros of generalized laguerre polynomials

nonintersecting string model and graphical approach equivalence with a potts model using a graphical method we establish the exact equivalence of the partition function of aq state nonintersecting string nis model on an arbitrary planar even valenced lattice with that of a q2 state potts model on a related lattice the nis model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types and the resulting potts model generally has multispin interactions for the square and kagome lattices this leads to the equivalence of a staggered nis model with potts models with anisotropic pair interactions indicating that these nis models have a first order transition forq 2 for the triangular lattice the nis model turns out to be the five vertex model of wu and lin and it relates to a potts model with two and three site interactions the most general model we discuss is an oriented nis model which contains the six vertex model and the nis models of stroganov and schultz as special cases