theta function parametrization and fusion for 3d integrable boltzmann weights we report progress in constructing boltzmann weights for integrable threedimensional lattice spin models we show that a large class of vertex solutions to the modified tetrahedron equation mte can be conveniently parametrized in terms of nth roots of theta functions on the jacobian of a compact algebraic curve fay s identity guarantees the fermat relations and the classical equations of motion for the parameters determining the boltzmann weights our parametrization allows us to write a simple formula for fused boltzmann weights r which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation imposing periodic boundary conditions we observe that the r satisfy the normal tetrahedron equation the scheme described contains the zamolodchikov baxter bazhanov model and the chessboard model as special cases

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

tridiagonal pairs and the askey wilson relations abstract the notion of a tridiagonal pair was introduced by ito tanabe and terwilliger let v denote a nonzero finite dimensional vector space over a field f a tridiagonal pair on v is a pair a a where a v v and a v v are linear transformations that satisfy some conditions assume a a is a tridiagonal pair on v recently terwilliger and vidunas showed that if a is multiplicity free on v then a a satisfy the following askey wilson relation for some scalars a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i in the present paper we show that if a tridiagonal pair a a satisfy the askey wilson relations then the eigenspaces of a and the eigenspaces of a have one common dimension and moreover if f is algebraically closed then that common dimension is 1

linear transformations that are tridiagonal with respect to both eigenbases of a leonard pair let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let x denote the set of linear transformations x v v such that the matrix representing x with respect to the basis i is tridiagonal and the matrix representing x with respect to the basis ii is tridiagonal we show that x is spanned by i a a aa a a and these elements form a basis for x provided the dimension of v is at least 3

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

the structure of some linear transformations abstract let f denote an algebraically closed field and let v denote a finite dimensional vector space over f recently ito and terwilliger considered a system of linear transformations a a a a on v which generalizes the notions of a tridiagonal pair and a q inverting pair in their paper they mentioned some open problems about this system in this paper we solve problem 1 2 with the following results let v i i 0 d denote the common eigenspaces of a a and let v i i 0 d denote the common eigenspaces of a a we show that each of a a a a is determined up to affine transformation by the sequences v i i 0 d v i i 0 d we also show that the following are equivalent i there exists a nonzero bilinear form on v such that a u v u a v and a u v u a v for all u v v ii there exist scalars in f with nonzero such that a a i and a a i and iii both a a and a a are tridiagonal pairs

the split decomposition of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering v0 v1 vd of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 vd 1 0 iii there exists an ordering v 0 v 1 v of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 v 1 0 iv there is no subspace w of v such that both aw w a w w other than w 0 and w v we call such a pair a tridiagonal pair on v in this note we obtain two results first we show that each of a a is determined up to affine transformation by the vi and v i secondly we characterize the case in which the vi and v i all have dimension one we prove both results using a certain decomposition of v called the split decomposition

new solutions of distinct physical structures to high dimensional nonlinear evolution equations abstract in this work we employ the tanh coth method for a reliable treatment on nonlinear evolution equations three models namely the 3 1 dimensional jimbo miwa equation the 2 1 dimensional calogero bogoyavlenskii schiff equation and the 3 1 dimensional potential ytsf equation are use as vehicles to conduct the analysis new travelling wave solutions are formally derived the work shows the power of the proposed scheme and the variety of the obtained solutions

a family of tridiagonal pairs abstract let f denote a field and let v denote a vector space of finite positive dimension over f let a a denote a tridiagonal pair on v of diameter d 3 assume the eigenvalue and dual eigenvalue sequences of a a satisfy i q2i i q 2 d 2 i for some nonzero scalars q f where q is not a root of unity assume that not all eigenvalues of a and a have multiplicity one let m and m denote the subalgebras of end v generated by a and a respectively and assume that v mv m v for some eigenvectors v of a associated with 0 and v of a associated with d we find a nice basis for v and describe the action of a a on this basis in terms of six parameters

matrix units associated with the split basis of a leonard pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v it is known that there exists a basis for v with respect to which the matrix representing a is lower bidiagonal and the matrix representing a is upper bidiagonal in this paper we give some formulae involving the matrix units associated with this basis

sharp tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 a conjecture of tatsuro ito and the second author states that if k is algebraically closed then a a is sharp in order to better understand and eventually prove the conjecture in this paper we begin a systematic study of the sharp tridiagonal pairs our results are summarized as follows assuming a a is sharp and using the data a vi i 0d a vi i 0d we define a finite sequence of scalars called the parameter array we display some equations that show the geometric significance of the parameter array we show how the parameter array is affected if is replaced by a vi i 0d a vi i 0d or a vd i i 0d a vi i 0d or a vi i 0d a vd i i 0d we prove that if the isomorphism class of is determined by the parameter array then there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

irreducible modules for the quantum affine algebra uq slˆ2 and its borel subalgebra abstract let u q b denote the standard borel subalgebra of the quantum affine algebra u q sl 2 we show that the following hold for any choice of scalars 0 1 from the set 1 1 i let v be a finite dimensional irreducible u q b module of type 0 1 then the action of u q b on v extends uniquely to an action of u q sl 2 on v the resulting u q sl 2 module structure on v is irreducible and of type 0 1 ii let v be a finite dimensional irreducible u q sl 2 module of type 0 1 when the u q sl 2 action is restricted to u q b the resulting u q b module structure on v is irreducible and of type 0 1

the equitable presentation for the quantum group uq g associated with a symmetrizable kac moody algebra g abstract we consider the quantum group u q g associated with a symmetrizable kac moody algebra g we display a presentation for u q g that we find attractive we call this the equitable presentation for g sl 2 the equitable presentation has generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1

a bilinear form for tridiagonal pairs of q serre type abstract let v denote a vector space of finite positive dimension and let v denote the vector space dual of v let a a be a tridiagonal pair on v and let a v v a v v be linear transformations such that for all v v and f v a f v f av and a f v f a v we show that a a is a tridiagonal pair on v we then show that if a a is of q serre type then a a is isomorphic to a a we also show that in this case there exist a unique bilinear form on v such that for all u v v au v u av and a u v u a v moreover this bilinear form is nondegenerate and symmetric

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

nonlinear quantum dynamical system based on the kadomtsev petviashvili ii equation the structure of soliton solutions of classical integrable nonlinear evolution equations which can be solved through the hirota transformation suggests a new way for the construction of nonlinear quantum dynamical systems that are based on the classical equations in the new approach the classical soliton solution is mapped into an operator u which is a nonlinear functional of the particle number operators over a fock space of quantum particles u obeys the evolution equation the classical soliton solutions are the eigenvalues of u in multi particle states in the fock space the construction easily allows for the incorporation of particle interactions which generate soliton effects that do not have a classical analog in this paper this new approach is applied to the case of the kadomtsev petviashvili ii equation the nonlinear quantum dynamical system describes a set of m 2s 1 particles with intrinsic spin s which interact in clusters of 1 n m 1 particles

balanced leonard pairs let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal let v0 v1 vd respectively v0 v1 vd denote a basis for v that satisfies i respectively ii for 0 i d let ai denote the coefficient of vi when we write avi as a linear combination of v0 v1 vd and let ai denote the coefficient of vi when we write a vi as a linear combination of v0 v1 vd in this paper we show a0 ad if and only if a0 ad moreover we show that for d 1 the following are equivalent i a0 ad and a1 ad 1 ii a0 ad and a1 ad 1 iii ai ad i and ai ad i for 0 i d these give a proof of a conjecture by the second author we say a a is balanced whenever ai ad i and ai ad i for 0 i d we say a a is essentially bipartite respectively essentially dual bipartite whenever ai respectively ai is independent of i for 0 i d observe that if a a is essentially bipartite or dual bipartite then a a is balanced for d 2 we show that if a a is balanced then a a is essentially bipartite or dual bipartite

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

algebro geometric solution of the discrete kp equation over a finite field out of a hyperelliptic curve we transfer the algebro geometric method of construction of solutions of the discrete kp equation to the finite field case we emphasize the role of the jacobian of the underlying algebraic curve in construction of the solutions we illustrate in detail the procedure on example of a hyperelliptic curve

affine transformations of a sharp tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension by a tridiagonal pair we mean an ordered pair a a of k linear transformations from v to v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 0 i d where v 1 0 v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 0 i where v 1 0 v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v it is known that a i a i is also a tridiagonal pair on v where are scalars in k with nonzero in this paper we give the necessary and sufficient conditions for these tridiagonal pairs to be isomorphic to a a or a a we do this under a mild assumption called the sharp condition

tridiagonal pairs of height one let a a denote a tridiagonal pair on a vector space v over a field k let v0 vd denote a standard ordering of the eigenspaces of a on v and let 0 d denote the corresponding eigenvalues of a we assume d 3 let q denote a scalar taken from the algebraic closure of k such that q2 q 2 1 3 0 2 1 we assume q is not a root of unity let i denote the dimension of vi the sequence 0 1 d is called the shape of the tridiagonal pair it is known there exists a unique integer h 0 h d 2 such that i 1 i for d h i d the integer h is known as the height of the tridiagonal pair in this paper we show that the shape of a tridiagonal pair of height one with 0 1 is either 1 2 2 2 1 or 1 3 3 1 in each case we display a basis for v and give the action of a a on this basis

digital speech coder having improved vector excitation source an improved excitation vector generation and search technique fig 1 is described for a code excited linear prediction celp speech coder 100 using a codebook of excitation code vectors a set of m basis vectors vm n are used along with the excitation signal codewords i to generate the codebook of excitation vectors ui n according to a vector sum technique 120 of converting the selector codewords into a plurality of interim data signals multiplying the set of m basis vectors by the interim data signals and summing the resultant vectors to produce the set of 2m codebook vectors the entire codebook of 2m possible excitation vectors is efficiently searched by using the vector sum generation technique with the m basis vectors without ever having to generate and evaluate each of the 2m code vectors themselves furthermore only m basis vectors need to be stored in memory 114 as opposed to all 2m code vectors

a refinement of the split decomposition of a tridiagonal pair let v denote a nonzero finite dimensional vector space over a field k and let a a denote a tridiagonal pair on v of diameter d let v u0 ud denote the split decomposition and let i denote the dimension of ui in this paper at first we show there exists a unique integer h 0 h d 2 such that i 1 i for d h i d we call h the height of the tridiagonal pair for 0 r h we define subspaces ui r r i d r by ui r ri r ur kerrd 2r 1 where r denotes the rasing map we show v is decomposed as a direct sum v r 0h i rd rui r this gives a refinement of the split decomposition define u r i rd rui r and observe v r 0hu r we show lu r u r 1 u r u r 1 for 0 r h where we set u 1 u h 1 0 let f r v u r denote the projection we show the lowering map l is decomposed as l l l 0 l where l r 1hf r 1 lf r l 0 r 0hf r lf r and l r 0h 1f r 1 lf r these maps satisfy l u r u r 1 l 0 u r u r and l u r u r 1 for 0 r h the main results of this paper are the following i for 0 r h 1 and r 2 i d r 1 rl l r holds on ui r for some scalar ii for 1 r h and r i d r 1 rl l r holds on ui r for some scalar iii for 0 r h and r 1 i d r 1 rl 0 l 0 r i holds on ui r for some scalars moreover we give explicit expressions of

towards a classification of the tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension let end v denote the k algebra consisting of all k linear transformations from v to v we consider a pair a a end v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v let e0 denote the element of end v such that e0 i v0 0 and e0 vi 0 for 1 i d let d resp d denote the k subalgebra of end v generated by a resp a in this paper we prove that the span of e0 dd de0 equals the span of e0 de0 de0 and that the elements of e0 de0 mutually commute we relate these results to some conjectures of tatsuro ito and the second author that are expected to play a role in the classification of tridiagonal pairs

two linear transformations each tridiagonal with respect to an eigenbasis of the other comments on the split decomposition let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and a v v that satisfy both conditions below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v referring to the above leonard pair it is known there exists a decomposition of v into a direct sum of one dimensional subspaces on which a acts in a lower bidiagonal fashion and a acts in an upper bidiagonal fashion this is called the split decomposition in this paper we give two characterizations of a leonard pair that involve the split decomposition

three mutually adjacent leonard pairs abstract let k denote a field of characteristic 0 and let v denote a vector space over k with positive finite dimension consider an ordered pair of linear transformations a v v and a v v that satisfies both conditions below i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v let a a denote a leonard pair on v a basis for v is said to be standard for a a whenever it satisfies i or ii above a basis for v is said to be split for a a whenever with respect to this basis the matrix representing one of a a is lower bidiagonal and the matrix representing the other is upper bidiagonal let a a and b b denote leonard pairs on v we say these pairs are adjacent whenever each basis for v which is standard for a a resp b b is split for b b resp a a our main results are as follows theorem 1 there exist at most 3 mutually adjacent leonard pairs on v provided the dimension of v is at least 2 theorem 2 let a a b b and c c denote three mutually adjacent leonard pairs on v then for each of these pairs the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression theorem 3 let a a denote a leonard pair on v whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression then there exist leonard pairs b b and c c on v such that a a b b and c c are mutually adjacent

a differential difference system related to toroidal lie algebra we present a novel differential difference system in 2 1 dimensional space time one discrete two continuum arising from bogoyavlensky s 2 1 dimensional korteweg de vries hierarchy our method is based on the bilinear identity of the hierarchy which is related to the vertex operator representation of the toroidal lie algebra tor2

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

a method of coding a sampled speech signal vector the invention relates to a method of coding a sampled speech signal vector by selecting an optimal axcitation vector in an adaptive code book 100 this optimal excitation vector is obtained by maximizing the energy normalized square of the cross correlation between the convolution 102 of the excitation vectors with the impulse response h w n of a linear filter and the speech signal vector before the convolution the vectors of the code book 100 are block normalized 200 with respect to the vector component largest in magnitude in a similar way the speech signal vector s n is block normalized 202 with respect to its component largest in magnitude calculated values for the squared cross correlation c i and the energy e i and corresponding values c m e m for the best excitation vector so far are divided into a mantissa and a scaling factor with a limited number of scaling levels the number of levels can be different for squared cross correlation and energy during the calculation of the products c i e m and e i c m which are used for determining the optimal excitation vector the respective mantissas are multiplied and a seperate scaling factor calculation is performed

analyzing the combined multi waves polynomial solutions in a two layer liquid medium abstract in this work we investigate the behavior of mixed wave solutions in two layer liquid with dispersive waveguide these combined multi wave solutions were obtained in polynomial types for the 3 1 dimensional yu toda sasa fukuyama ytsf equation with variable coefficients by virtue of the generalized unified method and symbolic computations we formally derive polynomial solutions for this equation the obtained solutions include multi soliton solutions multi periodic solutions and multi elliptic solutions furthermore the physical insight and the movement role of the waves in the two layers are discussed and analyzed graphically for different selections of the arbitrary functions of the obtained solutions

the traces associated with a sharp tridiagonal system let kk denote a field and let v denote a vector space over kk with finite positive dimension let a a a a denote a tridiagonal pair on v let i i 0d resp i i 0d denote a standard ordering of the eigenvalues of a resp a a and for 0 i d0 i d let vivi resp vi be the eigenspace of a resp a a associated with i i resp i it is known that vi vi have the same dimension the tridiagonal pair a a a a is said to be sharp whenever dim v0 1dim v0 1 for 0 i d0 i d let eiei resp ei denote the primitive idempotent of a resp a a associated with i i resp i then a e0 e1 ed a e0 e1 ed is a tridiagonal system on v we say is sharp whenever the tridiagonal pair a a a a is sharp assume is sharp and let i i 0d denote the split sequence of the sequence i i 0d i i 0d i i 0d is called the parameter array of recently in 3 k nomura and p terwilliger proposed the following problem let denote a sharp tridiagonal system for 0 i d0 i d find each of tr eie0 tr eied tr ei e0 tr ei ed in terms of the parameter array of in the present paper we solve this problem

on elliptic lax systems on the lattice and a compound theorem for hyperdeterminants a general elliptic n n matrix lax scheme is presented leading to two classes of elliptic lattice systems one which we interpret as the higher rank analogue of the landau lifschitz equations while the other class we characterize as the higher rank analogue of the lattice krichever novikov equation or adler s lattice we present the general scheme but focus mainly on the latter type of models in the case n 2 we obtain a novel lax representation of adler s elliptic lattice equation in its so called 3 leg form the case of rank n 3 is analyzed using cayley s hyperdeterminant of format yielding a multi component system of coupled 3 leg quad equations

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

multiple addition theorem for discrete and continuous nonlinear problems the addition relation for the riemann theta functions and for its limits which leads to the appearance of exponential functions in soliton type equations is discussed the form of addition property presented resolves itself to the factorization of the n tuple product of the shifted functions and it seems to be useful for analysis of soliton type continuous and discrete processes in the n 1 space time a close relation with the natural generalization of bi and tri linear operators into multiple linear operators concludes the paper

tridiagonal pairs of krawtchouk type let f denote an algebraically closed field with characteristic 0 and let v denote a vector space over f with finite positive dimension let a a denote a tridiagonal pair on v with diameter d we say that a a has krawtchouk type whenever the sequence d 2i i 0d is a standard ordering of the eigenvalues of a and a standard ordering of the eigenvalues of a assume a a has krawtchouk type we show that there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for u v v we show that the following tridiagonal pairs are isomorphic i a a ii a a iii a a iv a a we give a number of related results and conjectures

a family of tridiagonal pairs and related symmetric functions a family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in detail the corresponding eigenvalue sequences eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described the overlap functions between the two dual bases are shown to satisfy a coupled system of recurrence relations and a set of discrete second order q difference equations which generalize those associated with the askey wilson orthogonal polynomials with a discrete argument normalizing the fundamental solution to unity the hierarchies of solutions are rational functions of one discrete argument explicitly derived in some simplest examples the weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given

the shape of some linear transformations abstract recently ito and terwilliger proposed a problem about some linear transformations a a a a on a finite dimensional vector space v over a field f which can be seen as a generalization of the notions of tridiagonal pair and q inverting pair see problem 1 2 in this paper we will introduce the notion of the shape of the linear transformations a a a a our main results are the properties of the shape i the shape is symmetric and unimodal ii let denote an anti isomorphism from end v to end v with v a vector space over f with the same dimension as v assume that the linear transformations a a a a on v satisfy all conditions of problem 1 2 then the linear transformations a a a a also satisfy all conditions of problem 1 2 moreover a a a a have the same shape as a a a a

the switching element for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let v i i 0 d resp w i i 0 d denote a basis for v referred to in i resp ii we show that there exists a unique linear transformation s v v that sends v 0 to a scalar multiple of v d fixes w 0 and sends w i to a scalar multiple of w i for 1 i d we call s the switching element we describe s from many points of view

periodic multiwave solutions to the toda type cellular automaton an ultradiscretization of the riemann theta function is proposed the ultradiscretization satisfies an addition formula which is an ultradiscrete analogue of an addition formula for the riemann theta function using the addition formula periodic multiwave solutions to the toda type cellular automaton are obtained

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

mock tridiagonal systems we introduce the notion of a mock tridiagonal system this is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition we show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications this paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

tridiagonal pairs of q racah type and the μ conjecture abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v d i v i v d i coincide we say the pair a a is sharp whenever dim v 0 1 it is known that if k is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by ito and the second author shortly afterwards we introduced a conjecture called the conjecture which implies the classification conjecture in this paper we show that the conjecture holds in a special case called q racah

the tetrahedron algebra and its finite dimensional irreducible modules abstract recently terwilliger and the present author found a presentation for the three point sl 2 loop algebra via generators and relations to obtain this presentation we defined a lie algebra by generators and relations and displayed an isomorphism from to the three point sl 2 loop algebra in this paper we classify the finite dimensional irreducible modules

two linear transformations each tridiagonal with respect to an eigenbasis of the other the td d canonical form and the lb ub canonical form abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and b v v which satisfy both i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we call such a pair a leonard pair on v we introduce two canonical forms for leonard pairs we call these the td d canonical form and the lb ub canonical form in the td d canonical form the leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix subject to a certain normalization in the lb ub canonical form the leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix subject to a certain normalization we describe the two canonical forms in detail as an application we obtain the following results given square matrices a b over k with a tridiagonal and b diagonal we display a necessary and sufficient condition for a b to represent a leonard pair given square matrices a b over k with a lower bidiagonal and b upper bidiagonal we display a necessary and sufficient condition for a b to represent a leonard pair we briefly discuss how leonard pairs correspond to the q racah polynomials and some related polynomials in the askey scheme we present some open problems concerning leonard pairs

painleve property auto backlund transformation lax pairs and reduction to the standard form for the korteweg de vries equation with nonuniformities it is demonstrated that the kdv equation with nonuniformities ut a t u b x t u x c t uux d t uxxx e x t 0 has the painleve property if the compatibility condition among the coefficients of it holds bt a lc b bbx dbxxx 2ah hl d c2 dh dt ce x 2a2 al d3 c4 da dt l d c l d c2 d dt l d c where l d dt lg and h t is an arbitrary function of t the auto backlund transformation and lax pairs for this equation are obtained by truncating the laurent expansion furthermore assuming the compatibility condition then the kdv equation with nonuniformities is transformable via suitable variable transformations to the standard kdv

exact periodic cross kink wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional potential ytsf equation abstract in this paper the 3 1 dimensional potential ytsf equation is investigated exact solutions with three wave form including periodic cross kink wave periodic two solitary wave and breather type of two solitary wave solutions are obtained using hirota s bilinear form and generalized three wave approach with the aid of symbolic computation moreover the properties for some new solutions are shown with some figures

time discretization of the spin calogero moser model and the semi discrete matrix kp hierarchy we introduce the discrete time version of the spin calogero moser system the equations of motion follow from the dynamics of poles of rational solutions to the matrix kadomtsev petviashvili hierarchy with discrete time the dynamics of poles is derived using the auxiliary linear problem for the discrete flow

tridiagonal pairs and the μ conjecture let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by t ito and the second author we present a result which supports the conjecture given scalars i i 0d i i 0d in f that satisfy the known constraints on the eigenvalues of a tridiagonal pair we define an f algebra t by generators and relations we consider the f algebra e0 te0 for a certain idempotent e0 t let f x1 xd denote the polynomial algebra over f involving d mutually commuting indeterminates we display a surjective f algebra homomorphism f x1 xd e0 te0 we conjecture that is an isomorphism we show that this conjecture implies the classification conjecture and that the conjecture holds for d 5

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

a new variable coefficient korteweg de vries equation based sub equation method and its application to the 3 1 dimensional potential ytsf equation with the aid of maple we present variable coefficient korteweg de vries equation based sub equation method the key idea of our method is to take advantage of the variable coefficient korteweg de vries equation and its various solutions to generate various solutions of nonlinear evolution equations the efficiency of the method can be demonstrated on the 3 1 dimensional potential ytsf equation and we construct successfully its new styles of solutions

the hirota equation over finite fields algebro geometric approach and multisoliton solutions we consider the hirota equation the discrete analogue of the generalized toda system over a finite field we present the general algebro geometric method of construction of solutions of the equation as an example we construct analogues of the multisoliton solutions for which the wavefunctions and the function can be found using rational functions within the class of multisoliton solutions we isolate generalized breather type solutions which have no direct counterparts in the complex field case

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

some properties of sharp hessenberg pairs abstract let f denote a field and let v denote a vector space over f with finite positive dimension we consider an ordered pair of f linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable on v ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v 0 v 1 v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v 0 v 1 v i 1 for 0 i where v 1 0 and v 1 0 we call such a pair a hessenberg pair on v it is known that if the hessenberg pair a a on v is irreducible then d and for 0 i d the dimensions of v i and v d i coincide we say a hessenberg pair a a on v is sharp whenever it is irreducible and dim v 0 dim v d 1 in this paper we give the definitions of a hessenberg system and a sharp hessenberg system we discuss the connection between a hessenberg pair and a hessenberg system we also define a finite sequence of scalars called the parameter array for a sharp hessenberg system which consists of the eigenvalue sequence the dual eigenvalue sequence and the split sequence we calculate the split sequence of a sharp hessenberg system we show that a sharp hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on v that satisfies au v u av and a u v u a v for all u v v

q inverting pairs of linear transformations and the q tetrahedron algebra abstract as part of our study of the q tetrahedron algebra q we introduce the notion of a q inverting pair roughly speaking this is a pair of invertible semisimple linear transformations on a finite dimensional vector space each of which acts on the eigenspaces of the other according to a certain rule our main result is a bijection between the following two sets i the isomorphism classes of finite dimensional irreducible q modules of type 1 ii the isomorphism classes of q inverting pairs

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

new kink multi soliton solutions for the 3 1 dimensional potential yu toda sasa fukuyama equation by means of transformation of unknown function the non integrable system of 3 1 dimensional potential yu toda sasa fukuyama ytsf equation is converted into the combined equation of differently two bilinear forms by the darvishi s idea some new kink multi soliton solutions are obtained by using homoclinic test approach and three wave method respectively

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

leonard pairs and the q racah polynomials abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal wecall such a pair a leonard pair on v in the appendix to linear algebra appl 330 2001 p 149 we outlined a correspondence between leonard pairs and a class of orthogonal polynomials consisting of the q racah polynomials and some related polynomials of the askey scheme we also outlined how for the polynomials in this class the 3 term recurrence difference equation askey wilson duality and orthogonality can be obtained in a uniform manner from the corresponding leonard pair the purpose of this paper is to provide proofs for the assertions which we made in that appendix

method of and device for speech signal coding and decoding by parameter extraction and vector quantization techniques this method provides a filtering of blocks of digital samples of speech signal by a linear prediction inverse filter followed by a shaping filter whose coefficients are chosen out of a codebook of quantized filter coefficient vectors obtaining a residual signal subdivided into vectors each vector is classified by an index q depending on the zero crossing frequency and r m s value it is then normalized on the basis of the quantized r m s value and then of a vector of quantized short term mean values the mean square error made in quantizing said vectors with vectors contained in a codebook and forming excitation waveforms in computed in this codebook the search is limited to a subset of vectors determined by index q and p of short term mean vector the coding signal consists of the index of the filter coefficient vector of indices q p of quantization index m of the r m s value and of the index of the vector of the excitation waveform which has generated minimum weighted mean square error fig 1

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

some trace formulae involving the split sequences of a leonard pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let diag 0 1 d denote the diagonal matrix referred to in ii above and let diag 0 1 d denote the diagonal matrix referred to in i above it is known that there exists a basis u0 u1 ud for v and there exist scalars 1 2 d in k such that aui iui ui 1 0 i d 1 aud dud a ui iui 1 i ui 1 i d a u0 0 u0 the sequence 1 2 d is called the first split sequence of the leonard pair it is known that there exists a basis v0 v1 vd for v and there exist scalars 1 2 d in k such that avi d ivi vi 1 0 i d 1 avd 0vd a vi ivi 1 i vi 1 i d a v0 0 v0 the sequence 1 2 d is called the second split sequence of the leonard pair we display some attractive formulae for the first and second split sequence that involve the trace function

n soliton solutions to the bogoyavlenskii schiff equation and a quest for the soliton solution in 3 1 dimensions we study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the hirota s bilinear form we explicitly discuss the bogoyavlenskii schiff equation in 2 1 dimensions its analytical proof of multisoliton solution and a new feature are given being guided by the strong symmetry we also propose a new equation in 3 1 dimensions

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

multiple soliton solutions for the calogero bogoyavlenskii schiff jimbo miwa and ytsf equations abstract in this work we employ the hirota s bilinear method to derive multiple front solutions for nonlinear evolution equations three models namely the 2 1 dimensional calogero bogoyavlenskii schiff equation the 3 1 dimensional jimbo miwa equation and the 3 1 dimensional potential ytsf equation are used as vehicles to conduct the analysis the work shows the power of the bilinear method and the variety of the obtained front solutions

lie symmetry analysis and exact solutions of 3 1 dimensional yutodasasafukuyama equation in mathematical physics in this paper the lie symmetry analysis method has been proposed for finding similarity reduction and exact solutions of nonlinear evolution equation here for illustrating the effectiveness and accuracy of proposed method we have taken 3 1 dimensional yutodasasafukuyama ytfs equation also by using symmetry reduction method we have reduced nonlinear partial differential equation npde to nonlinear ordinary differential equation which has advantage to provide semi analytical solution we have obtained infinitesimal generators the entire geometric vector field commutator table of lie algebra and symmetry group by using lie symmetry analysis method then we have used tanh method for finding new analytical exact solutions of some reduced transform equations

the determinant of aa a a for a leonard pair a a abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in this paper we investigate the commutator aa a a our results are as follows abbreviate d dim v 1 and first assume d is odd we show aa a a is invertible and display several attractive formulae for the determinant next assume d is even we show that the null space of aa a a has dimension 1 we display a nonzero vector in this null space we express this vector as a sum of eigenvectors for a and as a sum of eigenvectors for a

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

dispersion equations and a comparison of different quasi periodic solutions of the sine gordon equation a system of dispersion equations for quasi periodic solutions of the multidimensional sine gordon equation is discussed this system of algebraical equations determines the parameters appearing in the solution which involves abstract theta functions in the case of the two phase quasi periodic solutions it is shown that the form involving theta functions represents a broader class than the class of solutions given by the expression 4 tan 1 fg the condition for the equivalence of both classes is also reported