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

dynamics of complex interfaces complex interfacial dynamics is studied in an oscillatory medium described by a deterministic coupled map lattice this dynamical system supports only stable periodic attractors the interfaces that separate the stable homogeneous phases exhibit different types of behavior ranging from simple planar fronts with low periodicity to highly irregular fronts with complex spatiotemporal transients a dynamical analysis of the system is carried out for a small interface length l in which the probabilities of occurrence of given periodic orbits the velocities of the corresponding interfaces and lyapunov exponents are calculated the importance of transient dynamics for large l is demonstrated in the large l regime the interfacial evolution and structure are characterized in statistical terms and the simulation results are compared with phenomenological stochastic models such as edwards wilkinson and kardar parisi zhang equations in some parameter regions the deterministic transient interfacial dynamics of the coupled map model is described well by such models if finite size effects are taken into account nucleation and growth dynamics are also investigated the system provides a framework in which to study complex interfacial structures

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

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

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

the classification of topologically expansive lorenz maps we show that topologically expansive lorenz maps can be described up to topological conjugacy by their kneading invariants we also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive lorenz map a simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described

prime and renormalisable kneading invariants and the dynamics of expanding lorenz maps abstract we study the dynamics and kneading theory of topologically expansive piecewise increasing maps of the interval with a single discontinuity lorenz maps we characterise those maps which are locally eventually onto locally onto and neither of the above in terms of the renormalisability or otherwise of the kneading invariant we also show how the decomposition of the non wandering set is related to the renormalisability of the kneading invariant we discuss our results in the context of other authors results on similar classes of maps

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

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

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

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

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

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

soliton solutions of a few nonlinear wave equations this paper carries out the integration of a few nonlinear wave equations to obtain topological as well as non topological soliton solutions the mathematical techniques used to obtain the soliton solutions are he s variational iteration method the tanh method and the ansatz method the nonlinear wave equations that are studied are coupled mkdv equations drinfeld sokolov equation and its generalized version finally some numerical simulations are given to support the analytical solutions

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

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

one dimensional dynamics for traveling fronts in coupled map lattices multistable coupled map lattices typically support traveling fronts separating two adjacent stable phases we show how the existence of an invariant function describing the front profile allows a reduction of the infinitely dimensional dynamics to a one dimensional circle homeomorphism whose rotation number gives the propagation velocity the mode locking of the velocity with respect to the system parameters then typically follows we study the behavior of fronts near the boundary of parametric stability and we explain how the mode locking tends to disappear as we approach the continuum limit of an infinite density of sites

the emergence of coherent structures in coupled map lattices we show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices this phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics if we further increase the coupling then new ground states appear showing the coexistence of a large scale coherent structure with a small scale chaotic motion this allows us to propose a generalization of the notion of spatial intermittency

explicit exact solutions for the generalized zakharov equations with nonlinear terms of any order in this paper an auxiliary ordinary differential equation with nonlinear terms of any order is introduced and its exact solutions are obtained by means of the auxiliary equation and its solutions abundant explicit exact solutions to the generalized zakharov equations with nonlinear terms of any order are obtained in a concise manner that include new solitary wave solutions and triangular periodic wave solutions

the sine cosine and the tanh methods reliable tools for analytic treatment of nonlinear dispersive equations the sine cosine method and the tanh method are used for analytic treatment of nonlinear dispersive equations nonlinear variants of boussinesq equation are used as vehicles to show the strength of these methods solutions of distinct physical structures solitons compactons solitary patterns solutions and periodic solutions are formally derived the results show that the change in physical structures of the obtained solutions depends mainly on exponents and on the coefficients of the derivatives involved

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

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

variational approach to solitary wave solution of the generalized zakharov equation he s semi inverse method is applied to search for the solitary wave solution of the generalized zakharov equation gze the solution process reveals that the method is easy and straightforward

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

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

robust space time intermittency and 1 f noise abstract we investigate a type of intermittency that occurs in space as well as time studying a one dimensional lattice of coupled quadratic maps this system naturally forms spatial domains motion of the domain walls causes spatially localized changes from chaotic to almost periodic behavior the almost periodic phases have eigenvalues quite close to one resulting in long lived laminar bursts with a 1 f low frequency spectrum this behavior has aspects of both crutchfield and pomeau manneville intermittency unlike pomeau manneville however the behavior that we observe here is quite robust under changes of parameters

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

1 soliton solution of the d m n equation with generalized evolution abstract this paper obtains the 1 soliton solution of the nonlinear dispersive drinfel d sokolov equation with power law nonlinearity in the first case the soliton solution is without the generalized evolution the solitary wave ansatz method is used to carry out the integration subsequently the he s semi inverse variational principle is used to integrate the equation with power law nonlinearity parametric conditions for the existence of envelope solitons are given

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

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

1 soliton solution of the generalized zakharov equation in plasmas by he s variational principle this paper obtains the 1 soliton solution to zakharov equation with power law and dual power law nonlinearities the he s variational principle is used to carry out the integration of this equation

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

chaotic traveling waves in a coupled map lattice traveling waves triggered by a phase slip are studied in a coupled map lattice a local phase slip affects globally the system which is in strong contrast with kink propagation attractors with different velocities coexist and form quantized bands determined by the number of phase slips if the system size is not far from an integer multiple of the selected wavelength attractors are tori otherwise weak chaos induces modulation of waves or a chaotic itinerancy of traveling states with a long ranged temporal correlation supertransients before the formation of traveling waves are noted in the high nonlinearity regime in the weaker nonlinearity there are fluctuations of domain sizes and brownian like motion of domains propagation of chaotic domains by phase slips is also found

mode locking in coupled map lattices abstract we study propagation of pulses along one way coupled map lattices which originate from the transition between two superstable states of the local map the velocity of the pulses exhibits a staircase like behaviour as the coupling parameter is varied for a piecewise linear local map we prove that the velocity of the wave has a devil s staircase dependence on the coupling parameter a wave travelling with rational velocity is found to be stable to parametric perturbations in a manner akin to rational mode locking for circle maps we provide evidence that mode locking is also present for a broader range of maps and couplings

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

synchronization of spatiotemporal chaotic systems by feedback control we demonstrate that two identical spatiotemporal chaotic systems can be synchronized by 1 linking one or a few of their dynamical variables and 2 applying a small feedback control to one of the systems numerical examples using the diffusively coupled logistic map lattice are given the effect of noise and the limitation of the technique are discussed

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

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

dynamical survey of a generalized zakharov equation and its exact travelling wave solutions we investigate the dynamical behavior of a generalized zakharov equation for the complex envelope of the high frequency wave and the real low frequency field by analyzing its phase portraits following a dynamical system approach in different parameter regions we depict phase portraits of a travelling wave system through discussing the bifurcation of phase portraits we unwrap explicit miscellaneous travelling waves including localized and periodic ones

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

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

spatially periodic orbits in coupled map lattices we obtain the conditions that ensure the stability of spatially and temporally periodic orbits of coupled map lattices the stability matrices can be put in a circulant and block circulant form this allows us to reduce the problem to smaller matrices corresponding to the building blocks of spatial periodicity we find that additional conditions are imposed as we expand the size of the lattice for the traveling wave solution the analysis is considerably simplified we have analyzed both the one dimensional and higher dimensional lattices

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

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