a generalization of the caginalp phase field system based on the cattaneo law abstract we consider in this article a generalization of the caginalp model for phase transitions based on the maxwell cattaneo law instead of the classical fourier law in particular we prove the existence and uniqueness of solutions we finally study the spatial behavior of the solutions in a semi infinite cylinder

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

leonard pairs in classical mechanics leonard pairs lp are matrices with the property of mutual tri diagonality we introduce and study a classical analogue of lp we show that corresponding classical leonard dynamical variables satisfy non linear relations of the aw type with respect to poisson brackets

quasilinear elliptic equations with quadratic growth in unbounded domains on donne des resultats d existence et de regularite pour des equations elliptiques quasilineaires a croissance quadratique par rapport au gradient du type au f x u grad u 0 dans u 0 sur

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 q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

quasilinear elliptic problems with singular and homogeneous lower order terms abstract we deal with singular quasilinear elliptic equations namely u u x u q u q 1 f x in u 0 in u 0 on where is a bounded smooth domain of r n n 3 r 1 q 2 0 l and 0 f l p for some p n 2 we completely describe the set of values of the parameter for which the problem admits solution thus we study existence nonexistence and uniqueness of bounded weak solutions in both cases f 0 and f 0

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

uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions in this paper we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions

multiplicity results in the non coercive case for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u c x u x u 2 h x u h 0 1 l where r n n 3 is a bounded domain with smooth boundary it is assumed that c h belong to l p for some p n with c 0 as well as l and 1 0 for some 1 r it is known that when 0 problem p has at most one solution in this paper we study under various assumptions the structure of the set of solutions of p assuming that 0 our study unveils the rich structure of this problem we show in particular that what happen for 0 influences the set of solutions in all the half space 0 h 0 1 l most of our results are valid without assuming that h has a sign if we require h to have a sign we observe that the set of solutions differs completely for h 0 and h 0 we also show when h has a sign that solutions not having this sign may exists some uniqueness results of signed solutions are also derived the paper ends with a list of open problems

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

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

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

on q orthogonal polynomials dual to little and big q jacobi polynomials we derive discrete orthogonality relations for polynomials dual to little and big q jacobi polynomials this derivation essentially requires use of bases consisting of eigenvectors of certain self adjoint operators which are representable by a jacobi matrix recurrence relations for these polynomials are also given

a unified procedure for construction of theories of deformable media i classical continuum physics a new unified procedure for constructing continuum theories of deformable media is presented and used in this and a companion paper the procedure starts with a balance of energy and derives from it all the relevant balance laws that may also include those that are associated with thermal electrical and magnetic effects the basic energetic ingredients that are included in the balance of energy depend of course on the nature of the particular theory of material behaviour desired the advantage of the new procedure becomes especially apparent when one considers formulation of a new theory of material behaviour for which additional balance laws involving new kinetic quantities are required to accompany any additional basic kinematic and thermal variables additional to those in the classical formulation indeed in the formulation of such new theories usually little or no previous information is available concerning properties of the new kinetic quantities in the additional balance laws and in this connection the unified procedure of this paper provides a simple attractive setting for deriving the basic equations that are automatically consistent with the energy balance in this paper first the basic features of the new procedure are illustrated in the context of classical thermomechanics generalizations of this thermomechanical theory are then discussed in two cases 1 in the presence of an additional kinematic variable and 2 in the presence of full electromagnetic effects both of these generalizations bring out some interesting novel features when new theories are being constructed

continuum of solutions for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u h 0 1 l u c x u x u 2 h x where r n n 3 is a bounded domain with smooth boundary it is assumed that c 0 c h belong to l p for some p n 2 and that l we explicitly describe a condition which guarantees the existence of a unique solution of p when 0 and we show that these solutions belong to a continuum the behaviour of the continuum depends in an essential way on the existence of a solution of p 0 it crosses the axis 0 if p 0 has a solution otherwise it bifurcates from infinity at the left of the axis 0 assuming that p 0 has a solution and strengthening our assumptions to x 1 0 and h 0 we show that the continuum bifurcates from infinity on the right of the axis 0 and this implies in particular the existence of two solutions for any 0 sufficiently small

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

nonlinear elliptic systems in stochastic game theory on etudie des systemes d equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de nash

growth and uniqueness in thermoelasticity a uniqueness theorem is proved for two theories of thermoelasticity capable of admitting finite speed thermal waves the theories having been proposed by green naghdi uniqueness is proved under

onsager s algebra and the dolan grady condition in the non self dual case in this paper it is shown that a pair of operators that satisfy the dolan grady conditions generate an onsager algebra even in the case that they are not self dual briefly considered will be the implications for the spectra of superintegrable spin chains in various sectors

a priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient abstract we study nonnegative solutions of the boundary value problem p u c x u x u 2 h x u h 0 1 l where is a smooth bounded domain of r n c l h l r for some r n 2 and c h 0 our main motivation is to study the semidefinite case namely unlike in previous work on the subject we do not assume to be uniformly positive in nor even positive everywhere in space dimensions up to n 5 we establish uniform a priori estimates for weak solutions of p when 0 is bounded away from 0 this is proved under the assumption that the supports of and c intersect a condition that we show to be actually necessary and in some cases we further assume that is uniformly positive on the support of c and or some other conditions as a consequence of our a priori estimates assuming that p 0 has a solution we deduce the existence of a continuum c of solutions such that the projection of c onto the axis is an interval of the form 0 a for some a 0 and that the continuum c bifurcates from infinity to the right of the axis 0 in particular for each 0 small enough problem p has at least two distinct solutions

onsager s algebra and superintegrability the author considers the irreducible representations of the onsager algebra and shows that for a finite system possessing such an algebra leads to an ising like structure in the spectra of associated hamiltonians and transfer matrices the chiral potts model is considered as an example for transfer in the diagonal direction it is known to be superintegrable for transfer in the principal direction a new superintegrable solution manifold is found

energy decay of type iii linear thermoelastic plates with memory abstract we analyze the decay properties of the solution semigroups generated by an abstract version of the linear systems u t t 2 u t 0 t t t u t 0 and u t t 2 u t 0 t t 0 s t t s d s u t 0 ruling the evolution of linear thermoelastic plates within the theory of heat conduction of type iii

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

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

uniqueness results for nonlinear elliptic equations with a lower order term abstract we prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u w 0 1 p div a x 1 u 2 p 2 2 u b x 1 u 2 1 2 f in d where is a bounded open subset of r n n 2 p satisfies p 2 n n 1 a is a function belonging to l such that a x 0 f belongs to the dual space w 1 p b belongs to some lebesgue space l r with r r n p and belongs to the interval 0 n p r with n p r and r n p functions which are specified below

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

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

the terwilliger algebra of a distance regular graph of negative type abstract let denote a distance regular graph with diameter d 3 assume has classical parameters d b with b 1 let x denote the vertex set of and let a mat x c denote the adjacency matrix of fix x x and let a mat x c denote the corresponding dual adjacency matrix let t denote the subalgebra of mat x c generated by a a we call t the terwilliger algebra of with respect to x we show that up to isomorphism there exist exactly two irreducible t modules with endpoint 1 their dimensions are d and 2 d 2 for these t modules we display a basis consisting of eigenvectors for a and for each basis we give the action of a

convergence to equilibrium for a parabolic hyperbolic phase field model with cattaneo heat flux law abstract in this paper we consider the well posedness and the asymptotic behavior of solutions to the following parabolic hyperbolic phase field system 0 1 t 3 0 t t div q 0 q t q 0 in 0 subject to the homogeneous neumann boundary condition for 0 2 n 0 on 0 and no heat flux boundary condition for q 0 3 q n 0 on 0 and the initial conditions 0 4 0 0 0 0 q 0 q 0 in where r 3 is a bounded domain with a smooth boundary and n is the outward normal direction to the boundary in this paper we first establish the existence and uniqueness of a global strong solution to 0 1 0 2 0 3 0 4 then we prove its convergence to an equilibrium as time goes to infinity

on a phase field system based on the cattaneo law abstract our aim in this paper is to study a generalization of the caginalp phase field system based on the maxwell cattaneo law for heat conduction and endowed with neumann boundary conditions in particular we obtain well posedness results and study the dissipativity of the associated solution operators we also prove when the enthalpy is conserved the existence of the global attractor we finally study the spatial behavior of solutions in a semi infinite cylinder assuming that such solutions exist and have a proper spatial decay at infinity

a generalization of the caginalp phase field system with neumann boundary conditions abstract we study a generalized caginalp phase field system based on the theory of type iii heat conduction proposed by green and naghdi and supplemented with neumann boundary conditions in contrast to the dirichlet case the system exhibits a lack of dissipation on the thermal displacement variable however minus its spatial average is dissipative and we are able to prove the existence of the global attractor with optimal regularity for the associated semigroup

energy bounds for some non standard problems in thermoelasticity a e green f r s and p m naghdi developed two theories of thermoelasticity called type ii and type iii which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory we here derive energy bounds for a class of problem in which the initial data are given as a combination of data at time t 0 and at a later time t t such problems are in vogue in the mathematical literature and may be used for example to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time

radial solutions of the dirichlet problem for a class of quasilinear elliptic equations arising in optometry abstract this paper deals with the quasilinear elliptic problem div u 1 u 2 a x u b x 1 u 2 in b u 0 on b where b is an open ball in r n with n 2 and a b c 1 b are given radially symmetric functions with a x 0 in b this class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea as well as in the modeling theory of capillarity phenomena for compressible fluids unlike all previous works published on these subjects existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a b are not necessarily constant and no sign condition is assumed on b

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

existence of solutions for some doubly degenerate parabolic equations with natural growth terms abstract this paper deals with a class of doubly degenerate parabolic equations of the type b u t div a x t u u f x t u u f where b is a strictly increasing c 1 function under some suitable hypothesis we prove that there exists at least a bounded solution to this problem in standard weak sense

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

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

a note on discontinuity waves in type iii thermoelasticity two recent nonlinear theories of thermoelasticity developed by green and naghdi are examined it is shown that in type ii theory second sound is permissible and both mechanical and temperature waves may propagate in type iii theory we show that the situation is more analogous to that in classical nonlinear thermoelasticity one wave propagates and a homothermal temperature wave is allowed

a unified procedure for construction of theories of deformable media ii generalized continua in contrast to classical theories discussed in the accompanying paper part i the present paper deals exclusively with generalized continua a term that refers to a body embedded in a euclidean three dimensional space with each of its material points endowed with additional kinematic structure such generalized continua are conveniently identified here by cosserat or directed continua in which the additional structure is represented by independent director fields and the material domain of the body manifold is classified according to the four categories a a three dimensional volume b a two dimensional surface c a one dimensional space curve and d a point in the derivation of a thermomechanical theory for each of these categories we use the same unified procedure as in part i but now the various energies that enter the balance of energy must be modified to include additional kinetic ingredients new theories of this kind have increasingly provided over the last three decades effective means of formulating and studying new behaviour of materials in both fluid and solid mechanics that were not previously possible by the classical theories our attention is first focused in the development of the theory for the first category and this is discussed in two stages first for a cosserat continuum with a single director and then for n directors our derived results include for the first time a derivation from the balance of energy of balances of director momenta and balances of entropies in the presence of more than one independent temperature field the treatments of categories b to d are somewhat shorter since their developments are formally similar to that of category a

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

on the impossibility of localization in linear thermoelasticity in the middle of the 1990s green naghdi proposed three theories of thermoelasticity that they labelled as types i ii and ii the type ii theory which is also called thermoelasticity without energy dissipation is conservative and the solutions cannot decay with respect to time it is well known that in general in the linear theories of thermoelasticity of types i and iii the solutions decay with respect to time in many situations this decay is at least exponential in this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time we investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of green naghdi this means that the only solution that vanishes after a finite time is the null solution the main idea is to show the uniqueness of solutions for the backward in time problem to be precise for type iii thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains and for the type i thermoelasticity in the case of exterior domains even when the solutions can be unbounded whether the spatial variable goes to infinity

remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions abstract in this note we present some uniqueness and comparison results for a class of problem of the form 0 1 l u h x u u h x u h 0 1 l where r n n 2 is a bounded domain l is a general elliptic second order linear operator with bounded coefficients and h is allowed to have a critical growth in the gradient in some cases our assumptions prove to be sharp