multiple results of p laplacian dynamic equations on time scales in this paper a new fixed point theorem introduced by avery and peterson and another fixed point theorem are applied to boundary value problems for p laplacian dynamic equations on time scales by imposing some growth conditions on the nonlinearity f we establish the existence results of at least three positive solutions as an application some examples are included to illustrate the main results

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

impulsive boundary value problems with integral boundary conditions and one dimensional p laplacian abstract by using the classical fixed point index theorem for compact maps some sufficient conditions for the existence of multiple positive solutions for a class of second order impulsive differential equations with p laplacian and integral boundary conditions are established which are the complement of previously known results meanwhile the existence of many solutions is also studied

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

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

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

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

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

solvability of the φ laplacian with nonlocal boundary conditions rather mild sufficient conditions are provided for the existence of positive solutions of a boundary value problem of the form f x t c t fx t 0 email protected 0 1 x 0 l 0 x x 1 l 1 x 0 which unify several cases discussed in the literature in order to formulate these conditions one needs to know only properties of the homeomorphism f r r and have information about the level of growth of the response operator f no metric information concerning the linear operators l 0 l 1 in the boundary conditions is used except that they are positive and continuous and such that l j 1 email protected 0 1

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on time scales in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value pr

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

convergence of the domain decomposition finite element boundary element coupling methods in this work we analyze three available domain decomposition methods we also establish the convergence conditions the theoretical analysis provides an interval in which a relaxation parameter has to be chosen in order to achieve convergence moreover it allows the selection of the relaxation parameter so that convergence is achieved with the minimum number of iterations several example problems are given for elaboration

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

a domain decomposition approach for coupled modelling of nonlinear soil structure interaction abstract this paper proposes a new approach for modelling coupled nonlinear soil structure interaction problems by domain decomposition it is assumed that the soil structure coupled system is physically partitioned into independently modelled soil and structure sub domains a coupling procedure based on the sequential iterative dirichlet neumann coupling algorithm is presented which utilizes the condensed tangent stiffness matrices at the soil structure interface to ensure and accelerate convergence to compatibility in successive update of the boundary conditions as a special case of the proposed approach the condensed tangent stiffness matrix is approximated via reduced order models of the partitioned sub domains throughout the coupling iterations a simulation environment has been developed utilizing discipline oriented solvers for nonlinear structural and geotechnical analysis and is discussed this tool is used to demonstrate the applicability of the presented coupling algorithm in modelling nonlinear soil structure interaction problems highlighting its relative merits compared to interface relaxation algorithms furthermore the developed tool is employed for a case study involving nonlinear soil structure interaction analysis between a plane frame and soil subjected to ground excavation where the computational and modelling benefits of the proposed approach and the developed tool are clearly established

baxter s relations and spectra of quantum integrable models generalized baxter s relations on the transfer matrices also known as bax ter s tq relations are constructed and proved for an arbitrary untwisted quantum affine algebra moreover we interpret them as relations in the grothendieck ring of the cate gory o introduced by jimbo and the second author in hj involving infinite dimensional representations constructed in hj which we call here prefundamental we define the transfer matrices associated to the prefundamental representations and prove that their eigenvalues on any finite dimensional representation are polynomials up to a universal factor these polynomials are the analogues of the celebrated baxter polynomials com bining these two results we express the spectra of the transfer matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized baxter polynomials this proves a conjecture of reshetikhin and the first author formulated in 1998 fr1 we also obtain generalized bethe ansatz equations for all untwisted quantum affine algebras

positive solutions for a fourth order p laplacian boundary value problem abstract in this paper we study the existence multiplicity and uniqueness of positive solutions for the fourth order p laplacian boundary value problem u p 1 u f t u u 2 i 0 u 2 i 1 0 i 0 1 here p 0 and f c 0 1 r r r 0 based on a priori estimates achieved by utilizing properties of concave functions we use fixed point index theory to establish our main results

a unified approach to some two point three point and four point boundary value problems with caratheodory functions this paper deals with boundary value problems formula and formula wherea b 0 0 aandbare given real numbers gis a continuous strictly increasing function withg r r andfis a caratheodory function satisfying a nagumo condition the problems have solutions if and only if there exist a lower solutionxand an upper solutionyandx t y t for alla t b

existence of countably many positive solutions of nth order m point boundary value problems the fixed point index theory and a new fixed point theorem in cones are used to prove the existence of countably many positive solutions of nth order m point nonlinear boundary value problems

three positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales abstract the five functionals fixed point theorem is applied to investigate the existence of at least three positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on the half line in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value problems on the half line f u a t f u t 0 0r is the increasing homeomorphism and positive homomorphism and f 0 0 we show the sufficient conditions for the existence of countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

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

triple positive solutions of boundary value problems for p laplacian dynamic equations on time scales a new triple fixed point theorem is applied to investigate the existence of at least three positive solutions of boundary value problems for p laplacian dynamic equations on time scales

multiplicity of positive solutions for second order three point boundary value problems we study the multiplicity of positive solutions for the second order three point boundary value problem u h t f u 0 t 0 1 u 0 0 u u 1 where 0 1 0 1 the methods employed are fixed point index theorems and leray schauder degree and upper and lower solutions

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

existence and iteration of positive solutions for some p laplacian boundary value problems abstract this paper deals with the existence and iteration of positive solutions for the following p laplacian differential equation p u t q t f t u t u t 0 t 0 1 subject to one of the following boundary conditions p u 0 p u 0 0 p u 1 p u 1 0 or u 0 g 1 u 0 0 u 1 g 2 u 1 0 the main tool is the monotone iterative technique we obtain not only the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

impulsive boundary value problems involving the one dimensional p laplacian abstract by using the classical fixed point index theorem for compact maps some sufficient conditions for the existence of multiple positive solutions for a class of second order impulsive differential equations with p laplacian are established which are complement of previously known results

multiple positive solutions to a three point boundary value problem with p laplacian abstract sufficient conditions are obtained that guarantee the existence of at least two positive solutions for the equation g u t a t f u 0 subject to boundary conditions by a simple application of a new fixed point theorem due to avery and henderson

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

positive solutions for a system of second order quasilinear boundary value problems abstract in this paper we study the existence and multiplicity of positive solutions for the system of second order quasilinear boundary value problems u p 1 f t u v v q 1 g t u v u 0 u 1 0 v 0 v 1 0 where p q 1 and f g c 0 1 r 2 r r 0 based on a priori estimates achieved by utilizing jensen s inequality for nonnegative concave functions and homogeneous operators we use fixed point index theory to establish the main results

triple positive pseudo symmetric solutions of three point bvps for p laplacian dynamic equations on time scales abstract let t be a time scale such that 0 t t we consider the three point boundary value problem for p laplacian dynamic equations on time scales t of the form p u t h t f t u t 0 for t 0 t t with boundary conditions u 0 0 u u t where t is symmetric in t t and p u u p 2 u with p 1 by using a pseudo symmetric technique and the five functionals fixed point theorem we prove that the boundary value problem has at least three positive pseudo symmetric solutions under some assumptions as an application an example is given to illustrate the result

existence of two solutions of m point boundary value problem for second order dynamic equations on time scales in this paper by means of a new twin fixed point theorem in a cone the existence of at least two positive solutions of m point boundary value problem for second order dynamic equations on time scales is considered

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

triple positive solutions for p laplacian m point boundary value problem on time scales in this paper we discuss the following p laplacian m point boundary value problem on time scales t f p u d t h t f t u t u d t 0 email protected 0 t t u 0 0 f p u d t email protected i 1m 2a email protected p u d x i where f p u u p 2u for p 1 some new existence criteria for at least three positive solutions are established by using a generalization of the leggett williams fixed point theorem due to bai and ge an example is also given to demonstrate the main result

coupled finite element hierarchical boundary element methods for dynamic soil structure interaction in the frequency domain summary this paper discusses the coupling of finite element and fast boundary element methods for the solution of dynamic soil structure interaction problems in the frequency domain the application of hierarchical matrices in the boundary element formulation allows considering much larger problems compared to classical methods three coupling methodologies are presented and their computational performance is assessed through numerical examples it is demonstrated that the use of hierarchical matrices renders a direct coupling approach the least efficient as it requires the assembly of a dynamic soil stiffness matrix iterative solution procedures are presented as well and it is shown that the application of such schemes to dynamic soil structure interaction problems in the frequency domain is not trivial as convergence can hardly be achieved if no relaxation procedure is incorporated aitken s 2 method is therefore employed in sequential iterative schemes for the calculation of an optimized interface relaxation parameter while a novel relaxation technique is proposed for parallel iterative algorithms it is demonstrated that the efficiency of these algorithms strongly depends on the boundary conditions applied to each subdomain the fastest convergence is observed if neumann boundary conditions are imposed on the stiffest subdomain the use of a dedicated solver for each subdomain hence results in a reduced computational effort a monolithic coupling strategy often used for the solution of fluid structure interaction problems is also introduced the governing equations are simultaneously solved in this approach while the assembly of a dynamic soil stiffness matrix is avoided copyright 2013 john wiley sons ltd

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

interface relaxation algorithms for bem bem coupling and fem bem coupling this paper presents several interface relaxation algorithms for boundary element boundary element coupling bem bem and for finite element boundary element coupling fem bem the domain of the original problem is sub divided into sub domains which are modeled by the finite element or boundary element methods the multi domain system is coupled using smoothing operators on the inter domain boundaries separate computations for the bem and fem sub domains and successive update of the boundary conditions at the interfaces are performed until convergence is achieved the interface relaxation coupling algorithms preserve the nature of the fem and bem further they do not require any access to the matrices generated by the fem or bem and make it easier to utilize different software in different sub domains

double positive solutions for a nonlinear four point boundary value problem with a p laplacian operator in this paper we study the existence of positive solutions for a nonlinear four point boundary value problem with a p laplacian operator by using a three functionals fixed point theorem in a cone the existence of double positive solutions for the nonlinear four point boundary value problem with ap laplacian operator is obtained this is different than previous results

asymptotic solutions for a second order differential equation with three point boundary conditions abstract in this paper we are concerned with the following singularly perturbed second order three point boundary value problem 1 1 y 2 q 1 x q 2 x y 0 1 1 2 y 0 a y 1 y where 0

positive solutions for a system of p laplacian boundary value problems this paper is concerned with the existence and multiplicity of positive solutions for the system of p laplacian boundary value problems u i p i 1 f i t u 1 u n u i 0 u i 1 0 i 1 n where n 2 p i 1 f email protected c 0 1 xr n r i 1 n r 0 based on a priori estimates achieved by utilizing the jensen integral inequalities and r n monotone matrices we use fixed point index theory to establish the existence and multiplicity of positive solutions for the above problem

solving composite problems with interface relaxation this paper deals with a solution approach suitable for composite partial differential equations pdes with interface conditions we present a general framework based on interface relaxation which provides a uniform platform for building problem solving environments through efficient software integration and for implementing various relaxation schemes mathematically this framework contains many existing domain decomposition methods and also allows the extension to a variety of new relaxers in particular we describe a class of relaxers which are suitable for general and complicated interface conditions interface relaxation is more general than the traditional domain decomposition methods in that it allows unrelated pde problems on different subdomains convergence analysis error estimates and preconditioning strategies are presented which show that these relaxers are competitive with existing domain decomposition methods for model problems involving a single pde we present experimental results which demonstrate the wide applicability of this approach differences between this approach and other domain decomposition methods are also discussed

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

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

triple positive solutions of m point bvps for p laplacian dynamic equations on time scales abstract this paper is concerned with the existence of positive solutions of p laplacian dynamic equation p u t a 1 t f u t 0 subject to boundary conditions u 0 b 0 i 1 m 2 a i u i 0 u t 0 or u 0 0 u t b 1 i 1 m 2 b i u i 0 where p v v p 2 v with p 1 by using the five functionals fixed point theorem we prove that the boundary value problem has at least three positive solutions as an application an example is given to illustrate the result

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

existence of positive negative and sign changing periodic solutions for a class of integral equations in this paper the simultaneous existence of positive negative and sign changing periodic solutions for a class of integral equations of the form f x email protected x x w wk x y f y f email protected y dy email protected email protected is considered where w is a closed subset of r n with a periodic structure our main result is different from most existing results since they provide three constant sign periodic solutions only

a dirichelet neumann m point bvp with a p laplacian like operator abstract let be odd increasing homeomorphisms from r onto r satisfying 0 0 0 and let f a b r r r be a function satisfying carath e odory s conditions let i r i a b i 1 m 2 a 1 2 m 2 b be given we are interested in the problem of existence of solutions for the m point boundary value problem u f t u u t a b u a 0 u b i 1 m 2 i u i in the resonance and non resonance cases we say that this problem is at resonance if the associated problem u 0 t a b u a 0 u b i 1 m 2 i u i has a non trivial solutions this is the case if and only if i 1 m 2 i 1 our results use topological degree methods interestingly enough in the non resonance case i e when i 1 m 2 i 1 the sign of degree for the relevant operator depends on whether i 1 m 2 i 1 or i 1 m 2 i 1

positive solution of three point boundary value problem for the one dimensional p laplacian with singularities abstract in the paper we deal with positive solutions of the following nonlinear three point singular boundary value problem with a p laplacian operator p u q t f t u 0 0 t 1 subject to u 0 g u 0 0 u 1 u 0 or u 0 u 0 u 1 g u 1 0 where f t u may be singular at u 0 and q t may be singular at t 0 1 new existence principles are established

existence of three positive solutions for boundary value problems with p laplacian abstract by using fixed point theorem we study the following equation g u t a t f u 0 subject to boundary conditions where g v v p 2 v with p 1 the existence of at least three positive solutions is proved

existence of three positive pseudo symmetric solutions for a one dimensional p laplacian abstract we apply the five functionals fixed point theorem to verify the existence of at least three positive pseudo symmetric solutions for the three point boundary value problem g u a t f u 0 u 0 0 and u u 1 where g v v p 2 v with p 1 and 0 1

double positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales a new double fixed point theorem is applied to investigate the existence of at least double positive solutions of three point boundary value problems for p laplacian dynamic equations on a time scale