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

multisynchronization of coupled heterogeneous genetic oscillator networks via partial impulsive control this paper focuses on the collective dynamics of multisynchronization among heterogeneous genetic oscillators under a partial impulsive control strategy the coupled nonidentical genetic oscillators are modeled by differential equations with uncertainties the definition of multisynchronization is proposed to describe some more general synchronization behaviors in the real considering that each genetic oscillator consists of a large number of biochemical molecules we design a more manageable impulsive strategy for dynamic networks to achieve multisynchronization not all the molecules but only a small fraction of them in each genetic oscillator are controlled at each impulsive instant theoretical analysis of multisynchronization is carried out by the control theory approach and a sufficient condition of partial impulsive controller for multisynchronization with given error bounds is established at last numerical simulations are exploited to demonstrate the effectiveness of our results

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

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

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

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

synchronization control for nonlinear stochastic dynamical networks pinning impulsive strategy in this paper a new control strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling pinning state feedback controllers have been proved to be effective for synchronization control of state coupled dynamical networks we will show that pinning impulsive controllers are also effective for synchronization control of the above mentioned dynamical networks some generic mean square stability criteria are derived in terms of algebraic conditions which guarantee that the whole state coupled dynamical network can be forced to some desired trajectory by placing impulsive controllers on a small fraction of nodes an effective method is given to select the nodes which should be controlled at each impulsive constants the proportion of the controlled nodes guaranteeing the stability is explicitly obtained and the synchronization region is also derived and clearly plotted numerical simulations are exploited to demonstrate the effectiveness of the pinning impulsive strategy proposed in this paper

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

impulsive multiconsensus of second order multiagent networks using sampled position data a multiconsensus problem of multiagent networks is solved in this paper where multiconsensus refers to that the states of multiple agents in each subnetwork asymptotically converge to an individual consistent value when there exist information exchanges among subnetworks a distributed impulsive protocol is proposed to achieve multiconsensus of second order multiagent networks in terms of three categories 1 stationary multiconsensus 2 the first dynamic multiconsensus and 3 the second dynamic multiconsensus this impulsive protocol utilizes only sampled position data and is implemented at sampling instants for those three categories of multiconsensus the control parameters in the impulsive protocol are designed respectively moreover necessary and sufficient conditions are derived under which each multiconsensus can be reached asymptotically several simulations are finally provided to demonstrate the effectiveness of the obtained theoretical results

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

on hybrid impulsive and switching systems and application to nonlinear control in this note a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated using switched lyapunov functions some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established in addition a new hybrid impulsive and switching control strategy for nonlinear systems is developed a typical example the unified chaotic system is given to illustrate the theoretical results

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

exponential synchronization of complex networks of linear systems and nonlinear oscillators a unified analysis a unified approach to the analysis of synchronization for complex dynamical networks i e networks of partial state coupled linear systems and networks of full state coupled nonlinear oscillators is introduced it is shown that the developed analysis can be used to describe the difference between the state of each node and the weighted sum of the states of those nodes playing the role of leaders in the networks thus making it feasible to consider the error dynamics for the whole network system different from the other various methods given in the existing literature the analysis employed in this paper is demonstrated successfully in not only providing the consistent convergence analysis with much simpler form but also explicitly specifying the convergence rate

synchronization error estimation and controller design for delayed lur e systems with parameter mismatches this paper investigates the problem of master slave synchronization of two delayed lur e systems in the presence of parameter mismatches first by analyzing the corresponding synchronization error system synchronization with an error level which is referred to as quasi synchronization is established some delay dependent quasi synchronization criteria are derived an estimation of the synchronization error bound is given and an explicit expression of error levels is obtained second sufficient conditions on the existence of feedback controllers under a predetermined error level are provided the controller gains are obtained by solving a set of linear matrix inequalities finally a delayed chua s circuit is chosen to illustrate the effectiveness of the derived results

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

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

cluster consensus in discrete time networks of multiagents with inter cluster nonidentical inputs in this paper cluster consensus of multiagent systems is studied via inter cluster nonidentical inputs here we consider general graph topologies which might be time varying the cluster consensus is defined by two aspects intracluster synchronization the state at which differences between each pair of agents in the same cluster converge to zero and inter cluster separation the state at which agents in different clusters are separated for intra cluster synchronization the concepts and theories of consensus including the spanning trees scramblingness infinite stochastic matrix product and hajnal inequality are extended as a result it is proved that if the graph has cluster spanning trees and all vertices self linked then the static linear system can realize intra cluster synchronization for the time varying coupling cases it is proved that if there exists t 0 such that the union graph across any t length time interval has cluster spanning trees and all graphs has all vertices self linked then the time varying linear system can also realize intra cluster synchronization under the assumption of common inter cluster influence a sort of inter cluster nonidentical inputs are utilized to realize inter cluster separation such that each agent in the same cluster receives the same inputs and agents in different clusters have different inputs in addition the boundedness of the infinite sum of the inputs can guarantee the boundedness of the trajectory as an application we employ a modified non bayesian social learning model to illustrate the effectiveness of our results

a micropolar peridynamic theory in linear elasticity a state based micropolar peridynamic theory for linear elastic solids is proposed the main motivation is to introduce additional micro rotational degrees of freedom to each material point and thus naturally bring in the physically relevant material length scale parameters into peridynamics non ordinary type modeling via constitutive correspondence is adopted here to define the micropolar peridynamic material along with a general three dimensional model homogenized one dimensional timoshenko type beam models for both the proposed micropolar and the standard non polar peridynamic variants are derived the efficacy of the proposed models in analyzing continua with length scale effects is established via numerical simulations of a few beam and plane stress problems c 2015 elsevier ltd all rights reserved

a time varying complex dynamical network model and its controlled synchronization criteria today complex networks have attracted increasing attention from various fields of science and engineering it has been demonstrated that many complex networks display various synchronization phenomena in this note we introduce a time varying complex dynamical network model we then further investigate its synchronization phenomenon and prove several network synchronization theorems especially we show that synchronization of such a time varying dynamical network is completely determined by the inner coupling matrix and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network

synchronization of complex dynamical networks with switching topology a switched system point of view in this paper we study the synchronization problem for complex dynamical networks with switching topology from a switched system point of view the synchronization problem is transformed into the stability problem for time varying switched systems we address two basic problems synchronization under arbitrary switching topology and synchronization via design of switching within a pre given collection of topologies when synchronization cannot be achieved by using any topology alone in this collection for the both problems we first establish synchronization criteria for general connection topology then under the condition of simultaneous triangularization of the connection matrices a common lyapunov function for the first problem and a single lyapunov and multiple lyapunov functions for the second problem are systematically constructed respectively by those of several lower dimensional dynamic systems in order to achieve synchronization using multiple lyapunov functions a stability condition and switching law design method for time varying switched systems are also presented which avoid the usual non increasing condition

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

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