the universal r matrix for u_q rm sl 3 and beyond ther matrices for the quantised lie algebrasa n are constructed through the quantum double procedure given by drinfel d 6 the case ofu q sl 3 is thoroughly analysed initially to demonstrate the more subtle points of the calculation the ease of the calculation fora n is very dependent on a choice of generators for the borel subalgebrau q b and its dual and a certain ordering imposed on these generators which is related to the length of a certain word in the weyl group

unusual solutions to the yang baxter equation the complete list of non symmetric eight vertex constant solutions to the yang baxter equation is obtained and new solutions dependent on more than one variable are presented the corresponding spin hamiltonians are derived by sutherland s method 1970

colored solutions of the yang baxter equation from representations of uqgl 2 we study the hopf algebra structure and the highest weight representation of a multiparameter version of uqgl 2 the hopf algebra maps of this algebra are explicitly given we show that the multiparameter universal r matrix can be constructed directly as a quantum double intertwiner without using reshetikhin s twisting transformation we find there are two types highest weight representations for this algebra type a corresponds to the qeneric q and type b corresponds to the case that q is a root of unity when applying the representation theory to the multi parameter universal r matrix both standard and nonstandard colored solutions of the yang baxter equation are obtained

on the quantum symmetries associated with the two parameter free fermion model the quantum algebras associated with the r matrix obtained from the two parameter free fermion model are examined it is shown that in addition to uq s gl 1 1 there exists another algebra which is isomorphic to uq s gl 1 1 as an algebra but has a different hopf structure it does not have a classical limit and is related to up sl 2 c at a root of unity p i i2 1

duals of colored quantum universal enveloping algebras and colored universal t matrices we extend the notion of dually conjugate hopf super algebras to the colored hopf super algebras hc that we recently introduced we show that if the standard hopf super algebras hq that are the building blocks of hc have hopf duals hq then the latter may be used to construct coloured hopf duals hc endowed with colored algebra and antipode maps but with a standard coalgebraic structure next we review the case where the hq s are quantum universal enveloping algebras of lie super algebras uq g so that the corresponding hq s are quantum super groups gq we extend the fronsdal and galindo universal t matrix formalism to the colored pairs uc g gc by defining colored universal t matrices we then show that together with the colored universal r matrices previously introduced the latter provide an algebraic formulation of the colored rtt relations proposed by basu mallick this establishes a link between the colored extensions of drinfeld jimbo and faddeev reshetikhin takhtajan pictures of quant

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

the universal r matrix and the yang baxter equation with parameters the authors formulate and demonstrate with some examples a method for obtaining solutions of the yang baxter equation with parameters

coloured frt algebra and its yang baxterization leading to integrable models with non additive r matrices a colour representation of faddeev reshetikhin takhtajan frt algebra which in contrast to the standard case is related to the coloured braid group representation with generic values of q is presented explicit realizations of l or matrices occurring in this coloured variant of frt algebra are also obtained for the uq gl 2 quantized algebra though these realizations are found to depend manifestly on the colour parameters the underlying quantum group structure and associated co product are interestingly free from such dependence this allows us to perform the yang baxterization of the coloured frt algebra successfully which leads to the construction of an ancestor law operator associated with a new non additive type quantum r matrix through different realizations of this lax operator a new class of quantum integrable models representing colour generalizations of the well known models such as the lattice sine gordon model the ablowitz ladik model lattice and the derivative nonlinear schrodinger model etc is generated

epidemic dynamics discrete time and cellular automaton models abstract we present a simple model of population dynamics in the presence of an infection the model is based on discrete time equations for sane and infected populations in interaction and correctly describes the dynamics of the epidemic we find that for some choices of the parameters the model can possess conserved quantities we also propose an ultra discrete cellular automaton version of the model which despite its extremely simple structure still captures the essence of the epidemic dynamics

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

strong mixing gaussian measures for chaotic semigroups abstract in this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable banach spaces we first prove an identity characterizing invariant gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup this gives an answer to a question raised by rudnicki in his inspiring review paper 32 under suitable conditions we use the proved identity to give an invariant mixing gaussian measure as distribution of a wiener integral

chaos and hypercyclicity for solution semigroups to some partial differential equations abstract we investigate the solution semigroup to some partial differential equations and give some sufficient conditions for the solution semigroup to be hypercyclic or chaotic

euler lagrange correspondence of cellular automaton for traffic flow models we propose a euler lagrange transformation for cellular automata ca by developing new explicit transformation formulas this transformation is done in the fully discrete level of variables and corresponds to the well known continuous version of it which appears in continuous mechanics such as fluid dynamics and plasma physics applying this method to the traffic problem we have obtained the lagrange representation of a traffic model and also succeeded in clarifying the relation between different types of traffic models it is shown that the burgers ca which is a corresponding ca of the continuous burgers equation plays a central role in considering this relation

anti synchronization of uncertain unified chaotic systems with dead zone nonlinearity abstract this paper addresses chaos anti synchronization of uncertain unified chaotic systems with dead zone input nonlinearity using the sliding mode control technique and lyapunov stability theory a proportional integral pi switching surface is proposed to ensure the stability of the closed loop error system in sliding mode then a sliding mode controller smc is proposed to guarantee the hitting of the switching surface even with uncertainties and the control input containing dead zone nonlinearity some simulation results are included to demonstrate the effectiveness and feasibility of the proposed synchronization scheme

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

chaotic c0 semigroups induced by semiflows in lebesgue and sobolev spaces abstract we give characterizations of chaos for c 0 semigroups induced by semiflows on l p for open r similar to the characterizations of hypercyclicity and mixing of such c 0 semigroups proved in 18 moreover we characterize hypercyclicity mixing and chaos for these classes of c 0 semigroups on w 1 p i for a bounded interval i r and prove that these c 0 semigroups are never hypercyclic on w 1 p i we apply our results to concrete first order partial differential equations such as the von foerster lasota equation

new hierarchy of colored braid group representations novel solutions of the yang baxter relation for a series of multistate vertex models are presented in the braid group limit the obtained family of solutions gives a sequence of colored braid group representations possible generalizations of the multivariable alexander polynomial are discussed

universal r matrix of reductive lie algebras and quantum integrable systems from its colour representation the universal r matrix intertwining between the coproduct structures related to the universal deformations of reductive lie algebras has been found in an explicit form using twisting a possible colour representation of this r matrix and its application to the integrable systems are shown using the example of a deformed gl n algebra

yang baxterization of braid group representations for a given braid group representation bgr a process of the yang baxterization is formulated to generate solutions of the yang baxter equation ybe when a bgr admits the birman wenzl bw algebraic structure this process can be explicitly passed through and two types of trigonometric solutions of ybe are generated from such a bgr these two solutions have the essential difference to each other and both of them preserve the crossing symmetry property if the given bgr has by taking certain reduction on the bw algebra the rational solution is also generated a practical condition to judge whether a bgr satisfies the bw algebra is given from which one finds that not only the familiar bgrs of 5 7 9 but also some new ones obtained recently in 12 have the bw structure thus they can be explicitly yang baxterized to solutions of the ybe

do ultradiscrete systems with parity variables satisfy the singularity confinement criterion ultradiscrete singularity confinement test which is an integrability detector for ultradiscrete equations with parity variables is applied to various ultradiscrete equations the ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures

exact solutions for discrete and ultradiscrete modified kdv equations and their relation to box ball systems a new class of solutions is proposed for discrete and ultradiscrete modified kdv equations these are directly related to solutions of the box and ball system with a carrier moreover an extended box and ball system and its exact solutions are discussed

problem of stability and chaotic properties of multidimensional lasota equation in the context of matuszewska orlicz indices abstract we study asymptotic properties of the dynamical system generated by the multidimensional lasota equation we give the conditions of its stability and chaos in the sense of devaney in orlicz spaces we apply matuszewska orlicz indices to a description of asymptotic behavior of the considered semigroup we analyze also the conditions under which asymptotic behavior for more general form of the equation is provided an example illustrates the criteria when the semigroups generated by the equations have not asymptotic behavior

universal r matrix for a two parametric quantization of gl 2 for the two parametric quantized group glq s 2 the authors calculate the explicit form of the universal r matrix

a nonlinear deformed su 2 algebra with a two color quasitriangular hopf structure nonlinear deformations of the enveloping algebra of su 2 involving two arbitrary functions of j0 and generalizing the witten algebra were introduced some time ago by delbecq and quesne in the present paper the problem of endowing some of them with a hopf algebraic structure is addressed by studying in detail a specific example referred to as aq 1 this algebra is shown to possess two series of n 1 dimensional unitary irreducible representations where n 0 1 2 to allow the coupling of any two such representations a generalization of the standard hopf axioms is proposed by proceeding in two steps in the first one a variant and extension of the deforming functional technique is introduced variant because a map between two deformed algebras suq 2 and aq 1 is considered instead of a map between a lie algebra and a deformed one and extension because use is made of a two valued functional whose inverse is singular as a result the hopf structure of suq 2 is carried over to aq 1 ther

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

direct ultradiscretization of ai and bi functions and special solutions for the painleve ii equation ultradiscrete ai and bi functions are directly derived through the ultradiscrete limit from q difference analogues of the ai and bi functions respectively an infinite number of identities among the number of restricted partitions are obtained as by products a direct relationship between a class of special solutions for the ultradiscrete painlev ii equation and those of the q painlev ii equation which have a determinantal structure is also established

ultradiscretization without positivity we present a new ultradiscretization approach which can be applied to discrete systems the solutions of which are not positive definite this was made possible thanks to an ansatz involving the hyperbolic sine function we apply this new procedure to simple mappings for the linear and homographic mappings we obtain ultradiscrete forms and explicitly construct their solutions two discrete painleve ii equations are also analysed and ultradiscretized we show how to construct the ultradiscrete analogues of their rational and airy type solutions