representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

continuous dependence results for a general model of size dependent population dynamics we are concerned with a general model of size structured population dynamics with the growth rate depending on the individual s size and time in this paper we shall study the continuous dependence of the solution on all given data such as aging and birth functions growth rate functions and initial data

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

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

on the clebsch gordan coefficients for the two parameter quantum algebra su 2 p q we show that the clebsch gordan coefficients for the su 2 p q algebra depend on a single parameter q square root pq contrary to the explicit calculation of smirnov and wehrhahn 1992

quantum algebra deforming maps clebsch gordan coefficients coproducts r and u matrices quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts map induced coproducts are connected to the usual ones by similarity transformations u that may be expressed either in terms of clebsch gordan coefficients or in a universal operator form the product of two such u matrices yields the r matrix for a fixed value of the spectral parameter which bears on the yang baxterization of u as well as r all this is explicitly illustrated for the tensor product 1 2 j of su 2 q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two parameter manifold some observations are made on the general structure of u and r matrices and of coproduct manifolds based on the solutions of hierarchies of partial difference equations applications of deforming maps and u matrices to the physics of spin chains are outlined

noncommutative phase and unitarization of glp q 2 in this article imposing hermitian conjugate relations on the two parameter deformed quantum group glp q 2 is studied this results in a noncommutative phase associated with the unitarization of the quantum group after the achievement of the quantum group up q 2 with pq real via a noncommutative phase the representation of the algebra is built by means of the action of the operators constituting the up q 2 matrix on states

steady states of a selection mutation model for an age structured population abstract we introduce a selection mutation model for an age and age at maturity structured population the age at maturity is considered an evolutionary trait on which the vital rates depend subject to selection by competition and to mutation modelled by a birth term which takes the form of a convolution like operator we prove well posedness of the initial value problem and the existence of non trivial steady states under suitable hypotheses which yield uniqueness when the interaction variable is one dimensional

universal t matrix representations of ospq 1 2 and little q jacobi polynomials we obtain a closed form expression of the universal t matrix encapsulating the duality between the quantum superalgebra uq osp 1 2 and the corresponding supergroup ospq 1 2 the classical q 1 limit of this universal t matrix yields the group element of the undeformed ospq 1 2 supergroup the finite dimensional representations of the quantum supergroup ospq 1 2 are readily constructed employing the above mentioned universal t matrix and the known finite dimensional representations of the dually related deformed uq osp 1 2 superalgebra proceeding further we derive the product law the recurrence relations and the orthogonality of the representations of the quantum supergroup ospq 1 2 it is shown that the entries of these representation matrices are expressed in terms of the little q jacobi polynomials with q q two mutually complementary singular maps of the universal t matrix on the universal r matrix are also presented

representations of two parameter quantum algebras and p q special functions abstract in this paper the connection between p q analogue of special functions and representations of certain two parameter quantum algebras has been developed the realizations of the representations of these quantum algebras are in terms of p q derivative operators the matrix elements of these representations are computed in terms of the generalized p q hypergeometric series this leads to the developments in the theory of p q special functions

on up q gl 2 and a p q virasoro algebra the quantum algebra up q gl 2 with two independent deformation parameters p q is studied and in particular its universal r matrix is constructed using reshetikhin s method a contraction procedure then leads to the p q deformed heisenberg algebra up q h 1 and its universal r matrix using a sugawara construction employing an infinite number of copies of these heisenberg modes a p q deformed virasoro algebra is obtained the closure property of the p q virasoro algebra necessitates two parameters alpha beta for the generators lm alpha beta while the parameter alpha may be taken as an integer the parameter beta is continuous on a complex path and imparts an integral equation structure to the p q deformed virasoro algebra

duality for exotic bialgebras in the classification of hietarinta three triangular 4 4 r matrices lead via the frt formalism to matrix bialgebras which are not deformations of the trivial one in this paper we find the bialgebras which are in duality with these three exotic matrix bialgebras we note that the l t duality of frt is not sufficient for the construction of the bialgebras in duality we find also the quantum planes corresponding to these bialgebras both by the wess zumino r matrix method and by manin s method

abstract linear partial differential equations related to size structured population models with diffusion abstract we study abstract linear partial differential equations in banach spaces and or banach lattices related to size structured population models with spatial diffusion and their dual problems we introduce mild solutions through semigroup theory and characteristic method and investigate differentiability of mild solutions existence of a unique mild solution is shown also a comparison result is obtained as well as the boundedness of mild solutions is investigated in the banach lattice setting furthermore we consider the dual problems and then we introduce weak solutions and establish their uniqueness

duality for the matrix quantum group glp q 2 c it is shown that the algebra up q dual to glp q 2 c is isomorphic to u pq 1 2 sl 2 c z as a commutation algebra where z is a subalgebra central in up q the subalgebra z is a hopf subalgebra of up q while the commutation subalgebra u pq 1 2 sl 2 c is not a hopf subalgebra

on clebsch gordan coefficients and matrix elements of representations of the quantum algebra uq su2 clebsch gordan coefficients and matrix elements of irreducible representations of the quantum algebra uq su2 were considered in several papers in particular a few expressions for them were derived an approach to clebsch gordan coefficients and to matrix elements of representations of uq su2 on the base of the theory of basic hypergeometric functions is given this approach allows one to obtain q analogs of all well known classical expressions for clebsch gordan coefficients most of them were absent new symmetry relations generating functions and recurrence formulas for clebsch gordan coefficients of uq su2 are obtained unlike other papers clebsch gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions in fact without using the notion of a c algebra and of a hopf algebra

canonical b series b series provide a powerful general tool to express numerical methods for differential equations many differential equations are of hamiltonian form and there has been much recent interest in constructing so called canonical or symplectic integrators for the hamiltonian case in this paper we provide a necessary and sufficient condition for a b series to correspond to a canonical method

on the representations of glp q 2 glp q 1 mod 1 and non commutative spaces explicit realizations of the quantum groups glp q 2 and glp q 1 mod 1 corresponding to unimodular values of the deformation parameters p and q are given in terms of the canonically conjugate x p operators using the heisenberg weyl commutation relations matrix representations are also discussed some observations are made on similar realizations of the non commutative coordinate spaces on which the quantum groups act as endomorphisms

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

a general model of size dependent population dynamics with nonlinear growth rate we study a general model of size dependent population dynamics with nonlinear growth rate the existence of a local solution is shown by using schauder s fixed point theorem uniqueness and continuous dependence on initial data are also established under additional conditions

unitarizable representations of the deformed para bose superalgebra uq osp 1 2 at roots of 1 the unitarizable irreps of the deformed para bose superalgebra pbq which is isomorphic to uq osp 1 2 are classified at q being root of 1 new finite dimensional irreps of uq osp 1 2 are found explicit expressions for the matrix elements are written down

representation functions for jordanian quantum group slh 2 and jacobi polynomials the explicit expressions of the representation functions d functions for jordanian quantum group slh 2 are obtained by combination of tensor operator technique and drinfeld twist it is shown that the d functions can be expressed in terms of jacobi polynomials as the undeformed d functions can some of the important properties of the d functions for slh 2 such as wigner s product law recurrence relations and rtt type relations are also presented

finite dimensional representations of the quantum group glp q 2 using the exponential map from up q gl 2 using the fronsdal galindo formula for the exponential mapping from the quantum algebra up q gl 2 to the quantum group glp q 2 we show how the 2j 1 dimensional representations of glp q 2 can be obtained by exponentiating the well known 2j 1 dimensional representations of up q gl 2 for j 1 3 2 j 1 2 corresponds to the defining two dimensional t matrix the earlier results on the finite dimensional representations of glq 2 and slq 2 or suq 2 are obtained when p q representations of uq q 2 q in c mod r and uq 2 q in r mod 0 are also considered the structure of the clebsch gordan matrix for up q gl 2 is studied the same clebsch gordan coefficients are applicable in the reduction of the direct product representations of the quantum group glp q 2

exponential mapping for non semisimple quantum groups the concept of a universal t matrix introduced by fronsdal and galindo 1993 in the framework of quantum groups is discussed here as a generalization of the exponential mapping new examples related to inhomogeneous quantum groups of physical interest are developed the duality calculations are explicitly presented and it is found that in some cases the universal t matrix as for lie groups is expressed in terms of usual exponential series