the quantum deformation of su 1 1 as the dynamical symmetry of the anharmonic oscillator as an application of q deformed algebras to standard quantum mechanics the author shows that the su 1 1 dynamical symmetry of the quantum harmonic oscillator is deformed in the first order of approximation to the dynamical symmetry defined by the quantized universal enveloping algebra of su 1 1 when the harmonic potential is perturbed with a potential in x4 the resolution of the anharmonic oscillator is carried out algebraically in terms of generalized lowering and rising operators

on the theorem of reichert abstract this paper reworks and amplifies reichert s proof of his theorem 1969 which asserts that any impedance function of a one port electrical network which can be realised with two reactive elements and an arbitrary number of resistors can be realised with two reactive elements and three resistors

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

an su 2 analogue of the azbel hofstadter hamiltonian motivated by the recent findings of wiegmann zabrodin and faddeev kashaev concerning the appearance of the quantum symmetry in the problem of a bloch electron on a two dimensional magnetic lattice we introduce a modification of the tight binding azbel hofstadter hamiltonian that is a specific spin s euler top that can be considered as its classical analogue the eigenvalue problem for the proposed model in the coherent state representation is described by the s gap lame equation and thus is completely solvable we observe a striking similarity between the shapes of the spectra of the two models for various values of the spin s

chiral potts rapidity curve descended from six vertex model and symmetry group of rapidities in this paper we present a systematical account of the descending procedure from the six vertex model to the n state chiral potts model through fusion relations of j operators following the works of bazhanov stroganov and baxter bazhanov perk a careful analysis of the descending process leads to the appearance of the high genus curve as the rapidity constraint for the chiral potts models full symmetries of the rapidity curve are identified as is its symmetry group structure by normalized transfer matrices of the chiral potts model the 2 t relation can be reduced to functional equations over a hyperelliptic curve associated with rapidities by which the degeneracy of 2 eigenvalues is revealed in the case of the superintegrable chiral potts model

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

representations of the q deformed algebra uq so4 we study the nonstandard q deformation uq so4 of the universal enveloping algebra u so4 obtained by deforming the defining relations for skew symmetric generators of u so4 this algebra is used in quantum gravity and algebraic topology we construct a homomorphism of uq so4 to the certain nontrivial extension of the drinfeld jimbo quantum algebra uq sl2 2 and show that this homomorphism is an isomorphism by using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra uq so4 it is proved that for q not a root of unity each irreducible finite dimensional representation of uq so4 is equivalent to one of these representations we prove that every finite dimensional representation of uq so4 for q not a root of unity is completely reducible it is shown how to construct by using the homomorphism tensor products of irreducible representations of uq so4 note that no hopf algebra structure is known fo

superintegrable chiral potts model thermodynamic properties an inverse model and a simple associated hamiltonian the partition function of then state superintegrable chiral potts model is obtained exactly and explicitly if not completely rigorously for a finite lattice with particular boundary conditions this yields the bulk and surface free energies and horizontal and vertical correlation lengths and interfacial tensions the critical exponents are 1 2 n hor hor 2 n and vert vert 1 and the finite size corrections are obtained at criticality the eigenvalue spectrum of the column to column transfer matrix is that of a direct product ofn byn matrices inverting this matrix gives a related solvable model which is a generalization of the free fermion model the associated hamiltobian has a very simple form suggesting there may be a more direct algebraic method perhaps a generalized clifford algebra for obtaining its eigenvalues

the q hermite polynomial and the representations of heisenberg and quantum heisenberg algebras the authors introduce a pair of canonical conjugate q deformed operators d x and discuss the relations between the deformed operators d x and q series the realizations of some lie symmetries heisenberg and quantum heisenberg algebras are given for the operators d and x they show that the q analogous hermite polynomials are representations of heisenberg and quantum heisenberg algebras realized in this way when q is a root of unity the properties of the q analogous hermite polynomials are also discussed

quantum groups and lie admissible time evolution the time evolution of operators for q oscillators is derived for the first time by exploiting the connection between q deformation algebras and lie admissible algebras

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

passive controller realization of a biquadratic impedance with double poles and zeros as a seven element series parallel network for effective mechanical control this work is concerned with the passive realizability problem for a biquadratic impedance with double poles and zeros as a seven element series parallel rlc damper spring inerter network which is a series or a parallel connection of two networks a three element series parallel network and a four element series parallel network first a necessary and sufficient condition is obtained for the realizability of a biquadratic impedance with double poles and zeros as a series parallel network containing no more than six elements see theorem 1 then the main theorem is presented see theorem 2 which is derived for three reactive element four reactive element and five reactive element settings respectively see lemmas 2 4 the results can be directly utilized to physically construct a passive mechanism containing dampers springs and inerters which can be regarded as a passive controller for some mechanical control systems

representations of the q deformed algebra uq iso2 an algebra homomorphism from the q deformed algebra uq iso2 with generating elements i t1 t2 and defining relations i t2 q t1 t1 i q t2 t2 t1 q 0 where a b q q1 2ab q 1 2ba to the extension uq m2 of the hopf algebra uq m2 is constructed the algebra uq iso2 at q 1 leads to the lie algebra iso2 m2 of the group iso 2 of motions of the euclidean plane the hopf algebra uq m2 which is not isomorphic to uq iso2 is treated as a hopf q deformation of the universal enveloping algebra of iso2 and is well known in the literature not all irreducible representations of uq m2 can be extended to representations of the extension uq m2 composing the homomorphism with irreducible representations of uq m2 we obtain representations of uq iso2 not all of these representations of uq iso2 are irreducible the reducible representations of uq iso2 are decomposed into irreducible components in this way we obtain all irreducible representations of uq iso2 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra iso2 when q1 representations of the other part have no classical analogue

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

representations of the cyclically symmetric q deformed algebra soq 3 an algebra homomorphism from the nonstandard q deformed cyclically symmetric algebra uq so3 to the extension uq sl2 of the hopf algebra uq sl2 is constructed not all irreducible representations of uq sl2 can be extended to representations of uq sl2 composing the homomorphism with irreducible representations of uq sl2 we obtain representations of uq so3 not all of these representations of uq so3 are irreducible reducible representations of uq so3 are decomposed into irreducible components in this way we obtain all irreducible representations of uq so3 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra so3 when q 1 representations of the other part have no classical analog using the homomorphism it is shown how to construct tensor products of finite dimensional representations of uq so3 irreducible representations of uq so3 when q is a root of unity are constructed some of them are obtained from irreducible re

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

generalized chiral potts models and minimal cyclic representations of u_q widehat mathfrak g mathfrak l n c finite dimensional cyclic representations of the quantized universal enveloping algebra u q g exist only when the deformation parameter q is a root of unity q n 1 in such representations the n th power of each chevalley generator acts as a non zero scalar these values of e i n f i n and t i n are part of a continuous family of parameters upon which the cyclic representations depend

on α and d negative imaginary systems this paper is concerned with and d negative imaginary systems the definition of negative imaginary transfer functions is firstly introduced then the relationship between negative imaginary and negative imaginary transfer functions is studied by means of the generalised inverse an negative imaginary lemma is proposed to test the negative imaginary property of transfer functions also a necessary and sufficient condition is provided for the stability of interconnection of negative imaginary systems a state feedback controller design condition is established such that the resulting system is negative imaginary moreover the concept of negative imaginary transfer functions is extended to that of d negative imaginary transfer functions finally the developed results are illustrated by numerical examples

an improved inversion free method for solving the matrix equation x a x α a q in this paper based on the damped newton method we present an inversion free iterative method with a stepsize parameter for solving the nonlinear matrix equation x a x aa q where q is a hermitian positive definite matrix and a 1 the proposed method is an extension of algorithm 2 2 in z peng s m el sayed x zhang iterative methods for the extremal positive definite solution of the matrix equation x a x aa q j comput appl math 200 2007 520 527 preliminary numerical results show that the proposed method can improve numerical effect

a new inversion free method for a rational matrix equation abstract motivated by the classical newton schulz method for finding the inverse of a nonsingular matrix we develop a new inversion free method for obtaining the minimal hermitian positive definite solution of the matrix rational equation x a x 1 a i where i is the identity matrix and a is a given nonsingular matrix we present convergence results and discuss stability properties when the method starts from the available matrix aa we also present numerical results to compare our proposal with some previously developed inversion free techniques for solving the same rational matrix equation

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

a characterization of functional realizations of three dimensional quantum groups the authors study in an exhaustive and systematic way the functional realizations of quantum and classical algebras related with su 2 su 1 1 and the oscillator group os 1 in order to achieve this purpose they have adopted a mapping that is a generalization of an algebra expansion procedure they have not transformed the algebras themselves but their representations this method enables them not only to relate classical algebras and their q analogues but also different type algebras whether they are quantum or not they have obtained and classified all possible q realizations including as particular cases all those already known in the literature

a new inversion free iteration for solving the equation x a x 1 a q in this paper we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation x a x 1a q it is more accurate than zhan s algorithm j sci comput 17 1996 1167 and has less number of operations than the algorithm of guo and lancaster math comput 68 1999 1589 we derive convergence conditions of the iteration and existence conditions of a solution to the problem finally we give some numerical results to illustrate the behavior of the considered algorithm

the embedding u q so3 uq sl3 we study the embedding u q so3 uq sl3 where uq sl3 is a well known drinfeld jimbo quantum algebra and the algebra u q so3 is the cyclically symmetric q deformation of the universal enveloping algebra u so3 of the lie algebra so3 which is not a drinfeld jimbo quantum algebra finite dimensional irreducible representations of uq sl3 are decomposed into irreducible representations of u q so3 an explicit expression for the matrix of the transition from the gel fand tsetlin basis for uq sl3 to the bases of irreducible representations of u q so3 is calculated for representations of uq sl3 with highest weights l 0 0 entries of this matrix are expressed in terms of products of dual q krawtchouk polynomials and dual q hahn polynomials expressions for representation operators of uq sl3 in the u q so3 basis are given

representations of the q deformed algebras uq so2 1 and uq so3 1 representations of the algebra uq so2 1 are studied this algebra is a q deformation of the universal enveloping algebra u so2 1 of the lie algebra of the group so0 2 1 and differs from the quantum algebra uq su1 1 classifications of irreducible representations and of infinitesimally unitary irreducible representations of uq so2 1 with simple and discrete spectrum of one of generators are derived the set of infinitesimally unitary representations of uq so2 1 does not coincide with that for the algebra uq su1 1 the sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra uq so3 1 are given we also consider representations of uq son 1 which are of class 1 with respect to the subalgebra uq son

quantum deformations of su 3 and subalgebras quantum deformations of su 3 is discussed special attention is given to soq 3 and so 3 subalgebras of suq 3 and su 3 various chains of suq 3 and su 3 are realized by using the deforming functionals both boson and q boson realizations are also discussed

thermodynamic limit for a spin lattice an integrable spin lattice is a higher dimensional generalization of integrable spin chains in this paper we consider a special spin lattice related to quantum mechanical interpretation of the three dimensional lattice model in statistical mechanics zamolodchikov and baxter the integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables the significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters there is a specific bilinear functional equation for the eigenvalues of this transfer matrix the spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity the limiting form of bilinear equation is derived it allows to analyze the distributions of eigenvalues of the whole commuting set the ground state distribution is obtained explicitly a structure of excited states is discussed

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

quantum double structure for heisenberg weyl algebra without q deformation in this paper a new quasi triangular hopf algebra is presented as the quantum double of the heisenberg weyl algebra without q deformation its universal r matrix is built and the corresponding representation theory is studied with the explicit construction for the representations of this quantum double

a note on the generalized lie algebra in a recent paper dobrev and sudbery classified the highest weight and lowest weight finite dimensional irreducible representations of the quantum lie algebra introduced by lyubashenko and sudbery the aim of this note is to add to this classification all the finite dimensional irreducible representations which have no highest weight and or no lowest weight in the case when q is a root of unity for this purpose we give a description of the enlarged centre

solutions and perturbation analysis for the nonlinear matrix equation x i 1mai x 1ai i abstract in this paper we study the nonlinear matrix equation x i 1 m a i x 1 a i i arising in some physical calculations some new necessary and sufficient conditions for the existence of positive definite solutions are given based on the matrix differentiation a precise perturbation bound for the maximal positive definite solution is derived and the new result is illustrated by a numerical example in the end

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

form factors in the baxter bazhanov stroganov model i norms and matrix elements we continue our investigation of the baxter bazhanov stroganov model using the method of separation of variables 1 in this paper we calculate the norms and matrix elements of a local spin operator between eigenvectors of the auxiliary problem for the norm the multiple sums over the intermediate states are performed explicitly in the case n 2 we solve the baxter equation and obtain form factors of the spin operator of the periodic ising model on a finite lattice

sl n onsager s algebra and integrability we define ansl n analog of onsager s algebra through a finite set of relations that generalize the dolan grady defining relations for the original onsager s algebra this infinite dimensional lie algebra is shown to be isomorphic to a fixed point subalgebra ofsl n loop algebra with respect to a certain involution as the consequence of the generalized dolan grady relations a hamiltonian linear in the generators ofsl n onsager s algebra is shown to posses an infinite number of mutually commuting integrals of motion

parallel matrix function evaluation via initial value ode modeling the purpose of this article is to propose ode based approaches for the numerical evaluation of matrix functions f a a question of major interest in the numerical linear algebra for that we model f a as the solution at a finite time t of a time dependent equation we use parallel algorithms such as the parareal method on the time interval 0 t in order to solve the obtained evolution equation when f a is reached as a stable steady state it can be computed by combining parareal algorithms and optimal control techniques numerical illustrations are given

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

generalized chiral potts models and minimal cyclic representations of u_q widehat germ g germ l n bf c we present for odd n a construction of the n 1 state generalization of the chiral potts model proposed recently by bazhanov et al the yang baxter equation is proved

cyclic l operator related with a 3 state r matrix we consider the problem of constructing a cyclicl operator associated with a 3 stater matrix related to theuq sl 3 algebra atqn 1 this problem is reduced to the construction of a cyclic i e with no highest weight vector representation of some twelve generating element algebra which generalizes theuq sl 3 algebra we found such representation acting incn cn cn the necessary conditions of the existence of the intertwining operator for two representations are also discussed

cyclic representations of a q deformation of the virasoro algebra finite dimensional representations of a recently proposed q deformation vq of the virasoro algebra v0 for q a root of unity are constructed the representations have a cyclic structure and only in some special cases are they simultaneously highest weight representations of the su 1 1 like subalgebra of vq in the classical limit only those cyclic representations that are related to regular similarity transformations that become singular in the limit q to 1 reproduce highest weight representations of the standard virasoro algebra v0

representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the yang baxter equation quantum doubles of finite group algebras form a class of quasitriangular hopf algebras that algebraically solve the yang baxter equation each representation of the quantum double then gives a matrix solution of the yang baxter equation such solutions do not depend on a spectral parameter and to date there has been little investigation into extending these solutions such that they do depend on a spectral parameter here we first explicitly construct the matrix elements of the generators for all irreducible representations of quantum doubles of the dihedral groups dn these results may be used to determine constant solutions of the yang baxter equation we then discuss baxterization ansatze to obtain solutions of the yang baxter equation with a spectral parameter and give several examples including a new 21 vertex model we also describe this approach in terms of minimal dimensional representations of the quantum doubles of the alternating group a4 and the symmetric group s4

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

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

a note on tests for positive real functions this technical note presents a modified test for positive realness of scalar real rational functions which avoids the need to test residue conditions necessary and sufficient conditions for the positive realness of some classes of low order real rational functions are given by making use of the modified test

two structure preserving doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation in this paper we present two structure preserving doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation x a h x 1 a q where x c n i n is an unknown matrix and q c n i n is a hermitian positive definite matrix we prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation r quadratically in addition we also present some numerical results to illustrate the behavior of the considered algorithm

the q deformed poisson bracket levi civita symbol and poincare algebra in this paper we use a special choice of the glx qij 2n quantum plane and its differential calculus for a q deformed phase space to define a modified poisson bracket and construct the contraction rule for the q deformed levi civita symbol we find the q deformed phase space variable realization of the soq 3 and q deformed poincare algebras

positive definite solutions of the matrix equations x a x q a q q 1 in this paper we investigate nonlinear matrix equations x a x qa q where q 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x qa q to have two different positive definite solutions and several sufficient conditions for the equation x a x qa q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions of these equations numerical examples are given to illustrate the effectiveness of the methods

the simplified topological varepsilon algorithms for accelerating sequences in a vector space when a sequence of numbers is slowly converging it can be transformed into a new sequence which under some assumptions converges faster to the same limit one of the best known sequence transformations is the shanks transformation which can be recursively implemented by the varepsilon algorithm of wynn this transformation and this algorithm have been extended in two different ways to a sequence of elements of a topological vector space e in this paper we present new algorithms for implementing these topological shanks transformations they no longer require the manipulation of elements of the algebraic dual space e of e nor do they use the duality product inside the rules of the algorithms they need the storage of fewer elements of e and the stability is improved they also allow us to prove convergence and acceleration results for some types of sequences various applications involving sequences of vectors or matrices show the interest of the new algorithms

the quantum group suq 2 and the q analogue of angular momenta in classical mechanics a set of classical quantities that obey the poisson brackets corresponding to lie commutation relations of the quantum group suq 2 are obtained in classical mechanics they are constructed in terms of a function q l2 and conventional angular momenta which satisfy the classical correspondence of the lie commutation relation of the lie algebra su 2 these quantities are called the q analogue angular momenta and their conservation law is mentioned the authors give an example of q l2 that causes the classical q analogue angular momenta to reduce to the conventional angular momenta while the quantum deformation parameter gamma ln q vanishes

on tensor products of representations of the non standard q deformed algebra u q son the tensor product of vector and arbitrary representations of the non standard q deformation u q son of the universal enveloping algebra u son of lie algebra son is defined the clebsch gordan coefficients of the tensor product of vector and arbitrary classical type representations of q algebra u q son are found in an explicit form some important corollaries are considered in particular the wigner eckart theorem for vector operators is proved

two parametric solution to graded yang baxter equation and two parametric uuvgl 1 mod 1 algebra as a hopf algebra the authors discuss a two parametric solution of the graded yang baxter equation ybe and perform the yang baxterization to obtain the solution to the quantum ybe in the formalism developed previously they present the two parametric quantized superalgebra uuvgl 1 mod 1 and prove that this algebra is a hopf algebra with the hopf operations explicitly provided