a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

equivalence of q bosons using the exponential phase operator various forms of the q boson are explained and their hidden symmetry revealed by transformations using the exponential phase operator both the one component and the multicomponent q bosons are discussed as a byproduct we obtain a new boson algebra having a shifted vacuum structure and define a global operator u 1 gauge transformation

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

on the constrained b type kadomtsev petviashvili hierarchy hirota bilinear equations and virasoro symmetry we derive the bilinear equations of the constrained bkp hierarchy from the calculus of pseudodifferential operators the full hierarchy equations can be expressed in hirota s bilinear form characterized by the functions and besides we also give a modification of the original orlov schulman additional symmetry to preserve the constrained form of the lax operator for this hierarchy the vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless virasoro algebra

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

generalized q bosons and their squeezed states generalized q boson operators associated with the simultaneous creation of several q bosons are introduced these operators give rise to a realization of the quantum weyl heisenberg algebra and are applied to the construction of corresponding holstein primakoff realizations of the quantum algebras suq 2 and suq 1 1 coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states and some of their properties are studied particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states

energy spectrum potential and inertia functions of a generalized f oscillator we consider a generalized four parameter q algebra aa q a a q n n a a n a a associating with operators a and a the nonlinear f oscillator operators defined in terms of the usual harmonic oscillator operators as a af n and a f n a where a and a are operators of the weyl heisenberg algebra and n a a the function f n is determined from the commutation relations we write the hamiltonian for the free f oscillator and obtain its energy spectrum besides expressing the hamiltonian in terms of coordinate and momentum we determine the potential and inertia functions coordinate dependent mass and analyse their behaviour by varying the parameters

double squeezing in generalized q coherent states using a generalization of the q commutation relations we develop a formalism in which we define generalized q bosonic operators this formalism includes both types of the usual q deformed bosons as special cases the coherent states of these operators show interesting and novel noise reduction properties including simultaneous squeezing in both field components unlike the conventional case in which squeezing is permitted in only one component this also contrasts with the usual quantum group deformation which also only permits one component squeezing

a q deformed oscillator system with quantum group slq l symmetry a q deformed oscillator system with quantum group slq l symmetry is constructed in terms of the q deformed boson operators the energy level degeneracy associated with the irreducible representation of slq l is analysed especially for the case that q is a root of unity

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are 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

preconditioning legendre spectral collocation methods for elliptic problems i finite difference operators in 1979 orszag and morchoisne independently proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation over the years there has been intensive research both experimental and theoretical on the finite difference and finite element preconditioning of both chebyshev and legendre spectral collocation methods for this problem in this work we present results on a frequently employed variant of finite element preconditioning of legendre spectral collocation methods for the helmholtz equation with dirichlet boundary conditions we show that there exist two positive constants 0 0 and w n m a n m u u ell 2 w n m m 1 n m s n m u u ell 2 leq lambda 1 where w n m rm diagonal w k tilde w j the diagonal matrix of legendre gauss lobatto lgl weights and a n m is the collocation matrix m n m is the mass matrix and s n m is the stiffness matrix of the finite element method these estimates imply that the eigenvalues gamma k of s 1 n m m n m a n m satisfy rm re gamma k geq lambda 0 0 gamma k leq lambda 1 these results depend on a thorough study of the ratio 2w k x k 1 x k 1 where x k are lgl collocation points

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

real de sitter and conformal algebras in terms of q oscillators we express the cartan weyl basis of all quantum real d 4 de sitter and d 4 conformal algebras in terms of q deformed oscillators

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

time quantization and q deformations we extend to quantum mechanics the technique of stochastic subordination by means of which one can express any semi martingale as a time changed brownian motion as examples we considered two versions of the q deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

quantum and classical mechanics of q deformed systems quantum and classical mechanics of a system of q deformed bosonic oscillators are considered the q deformed heisenberg weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables the corresponding hilbert space is constructed in this approach the deformation parameter turns out to be a function of the planck constant an oscillator frequency and a parameter with dimension of length the hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q deformed classical theory the phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory the dimensional parameter introduced determines its radius it is argued that the q deformation can be associated with a special non canonical transformation the principle of least action for the classical q deformed system is formulated a representation of uq m in a space of functions on a phase space spanned by commutative coordinates is constructed

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

multi parametric deformed heisenberg algebras a route to complexity we introduce a generalization of the heisenberg algebra which is written in terms of a functional of one generator of the algebra f j0 that can be any analytical function when f is linear with slope we show that the algebra in this case corresponds to q oscillators for q2 tan the case where f is a polynomial of order n in j0 corresponds to an n parameter deformed heisenberg algebra the representations of the algebra when f is any analytical function are shown to be obtained through the study of the stability of the fixed points of f and their composed functions the case when f is a quadratic polynomial in j0 the simplest nonlinear scheme which is able to create chaotic behaviour is analysed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of heisenberg algebra

analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations abstract in the work of solving a uniformly elliptic differential equations au u a 1 u x a 2 u y a 0 u f in the unit square with boundary conditions by the c 1 cubic spline collocation method one may need to investigate efficient preconditioning techniques for this purpose using the generalized field of values argument we show the uniform bounds of the eigenvalues of the preconditioned matrix when a full finite element preconditioning is considered

generalized q oscillators and their hopf structures we study the relationships among the various forms of the q oscillator algebra and consider the conditions under which it supports a hopf structure we also present a generalization of this algebra together with its corresponding hopf structure its multimode extensions are also considered

dressing method darboux transformation and generalized restricted flows for the kdv hierarchy abstract we define generalized restricted flows by restricting general symmetry of the kdv hierarchy lax representation for generalized restricted flows is derived with the use of infinitesimal version of the zakharov shabat method finite form of this method is employed for deriving darboux transformation for restricted flows this darboux transformation adds one degree of freedom

two binary darboux transformations for the kdv hierarchy with self consistent sources two binary integral type darboux transformations for the kdv hierarchy with self consistent sources are proposed in contrast with the darboux transformation for the kdv hierarchy one of the two binary darboux transformations provides non auto backlund transformation between two nth kdv equations with self consistent sources with different degrees the formula for the m times repeated binary darboux transformations are presented this enables us to construct the n soliton solution for the kdv hierarchy with self consistent sources

on boson algebras as hopf algebras certain types of generalized undeformed and deformed boson algebras which admit a hopf algebra structure are introduced together with their fock type representations and their corresponding r matrices it is also shown that a class of generalized heisenberg algebras including those underlying physical models such as that of calogero sutherland is isomorphic with one of the types of boson algebra proposed and can be formulated as a hopf algebra

new deformed heisenberg oscillator the discrete spectrum of a deformed oscillator is calculated here for the first time according to the noncanonical heisenberg algebra the spectrum extends from omega 2 1 mu to omega mu where mu is a positive deformation parameter

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

recursively minimally deformed oscillators a recursive deformation of the boson commutation relation is introduced each step consists of a minimal deformation of a commutator a a fk n into a a qk 1 fk n where stands for the set of deformation parameters that fk depends on followed by a transformation into the commutator a a fk 1 qk 1 n to which the deformed commutator is equivalent within the fock space starting from the harmonic oscillator commutation relation a a 1 we obtain the arik coon and macfarlane biedenharn oscillators at the first and second steps respectively followed by a sequence of multiparameter generalizations several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained the generic form consists of a linear combination of exponentials of the number operator and the various recursive families can be classified according to the number of free linear parameters involved that depends on the form of the initia

the distribution of zeros of general q polynomials a general system of q orthogonal polynomials is defined by means of its three term recurrence relation this system encompasses many of the known families of q polynomials among them the q analogue of the classical orthogonal polynomials the asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation this result is applied to specific classes of polynomials known by the names q hahn q kravchuk q racah q askey and wilson al salam carlitz and the celebrated little and big q jacobi

on exact solutions of modified complex ginzburg landau equation abstract the one dimensional modified complex ginzburg landau equation has been studied by the use of the conte and musette method this method permits us to derive all the known exact solutions in a unified natural scheme these solutions are expressed in terms of solitary wave periodic unbounded wave and shock type wave we also find previously unknown exact propagating hole the degeneracies of modified complex ginzburg landau equation have also been examined as well as several of their solutions

restricted flows of soliton hierarchies coupled kdv and harry dym case restricted flows of soliton hierarchies associated with the energy dependent schrodinger spectral problem are determined explicitly a remarkable connection with separable potentials is used for proving complete integrability of the restricted flows a previously unknown lagrangian and hamiltonian formulation of the neumann system is found whole families of generalizations of the neumann and garnier systems are given

finite element preconditioning of g ni spectral methods several old and new finite element preconditioners for nodal based spectral discretizations of delta u f in the domain omega 1 1 d d 2 or 3 with dirichlet or neumann boundary conditions are considered and compared in terms of both condition number and computational efficiency the computational domain covers the case of classical single domain spectral approximations see c canuto et al spectral methods fundamentals in single domains springer heidelberg 2006 as well as that of more general spectral element methods in which the preconditioners are expressed in terms of local upon every element algebraic solvers the primal spectral approximation is based on the galerkin approach with numerical integration g ni at the legendre gauss lobatto lgl nodes in the domain the preconditioning matrices rely on either mathbb p 1 mathbb q 1 or mathbb q 1 ni i e with numerical integration finite elements on meshes whose vertices coincide with the lgl nodes used for the spectral approximation the analysis highlights certain preconditioners which yield the solution at an overall cost proportional to n d 1 where n denotes the polynomial degree in each direction

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

legendre and chebyshev dual petrov galerkin methods for hyperbolic equations a legendre and chebyshev dual petrov galerkin method for hyperbolic equations is introduced and analyzed the dual petrov galerkin method is based on a natural variational formulation for hyperbolic equations consequently it enjoys some advantages which are not available for methods based on other formulations more precisely it is shown that i the dual petrov galerkin method is always stable without any restriction on the coefficients ii it leads to sharper error estimates which are made possible by using the optimal approximation results developed here with respect to some generalized jacobi polynomials iii one can build an optimal preconditioner for an implicit time discretization of general hyperbolic equations 2007 elsevier b v all rights reserved msc 65m70 35l50 65m12

a study of representations of the algebra of functions on the quantum group glq n a q analogue of the root system is constructed for this algebra which is similar to the root system of classical lie algebras it is then used to construct in detail a class of representations of the algebra of functions on the quantum group glq n and a q boson realization of the generators of glq n is given i also construct infinite dimensional hilbert space representations of this algebra the main result of this paper is stated in proposition 8

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

algebra of pseudodifferential operators and symmetries of equations in the kadomtsev petviashvili hierarchy point symmetries are obtained for all equations in the kp hierarchy the lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tn the symmetry algebra is a semidirect sum of a virasoro algebra and a kac moody one the positive part of this algebra is embedded into the known w algebra of kp symmetries and into the free fermion algebra l the corresponding action on the tau function is presented the negative part of the point symmetries does not fit into the free fermion algebra but is embedded into a p algebra based on the algebra of pseudodifferential operators

the ghost symmetry of the bkp hierarchy in this paper we systematically develop the ghost symmetry of the kadomtsev petviashvili sub hierarchy of b type bkp through its actions on the lax operator l the eigenfunctions and the function in this process the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the bkp hierarchy moreover the bilinear identity of the constrained bkp hierarchy and adler shiota van moerbeke formula of the bkp hierarchy are rederived compactly by means of the spectral representation and ghost symmetry

verma module of the quantum group gl n q and its q boson and heisenberg weyl relation realizations the structure and verma module of the quantum matrix element algebra a n q of the quantum group gl n q are studied the method used here is similar to that for studying the structure and verma module of semisimple lie algebras the concept of the cartan subalgebra raising and lowering operators and their pairs is defined the q boson and the corresponding heisenberg weyl relation realizations are generally studied and the cyclic representations of a n q are obtained the explicit examples a 2 q and a 3 q are discussed in detail

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

on position and momentum operators in the q oscillator algebra the aim of this paper is to study the position and momentum operators in q deformed oscillator algebras the natural form of the position operator is xp qpn a a qpn where p is a real number this operator is an operator representable by a jacobi matrix using the theory of jacobi matrices the theory of classical moment problem and the theory of basic hypergeometric functions it is shown that depending on values of q and p xp can be unbounded symmetric operator which has the deficiency indices 1 1 and hence is not self adjoint but has self adjoint extensions bounded self adjoint operator with continuous simple spectrum or self adjoint operator of trace class therefore with discrete spectrum with zero as the point of accumulation of eigenvalues the connection of the q deformed heisenberg relation px qxp 1 for the position and momentum operators with a q deformation of the quantum harmonic oscillator is also considered

q oscillators realization of fq 4 and gq 3 a realization of the quantum superalgebras corresponding to f 4 and g 3 is given in terms of q deformed bosonic and fermionic oscillators

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems spectral collocation approximations based on legendre gauss lobatto lgl points for helmholtz equations as well as for the linear elasticity system in rectangular domains are studied the collocati

the q boson realizations of the quantum group uq sl n 1 c the authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras uq sl n 1 c in these formulae the generators of the algebra uq sl n 1 c are expressed by means of n canonical q boson pairs one auxiliary representation of the algebra uq gl n c

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

preconditioning chebyshev spectral collocation method for elliptic partial differential equations in this paper we analyze a preconditioning technique for the solution of chebyshev spectral collocation equations with dirichlet boundary conditions we obtain bounds on the eigenvalues for the helmholtz equation these eigenvalue bounds are obtained as a consequence of estimates on the field of values tilde a n 2 u u l 2 tilde q n 2 u u l 2 where tilde a n 2 is the weighted collocation matrix and tilde q n 2 is the preconditioner the preconditioner tilde q n 2 is robust in the sense that it provides bounds on the h 1 0 w condition number of tilde q n 2 1 tilde l n 2 when tilde l n 2 is the weighted collocation matrix associated with the general elliptic operator lu delta u a 1u x a 2u y a 0u

generalized deformed oscillator corresponding to the modified poschl teller energy spectrum a generalized deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the modified poschl teller potential is constructed for special values of the potential depth the deformed oscillator algebra has a finite dimensional irreducible representation the polynomial representation and the associate deformed operations of integration and differentiation are studied the results of this study are general and they can be applied directly to the case of the q deformed oscillator with q being a root of unity