a rodriguez formula and integration measure for the quantum deformed legendre functions quantum deformed analogs of the legendre functions are constructed from shift operator representations of the quantum deformed orbital angular momentum operators a rodriguez formula is found as well as the measure used to orthonormalize the functions

necessary and sufficient conditions for the controllability and observability of a class of linear time invariant systems with interval plants we establish necessary and sufficient conditions for the controllability of single input multi output linear time invariant systems with interval plants and for the observability of multi input single output linear time invariant systems with interval plants in arriving at our results we determine allowable plant uncertainty bounds for controllability and observability two specific examples are considered to demonstrate the applicability of our results

contractions of lie algebras and separation of variables the wigner inonu contraction from the rotation group o 3 to the euclidean group e 2 is used to relate the separation of variables in the laplace beltrami operators on two corresponding homogeneous spaces different realizations of the contraction take the two separable coordinate systems on the sphere to the four on the plane

hylomorphic solitons for the benjamin ono and the fractional kdv equations abstract this paper concerns with the existence of solitons namely stable solitary waves for the benjamin ono and the fractional kdv equations

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

stability robustness of lq optimal regulators for the performance index with cross product terms suggested are the guaranteed stability margins of lq optimal regulators for the performance index with cross product terms using modified return difference equality sufficient conditions to guarantee the required stability margins are presented as well as the conditions to guarantee the stability margins of standard lq optimal regulators an example for lq optimal control of a dc motor system is given to demonstrate how the weighting matrices can be chosen to guarantee the required stability margins

optimal choice of time scaling factor for linear system approximations using laguerre models clowes 1965 illustrated how to select the optimal time scaling factor for approximations of rational transfer functions using a series expansion of laguerre functions the present paper extends this result to approximations for general l sub 2 stable linear systems as a direct application of this extended result we are able to obtain approximations for irrational transfer functions which minimize the frequency domain error in the l sub 2 sense also we give empirical solutions for the optimal time scaling factor for approximations of first order plus delay systems which can then be applied to a larger class of overdamped systems

q oscillator realizations of the metaplectic representation of quantum osp 3 2 q oscillator representations of the quantum superalgebra ospq 3 2 are discussed it is pointed out that the metaplectic representation of ospq 3 2 can be equivalently realized either in terms of one q boson and one q fermion or with one q boson and two q fermions

necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time varying systems in this paper we consider linear time varying systems subject to norm bounded one block structured uncertainties first given an unforced system we find necessary and sufficient conditions for quadratic stability then considered a controlled system we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case

h sub infinity control and quadratic stabilization of systems with parameter uncertainty via output feedback the article concerns linear systems which are subject to both time varying norm bounded parameter uncertainty and exogenous disturbance it addresses the robust h sub infinity control problem of designing a linear dynamic output feedback controller such that the closed loop system is quadratically stable and achieves a prescribed level of disturbance attenuation for all admissible parameter uncertainties it is shown that such a problem is equivalent to a scaled h sub infinity control problem

robustness of controllability and observability of continuous linear time varying systems with parameter perturbations the controllability and the observability of continuous linear time varying systems with norm bounded parameter perturbations are analyzed the sufficient conditions which ensure the controllability of the perturbed system are presented by using the derived state transition matrix relationship between the nominal system and the perturbed system the proposed sufficient conditions allow the evaluation of controllability under the norm bounded parameter perturbations the robustness of the observability is also studied for continuous linear time varying systems with norm bounded parameter perturbations and the corresponding sufficient conditions are provided

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

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

the robustness of controllability and observability of linear time varying systems fixed point methods from nonlinear analysis are used to establish conditions under which the uniform complete controllability of linear time varying systems is preserved under nonlinear perturbations in the state dynamics and the zero input uniform complete observability of linear time varying systems is preserved under nonlinear perturbation in the state dynamics and output read out map robustness of partial controllability observability and a specific kind of nonzero input observability are also proven

on controllability and observability for a class of impulsive systems many dynamic systems in physics chemistry biology engineering and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes these complex dynamic behaviors can be modeled by impulsive differential systems this paper studies the controllability and observability for a class of time varying impulsive control systems several sufficient and necessary conditions for state controllability and state observability of such systems are established and the corresponding criteria for time invariant impulsive control systems are also obtained meanwhile several new results associated with variation of parameters for time varying impulsive control systems are derived

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

brief paper a notion of possible controllability for uncertain linear systems with structured uncertainty this paper introduces a notion of possible controllability for a class of uncertain linear systems with structured uncertainty described by averaged integral quadratic constraints this notion relates to the question of when a state is controllable for some possible value of the uncertainty the notion of possible controllability is motivated by a desire to extend the theory of minimal realization for linear time invariant systems to the case of uncertain systems with structured uncertainty

the metaplectic representation of suq 1 1 and the q gegenbauer polynomials the metaplectic representation of the quantum algebra suq 1 1 is shown to provide a group theoretic setting for certain basic orthogonal polynomials generalizing the usual gegenbauer polynomials

diagonalization of operators and one parameter families of nonstandard bases for representations of suq 2 nonstandard bases for finite dimensional irreducible representations of the quantum algebra suq 2 are constructed by diagonalizing one parameter families of the operators qj3 4 j j qj3 4 cqj3 and iqj3 4 j j qj3 4 cqj3 c we derive explicit expressions for the eigenfunctions and the corresponding eigenvalues of these operators in an arbitrary irreducible representation of suq 2 it is shown that the matrix elements of the intertwining operator aj c which is a q extension of the classical su 2 operator aj j1aj ajj3 are expressed in terms of the dual q krawtchouk polynomials diagonalization of some other operators associated with the dual q hahn polynomials is also examined

robust kalman filtering for uncertain systems abstract this paper studies the problem of kalman filter design for uncertain systems the system under consideration is subjected to time varying norm bounded parameter uncertainties in both the state and measurement matrices the problem we address is the design of a state estimator such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties a riccati equation approach is proposed to solve the above problem furthermore a suboptimal covariance upper bound can be computed by a convex optimization

observability of linear stochastic uncertain systems a concept of observability of uncertain systems has received some attention in the recent literature in this note we extend this concept to stochastic uncertain systems we show that the observability of a stochastic uncertain system can be characterized using a relative entropy functional which plays the same role as the observability gramian plays in linear system analysis also we establish a connection between the observability of a linear stochastic uncertain system under consideration and risk sensitive performance of an associated reference system

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

a minimization method and applications to the study of solitons abstract roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time a soliton is a solitary wave which exhibits some strong form of stability so that it has a particle like behavior in this paper we prove a general abstract theorem theorem 26 which allows to prove the existence of a class of solitons such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons then we apply the abstract theory to problems related to the nonlinear schrodinger equation nse and to the nonlinear klein gordon equation nkg

unitary representations of the quantum algebra suq 2 on a real two dimensional sphere for q r or generic q s1 some time ago rideau and winternitz introduced a realization of the quantum algebra suq 2 on a real two dimensional sphere or a real plane and constructed a basis for its representations in terms of q special functions which can be expressed in terms of q vilenkin functions and are related to little q jacobi functions q spherical functions and q legendre polynomials in their study the values of q were implicitly restricted to q r in the present paper we extend their work to the case of generic values of q s1 i e q values different from a root of unity in addition we unitarize the representations for both types of q values q r and generic q s1 by determining some appropriate scalar products from the latter we deduce the orthonormality relations satisfied by the q vilenkin functions

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

symmetry techniques for the al salam chihara polynomials an algebraic interpretation for the al salam chihara q orthogonal polynomials is presented using an explicit realization in terms of divided difference operators their symmetry q algebra is determined and studied this algebraic structure allows one to derive a new expansion formula for these polynomials

robust controllability of linear stochastic uncertain systems the paper extends the concept of robust controllability via linear state feedback to stochastic uncertain systems we show that the controllability of a stochastic uncertain system can be characterized using solutions to a game type differential riccati equation

representations of the quantum algebra suq 2 on a real two dimensional sphere representations of the quantum algebra suq 2 are constructed in terms of q special functions defined on real o 3 spheres and real planes they include q vilenkin functions related to little q jacobi functions q spherical functions and q legendre polynomials

new connection formulae for some q orthogonal polynomials in q askey scheme new nonlinear connection formulae of the q orthogonal polynomials such as continuous q laguerre continuous big q hermite q meixner pollaczek and q gegenbauer polynomials in terms of their respective classical analogues are obtained using a special realization of the q exponential function as infinite multiplicative series of ordinary exponential function

nonlinear differential difference and difference equations integrability and exact solvability abstract a brief review on the recent results of nonlinear differential difference and difference equations toward its complete integrability and exact solvability is presented in particular we show how lie s theory of differential equations can be extended to differential difference and pure difference equations and illustrate its applicability through the discrete korteweg devries equation as an example also we report that an autonomous nonlinear difference equation of an arbitrary order with one or more independent variables can be linearised by a point transformation if and only if it admits a symmetry vector field whose coefficient is the product of two functions one of the dependent variable and of the independent variables this is illustrated by linearising several first and second order ordinary nonlinear difference equations a possible connection between the lie symmetry analysis and the onset of chaos with reference to first order mappings is explored

bases for representations of quantum algebras we derive an explicit expression for the eigenfunctions and the corresponding eigenvalues of the operator q1 4j q q 1 4j q qj3 q 2 in an arbitrary irreducible representation of the algebra suq 2 the general form of the intertwining operator aj q which is a q extension of the classical su 2 operator aj j1aj ajj3 is also found the matrix elements of aj q are expressed in terms of the dual q kravchuk polynomials

harmonic functions of suq 2 for q ℝ and q s1 we show in this paper that a particular family of askey wilson polynomials can be interpreted directly in the light of q deformed suq 2 algebras this approach allows us to correct previous results concerning the q legendre functions investigated in granovskii and zhedanov 1993 j phys a math gen 26 4331 we also establish the orthonormalization and the special cases q 1 classical limit and q asymptote we conclude that these q legendre functions differ significantly from their classical counterparts only when q is in the vicinity of the unitary circle where the singular points of the absolute value of these q functions undergo a series of bifurcations

robust controllability for a class of uncertain linear time invariant mimo systems this note studies the problem of robust controllability of linear time invariant multiple input multiple output systems with a being or a and b both being sign invariant interval matrices for controllability sufficient conditions as well as necessary and sufficient conditions based on interval division method are presented the note also provides a way to handle interval systems that are not sign invariant in the a matrix some examples illustrate the results

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

basic hypergeometric functions and covariant spaces for even dimensional representations of uq osp 1 2 representations of the quantum superalgebra uq osp 1 2 and their relations to the basic hypergeometric functions are investigated we first establish clebsch gordan decomposition for the superalgebra uq osp 1 2 in which the representations having no classical counterparts are incorporated formulae for these clebsch gordan coefficients are derived and is observed that they may be expressed in terms of the q hahn polynomials we next investigate representations of the quantum supergroup ospq 1 2 which are not well defined in the classical limit employing the universal matrix the representation matrices are obtained explicitly and found to be related to the little q jacobi polynomials characteristically the relation q q is satisfied in all cases using the clebsch gordan coefficients derived here we construct new noncommutative spaces that are covariant under the coaction of the even dimensional representations of the quantum supergroup ospq 1 2

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

models of q algebra representations tensor products of special unitary and oscillator algebras this paper begins a study of one and two variable function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions the algebras considered are the quantum algebra uq su2 and a q analog of the oscillator algebra not a quantum algebra in each case a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations is used to compute the clebsch gordan coefficients it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed it is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations it is also demonstrated that general q hypergeometric functions can occur as

the q coulomb problem in configuration space we formulate the q coulomb problem in configuration space with the aid of ladder operators for the radial wave function the highest angular momentum state corresponding to the principal quantum number n is found to be the monomial rn 1 multiplied by a q exponential the states of lower angular momentum are q associated laguerre polynomials multiplied by the same q exponential the state functions all lie in the complex plane and may be interpreted in the standard way the energy levels are again given by a balmer formula with n replaced by the basic n

controllability and observability of linear time invariant uncertain systems irrespective of bounds of uncertain parameters in this paper the controllability and observability of linear time invariant uncertain systems are investigated the systems under consideration contain time invariant uncertain parameters that may take arbitrarily large values in such a situation the locations of uncertain parameters in system matrices play an important role we examine the permissible locations of uncertain parameters in system matrices for a linear uncertain system to be controllable and observable independently of the bounds of the uncertain parameters the objective of this paper is to show that a linear uncertain system is controllable and observable irrespective of the bounds of uncertain parameters if and only if the system has a particular configuration called a complete generalized antisymmetric stepwise configuration cgasc furthermore the dual configuration of a cgasc is introduced and studied here