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

the quantum superalgebra uq osp 1 2n deformed para bose operators and root of unity representations we recall the relation between the lie superalgebra osp 1 2n and para bose operators the quantum superalgebra uq osp 1 2n defined as usual in terms of its chevalley generators is shown to be isomorphic to an associative algebra generated by so called pre oscillator operators satisfying a number of relations from these relations and the analogue with the non deformed case one can interpret these pre oscillator operators as deformed para bose operators some consequences for uq osp 1 2n cartan weyl basis poincare birkhoff win basis and its hopf subalgebra uq gl n are pointed out finally using a realization in terms of q commuting q bosons we construct an irreducible finite dimensional unitary fock representation of uq osp 1 2n and its decomposition in terms of uq gl n representations when q is a root of unity

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

transformations of some induced osp 3 2 modules in an so 3 sp 2 basis the modules of the orthosymplectic lie superalgebra osp 3 2 induced from finite dimensional irreducible submodules of the stability subalgebra so 3 gl 1 are investigated the corresponding infinite dimensional irreducible or indecomposable modules the kac modules and the related typical and atypical modules are studied in detail every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so 3 sp 2 for each of these infinite dimensional or finite dimensional irreducible or indecomposable modules relations are written down giving the transformations of the basis under the action of the algebra generators

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

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

robust discrete controllers guaranteeing l sub 2 and l sub infinity performances an algorithm is given for designing discrete time controllers which are robust with respect to parameter variations and which also guarantee specified l sub 2 performance constraints for stochastic inputs and l sub infinity performance constraints for deterministic inputs this is not an analysis but a design method guaranteeing the specified robust performances also an upper bound on the covariance matrix is provided

upper and lower covariance bounds for perturbed linear systems both upper and lower bounds are established for state covariance matrices under parameter perturbations of the plant the motivation for this study lies in the fact that many robustness properties of linear systems are given explicitly in terms of the state covariance matrix moreover there exists a theory for control by covariance assignment the results provide robustness properties of these covariance controllers

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

finite dimensional representations of the quantum superalgebra uq gl 2 2 ii nontypical representations at generic q the construction approach proposed in the previous paper n a ky j math phys 35 2583 1994 allows us there and in the present paper to construct at generic deformation parameter q all finite dimensional representations of the quantum lie superalgebra uq gl 2 2 the finite dimensional uq gl 2 2 modules wq constructed in the previous paper are either irreducible or indecomposable if a module wq is indecomposable i e when the condition 4 41 in the previous paper does not hold there exists an invariant maximal submodule of wq say iqk such that the factor representation in the factor module wq iqk is irreducible and called nontypical here in this paper indecomposable representations and nontypical finite dimensional representations of the quantum lie superalgebra uq gl 2 2 are considered and classified as their module structures are analyzed and the matrix elements of all nontypical representations are written down explicitly

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

covariance bounds for discrete time linear systems with time varying parameter uncertainty this paper deals with the computation of upper bounds for the state covariance matrix of discrete time linear systems subject to stochastic excitation and additive time varying uncertainty in the system dynamic matrix such upper bounds are obtained as the stabilizing solutions of suitable h type riccati equations a necessary and sufficient condition for the existence of such solutions is given in terms of the h norm of a suitable transfer function as for the computation of the optimal bound it is demonstrated that the bounds are a convex function of a scalar parameter so that efficient numerical schemes can be worked out

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

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

on a direct approach to quasideterminant solutions of a noncommutative kp equation a noncommutative version of the modified kp equation and a family of its solutions expressed as quasideterminants are discussed the origin of these solutions is explained by means of darboux transformations and the solutions are verified directly we also verify directly an explicit connection between quasideterminant solutions of the noncommutative mkp equation and the noncommutative kp equation arising from the miura transformation

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

the optimal projection equations with petersen hollot bounds robust stability and performance via fixed order dynamic compensation for systems with structured real valued parameter uncertainty a feedback control design problem involving structured real valued plant parameter uncertainties is considered a quadratic lyapunov bound suggested by recent work of i r petersen and c v hollot 1986 is utilized in conjunction with the guaranteed cost approach of s s l chang and t k c peng 1972 to guarantee robust stability with robust performance bound necessary conditions that generalize the optimal projection equations for fixed order dynamic compensation are used to characterize the controller that minimizes the performance bound the design equations thus effectively serve as sufficient conditions for synthesizing dynamic output feedback controllers that provide robust stability and performance

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

quasideterminant multisoliton solutions of a supersymmetric chiral field model in two dimensions the darboux transformation of a supersymmetric principal chiral field model in two dimensions is studied we obtain exact superfield multisoliton solutions of the model by means of iterated darboux transformation and express them in terms of quasideterminants

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

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

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

a superalgebra uq osp 3 2 generated by deformed paraoperators and its morphism onto a wq 1 1 clifford weyl algebra a deformed universal enveloping superalgebra uq of the lie superalgebra osp 3 2 which is a hopf algebra is defined as freely generated by deformed para bose operators b and deformed para fermi operators f the q analog of the cartan weyl basis and all deformed supercommutation relations between the generators are written down a morphism of uq onto a deformed clifford weyl superalgebra wq 1 1 a is constructed where wq 1 1 a is generated by q bosons b and q fermions f under the condition that the q bosons anticommute with the q fermions the corresponding oscillator representation is studied

quasideterminant solutions of a non abelian toda lattice and kink solutions of a matrix sine gordon equation two families of solutions of a generalized non abelian toda lattice are considered these solutions are expressed in terms of quasideterminants constructed by means of darboux and binary darboux transformations as an example of the application of these solutions we consider the 2 periodic reduction to a matrix sine gordon equation in particular we investigate the interaction properties of polarized kink solutions

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

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

superposition of solutions to backlund transformations for the su n principal σ model we show that the backlund transformations for the su n principal model may be linearized using a geometrical interpretation of these equations involving the minimal orbit of su n n in the grassmann manifold gn c2n linearization puts the equations in zakharov mikhailov shabat zms form using this form of the equations we prove inductively a nonlinear superposition law and a permutability theorem for iterated backlund transformations analogous to known results in the theory of the sine gordon and kdv equations from the superposition law we get an explicit form for multisoliton solutions to the model

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

discrete time robust control via state feedback for single input systems a nonlinear robust state feedback controller for discrete time uncertain systems is proposed the uncertainties in the system must satisfy the matching condition but only their bounds need to be known the controller is designed via the second method of lyapunov the detailed derivation of the controller for single input systems is given the system response remains in the neighborhood of the zero state if uncertainties are small the uncertainty bound for the stability condition is derived

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

l2 to l gains for sampled data systems four kinds of 2 to gains for sampled data systems are defined and derived in this paper it turns out that these gains can be expressed by analytical formulae which are easy to compute only a discrete lyapunov equation is involved

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

darboux transformation of the generalized coupled dispersionless integrable system the darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on a non abelian lie group is studied and the solutions are shown to be expressed in terms of quasideterminants as an explicit example the darboux transformation on scalar solutions to the system based on the lie group su 2 is discussed in detail and the solutions are shown to be expressed as ratios of determinants

state space formulae for discrete time h optimization abstract state space formulae and proofs are given for ail the important steps in discrete time h 8 optimization the steps closely follow the algorithms for continuous time systems but there are some technically involved differences in the detail that make their derivation non trivial

quasideterminant solutions of an integrable chiral model in two dimensions the darboux transformation is used to obtain multisoliton solutions of the chiral model in two dimensions the matrix solutions of the principal chiral model and its lax pair are expressed in terms of quasideterminants the iteration of the darboux transformation gives the quasideterminant multisoliton solutions of the model it has been shown that the quasideterminant multisoliton solution of the chiral model is the same as obtained by zakharov and mikhailov using the dressing method based on the matrix riemann hilbert problem

algebraic structure of green s ansatz and its q deformed analogue the algebraic structure of green s ansatz 1953 is analysed in such a way that its generalization to the case of q deformed para bose and para fermi operators becomes evident to this end the underlying lie super algebraic properties of the parastatistics are essentially used

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states