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

indecomposable generalisations of the standard angular momentum representations of so 3 indecomposable representations of the lie algebra of the rotation group so 3 are studied on the heisenberg weyl basis which is the basis for a subspace of a quotient space of the space of the universal enveloping algebra of the heisenberg weyl algebra the author obtains an infinite dimensional indecomposable representation which induces the finite dimensional representation on certain quotient spaces and subduces the infinite dimensional indecomposable representations on invariant subspaces the standard angular momentum representations of so 3 are given as special cases

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

irreducible tensor operators for the quantum algebras su 2 q irreducible tensor operators of su 2 q are defined based on the coproduct and are constructed by using the jordan schwinger mapping

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

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

finite dimensional control of distributed parameter systems by galerkin approximation of infinite dimensional controllers the galerkin method is presented as a way to develop finite dimensional controllers for linear distributed parameter systems dps the direct approach approximates the open loop dps and then generates the controller from this approximation the indirect approach approximates the infinite dimensional stabilizing controller conditions under which they produce equivalent controllers in the sense of closed loop stability are presented the indirect approach is shown to converge to the stable closed loop system consisting of dps and infinite dimensional controller conditions are presented on the behavior of the galerkin method for the open loop dps which guarantee closed loop stability for large enough finite dimensional approximations

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

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

on the principal subalgebra of quantum enveloping algebras glq l 1 the existence of a principal subalgebra of type slq 2 for quantum enveloping algebras glq l 1 or slq l 1 is investigated surprisingly only when l 2 and when all relations are restricted to symmetric representations such a principal subalgebra happens to exist this case slq 3 slq 2 is the q deformation of the classical su 3 so 3 embedding for symmetric su 3 representations and is analysed in more detail giving a connection with q deformed spherical harmonics

boson realisation of virasoro and kac moody algebras and their indecomposable representations in this letter we give the boson realisation of virasoro and kac moody algebras without central terms and construct their indecomposable representations from some representations of the heisenberg weyl algebra the boson realisation of the kac moody algebra a 2 associated to the lie algebra su 2 is discussed in detail

brief paper linear distributed parameter systems closed loop exponential stability with a finite dimensional controller many engineering systems exhibit dynamical behavior which must be described by partial rather than ordinary differential equations these distributed parameter systems dps have infinite dimensional state space descriptions yet they must be controlled by finite dimensional feedback systems implemented with a limited number of control actuators and sensors and an on line digital computer the most basic problem to be solved is whether such a finite dimensional controller can produce a stable closed loop system in this paper results are presented which characterize the linear dps which can be exponentially stabilized by a finite dimensional controller this characterization makes use of stabilizing subspaces and the solvability of the nonlinear asymmetric riccati equation connections with the usual model reduction approaches to finite dimensional dps controller design are also developed

a mapping construction for the q deformed so 3 u 3 embedding anticipating subsequent applications in nuclear structure theory a first construction of a dyson mapping for a q deformed u 3 algebra relevant to this field is presented to achieve this a q deformed algebra is initially considered realized in terms of tensor operators with respect to the standard and containing a q deformed so 3 angular momentum algebra the desired mapping is then realized in terms of two boson type conjugated tensor operators of first rank a key problem is to determine the commutation relations between them our construction is based on the requirement that subsets of the commutation relations of the original so 3 algebra is preserved as a result the images of the so 3 subalgebra of close the same commutation relations as the initial subalgebra of the angular momentum in addition a q deformed u 3 algebra containing the so 3 subalgebra of the images is obtained its generators are the q deformed components of a quadrupole operator together with the images of the so 3 subalgebra in the limiting case the reduction crucial to nuclear structure physics is recovered

stabilization of distributed parameter systems by piecewise polynomial control in this paper we consider the stabilization of continuous time distributed parameter systems by piecewise polynomial controls we obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls

the optimal projection equations for fixed order sampled data dynamic compensation with computation delay for an lqg type sampled data regulator problem which accounts for computational delay and utilizes an averaging a d device the equivalent discrete time problem is shown to be of increased order due to the inclusion of delayed measurement states the optimal projection equations for reduced order discrete time compensation are applied to the augmented problem to characterize low order controllers the design results are illustrated on a tenth order flexible beam example

the realizations of quantum groups of an 1 and cn types in q deformed oscillator systems at classical and quantum levels in the classical q deformed oscillators system the poisson bracket realizations of the quantum enveloping algebras of an 1 and cn types are given in the symplectic space v without deformation when the oscillators system is canonically quantized the lie bracket realizations of the quantum enveloping algebras of an 1 and cn types are obtained the hopf structures of these quantum enveloping algebras are supplied

uq covariant oscillators and vertex operators uq sl 2 covariant bose and fermi oscillators are studied systematically and applied to constructing vertex operators and uq invariant blocks for qp 1 p 2 3 a p2 dimensional unitary fock space representation of the bose oscillator algebra a q is constructed this representation does not allow however for the construction of all uq invariant blocks associated with an indecomposable representation of uq an extension of a is needed in this case which involves operators creating states of q spin projection p 2 from the fock vacuum the extended algebra only has infinite dimensional indefinite metric space representations an algebraic notion of trace is introduced provided q2 1 in the bose case this allows the introduction of a family of in general mixed uq invariant temperature states such that the vacuum expectation value appears in the zero temperature limit

boson fermion realisation of indecomposable representations for lie superalgebras lndecomposable representations of lie superalgebras are studied on quotient spaces of the universal enveloping algebra of the heisenberg weyl superalgebra b b f j e by boson fermion realisation these representations are constructed from certain types of indecomposable representations of the heisenberg weyl superalgebra and induce usual irreducible representations on invariant subspaces of a quotient space as a physically significant example the explicit form of the boson fermion realisation of indecomposable representations of the classical lie superalgebra su 211 are obtained and discussed in this paper

optimal quantized control the optimal closed loop quantized control is derived for the linear quadratic gaussian formulation and shown to be separable in estimation control and quantization the optimal quantizer is time varying and minimizes a quadratic distortion measure with weighting matrix dependent upon the solution to the matrix riccati equation the optimal cost togo is shown to be the sum of the cost to go for the optimal continuous valued control solution and a term reflecting the quantizer distortion

tensorial structure of a q deformed sp 4 r superalgebra a supersymmetric quantum algebra is generated by irreducible tensor operators with respect to the algebra suq 2 the even generators are realized as tensor products of q boson creation and annihilation operators transforming as suq 2 spinors and acting as odd generators in this way the transformation properties of all the algebra s generators with respect to the q deformed algebra of the angular momentum are simultaneously preserved which is important in view of future applications in physics in the limit q to 1 the classical lie superalgebra sp 4 r or the algebra osp 1 mod 4 is recovered

the structure of discrete time finite dimensional control of distributed parameter systems abstract in many applications the distributed parameter nature of engineering systems cannot be ignored this leads to state space descriptions of these systems on infinite dimensional spaces yet feedback controllers for engineering applications must be implemented with on line digital computers hence the controllers must be discrete time finite dimensional systems the principal question is whether such a controller can achieve exponentially stable control of an infinite dimensional system the main result theorem 5 shows that under some technical restrictions all such stable controllers for linear distributed parameter systems dps can do no better than feedback based on a finite dimensional projection of the dps state such stabilizing subspaces associated with the controller reveal the mathematical structure of the dps control problem but are not easy to calculate explicitly the more natural approach is to select model reducing subspaces which approximate the open loop dps and synthesize the controller from the reduced order model stability bounds are presented theorem 7 and connections are made with the structural results theorem 5

tensor operators for quantum groups and applications tensor operators are defined for hopf algebras if the hopf algebra is quasitriangular the properly defined tensor product of tensor operators is a tensor operator in one application the problem of defining invariant correlation functions for the uq sl 2 symmetric spin 1 2 heisenberg chain is considered in another application it is shown that suitably modified bosonic and fermionic q creation and annihilation operators are tensor operators of uq sl n