coherent states on the circle a careful study of the physical properties of a family of coherent states on the circle introduced some years ago by de bievre and gonzalez in 1992 semiclassical behaviour of the weyl correspondence on the circle group theoretical methods in physics vol i madrid ciemat is carried out they were obtained from the weyl heisenberg coherent states in by means of the weil brezin zak transformation they are labelled by the points of the cylinder and they provide a realization of by entire functions similar to the well known fock bargmann construction in particular we compute the expectation values of the position and momentum operators on the circle and we discuss the heisenberg uncertainty relation

invariant analytic orthonormalization procedure with an application to coherent states we discuss a general strategy which produces an orthonormal set of vectors stable under the action of a given set of unitary operators aj j 1 2 n starting from a fixed normalized vector in h and from a set of unitary operators we discuss several examples of this procedure and in particular we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done

spaces of type w growth of hermite coefficients wigner distribution and bargmann transform abstract in their famous monograph on generalized functions gelfand and shilov introduce the function spaces w m w and w m these spaces consist of c functions and holomorphic functions respectively with growth behaviour specified by suitable convex functions m and in this paper we study the spaces wmmx where mx denotes young s dual function corresponding to m we characterize the hermite expansion coefficients of the functions in wmmx their fourier transforms wigner distributions and bargmann transforms in particular we prove that wmmx wm wmx

implicit discrete time systems and accessibility an intrinsic description for dynamic systems whose evolution along discrete time is governed by nonlinear implicit difference equations in one independent variable and zero order algebraic equations is presented by means of differential geometrical methods where systems are associated with appropriate geometric objects reflecting their dynamics dynamic systems given in implicit form have the peculiarity that they may contain so called hidden restrictions a normal form is presented which is characterized by the circumstances that there are no further restrictions in addition it is illustrated that such a normal form allows for an equivalent system representation in explicit form based on the geometric picture of implicit discrete time systems the qualitative property of accessibility along a fixed trajectory is discussed by applying symmetry groups of discrete time systems and studying invariants of these groups a formal approach is provided that allows us to gather local accessibility criteria successively which can be tested by computer algebra several examples illustrate the results

the reverse order law for the generalized inverse a 2 t s by using the ranks of matrices the necessary and sufficient condition is given for the reverse order law of the generalized inverse a 2 t s of matrix product a a 1a 2 a n to hold as the special cases similar results for moore penrose inverse a the weighted moore penrose inverse a m n drazin inverse a d and group inverse a g are also derived

newton s method and quasi newton sqp method for general lc1 constrained optimization in this paper we study the second order sufficient and necessary conditions for lc 1 constrained optimization problems present newton methods and quasi newton sqp methods for this kind of problems and establish locally linear and superlinear convergence this work extends the classical theory of convex smooth minimization to general lc 1 constrained optimization

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

quantum mechanics on p adic numbers a quantum system with positions in and momenta in is studied displacement operators in the phase space of this system and the corresponding heisenberg weyl group are studied it is shown that such a system can be constructed from a semi infinite chain of spins which are coupled in a particular way

computation of feedback control for time optimal state transfer using groebner basis abstract computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered unlike a recent paper by the authors the target final state is not limited to the origin of state space but is allowed to be in the set of constrained controllable states switching surfaces are formulated as semi algebraic sets using groebner basis based elimination theory using these semi algebraic sets a nested switching logic is synthesized to generate the time optimal feedback control however the optimal control law enforces an unavoidable limit cycle in the time optimal trajectory for most non origin target points the time period of this limit cycle is dependent on the target position this dependence is algebraically characterized and a method to compute the time period of the limit cycle is provided as a natural extension the set of constrained controllable states is also computed

on controllability and near controllability of discrete time inhomogeneous bilinear systems without drift in this paper controllability and near controllability of discrete time inhomogeneous bilinear systems without drift are studied necessary and sufficient conditions for controllability and near controllability of the systems are respectively presented in particular algorithms for computing the control inputs to achieve the transition of a given pair of states for the controllable systems and nearly controllable systems are also provided examples are shown to demonstrate the conceptions and results of the paper

norm controllability of nonlinear systems in this paper we introduce and study the notion of norm controllability for nonlinear systems this property captures the responsiveness of a system with respect to applied inputs which is quantified via the norm of an output as a main contribution we obtain several lyapunov like sufficient conditions for norm controllability some of which are based on higher order derivatives of a lyapunov like function various aspects of the proposed concept and the sufficient conditions are illustrated by several examples including a chemical reactor application furthermore for the special case of linear systems we establish connections between norm controllability and standard controllability

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

reverse order laws for 1 2 3 generalized inverses in this paper we present purely algebraic necessary and sufficient conditions for reverse order laws to hold for 1 2 3 and 1 2 4 generalized inverses on the set of matrices

szego limit theorems for the harmonic oscillator let h 1d2 dx1 x2 be the harmonic oscillator hamiltonian on z r and let a be a selfadjoint do of order o in the beals fefferman class with weights qp 1 1 11 2 x f l 2 form the measure fi f lim 1 rank 7ra tr 7rx lwa where n air is the compression of onto the span of the hermite functions with eigenvalue less than or equal to a then one has the following szego limit theorem

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

multiplicative perturbations of matrices and the generalized triple reverse order law for the moore penrose inverse abstract for a matrix a let a denote its moore penrose inverse a matrix m is called a multiplicative perturbation of t c m n if m e t f for some e c m m and f c n n based on the alternative expression for m as m e t t t f t t the generalized triple reverse order law for the moore penrose inverse is obtained as m f t t y y t z z l r 1 e t t where y y t z z l r 1 is the weighted moore penrose inverse for certain matrices y z l and r associated to the triple t e f furthermore it is proved that this weighted moore penrose inverse in the resulting expression for m can be really replaced with t if e t t e t t t t f t t f t t in the special case that rank m rank t or m is a weak perturbation of t a simplified version of m as well as m m and m m is also derived

simple sufficient conditions for reachability of discrete time polynomial systems we consider reachability of discrete time polynomial systems the reachability of such systems can be interpreted as the surjectivity of the polynomial mappings defined by the systems for a polynomial mapping between suitable spaces of the same dimensions it has been shown that the injectivity of the mapping also implies the surjectivity by utilizing this fact we derive sufficient conditions for the reachability of the polynomial systems which essentially check the injectivity of the polynomial mappings

reverse order laws in c algebras abstract in this paper we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in c algebras extending rank conditions for matrices and range conditions for hilbert space operators

the growth of bargmann functions and the completeness of sequences of coherent states the growth of bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of bargmann functions of an order not exceeding and of a type not exceeding if of order these results generalize known results on the completeness of von neumann lattices the practical significance of this formalism in the context of quantum optics is also discussed

gabor schauder bases and the balian low theorem the balian low theorem is a strong form of the uncertainty principle for gabor systems that form orthonormal or riesz bases for l2 r in this paper we investigate the balian low theorem in the setting of schauder bases we prove that new weak versions of the balian low theorem hold for gabor schauder bases but we constructively demonstrate that several variants of the blt can fail for gabor schauder bases that are not riesz bases we characterize a class of gabor schauder bases in terms of the zak transform and product a2 weights the riesz bases correspond to the special case of weights that are bounded away from zero and infinity

the bargmann analytic representation in signal analysis the bargmann analytic representation of quantum mechanics is used in the context of time frequency signal analysis transformations between the various representations are derived examples that show how this method can be implemented numerically and demonstrate its robustness to noise are discussed

commutative algebraic methods for controllability of discrete time polynomial systems abstractin this paper we consider controllability of discrete time polynomial systems first we present a forward accessibility local reachability condition that can be verified in finite time in contrast to conventional conditions second we give a backward accessibility local controllability condition for an invertible system and a condition to verify invertibility finally we derive sufficient conditions to test whether the forward accessible system is reachable and to test the backward accessible system is controllable

controllability of nonlinear discrete time systems a lie algebraic approach this paper presents a geometric study of controllability for discrete time nonlinear systems various accessibility properties are characterized in terms of lie algebras of vector fields some of the results obtained are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

stabilization of polynomial nonlinear systems by algebraic geometry techniques this technical note applies methods from algebraic geometry namely polynomial ideals and sub modules to find parameterized sets of state feedback stabilizing control laws for polynomial continuous time affine nonlinear systems the proposed method is illustrated by simple examples

symmetry and the chazy equation there are three different actions of the unimodular lie group sl 2 c on a two dimensional space in every case we show how an ordinary differential equation admitting sl 2 as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation a particular example is the chazy equation whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation the reduction method leads to an alternative formula in terms of solutions to the lame equation resulting in a surprising transformation between the lame and hypergeometric equations finally we discuss the painleve analysis of the singularities of solutions to the chazy equation

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

when is b a a generalized inverse of ab in practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted in addition to full rank factorizations normal factorizations and singular value decompositions svd there are other factorizations of particular matrices that are natural to certain problems for instance in statistics answers to questions like when is b a a generalized inverse of ab may thus be a computational tool besides they are of significant interest because they provide us with insights into the g inversion of matrix products

a note on observability tests for general polynomial and simple wiener hammerstein systems we propose an algorithm based on symbolic computation packages for testing observability conditions of general polynomial systems which were formulated in sontag siam j control optim 17 1979 139 151 computational complexity of the observability test can be reduced and the test simplified for classes of polynomial systems we illustrate this by considering the class of simple wiener hammerstein systems which consist of a series of two linear dynamic blocks between which a static nonlinearity is sandwiched we consider the case when the nonlinearity is a monomial n q q 1 simple necessary and sufficient conditions for observability are given and they resemble but are subtly different from the well known result on observability for the series connection of two linear systems c 1998 elsevier science b v all rights reserved

on the finiteness of accessibility test for nonlinear discrete time systems it is shown that for two large subclasses of discrete time nonlinear systems analytic systems defined on a compact state space and rational systems the minimum length r for input sequences ca

dead beat controllability of polynomial systems symbolic computation approaches state and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition qepcad symbolic computation program the method is unified for a very large class of systems and can handle one or two sided control constraints families of minimum time state output dead beat controllers are obtained the computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system we may beat the curse of complexity a computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented necessary and sufficient conditions are given they are still very difficult to check therefore a number of easier to check sufficient conditions are also provided the latter are based on the grobner basis method and qepcad it is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system

the reverse order law revisited abstract necessary and sufficient conditions are given for the triple reverse order law abc c b a to hold some special cases are considered

observability at an initial state for polynomial systems we consider observability at an initial state for polynomial systems when testing for local observability for nonlinear systems the observability rank condition based on the inverse function theorem is commonly used however the rank condition is a sufficient condition and we cannot test for global observability using the rank condition in this paper first we derive necessary and sufficient conditions for global observability at an initial state for continuous time polynomial systems then necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring using the same procedure we derive both global and local observability conditions for discrete time polynomial systems each condition is characterized by a finite set of equations since polynomial rings are noetherian finally examples demonstrate the proposed criteria for testing for observability

multi resolution analysis and fractional quantum hall effect an equivalence result in this article we prove that any multi resolution analysis of l2 r produces for some values of the filling factor a single electron wave function of the lowest landau level lll which together with its magnetic translation gives rise to an orthonormal set in the lll we also discuss the inverse construction moreover we extend this procedure to the higher landau levels and we discuss the analogies and the differences between this procedure and the one previously proposed by j p antoine and the author

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

an integral transform related to quantization ii some mathematical properties we study in more detail the mathematical properties of the integral transform relating the matrix elements between coherent states of a quantum operator to the corresponding classical function explicit families of hilbert spaces are constructed between which the integral transform is an isomorphism

controllability of structured polynomial systems two algorithms based on the grobner basis method which facilitate the controllability analysis for a class of polynomial systems are presented the authors combine these algorithms with some results on output dead beat controllability in order to obtain sufficient as well as necessary conditions for complete and state dead beat controllability for a surprisingly large class of polynomial systems our results are generically applicable to the class of polynomial systems in strict feedback form

feedback linearization and lattice theory abstract the tools of lattice theory are applied to readdress the static state feedback linearization problem for discrete time nonlinear control systems unlike the earlier results that are based on differential geometry the new tools are also applicable for nonsmooth systems in case of analytic systems close connections are established between the new results and those based on differential one forms the mathematica functions have been developed that implement the algorithms methods from this paper

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

triple reverse order law for weighted generalized inverses in this paper we present a new triple reverse order law of weighted generalized inverses which does not involve the calculation of any generalized inverse our result is an important improvement of the existing reverse order law of weighted generalized inverse

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two