parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

wigner weyl correspondence in quantum mechanics for continuous and discrete systems a dirac inspired view drawing inspiration from dirac s work on functions of non commuting observables we develop an approach to phase space descriptions of operators and the wigner weyl correspondence in quantum mechanics complementary to standard formulations this involves a two step process introducing phase space descriptions based on placing position dependences to the left of momentum dependences or the other way around then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators the method works uniformly for both continuous cartesian degrees of freedom and for systems with finite dimensional state spaces it is interesting that the kernel encountered is naturally expressible in terms of geometric phases and its removal involves extracting its square root in a suitable manner

the isotropic lines of z2d we show that the isotropic lines in the lattice z 2 d are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of sl 2 z d we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group e d m related to the topic

wigner function for discrete phase space exorcising ghost images abstract we construct using simple geometrical arguments a wigner function defined on a discrete phase space of arbitrary integer hilbert space dimension that is free of redundancies ghost images plaguing other wigner functions for discrete phase spaces are thus revealed as artifacts it allows to devise a corresponding phase space propagator in an unambiguous manner we apply our definitions to eigenstates and propagator of the quantum baker map scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution

convolutions for berezin quantization and berezin lieb inequalities concepts and results from quantum harmonic analysis such as the convolution between functions and operators or between two operators are identified as the appropriate setting for berezin quantization and berezin lieb inequalities based on this insight we provide a rigorous approach to the generalized phase space representation introduced by klauder skagerstam and their variants of berezin lieb inequalities in this setting hence our presentation of the results of klauder skagerstam gives a more conceptual framework which yields as a byproduct an interesting perspective on the connection between the berezin quantization and weyl quantization concepts and results from quantum harmonic analysis such as the convolution between functions and operators or between two operators are identified as the appropriate setting for berezin quantization and berezin lieb inequalities based on this insight we provide a rigorous approach to the generalized phase space representation introduced by klauder skagerstam and their variants of berezin lieb inequalities in this setting hence our presentation of the results of klauder skagerstam gives a more conceptual framework which yields as a byproduct an interesting perspective on the connection between the berezin quantization and weyl quantization

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

an extended weyl wigner transformation for special finite spaces abstract we extend the weyl wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by schwinger discrete transformation kernels are presented instead of continuous coordinate momentum pair system and systems such as the one dimensional canonical continuous coordinate momentum pair system and the two dimensional rotation system are described by special limits expressions are explicitly given for the spin one half case

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

limit laws for the maximum and minimum of stationary sequences the class of non degenerate joint limiting distributions for the maximum and minimum of stationary mixing sequences is determined these limit distributions are of the form h x h x y where h x y is a bivariate extreme value distribution unlike the classical result for i i d sequences the maximum and minimum of stationary mixing sequences may be asymptotically dependent we also give a sufficient condition for the asymptotic independence of the maximum and minimum finally an example of a stationary sequence satisfying the mixing condition d in leadbetter but which is not weakly mixing is constructed

convergent relaxations of polynomial matrix inequalities and static output feedback using a moment interpretation of recent results on sum of squares decompositions of nonnegative polynomial matrices we propose a hierarchy of convex linear matrix inequality lmi relaxations to solve nonconvex polynomial matrix inequality pmi optimization problems including bilinear matrix inequality bmi problems this hierarchy of lmi relaxations generates a monotone sequence of lower bounds that converges to the global optimum results from the theory of moments are used to detect whether the global optimum is reached at a given lmi relaxation and if so to extract global minimizers that satisfy the pmi the approach is successfully applied to pmis arising from static output feedback design problems

quantum harmonic analysis on phase space relative to an irreducible representation of the canonical commutation relations convolutions between quantum mechanical operators and between functions and operators are defined for which the usual weyl transform acts as a fourier transform basic properties of these operations are developed in close analogy to harmonic analysis on r2n using the quantum version of wiener s approximation theorem a natural one to one correspondence between the closed phase space translation invariant subspaces of classical and quantum observables is established

wigner distributions for finite state systems without redundant phase point operators we set up wigner distributions for n state quantum systems following a dirac inspired approach in contrast to much of the work in this study requiring a 2n 2n phase space particularly when n is even our approach is uniformly based on an n n phase space grid and thereby avoids the necessity of having to invoke a quadrupled phase space and hence the attendant redundance both n odd and even cases are analysed in detail and it is found that there are striking differences between the two while the n odd case permits full implementation of the marginal property the even case does so only in a restricted sense this has the consequence that in the even case one is led to several equally good definitions of the wigner distributions as opposed to the odd case where the choice turns out to be unique

instability analysis of uncertain systems via determinants and lmis this technical note investigates two instability measures in continuous time ct and discrete time dt uncertain systems the first given by the spectral abscissa ct case or radius dt case and the second given by the sum ct case or product dt case of the unstable eigenvalues it is supposed that the system depends polynomially on an uncertain vector constrained into a semi algebraic set the problem is to determine the largest instability measures over the admissible uncertainties it is shown that a sufficient condition for establishing an upper bound of the sought measures can be obtained in terms of linear matrix inequality lmi feasibility tests by exploiting the determinants of some specific matrices and that this condition is also necessary under some mild conditions on the semi algebraic set by using multipliers with degree sufficiently large moreover a condition is provided for establishing the tightness of the found best upper bounds lastly it is shown that in the special case where the semi algebraic set is an interval the degree of the multipliers is known a priori some numerical examples illustrate the proposed results

convex synthesis of robust controllers for linear systems with polytopic time varying uncertainty the design of robust controllers for linear systems with structured time varying uncertainty is a fundamental problem in control systems this problem is still open because the available methods for robust analysis lead to nonconvex optimization whenever design variables are present this paper proposes a novel approach that overcomes this difficulty specifically continuous time linear systems with polytopic time varying uncertainty are considered the basic problem consists of designing a robust static output feedback controller ensuring robust asymptotical stability a novel approach is proposed through the introduction of robust stabilizability functions rsfs i e functions able to establish whether a controller is robustly asymptotically stabilizing in particular rsfs based on homogeneous polynomial lyapunov functions hplfs are searched for the proposed approach requires the solution of two convex optimization problems with constraints expressed as linear matrix inequalities lmis for any size of the lmis the proposed approach provides a sufficient condition for the existence of a sought controller moreover this condition is also necessary when the size of the lmis is large enough several extensions of the proposed methodology are presented which deal with the synthesis of controllers ensuring desired decay rate fixed order dynamic output controllers and controllers with the minimum norm

blue functions probability and current density propagators in non relativistic quantum mechanics like a green function to propagate a particle s wavefunction in time a blue function is introduced to propagate the particle s probability and current density accordingly the complete blue function has four components they are constructed from path integrals involving a quantity like the action that we call the motion the blue function acts on the displaced probability density as the kernel of an integral operator as a result we find that the wigner density occurs as an expression for physical propagation we also show that in quantum mechanics the displaced current density is conserved bilocally in two places at one time as expressed by a generalized continuity equation

robust static output feedback controllers via robust stabilizability functions this technical note addresses the design of robust static output feedback controllers that minimize a polynomial cost and robustly stabilize a system with polynomial dependence on an uncertain vector constrained in a semialgebraic set the admissible controllers are those in a given hyper rectangle for which the system is well posed first the class of robust stabilizability functions is introduced i e the functions of the controller that are positive whenever the controller robustly stabilizes the system second the approximation of a robust stabilizability function with a controller dependent lower bound is proposed through a sums of squares sos program exploiting a technique developed in the estimation of the domain of attraction third the derivation of a robust stabilizing controller from the found controller dependent lower bound is addressed through a second sos program that provides an upper bound of the optimal cost the proposed method is asymptotically non conservative under mild assumptions

wigner distributions and quantum mechanics on lie groups the case of the regular representation we consider the problem of setting up the wigner distribution for states of a quantum system whose configuration space is a lie group the basic properties of wigner distributions in the familiar cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n dimensional euclidean space rn to a lie group g the notion of canonical momentum is carefully analyzed and the meanings of marginal probability distributions and their recovery from the wigner distribution are clarified for the case of compact g an explicit definition of the wigner distribution is proposed possessing all the required properties geodesic curves in g which help introduce a notion of the midpoint of two group elements play a central role in the construction

a generalized wigner function for quantum systems with the su 2 dynamical symmetry group we introduce a wigner like quasidistribution function to describe quantum systems with the su 2 dynamic symmetry group this function is defined in a three dimensional group manifold and can be used to represent the states defined in several su 2 invariant subspaces the explicit differential moyal like form of the star product is found and analyzed in the semiclassical limit

limit theory for moving averages of random variables with regularly varying tail probabilities let z k infty k infty be iid where the z k s have regularly varying tail probabilities under mild conditions on a real sequence c j j geq 0 the stationary process x n sum infty j 0 c jz n j n geq 1 exists a point process based on x n converges weakly and from this a host of weak limit results for functionals of x n ensue we study sums extremes excedences and first passages as well as behavior of sample covariance functions

a generalized wigner function on the space of irreducible representations of the weyl heisenberg group and its transformation properties a natural extension of the wigner function to the space of irreducible unitary representations of the weyl heisenberg group is discussed the action of the automorphisms group of the weyl heisenberg group onto wigner functions and their generalizations and onto symplectic tomograms is elucidated some examples of physical systems are considered to illustrate some aspects of the characterization of the wigner functions as solutions of differential equations

general approach to mathfrak su n quasi distribution functions we propose an operational form for the kernel of a mapping between an operator acting in a hilbert space of a quantum system with an symmetry group and its symbol in the corresponding classical phase space for symmetric irreps of this mapping is bijective we briefly discuss complications that will occur in the general case

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

characterization of informational completeness for covariant phase space observables in the nonrelativistic setting with finitely many canonical degrees of freedom a shift covariant phase space observable is uniquely characterized by a positive operator of trace one and in turn by the fourier weyl transform of this operator we study three properties of such observables and characterize them in terms of the zero set of this transform the first is informational completeness for which it is necessary and sufficient that the zero set has dense complement the second is a version of informational completeness for the hilbert schmidt class equivalent to the zero set being of measure zero and the third known as regularity is equivalent to the zero set being empty we give examples demonstrating that all three conditions are distinct the three conditions are the special cases for p 1 2 of a more general notion of p regularity defined as the norm density of the span of translates of the operator in the schatten p class we show that the relation between zero sets and p regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis

from the discrete weyl wigner formalism for symmetric ordering to a number phase wigner function the general weyl wigner formalism in finite dimensional phase spaces is investigated then this formalism is specified to the case of symmetric ordering of operators in an odd dimensional hilbert space a respective wigner function on the discrete phase space is found and the limit when the dimension of hilbert space tends to infinity is considered it is shown that this limit gives the number phase wigner function in quantum optics analogous results for the almost symmetric ordering in an even dimensional hilbert space are obtained relations between the discrete wigner functions introduced in our paper and some other discrete wigner functions appearing in literature are studied

parameter and controller dependent lyapunov functions for robust d stability and robust performance controller design this paper addresses the design of output controllers of chosen order in a given set for ensuring a number of desired properties such as robust d stability and robust performance it is shown that a sufficient condition for determining such controllers can be obtained by solving convex optimization problems with linear matrix inequality constraints through the introduction of a lyapunov function depending polynomially on both uncertainty and controller the proposed condition is also necessary for sufficiently large degrees of the polynomials involved under some assumptions

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

h guaranteed cost computation by means of parameter dependent lyapunov functions a linear matrix inequality approach to compute h guaranteed costs by means of parameter dependent lyapunov functions is presented in this paper the uncertain linear time invariant systems are supposed to belong to convex bounded domains polytope type uncertainty the conditions proposed are less conservative than other parameter dependent based methods from the literature providing better estimates of worst case h norms in both continuous and discrete time systems as illustrated by means of numerical examples

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

torus as phase space weyl quantization dequantization and wigner formalism the weyl quantization of classical observables on the torus as phase space without regularity assumptions is explicitly computed the equivalence class of symbols yielding the same weyl operator is characterized the heisenberg equation for the dynamics of general quantum observables is written through the moyal brackets on the torus and the support of the wigner transform is characterized finally a dequantization procedure is introduced that applies for instance to the pauli matrices as a result we obtain the corresponding classical symbols

informationally complete sets of gaussian measurements we prove the necessary and sufficient conditions for the informational completeness of an arbitrary set of gaussian observables on continuous variable systems with a finite number of degrees of freedom in particular we show that an informationally complete set either contains a single informationally complete observable or includes infinitely many observables we show that for a single informationally complete observable the minimal outcome space is the phase space and the corresponding probability distribution can always be obtained from the quantum optical q function by linear postprocessing and gaussian convolution in a suitable symplectic coordinatization of the phase space in the case of projection valued gaussian observables e g generalized field quadratures we show that an informationally complete set of observables is necessarily infinite finally we generalize the treatment to the case where the measurement coupling is given by a general linear bosonic channel and characterize informational completeness for an arbitrary set of the associated observables

sequential measurement of conjugate variables as an alternative quantum state tomography it is shown how it is possible to reconstruct the initial state of a one dimensional system by sequentially measuring two conjugate variables the procedure relies on the quasicharacteristic function the fourier transform of the wigner quasiprobability the proper characteristic function obtained by fourier transforming the experimentally accessible joint probability of observing position then momentum or vice versa can be expressed as a product of the quasicharacteristic function of the two detectors and that unknown of the quantum system this allows state reconstruction through the sequence 1 data collection 2 fourier transform 3 algebraic operation and 4 inverse fourier transform the strength of the measurement should be intermediate for the procedure to work

the isotropic lines of bb z _ d 2 we show that the isotropic lines in the lattice are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group related to the topic

on informational completeness of covariant localization observables and wigner coefficients an orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group from this a series of lemmas is derived showing the informational completeness of natural covariant localization operators as well as of the generalized wigner distributions matrix elements of the group some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators the results are applied to special phase space representations of the heisenberg affine and galilei groups

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case

optimal stopping on autoregressive schemes for j co 1 i i d random variables the autoregression x en axn 1 a2xn 2 yields a payoff yn en wkxk when stopped at time n 0 c as c oo the distributions of t and the remainder term rt xt c are asymptotically independent and determined for exponential and algebraic tailed distributions on en an asymptotic expression for the optimal payoff is given and c c y is calculated so that t yields a payoff asymptotically optimal and asymptotic to c as y 1

wiener s closure of translates problem and piatetski shapiro s uniqueness phenomenon n wiener characterized the cyclic vectors with respect to translations in p z and l p r p 1 2 in terms of the zero set of the fourier transform he conjectured that a similar characterization should be true for 1 p 2 our main result contradicts this conjecture