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

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

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

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

an assumption based tms abstract this paper presents a new view of problem solving motivated by a new kind of truth maintenance system unlike previous truth maintenance systems which were based on manipulating justifications this truth maintenance system is in addition based on manipulating assumption sets as a consequence it is possible to work effectively and efficiently with inconsistent information context switching is free and most backtracking and all retraction is avoided these capabilities motivate a different kind of problem solving architecture in which multiple potential solutions are explored simultaneously this architecture is particularly well suited for tasks where a reasonable fraction of the potential solutions must be explored

dynamic interaction between multiple cracks and a circular hole swept by sh waves abstract the focus of this contribution is to develop a hypersingular integral equation method hiem to solve the dynamic interaction between multiple cracks and a circular hole illuminated by plane sh waves the cracks can be in arbitrary positions in particular they can terminate at the boundary of the circular hole in order to solve the proposed problem the green s function of a point harmonic force applied at an infinite plane with a circular hole is obtained by the graf formula and the wave function expansion method in order to construct the hypersingular integral equations for the cracks the green s function is separated into a closed form singular part and a regular part on the basis of the reciprocity principle as well as the obtained green s function the hypersingular integral equations for the cracks are obtained numerical solution of the hypersingular integral equations yields the dynamic stress intensity factors dsif at the crack tips comparison of present results with known results verifies the proposed method some numerical examples are presented in the paper

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

testing equivalences for processes abstract given a set of processes and a set of tests on these processes we show how to define in a natural way three different equivalences on processes these equivalences are applied to a particular language ccs we give associated complete proof systems and fully abstract models these models have a simple representation in terms of trees

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

recursive t matrix algorithms for the solution of electromagnetic scattering from strip and patch geometries two recursive t matrix algorithms are presented and their reduced computational complexities and reduced memory requirements are demonstrated these algorithms are applied to the problem of electromagnetic scattering from conducting strips and patches with canonical geometries the geometries are reminiscent of finite sized frequency selective surfaces computational complexities of o n sup 2 and o n sup 7 3 and memory requirements of o n and o n sup 4 3 are shown to be feasible for two dimensional and three dimensional geometries respectively the formulation uses only two components of the electric field therefore the vector electromagnetic problem of scattering from three dimensional patch geometries can be solved using scalar addition theorems for spherical harmonic wave functions for a two dimensional strip problem both tm and te polarizations can be solved simultaneously using this formulation numerical scattering results in the form of radar cross sections rcs are validated by comparison with the method of moments

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

truly concurrent constraint programming we study causality relationships in concurrent constraint programming what is observed is not just the conjunction of constraints deposited in the store but also the causal dependencies between these constraints we describe a denotational semantics for cc that is fully abstract with respect to observing this causality relation on constraints this semantics preserves more fine grained structure of computation in particular the interleaving law is not verified is indeterminate choice relationships between such a denotational approach to true concurrency and different powerdomain constructions are explored

towards action refinement in process algebras abstract we present a simple process algebra which supports a form of refinement of an action by a process and address the question of an appropriate equivalence relation for it the main result of the paper is that an adequate equivalence can be defined in a very intuitive manner in fact we show that it coincides with the timed equivalence proposed by one of the authors we also show that it can be characterized equationally

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

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

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

transition matrix in acoustic scattering by a strip the transition matrix describing the scalar scattering by a strip has been obtained using a boundary integral equation technique it avoids the procedure of the extended boundary condition taking the strip as the limit of an elliptic cylinder our final results confirm the known results obtained from the extended boundary condition technique the method presented in this paper seems to be very attractive to similar problems in elastodynamic scattering