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

finite dimensional schwinger basis deformed symmetries wigner function and an algebraic approach to quantum phase schwinger s finite d dimensional periodic hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the wigner function these subalgebras are shown to be admissible endowed with the non negative norm of hilbert space vectors hence they provide the desired canonical basis for the algebraic formulation of the quantum phase problem certain equivalence classes in the space of labels are identified within each subalgebra and connections with area preserving canonical transformations are examined the generalized representations of the wigner function are examined in the finite dimensional cyclic schwinger basis these representations are shown to conform to all fundamental conditions of the generalized phase space wigner distribution as a specific application of the schwinger basis the number phase unitary operator pair in is studied and based on the admissibility of the underlying q oscillator subalgebra an algebraic approach to the unitary quantum phase operator is established this being the focus of this work connections with the susskind glogower carruthers nieto phase operator formalism as well as standard action angle wigner function formalisms are examined in the infinite period limit the concept of continuously shifted fock basis is introduced to facilitate the fock space representations of the wigner function

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

prime decomposition and correlation measure of finite quantum systems under the name prime decomposition pd a unique decomposition of an arbitrary n dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of n is introduced for a class of density matrices a complete tensor product factorization is achieved the construction is based on the chinese remainder theorem and the projective unitary representation of by the discrete heisenberg group the pd isomorphism is unitarily implemented and it is shown to be co associative and to act on as comultiplication density matrices with complete pd are interpreted as group like elements of to quantify the distance of from its pd a trace norm correlation index is introduced and its invariance groups are determined

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

symmetries and time evolution in discrete phase spaces a soluble model calculation abstract using the flexibility and constructive definition of the schwinger bases we developed different mapping procedures to enhance different aspects of the dynamics and of the symmetries of an extended version of the two level lipkin model the classical limits of the dynamics are discussed in connection with the different mappings discrete wigner functions are also calculated

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

generalized m estimators for errors in variables regression this paper discusses robust estimation for structural errors in variables ev linear regression models such models have important applications in many areas under certain assumptions including normality the maximum likelihood estimates for the ev model are provided by orthogonal regression or which minimizes the orthogonal distance from the regression line to the data points instead of the vertical distance used in ordinary regression or is very sensitive to contamination and thus efficient robust procedures are needed this paper examines the theoretical properties of bounded influence estimators for univariate gaussian ev models using a generalized m estimate approach the results include fisher consistency most b robust estimators and the or version of hampel s optimality problem

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

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

principal components and orthogonal regression based on robust scales both principal components analysis pca and orthogonal regression deal with finding a p dimensional linear manifold minimizing a scale of the orthogonal distances of the m dimensional data points to the manifold the main conceptual difference is that in pca p is estimated from the data to attain a small proportion of unexplained variability whereas in orthogonal regression p equals m 1 the two main approaches to robust pca are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust univariate dispersion measure this article is more akin to second approach but rather than finding the components one by one we directly undertake the problem of finding for a given p a p dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold the scale may be either a smooth m scale or a trimmed scale an iterative algorithm is developed that is shown to converge to a local minimum a

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

discrete generator coordinate symmetry restoration in finite dimensional spaces abstract group theoretical based techniques and fundamental results from number theory are used in order to allow for the construction of exact projectors in finite dimensional spaces these operators are shown to make use only of discrete variables which play the role of discrete generator coordinates and their application in the number symmetry restoration is carried out in a nuclear bcs wave function which explicitly violates that symmetry

preparata codes through lattices we define an elementary family of lattices from which we obtain a family of extended cyclic codes with coefficients in the modular integers the first nontrivial subfamily is the family of quaternary preparata codes the family of dual codes coincides with the extended low correlation sequences introduced by kumar helleseth and calderbank 1995

the algebra of weyl symmetrised polynomials and its quantum extension the algebra of weyl symmetrised polynomials in powers of hamiltonian operatorsp andq which satisfy canonical commutation relations is constructed this algebra is shown to encompass all recent infinite dimensional algebras acting on two dimensional phase space in particular the moyal bracket algebra and the poisson bracket algebra of which the moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra we propose the introduction of a second deformation by the replacement of the heisenberg algebra forp q with aq deformed commutator and construct algebras ofq symmetrised polynomials

quasiprobability distribution functions for periodic phase spaces i theoretical aspects an approach featuring s parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed for such an approach a suitable set of angle angular momentum coherent states must be constructed in an appropriate fashion

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

schwinger and pegg barnett approaches and a relationship between angular and cartesian quantum descriptions ii phase spaces following the discussion in state space language presented in a preceding paper we work on the passage from the phase space description of a degree of freedom described by a finite number of states without classical counterpart to one described by an infinite and continuously labelled number of states with this it is possible to relate an original schwinger idea to the pegg barnett approach to the phase problem in phase space language this discussion shows that one can obtain the weyl wigner formalism for both cartesian and angular coordinates as limiting elements of the discrete phase space formalism

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

on the structure of quantum phase space the space of labels characterizing the elements of schwinger s basis for unitary quantum operators is endowed with a structure of symplectic type this structure is embodied in a certain algebraic cocycle whose main features are inherited by the symplectic form of classical phase space in consequence the label space may be taken as the quantum phase space it plays in the quantum case the same role played by phase space in classical mechanics some differences coming inevitably from its nonlinear character

discrete quantum phase spaces and the mod n invariance instituto de fisica teorica universidade estadual paulista rua pamplona 145 cep 01405 sao paulo sp

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

formal series of generalized functions and their application to deformation quantization foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

graph states in phase space the phase space for a system of n qubits is a discrete grid of 2n 2n points whose axes are labeled in terms of the elements of the finite field to endow it with proper geometrical properties we analyze the representation of graph states in that phase space showing that these states can be identified with a class of nonsingular curves we provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach

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

the reference state for finite coherent states we propose a reference state for finite dimensional coherent states which is easy to deal with in comparison to former suggestions which we briefly review we also advance explicit calculations which shows that the phase of the overlap of finite coherent state has a structure analogous to the usual infinite dimensional continuous coherent states

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

robust estimation when more than one variable per equation of condition has error summary it is well known that in the errors in variables estimation problem the results will have unnecessary asymptotic bias unless the algorithm is properly formulated similar difficulties can be expected with robust estimation techniques that are based on extending least squares to a noneuclidean metric this paper presents an algorithm for robust estimation for the nonlinear model errors in variables case the properties of the estimates produced by the algorithm are investigated and a numerical example using data on galaxies is given

robust line estimation with errors in both variables abstract the estimator holding the central place in the theory of the multivariate errors in the variables ev model results from performing orthogonal regression on variables rescaled according to the covariance matrix of the errors our simulations on the univariate model essentially relegate this estimator to use only in large samples on data with no trace of outlier contamination a modification requiring a robust preliminary slope is proposed that essentially sets out the generalization to ev of the robust w estimator in regression it is demonstrated that the modification is robust to outlier contamination even in small samples given a sufficiently good preliminary estimator candidates for a preliminary slope estimator based on the data are discussed and the performance of one under simulation is examined

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

dynamics in discrete phase spaces and time interval operators abstract the von neumann liouville time evolution equation is represented in a discrete quantum phase space the mapped liouville operator and the corresponding wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found the propagator is also discussed and a time interval operator associated to a unitary operator which shifts the energy levels in the zeeman spectrum is introduced this operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution the pair of unitary operators which shifts the time and energy is shown to obey the weyl schwinger algebra

von neumann lattices and magnetic orbitals in finite phase space it is proven that in the finite phase space of dimensionality m the number of independent von neumann states is m r where r is the number of distinct zeros of the kq function when applied to magnetic orbitals this leads to a linear dependence between them for each zero of the corresponding kq function different interesting identities follow from the linear dependence of von neumann states

consistency of robust estimators in multi structural visual data segmentation a theoretical framework is presented to study the consistency of robust estimators used in vision problems involving extraction of fine details a strong correlation between asymptotic performance of a robust estimator and the asymptotic bias of its scale estimate is mathematically demonstrated where the structures are assumed to be linear corrupted by gaussian noise a new measure for the inconsistency of scale estimators is defined and formulated by deriving the functional forms of four recent high breakdown robust estimators for each estimator the inconsistency measures are numerically evaluated for a range of mutual distances between structures and inlier ratios and the minimum mutual distance between the structures for which each estimator returns a non bridging fit is calculated

quantum discrete phase space dynamics and its continuous limit the main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket suitable for the description of time evolution in finite dimensional spaces is discussed a set of operator bases is defined in such a way that the weyl wigner formalism is shown to be obtained as a limiting case in the same form the moyal bracket is shown to be the limiting case of the discrete dynamical bracket the dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case

measurement error model collinearities collinearities can occur among the error free true predictors in measurement error models just as they occur among predictors in traditional regression models the coefficient estimators for measurement error models suffer the same ill effects of collinearities as do least squares estimators for linear measurement error models collinearities are a property of the secondorder moment matrix of the unobservable error free predictors for nonlinear models they are a property of the second order moment matrix of derivatives of the nonlinear function in this article collinearities among the predictors in measurement error models are defined and diagnostics for their detection are discussed the collinearity diagnostics introduced are similar to those used in traditional regression models but they are applied to estimated second order moment matrices of the error free predictors rather than to the second order moment matrix of the observed predictors examples are discussed one of which demonstrates th

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

corrections to scaling for diffusion in disordered media the authors study the diffusion of a particle in a d dimensional lattice where disorder arises from a random distribution of waiting times associated with each site of the lattice using scaling arguments they derive in addition to the leading asymptotic behaviour the correction to scaling terms for the mean square displacement they also perform detailed monte carlo simulations for one two and three dimensions which give results in substantial agreement with the scaling argument predictions

a squeezed state holomorphic phase space representation of equations of motion instituto de fisica teorica universidade estadual paulista rua pamplona 145 01405 s paulo sp

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

a coordinate independent formulation of the weyl wigner transform theory abstract we present the weyl wigner ww transform theory in a much more compact way than usual by introducing the basis in an intrinsic form this permits the derivation of new identities and also leads to generalizations like the inclusion of finite dimensional systems in the ww theory which is also discussed we show in this case some striking differences in the structure of finite phase space depending on the underlying dimension of quantum space being an even or odd integer

cyclic codes over z sub 4 locator polynomials and newton s identities certain nonlinear binary codes contain more codewords than any comparable linear code presently known these include the kerdock 1972 and preparata 1968 codes that can be very simply constructed as binary images under the gray map of linear codes over z sub 4 that are defined by means of parity checks involving galois rings this paper describes how fourier transforms on galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes these methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of z sub 4 linear codes with length 2 sup m m odd and size 2 2 sup m 1 5m 2 the gray image of the code of length 32 is the best 64 2 sup 37 code that is presently known this paper also determines the exact minimum lee distance of the linear codes over z sub 4 that are obtained from the extended binary two and three error correcting bch codes by hensel lifting the gray image of the hensel lift of the three error correcting bch code of length 32 is the best 64 2 sup 32 code that is presently known this code also determines an extremal 32 dimensional even unimodular lattice

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

robust regression using repeated medians abstract the repeated median algorithm is a robustified u statistic in which nested medians replace the single mean unlike many generalizations of the univariate median repeated median estimates maintain the high 50 breakdown value and can resist the effects of outliers even when they comprise nearly half of the data because they are calculated directly not iteratively repeated median procedures can be used as starting values for iterative robust estimation methods for bivariate linear regression with symmetric errors repeated median estimates are unbiased and fisher consistent and their efficiency under gaussian sampling can be comparable to the efficiency of the univariate median author

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

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

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

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

robust estimation in the errors in variables model summary orthogonal regression analogues of m estimates of regression called here orthogonal regression m estimates are defined these estimates are shown to be consistent at elliptical errors in variables models and robust if the corresponding loss function is bounded the orthogonal regression analogues of regression scale estimates called here orthogonal regression s estimates are considered as well in particular they provide a robust estimate for the scale of the orthogonal residuals a crucial quantity in the computation of orthogonal regression m estimates finally we present an algorithm for computing orthogonal regression s and m estimates and the results of a small monte carlo experiment

estimation of nonlinear errors in variables models for computer vision applications in an errors in variables eiv model all the measurements are corrupted by noise the class of eiv models with constraints separable into the product of two nonlinear functions one solely in the variables and one solely in the parameters is general enough to represent most computer vision problems we show that the estimation of such nonlinear eiv models can be reduced to iteratively estimating a linear model having point dependent i e heteroscedastic noise process particular cases of the proposed heteroscedastic errors in variables heiv estimator are related to other techniques described in the vision literature the sampson method renormalization and the fundamental numerical scheme in a wide variety of tasks the heiv estimator exhibits the same or superior performance as these techniques and has a weaker dependence on the quality of the initial solution than the levenberg marquardt method the standard approach toward estimating nonlinear models

the wigner function associated with the rogers szego polynomials a wigner function associated with the rogers szego polynomials is proposed and its properties are discussed it is shown that from such a wigner function it is possible to obtain well behaved probability distribution functions for both angle and action variables defined on the compact support and for m 0 respectively the width of the angle probability density is governed by the free parameter q characterizing the polynomials

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

a note on nonlinear xing codes nonlinear xing codes are considered it is shown that xing codes of length p 1 where p is a prime are subcodes of cosets of reed solomon codes whose minimum distance equals xing s lower bound on the minimum distance this provides a straightforward proof for the lower bound on the minimum distance of the codes the alphabet size of xing codes is restricted not to be larger than the characteristic of the relevant finite field f sub r it is shown that codes with the same length and the same lower bounds on the size and minimum distance as xing codes exist for any alphabet size not exceeding the size r of the relevant finite field thus extending xing s results

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

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