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 schwinger su 3 construction i multiplicity problem and relation to induced representations the schwinger oscillator operator representation of su 3 is analyzed with particular reference to the problem of multiplicity of irreducible representations it is shown that with the use of an sp 2 r unitary representation commuting with the su 3 representation the infinity of occurrences of each su 3 irreducible representation can be handled in complete detail a natural generating representation for su 3 containing each irreducible representation exactly once is identified within a subspace of the schwinger construction and this is shown to be equivalent to an induced representation of su 3

universal formats for nonlinear ordinary differential systems it is shown that very general nonlinear ordinary differential systems embracing all that arise in practice may first be brought down to polynomial systems where the nonlinearities occur only as polynomials in the dependent variables by introducing suitable new variables into the original system second that polynomial systems are reducible to riccati systems where the nonlinearities are quadratic at most third that riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity the elemental systems have representations as novel types of matrix riccati equations different starting systems and their associated riccati systems differ from one another at the final elemental level in order and in initial data but not in format

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

tricritical point of the j1 j2 ising model on a hyperbolic lattice the ferromagnetic paramagnetic phase transition of the two dimensional frustrated ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method the model contains a ferromagnetic nearest neighbor interaction j 1 and a competing antiferromagnetic interaction j 2 a mean field like second order phase transition is observed when the ratio ensuremath kappa j 2 j 1 is less than 0 203 in the region 0 203l ensuremath kappa l1 4 the spontaneous magnetization is discontinuous at the transition temperature such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean field like

existence of an upper critical dimension in the majority voter model we study the critical properties of the majority voter model on d dimensional hypercubic lattices in two dimensions the majority voter model belongs to the same universality class as that of the ising model however the critical behaviors of the majority voter model on four dimensions do not exhibit mean field behavior using the monte carlo simulation on d dimensional hypercubic lattices we obtain the critical exponents up to d 7 and find that the upper critical dimension is 6 for the majority voter model we also confirm our results using mean field calculation

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

wigner distribution function for euclidean systems euclidean systems include poly and monochromatic wide angle optics acoustics and also infinite discrete data sets we use a recently defined wigner operator and quasiprobability distribution function to set up and study the phase space evolution of these models subject to differential and difference equations respectively infinite data sets and two dimensional monochromatic helmholtz fields are thus shown by their wigner function on a cylinder of direction and location the wigner function for polychromatic wavefields has c number coordinates of two dimensional wavenumber and position

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

wigner functions for curved spaces i on hyperboloids we propose a wigner quasiprobability distribution function for hamiltonian systems in spaces of constant curvature in this article on hyperboloids which returns the correct marginals and has the covariance of the shapiro functions under so d 1 transformations to the free systems obeying the laplace beltrami equation on the hyperboloid we add a conic oscillator potential in the hyperbolic coordinate as an example we analyze the one dimensional case on a hyperbola branch where this conic oscillator is the poschl teller potential we present the analytical solutions and plot the computed results the standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature

majority vote on directed erdős renyi random graphs abstract through monte carlo simulation the well known majority vote model has been studied with noise on directed random graphs in order to characterize completely the observed order disorder phase transition the critical noise parameter q c as well as the critical exponents and 1 have been calculated as a function of the connectivity z of the random graph

majority vote model on hyperbolic lattices we study the critical properties of a nonequilibrium statistical model the majority vote model on heptagonal and dual heptagonal lattices such lattices have the special feature that they only ca

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

phase transition of clock models on a hyperbolic lattice studied by corner transfer matrix renormalization group method two dimensional ferromagnetic n state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons the lattice lies on the hyperbolic plane with a constant negative scalar curvature we observe the spontaneous magnetization the internal energy and the specific heat at the center of sufficiently large systems where fixed boundary conditions are imposed for the cases n or 3 up to n 30 the model with n 3 which is equivalent to the three state potts model on the hyperbolic lattice exhibits a first order phase transition a mean field like phase transition of second order is observed for the cases n or 4 when n or 5 we observe a schottky type specific heat below the transition temperature where its peak height at low temperatures scales as n 2 from these facts we conclude that the phase transition of the classical xy model deep inside hyperbolic lattices is not of the berezinskii kosterlitz thouless type

the schwinger su 3 construction ii relations between heisenberg weyl and su 3 coherent states the schwinger oscillator operator representation of su 3 studied in a previous paper from the representation theory point of view is analyzed to discuss the intimate relationships between standard oscillator coherent state systems and systems of su 3 coherent states both su 3 standard coherent states based on choice of highest weight vector as fiducial vector and certain other specific systems of generalized coherent states are found to be relevant a complete analysis is presented covering all the oscillator coherent states without exception and amounting to su 3 harmonic analysis of these states

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

recurrent and feedforward polynomial modeling of coupled time series we present two methods for the prediction of coupled time series the first one is based on modeling the series by a dynamic system with a polynomial format this method can be formulated in terms of learning in a recurrent network for which we give a computationally effective algorithm the second method is a purely feedforward network procedure whose architecture derives from the recurrence relations for the derivatives of the trajectories of a ricatti format dynamic system it can also be used for the modeling of discrete series in terms of nonlinear mappings both methods have been tested successfully against chaotic series

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

statistical theory of learning curves under entropic loss criterion the present paper elucidates a universal property of learning curves which shows how the generalization error training error and the complexity of the underlying stochastic machine are related and how the behavior of a stochastic machine is improved as the number of training examples increases the error is measured by the entropic loss it is proved that the generalization error converges to h0 the entropy of the conditional distribution of the true machine as h0 m 2t while the training error converges as h0 m 2t where t is the number of examples and m shows the complexity of the network when the model is faithful implying that the true machine is in the model m is reduced to m the number of modifiable parameters this is a universal law because it holds for any regular machine irrespective of its structure under the maximum likelihood estimator similar relations are obtained for the bayes and gibbs learning algorithms these learning curves show the relation among the accuracy of learning the complexity of a model and the number of training examples