unitary rotation of square pixellated images the unitary rotation of square pixellated images is based on the finite su 2 oscillator model which describes systems whose values for position momentum and energy are discrete and finite in a two dimensional position space this allows the construction of angular momentum states orthonormal and complete for which rotations are defined as multiplication by phases that carry the rotation angle the decomposition of a digital square images in terms of these angular momentum states determines a unitary hence invertible rotation of the image whose kernel can be computed as a four dimensional array of real numbers

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

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

operator calculus approach to orthogonal polynomial expansions using techniques of operational calculus we show how to compute the generalized fourier coefficients for the meixner classes of orthogonal polynomials in particular krawtchouk polynomials are discussed in detail including an algorithm for computing krawtchouk transforms

linear canonical transformations and quantum phase a unified canonical and algebraic approach the algebra of generalized linear quantum canonical transformations is examined in the perspective of schwinger s unitary canonical operator basis formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space the representations of the wigner function in the generalized action angle unitary operator pair for certain hamiltonian systems with dynamical symmetry is examined this generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action angle wigner function

the action angle wigner function a discrete finite and algebraic phase space formalism the action angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space in this work a discrete and finite dimensional phase space formalism in which the phase space variables are discrete and the time is continuous is developed and the fundamental properties of the discrete weyl wigner moyal quantization are derived the action angle wigner function is shown to exist in the semi discrete limit of this quantization scheme a comparison with other formalisms which are not explicitly based on canonical discretization is made fundamental properties that an action angle phase space distribution respects are derived the dynamical properties of the action angle wigner function are analysed for discrete and finite dimensional model hamiltonians the limit of the discrete and finite dimensional formalism including a discrete analogue of the gaussian wavefunction spread viz the binomial wavepacket is examined and shown by examples that standard continuum quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity

dynamical emergence of markovianity in local time scheme recently we pointed out the so called local time scheme as a novel approach to quantum foundations that solves the preferred pointer basis problem in this paper we introduce and analyse in depth a rather non standard dynamical map that is imposed by the scheme on the one hand the map does not allow for introducing a properly defined generator of the evolution nor does it represent a quantum channel on the other hand the map is linear positive trace preserving and unital as well as completely positive but is not divisible and therefore non markovian nevertheless we provide quantitative criteria for dynamical emergence of time coarse grained markovianity for exact dynamics of an open system as well as for operationally defined approximation of a closed or open many particle system a closed system never reaches a steady state whereas an open system may reach a unique steady state given by the luders von neumann formula where the smaller the open system the faster a steady state is attained these generic findings extend the standard open quantum systems theory and substantially tackle certain cosmological issues

computation of several power indices by generating functions abstract in this paper we propose methods to compute the deegan packel the public good and the shift power indices by generating functions for the particular case of weighted voting games furthermore we define a new power index which combines the ideas of the shift and the deegan packel power indices and also propose a method to compute it with generating functions we conclude by some comments about the complexity to compute these power indices

continuous vs discrete fractional fourier transforms abstract we compare the finite fourier exponential and fourier kravchuk transforms both are discrete finite versions of the fourier integral transform the latter is a canonical transform whose fractionalization is well defined we examine the harmonic oscillator wavefunctions and their finite counterparts mehta s basis functions and the kravchuk functions the fractionalized fourier kravchuk transform was proposed in j opt soc amer a 14 1997 1467 1477 and is here subject of numerical analysis in particular we follow the harmonic motions of coherent states within a finite discrete optical model of a shallow multimodal waveguide

a local time induced unique pointer basis there is a solution to the problem of asymptotic completeness in many body scattering theory that offers a specific view of the quantum unitary dynamics which allows for the straightforward introdu

the golden number and fibonacci sequences in the design of voting structures abstract some distinguished types of voters as vetoes passers or nulls as well as some others play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power in this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters with this purpose in mind we enumerate these special classes of games and find out that its number always follows a fibonacci sequence with smooth polynomial variations as a consequence we find several families of games with the same asymptotic exponential behavior except for a multiplicative factor which is the golden number or its square from a more general point of view our studies are related with the design of voting structures with a predetermined importance ranking

properties of ladder tournaments abstract ladder tournaments use work trials to screen workers for positions in organizations despite being common in many industries their properties remain poorly understood especially in organizations characterized by complex complementarities and externalities in their production process in this study we formalize the ranking rule generated by a ladder tournament and derive measures of relative performance that can be used to compare workers when their true inputs cannot be observed we show that this rule is reflexive but not generally transitive unless the organization has fewer than three layers of hierarchy or is complete it follows that complete organizations have an embedded merit based hierarchical structure we also study the relationship between a worker s tournament rank and his utility or usefulness as measured by the frequency with which he plays a pivotal role we show that a worker s likelihood of being pivotal is a weakly increasing function of his rank this finding implies that in complete organizations ladder tournaments discriminate more effectively between workers than do mechanisms based on pivotability or marginal productivity

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 finite oscillator model related to mathfrak sl 2 1 we investigate a new model for the finite one dimensional quantum oscillator based upon the lie superalgebra in this setting it is natural to present the position and momentum operators of the oscillator as odd elements of the lie superalgebra the model involves a parameter p 0 p 1 and an integer representation label j in the 2j 1 dimensional representations wj of the hamiltonian has the usual equidistant spectrum the spectrum of the position operator is discrete and turns out to be of the form where k 0 1 j we construct the discrete position wavefunctions which are given in terms of certain krawtchouk polynomials these wavefunctions have appealing properties as can already be seen from their plots the model is sufficiently simple in the sense that the corresponding discrete fourier transform relating position wavefunctions to momentum wavefunctions can be constructed explicitly

achievable hierarchies in voting games with abstention abstract it is well known that he influence relation orders the voters the same way as the classical banzhaf and shapley shubik indices do when they are extended to the voting games with abstention vga in the class of complete games moreover all hierarchies for the influence relation are achievable in the class of complete vga the aim of this paper is twofold firstly we show that all hierarchies are achievable in a subclass of weighted vga the class of weighted games for which a single weight is assigned to voters secondly we conduct a partial study of achievable hierarchies within the subclass of h complete games that is complete games under stronger versions of influence relation

the canonical kravchuk basis for discrete quantum mechanics the well known kravchuk formalism of the harmonic oscillator obtained from the direct discretization method is shown to be a new way of formulating discrete quantum phase space it is shown that the kravchuk oscillator hamiltonian has a well defined unitary canonical partner which we identify with the quantum phase of the kravchuk oscillator the generalized discrete wigner function formalism based on the action and angle variables is applied to the kravchuk oscillator and its continuous limit is examined

weighted multiple majority games with unions generating functions and applications to the european union abstract an a priori system of unions or coalition structure is a partition of a finite set of players into disjoint coalitions which have made a prior commitment to cooperate in playing a game this paper provides ready to apply procedures based on generating functions that are easily implementable to compute coalitional power indices in weighted multiple majority games as an application of the proposed procedures we calculate and compare coalitional power indices under the decision rules prescribed by the treaty of nice and the new rules proposed by the council of the european union

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

a basic class of symmetric orthogonal polynomials of a discrete variable abstract by using a generalization of sturm liouville problems in discrete spaces a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters which generalizes all classical discrete symmetric orthogonal polynomials is introduced the standard properties of these polynomials such as a second order difference equation an explicit form for the polynomials a three term recurrence relation and an orthogonality relation are presented it is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class moreover moments corresponding to these weight functions can be explicitly computed finally a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail

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

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

closed form orthogonal number theoretic transform eigenvectors and the fast fractional ntt in this paper we propose a new method to find the closed form solution of number theoretic transform ntt eigenvectors we construct the complete generalized legendre sequence over the finite field cglsf and use it to solve the ntt eigenvector problem we derive the cglsf like ntt eigenvectors successfully including the case where the operation field is defined over the fermat and mersenne numbers the derived ntt eigenvector set is orthogonal and has a closed form it is suitable for constructing sub ntt building blocks for ntt implementation in addition with different eigenvalue assignment rule we can construct the fractional number theoretic transform fntt including the fractional fermat number transform ffnt the fractional complex mersenne number transform fcmnt and the fractional new mersenne number transform fnmnt they are the generalizations of the original transforms and all have the complexities of o nlog2n

power indices of simple games and vector weighted majority games by means of binary decision diagrams abstract a simple game is a pair consisting of a finite set n of players and a set w 2 n of winning coalitions vector weighted majority games v wmg are a special case of simple games in which an integer vector weight can be assigned to each player and there is a quota which a coalition has to achieve in order to win binary decision diagrams bdds are used as compact representations for boolean functions and sets of subsets this paper shows how a quasi reduced and ordered bdd qobdd of the winning coalitions of a v wmg can be build how one can compute the minimal winning coalitions and how one can easily compute the banzhaf shapley shubik holler packel and deegan packel indices of the players e g in case of weighted majority games it is shown that the banzhaf and holler packel indices of all players can be computed in expected time o nq and in general the banzhaf indices can be computed in time linear in the size of the qobdd representation of the winning coalitions other running times are proven as well the algorithms were tested on some real world games e g the international monetary fund and the eu treaty of nice

the maximum speed of dynamical evolution abstract we discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time its maximum speed of dynamical evolution previous analyses have given bounds in terms of e the standard deviation of the energy of the system here we give a strict bound that depends only on e e0 the system s average energy minus its ground state energy we also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution for example adding 1 j of energy to a given computer can never increase its processing rate by more than about 3 1033 operations per second

on the ordinal equivalence of the johnston banzhaf and shapley power indices abstract in this paper we characterize the games in which johnston shapley shubik and penrose banzhaf coleman indices are ordinally equivalent meaning that they rank players in the same way we prove that these three indices are ordinally equivalent in semicomplete simple games which is a newly defined class that contains complete games and includes most of the real world examples of binary voting systems this result constitutes a twofold extension of diffo lambo and moulen s result diffo lambo and moulen 2002 in the sense that ordinal equivalence emerges for three power indices not just for the shapley shubik and penrose banzhaf coleman indices and it holds for a class of games strictly larger than the class of complete games