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

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

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

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

average complexity of the jiang wang zhang pairwise tree alignment algorithm and of a rna secondary structure alignment algorithm we prove that the average complexity of the pairwise ordered tree alignment algorithm of jiang wang and zhang is in o nm where n and m stand for the sizes of the two trees respectively we show that the same result holds for the average complexity of pairwise comparison of rna secondary structures using a set of biologically relevant operations

compound poisson approximations for sums of random variables abstract this document shows that a sum of dependent random variables is approximately compound poisson when the variables are rarely nonzero and given they are nonzero their conditional distributions are nearly identical it give several upper bounds on the total variation distance between the distribution of such a sum and a compund poisson distribution included is an example for markovian occurrences of a rare event the bounds are consistent with those that are known for poisson approximations for sums of uniformly small random variables author

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

an algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group an algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group is formulated for the first time explicitly in detail for both the finite dimensional and infinite dimensional hilbert spaces for the finite dimensional hilbert space the su 2 algebraic representation is used while for the infinite dimensional hilbert space the heisenberg weyl algebraic representation is employed the fourier expansion technique is generalized to the operator space which is suitable for analysis of irregular spectra the polynomial operator basis is also used for the complement which is appropriate for analysis of some simple hamiltonians the proposed new approach is applied to solve the classical inverse sturm liouville problem and to study the problems of quantum regular and irregular spectra

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

the discrete harmonic oscillator harper s equation and the discrete fractional fourier transform certain solutions to harper s equation are discrete analogues of and approximations to the hermite gaussian functions they are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator and they lead to a definition of a discrete fractional fourier transform ft the discrete fractional ft is essentially the time evolution operator of the discrete harmonic oscillator

an algebraic view of the relation between largest common subtrees and smallest common supertrees the relationship between two important problems in tree pattern matching the largest common subtree and the smallest common supertree problems is established by means of simple constructions which allow one to obtain a largest common subtree of two trees from a smallest common supertree of them and vice versa these constructions are the same for isomorphic homeomorphic topological and minor embeddings they take only time linear in the size of the trees and they turn out to have a clear algebraic meaning

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

loop gauge theory and group cohomology a generalized fiber bundle model in which the fibers are hilbert spaces is studied unitary transformations are used to define a unitary isomorphism parallelism among them the weyl group is first used to connect the projective hilbert spaces ray spaces and to introduce a one form connection that defines which coset at a point y is parallel to a given coset at another point x then the central extension of the weyl group by u 1 is studied in order to introduce the most general mapping between the elements of these cosets this leads to a two form connection and makes the model a good candidate for a fiber bundle approach to string theories

rna secondary structure comparison exact analysis of the zhang shasha tree edit algorithm we are interested in rna secondary structure comparison using an approach which consists to represent these structures by labeled ordered trees following the problem considered this tree representation can be rough considering only the structural patterns or refined until an exact coding of the structure is obtained after some preliminary definitions and the description of the zhang shasha siam j comput 18 6 1989 1245 tree edit algorithm which is on the one hand the reference when dealing with ordered labeled trees comparison and on the other hand the starting point of our work this article will present an exact analysis of its complexity the purpose of this work is also to lead us to a better comprehension of the parameters of this algorithm in order to be able to modify it more easily without changing its time complexity to take into account biological constraints that occur when comparing rna secondary structures

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

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

simple fast algorithms for the editing distance between trees and related problems ordered labeled trees are trees in which the left to right order among siblings is significant the distance between two ordered trees is considered to be the weighted number of edit operations insert delete and modify to transform one tree to another the problem of approximate tree matching is also considered specifically algorithms are designed to answer the following kinds of questions 1 what is the distance between two trees 2 what is the minimum distance between t 1 and t 2 when zero or more subtrees can be removed from t 2 3 let the pruning of a tree at node n mean removing all the descendants of node n the analogous question for prunings as for subtrees is answered a dynamic programming algorithm is presented to solve the three questions in sequential time o t 1 times t 2 times min textit depth t 1 textit leaves t 1 times min textit depth t 2 textit leaves t 2 and space o t 1 times t 2 compared with o t 1 times t 2 t

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

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

poisson approximation for excursions of adaptive algorithms with a lattice state space this paper analyzes excursions of adaptive algorithms such as the lms with a lattice state space under certain conditions on the input and disturbance statistics the parameter estimate error forms a markov chain the approximations are valid if this chain has a strong tendency toward an equilibrium point the distribution of the number of excursions in n units of time is approximated by a poisson distribution the mean and distribution of the time of the occurrence of the first excursion are approximated by those of an exponential distribution expressions for the error in the approximations are also derived the approximations are shown to hold asymptotically as the excursion defining set converges to the empty set all the parameters required for the approximations and all expressions for the error in the approximations are calculable in a relatively straightforward manner