the mathfrak su 2 _ alpha hahn oscillator and a discrete fourier hahn transform we define the quadratic algebra which is a one parameter deformation of the lie algebra extended by a parity operator the odd dimensional representations of with representation label j a positive integer can be extended to representations of we investigate a model of the finite one dimensional harmonic oscillator based upon this algebra it turns out that in this model the spectrum of the position and momentum operator can be computed explicitly and that the corresponding discrete wavefunctions can be determined in terms of hahn polynomials the operation mapping position wavefunctions into momentum wavefunctions is studied and this so called discrete fourier hahn transform is computed explicitly the matrix of this discrete fourier hahn transform has many interesting properties similar to those of the traditional discrete fourier transform

yager s new class of implications jf and some classical tautologies recently yager r yager on some new classes of implication operators and their role in approximate reasoning information sciences 167 2004 193 216 has introduced a new class of fuzzy implications denoted j f called the f generated implications and has discussed some of their desirable properties such as neutrality exchange principle etc in this work we discuss the class of j f implications with respect to three classical logic tautologies viz distributivity law of importation and contrapositive symmetry necessary and sufficient conditions under which j f implications are distributive over t norms and t conorms and satisfy the law of importation with respect to a t norm have been presented since the natural negations of j f implications given by n j f x j f x 0 in general are not strong we give sufficient conditions under which they become strong and possess contrapositive symmetry with respect to their natural negations when the natural negations of j f are not strong we discuss the contrapositivisation of j f along the lines of j f implications a new class of implications called h generated implications j h has been proposed and the interplay between these two types of implications has been discussed notably it is shown that while the natural negations of j f are non filling those of j h are non vanishing properties which determine the compatibility of a contrapositivisation technique

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

discrete nonlinear equations and the fuci k spectrum abstract we consider matrix vector equations of the form ax f x that are motivated by nonlinear oscillating systems such as the tacoma narrows bridge we identify a particular set called the fuci k spectrum which is relevant to questions of solvability and we develop theorems to describe the spectrum and show how it relates to the solvability of the matrix equation

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

a generalization of the migrativity property of aggregation functions this paper brings a generalization of the migrativity property of aggregation functions suggested in earlier work of some of the present authors by imposing the a migrativity property of durante and sarkoci for all values of a instead of a single one replacing the algebraic product by an arbitrary aggregation function b naturally leads to the properties of a b migrativity and b migrativity this generalization establishes a link between migrativity and a particular case of aczel s general associativity equation already considered by cutello and montero as a recursive formula for aggregation following a basic investigation emphasis is put on aggregation functions that can be represented in terms of an additive generator more specifically strict t norms strict t conorms and representable uninorms

expansion formulas for terminating balanced 4 f 3 series from the biedenharn elliot identity for su 1 1 in a recent paper george gasper contemp math 254 2000 187 proved some expansion formulas for terminating balanced hypergeometric series of type 4f3 with unit argument in this article we show how one easily derives such expansion formulas from the biedenharn elliot identity for the lie algebra su 1 1 furthermore we give a rather systematic method for determining when two apparently different expansion formulas are the same up to transformation formulas this is a rather nice application of the so called invariance groups of hypergeometric series the method extends to other cases we briefly indicate how it works in the case of expansion formulas for 3f2 series we conclude with some basic analogues and show their relation with the askey wilson polynomials

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

mutually exchangeable fuzzy implications recently vemuri and jayaram 2012 have proposed a novel generating method of fuzzy implications called the composition further they have also proposed mutual exchangeability me a generalization of the exchange principle ep to a pair of fuzzy implications and have shown that me plays a central role in the preservation of basic properties functional equations and families of fuzzy implications w r t the composition vemuri and jayaram 2014 due to the important role played by me in this work we investigate the pairs i j of fuzzy implications that satisfy me towards this we show first that there exist pairs i j of fuzzy implications that satisfy me and determine some necessary conditions on such fuzzy implications following this for a given i i we find the set j i of all fuzzy implications j such that the pair i j satisfies me keeping in view the variety of fuzzy implications and the complexity of the functional equation me we restrict our investigations to four important families of fuzzy implications namely s n r f and g implications further we discuss a generalization of the cauchy multiplicative equation whose solutions help us in obtaining the set j i for some families of fuzzy implications

mean value and uncertainty of optical phase a simple mechanical analogy there exist several definitions of the mean value and the phase uncertainty of an optical field we show that one of these definitions is especially simple and intuitive because of its mechanical analogy on this basis we also derive the summation rule and the tchebyshev inequality for random angular and phase variables and a number phase uncertainty relation

on the law of importation x wedge y longrightarrow z equiv x longrightarrow y longrightarrow z in fuzzy logic the law of importation given by the equivalence x lambda y rarr z equiv xrarr y rarr z is a tautology in classical logic in a implications defined by turksen et al the above equivalence is taken as an axiom in this paper we investigate the general form of the law of importation j t x y z j x j y z where t is a t norm and j is a fuzzy implication for the three main classes of fuzzy implications i e r s and ql implications and also for the recently proposed yager s classes of fuzzy implications i e f and g implications we give necessary and sufficient conditions under which the law of importation holds for r s f and g implications in the case of ql implications we investigate some specific families of ql implications also we investigate the general form of the law of importation in the more general setting of uninorms and t operators for the above classes of fuzzy implications following this we propose a novel modified scheme of compositional rule of inference cri inferencing called the hierarchical cri which has some advantages over the classical cri following this we give some sufficient conditions on the operators employed under which the inference obtained from the classical cri and the hierarchical cri become identical highlighting the significant role played by the law of importation

lattice operations on fuzzy implications and the preservation of the exchange principle in this work we solve an open problem related to the preservation of the exchange principle ep of fuzzy implications under lattice operations 3 problem 3 1 we show that generalizations of the commutativity of antecedents ca to a pair of fuzzy implications i j viz the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem further we determine conditions under which these become necessary too finally we investigate the pairs of fuzzy implications from different families such that ep is preserved by the join and meet operations

resonance with respect to the fucik spectrum for non selfadjoint operators abstract in this paper we consider a real linear non selfadjoint operator l dom l l 2 l 2 and study the solvability of the problem l u u u g u f 0 in the resonance case with respect to the fucik spectrum l we extend the standard landesman lazer type conditions and compare our result with results for selfadjoint operators as well as with results for the resonance in the point spectrum i e for l

oscillator wigner functions a closed expression is obtained for the n dimensional wigner oscillator function and a diagrammatic technique is developed for finding the explicit dependence of the function on euler angles at any dimension n

number phase uncertainty relations the minimization problem of finding the number phase minimum uncertainty states mus is considered and its solutions are found either numerically or under some special conditions analytically the phase uncertainty measure is based on the bandilla paul dispersion the problem is treated i in a finite dimensional hilbert space and ii for a countably infinite dimensional hilbert space i e the standard quantum harmonic oscillator with the constraint of a given mean photon number the mus relations between the photon number uncertainty and phase uncertainty are presented connections to some other minimization problems are discussed

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

a finite oscillator model with equidistant position spectrum based on an extension of mathfrak su 2 we consider an extension of the real lie algebra su 2 by introducing a parity operator p and a parameter c this extended algebra is isomorphic to the bannai ito algebra with two parameters equal to zero for this algebra we classify all unitary finite dimensional representations and show their relation with known representations of su 2 moreover we present a model for a one dimensional finite oscillator based on the odd dimensional representations of this algebra for this model the spectrum of the position operator is equidistant and coincides with the spectrum of the known su 2 oscillator in particular the spectrum is independent of the parameter c while the discrete position wavefunctions which are given in terms of certain dual hahn polynomials do depend on this parameter

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

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

the oscillator model for the lie superalgebra sh 2 2 and charlier polynomials we investigate an algebraic model for the quantum oscillator based upon the lie superalgebra sh 2 2 known as the heisenberg weyl superalgebra or the algebra of supersymmetric quantum mechanics and its fock representation the model offers some freedom in the choice of a position and a momentum operator leading to a free model parameter using the technique of jacobi matrices we determine the spectrum of the position operator and show that its wavefunctions are related to charlier polynomials cn with parameter 2 some properties of these wavefunctions are discussed as well as some other properties of the current oscillator model

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

nonlinear resonance set for nonlinear matrix equations abstract we study nonlinear matrix equations of the form a x f x where f is mildly nonlinear we identify a particular set called the fucik spectrum or resonance set which is relevant to questions of solvability of the equation we develop theorems to describe the spectrum and show how it relates to the solvability of the matrix equation we see that a small perturbation in the constant coefficients in f may cause a radical change in the character of the solutions of a x f x

position and momentum bases for the monochromatic maxwell fish eye and the sphere in geometric optics the maxwell fish eye is a medium where light rays follow circles while in scalar wave optics this medium can only trap fields of certain discrete frequencies in the monochromatic case characterized by a positive integer l there are independent fields we identify two bases of functions one known as the sherman volobuyev functions is characterized as of most definite momenta the other is new and composed of most definite positions and normal derivatives for the fish eye scalar wavefields their construction uses the stereographic projection of the sphere and their identification is corroborated in the contraction limit to the homogeneous helmholtz medium

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

on the spectrum of a hamiltonian defined on suq 2 and quantum optical models analytical expressions are given for the eigenvalues and eigenvectors of a hamiltonian with suq 2 dynamical symmetry the relevance of such an operator in quantum optics is discussed as an application the ground state energy in the dicke model is studied through suq 2 perturbation theory

nonlinear perturbations of linear equations in rn abstract the main goal of this note is to prove an existence theorem for the semilinear equation 0 1 a x x g x x r n for the case in which a is a symmetric n n matrix with real entries the vector field g r n r n is a bounded continuous function and is a simple eigenvalue of the matrix a in particular let u denote an eigenvector of a corresponding to with u 1 and assume that the real valued functions p u lim r g r u for all s 1 and p u lim r g r u for all s 1 where denotes the euclidean inner product in r n are defined on the unit sphere s 1 b 1 0 where b 1 0 denotes the open unit ball in r n the main theorem in this work states that if p u and p u are continuous in a neighborhood of the point u in s 1 and p u u 0 p u u then the equation in 0 1 has at least one solution

the clebsch gordan coefficients of the three dimensional lorentz algebra in the parabolic basis starting from the oscillator representation of the three dimensional lorentz algebra so 2 1 we build a lie algebra of second order differential operators which realizes all series of self adjoint irreducible representations the choice of the common self adjoint extention over a two chart function space determines whether they lead to single or multivalued representations over the corresponding lie group the diagonal operator defining the basis is the parabolic subgroup generator the direct product of two such algebras allows for the calculation of all clebsch gordan coefficients explicitly as solutions of schrodinger equations for poschl teller potentials over one d d two d c or three c c charts all coefficients are given in terms of up to two 2f1 hypergeometric functions

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

representations through a monoid on the set of fuzzy implications fuzzy implications are one of the most important fuzzy logic connectives in this work we conduct a systematic algebraic study on the set ii of all fuzzy implications to this end we propose a binary operation denoted by which makes i i a non idempotent monoid while this operation does not give a group structure we determine the largest subgroup ss of this monoid and using its representation define a group action of ss that partitions ii into equivalence classes based on these equivalence classes we obtain a hitherto unknown representations of the two main families of fuzzy implications viz the f and g implications

r implications and the exchange principle the case of border continuous t norms it is well known that the residual of a left continuous t norm satisfies the exchange principle ep however the left continuity of a t norm is only sufficient and not necessary as many examples in the literature illustrate in this work we study the necessary and sufficient conditions on a t norm for its residual to satisfy ep we present a complete characterization of two classes of t norms whose residuals satisfy ep viz t norms that are border continuous and those that have an ordinal sum representation based on the obtained results we characterize t norms whose residuals satisfy both the exchange principle and the ordering property

finite q oscillator the finite q oscillator is a model that obeys the dynamics of the harmonic oscillator with the operators of position momentum and hamiltonian being functions of elements of the q algebra suq 2 the spectrum of position in this discrete system in a fixed representation j consists of 2j 1 sensor points and similarly for the momentum observable the spectrum of energies is finite and equally spaced so the system supports coherent states the wavefunctions involve dual q kravchuk polynomials which are solutions to a finite difference schrodinger equation time evolution times a phase defines the fractional fourier q kravchuk transform in the classical limit as q 1 we recover the finite oscillator lie algebra the n 2j limit returns the macfarlane biedenharn q oscillator and both limits contract the generators to the standard quantum mechanical harmonic oscillator