small systems of generators of isotropic unitary groups over finite fields of characteristic not two abstract some sets of generators of unitary groups on a nonsingular hermitian space with hyperbolic rank at least one over a finite field of characteristic not 2 with an involution are studied all unitary groups are shown to be generated by 3 elements and all symplectic groups are generated by 2 elements

response distributions in intensity resolution and speech discrimination in this paper the assumption of an equal gaussian distribution of the response to each stimulus in an experiment an assumption which has to be met if d is to be estimated by calculating the difference between z h and z fa is tested for two different sets of stimuli 1000 hz tones differing in level only and a continuum of stop consonants obtained by full spectral interpolation between p t and k response distributions were measured directly by means of a form of non numerical magnitude estimation in which subjects had to indicate the position of each stimulus on a quasi continuous rating scale it could be shown that in general all distributions were sufficiently unimodal but that their variances differed the consequences for the calculation of d are unlikely to be serious

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

integrable maps from galois differential algebras borel transforms and number sequences abstract a new class of integrable maps obtained as lattice versions of polynomial dynamical systems is introduced these systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation in particular symmetries and integrability see tempesta 2010 40 our approach is based on the properties of a suitable galois differential algebra that we shall call a rota algebra a formulation of the procedure in terms of category theory is proposed in order to render the lattice dynamics confined a borel regularization is also adopted as a byproduct of the theory a connection between number sequences and integrability is discussed

signal densities and criterion variance in speech and nonspeech perception the actual shape of signal densities has become an important issue when studying speech perception within the framework of signal detection theory sdt using an sdt model that allowed unequal criterion variances lopez bascuas proc eurosp 3 2281 2283 1995 found that speech signals did not accommodate to the standard gaussian assumption however schouten and van hessen j acoust soc am 104 2980 2990 1998 measured response distributions directly and assuming an interval scale concluded that the gaussian assumption held for both continua nevertheless pastore and macmillan j acoust soc am 111 2432 2002 applied roc analysis to schouten and van hessen s data and their curves supported the gaussian assumption for the nonspeech signals only later lopez bascuas et al j acoust soc am 115 2465 2004 showed that non linear z transformed rocs are not sufficient evidence for postulating non gaussian signal densities in this paper we try to figure out whether unequal criterion variances could underlie non linear z transformed rocs by fitting a restricted thurstonian sdt like model to a nonspeech continuum composed by white noise and a square wave the results indicate that unequal criterion variances is the possible cause of the deviant rocs for nonspeech

frequency and frequency ratio resolution by possessors of absolute and relative pitch examples of categorical perception the methodology derived from the trace context theory of intensity resolution durlach and braida 1969 macmillan et al 1988 was applied to resolution over an octave range along two continua a sequential frequency ratio continuum for possessors of relative pitch rp and a pure tone frequency continuum for possessors of absolute pitch ap the performance of both rp and ap possessors was exceptional in that total identification sensitivity along both continua was much greater than identification sensitivity along unidimensional psychophysical continua characterized by the 7 2 rule in addition the performance of rp possessors was exceptional in that on average total sensitivity for identification resolution was greater than sensitivity for resolution in discrimination finally identification sensitivity between category prototypes chromatic semits along both continua was approximately the same as identification sensitivity between phonemic category prototypes along speech continua despite the fact that both the discrimination ranges and the total number of categories are much larger for the two pitch continua

modeling phoneme perception i categorical perception on the basis of a number of vowel and stop consonant discrimination experiments ax and 2ifc fixed and roving with natural stimuli it is concluded that stop consonant perception is highly categorical there were few significant differences between the discrimination results and the phoneme identification results moreover the discrimination and identification response maxima differed significantly from the other data points vowel perception was much less categorical the maxima in the functions were much less significant and there were significant differences between the various paradigms consonant discrimination was much less if at all subject to range effects than vowel discrimination all these results point to different memory types for stop consonants and vowels and consequently to a combination of two different theories of speech sound discrimination dual process theory dpt for consonants and trace context theory tct for vowels

umbral calculus discretization and quantum mechanics on a lattice umbral calculus deals with representations of the canonical commutation relations we present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of lie algebras on a lattice related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus this observation then suggests various generalizations of these examples a special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

on pairs of difference operators satisfying d x id dedicated to the memory of gerda dorfmeister different finite difference replacements for the derivative are analyzed in the context of the heisenberg commutation relation the type of the finite difference operator is shown to be tied to whether one can naturally consider d and x to be self adjoint and skew self adjoint or whether they have to be viewed as creation and annihilation operators the first class generalizing the central difference scheme is shown to give unitary equivalent representations for the second case we construct a large class of examples generalizing previously known difference operator realizations of d x id

on polynomial solutions of differential equations a general method of obtaining linear differential equations having polynomial solutions is proposed the method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some lie algebra in the projectivized representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients it is shown in general that polynomial solutions of partial differential equations occur in the case of lie superalgebras there are polynomial solutions of some matrix differential equations quantum algebras give rise to polynomial solutions of finite difference equations particularly known classical orthogonal polynomials will appear when considering sl 2 r acting on rp1 as examples some polynomials connected to projectivized representations of sl2 r sl2 r q osp 2 2 and so3 are briefly discussed

generation of symplectic groups by transvections over local rings with at least 3 residue classes let r be a commutative local ring with identity 1 and a unique maximal ideal a let v denote a free module of rank iz over r with an alternating bilinear form f vx v r we assume r a f however v may have rad v qr a linear automorphism e of v is called an isometry on v if t satisfies f ax oy f x y for all x y in v the set of all isometries is a subgroup of the automorphism group of v which is called a symplectic group denoted by sp v for a submodule u of v we define u u e vi f v u 0 and rad u un u then u is called the orthogonal complement of u and rad u the radical of u for submodules u and w of v by u i w we mean u w with f u w 0 we define

classification of certain pairs of operators p q satisfying p q i id abstract in a separable hilbert space a certain class of pairs of operators p q satisfying the born heisenberg commutation relation p q i id on a dense domain is investigated this class is essentially defined by requiring q bounded and self adjoint p symmetric and p q we show that q is absolutely continuous and that p can be thought of as a first order differential operator the class considered contains the pair angle and angular momentum l z it is expected that the methods of this paper can be applied to more general classes of operators p q including the schrodinger case

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

quantum mechanics as noncommutative symplectic geometry the authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation this leads to a mathematical framework in which quantum mechanics can be understood as noncommutative symplectic geometry they point out the possibility to describe interactions geometrically as deformations of the differential calculus

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

groups generated by symplectic transvections over local rings abstract let r be a commutative local ring with the unique maximal ideal a let v be a free module of rank n over r and let spn v be the symplectic group on v with an alternating bilinear form f v v r we study the generation of a subgroup tr m of spn v where m x v f x v r and tr m is defined as the subgroup generated by all symplectic transvections with axis x for x m our main goal is to get a nice necessary and sufficient condition for any subset n m satisfying tr n tr m where tr n is the group generated by all symplectic transvections with axis x for x n in particular if f is nonsingular we have tr m spn v and therefore our necessary and sufficient condition gives us a criterion for an arbitrarily given n m satisfying tr n spn v also we shall investigate the tr n orbit of each x m find some small sets of generators of tr m consisting of transvections in tr n and as a result solve the author s conjecture in generators and relations in groups and geometries a barlotti et al eds pp 47 67 proc nato asi c

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

canonical commutation relation preserving maps we study maps preserving the heisenberg commutation relation ab ba 1 we find a one parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative it involves a non local coordinate operator while the dual derivative is just the jackson finite difference operator substitution of this realization into any differential operator involving x and d dx results in an isospectral deformation of a continuous differential operator into a finite difference one we extend our results to the deformed heisenberg algebra ab qba 1 as an example of potential applications various deformations of the hahn polynomials are briefly discussed

multiple scale analysis of dynamical systems on the lattice we propose a new approach to the multiple scale analysis of difference equations in the context of the finite operator calculus we derive the transformation formulae that map any given dynamical system continuous or discrete into a rescaled discrete system by generalizing a classical result due to jordan under suitable analytical hypotheses on the function space we consider the rescaled equations are of finite order our results are applied to the study of multiple scale reductions of dynamical systems and in particular to the case of a discrete nonlinear harmonic oscillator

classes of hypercomplex polynomials of discrete variable based on the quasi monomiality principle with the aim of derive a quasi monomiality formulation in the context of discrete hypercomplex variables one will amalgamate through a clifford algebraic structure of signature 0 n the umbral calculus framework with lie algebraic symmetries the exponential generating function egf carrying the continuum dirac operator d j 1 n e j x j together with the lie algebraic representation of raising and lowering operators acting on the lattice h z n is used to derive the corresponding hypercomplex polynomials of discrete variable as appell sets with membership on the space clifford vector valued polynomials some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the poisson charlier polynomials are introduced certain applications from the view of interpolation theory and integral transforms are also discussed