the classification of differential structures on quantum 2 spheres exterior algebras of differential forms on quantum 2 spheress qc 2 q 1 1 0 c 0 c 0 forq 1 are classified in the definition of exterior algebras we assume the invariance w r t the action of the quantumsu 2 group and dimensionality conditions which imply that we deal with two dimensional manifolds the exterior algebras exist only forc 0 and are unique in that case the corresponding generalized directional derivatives are provided

asymptotics of orthogonal polynomial s entropy this is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree this study is motivated primarily by quantum mechanics where the wave functions and the densities of the states of solvable quantum mechanical systems are expressed by means of orthogonal polynomials moreover the uncertainty principle lying in the ground of quantum mechanics is best formulated by means of position and momentum entropies in this sense the behavior for large values of the degree is intimately connected with the information characteristics of high energy states but the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials we describe some results obtained in the last 15 years as well as sketch the ideas behind their proofs

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

discrete entropy of generalized jacobi polynomials abstract given a sequence of orthonormal polynomials on r p n n 0 with p n of degree n we define the discrete probability distribution n x n 1 x n n x with n j x j 0 n 1 p j 2 x 1 p j 1 2 x j 1 n in this paper we study the asymptotic behavior as n of the shannon entropy s n x j 1 n n j x log n j x x 1 1 when the orthogonality weight is 1 x 1 x h x 1 and where h is real analytic and positive on 1 1 we show that the limit lim n s n x log n exists for all x 1 1 but its value depends on the rationality of arccos x for the particular case of the chebyshev polynomials of the first and second kinds we compare our asymptotic result with the explicit formulas for s n j n where j n are the zeros of p n obtained previously in 2

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

entropic functionals of laguerre and gegenbauer polynomials with large parameters the determination of the physical entropies renyi shannon tsallis of high dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states for the d dimensional hydrogenic and oscillator like systems the wavefunctions of the corresponding bound states are controlled by the laguerre and gegenbauer

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

classification of bicovariant differential calculi on the jordanian quantum groups and and quantum lie algebras assuming that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators we classify all four dimensional first order bicovariant calculi on the jordanian quantum group and all three dimensional first order bicovariant calculi on the jordanian quantum group it is found that there are three one parameter families of four dimensional bicovariant first order calculi on and that there is a single unique three dimensional bicovariant calculus on this three dimensional calculus may be obtained through a classical like reduction from any one of the three families of four dimensional calculi on details of the higher order calculi and also the quantum lie algebras are presented for all calculi the quantum lie algebra obtained from the bicovariant calculus on is shown to be isomorphic to the quantum lie algebra we obtain as an ad submodule within the jordanian universal enveloping algebra and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module we also obtain the quantum killing form for this quantum lie algebra

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

discrete differential calculus graphs topologies and gauge theory differential calculus on discrete sets is developed in the spirit of noncommutative geometry any differential algebra on a discrete set can be regarded as a reduction of the universal differential algebra and this allows a systematic exploration of differential algebras on a given set associated with a differential algebra is a di graph where two vertices are connected by at most two antiparallel arrows the interpretation of such a graph as a hasse diagram determining a locally finite topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure it is shown that field theories and in particular gauge theories can be formulated on a discrete set in close analogy with the continuum case the framework presented generalizes ordinary lattice theory which is recovered from an oriented hypercubic lattice graph it also includes e g the two point space used by connes and lott and others in models of elementary particle physics the formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an internal discrete space a la connes et al to models of dimensionally reduced gauge fields furthermore a symmetric lattice is also studied which in a certain continuum limit turns out to be related to a noncommutative differential calculus on manifolds

mean convergence of prolate spheroidal series and their extensions abstract the aim of this paper is to establish the range of p s for which the expansion of a function f l p in a generalized prolate spheroidal wave function pswfs basis converges to f in l p two generalizations of pswfs are considered here the circular pswfs introduced by d slepian and the weighted pswfs introduced by wang and zhang both cases cover the classical pswfs for which the corresponding results have been previously established by barcelo and cordoba to establish those results we prove a general result that allows to extend mean convergence in a given basis e g jacobi polynomials or bessel basis to mean convergence in a second basis here the generalized pswfs

bicovariant differential calculus on the two parameter quantum group glp q 2 the quantum lie algebra and quantum de rham complex of the two parameter quantum group glp q 2 are presented it is shown that the differential calculus on the two parameter quantum group glp q 2 given in this paper is bicovariant

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

woronowicz construction of compact quantum groups for functions on permutations classification result for n 3 abstract we provide a classification of compact quantum groups which can be obtained by the woronowicz construction when the arrays used in the twisted determinant condition are extensions of functions on permutations general properties of such quantum groups are revealed with the aid of operators intertwining tensor powers of the fundamental corepresentation two new families of quantum groups appear three versions of the quantum group su q 3 with complex q and a non commutative analogue of the semi direct product of two dimensional torus with the alternating group a 3

tool to recover scalar time delay systems from experimental time series we propose a method that is able to analyze chaotic time series gained from exp erimental data the method allows to identify scalar time delay systems if the dynamics of the system under investigation is governed by a scalar time delay differential equation of the form dy t dt h y t y t tau 0 the delay time tau 0 and the functi on h can be recovered there are no restrictions to the dimensionality of the chaotic attractor the method turns out to be insensitive to noise we successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

asymptotic upper bounds for the entropy of orthogonal polynomials in the szegő class we give an asymptotic upper bound as n for the entropy integral en w pn2 x log pn2 x w x dx where pn is the nth degree orthonormal polynomial with respect to a weight w x on 1 1 which belongs to the szeg class we also study two functionals closely related to the entropy integral first their asymptotic behavior is completely described for weights w in the bernstein class then as for the entropy we obtain asymptotic upper bounds for these two functionals when w x belongs to the szeg class in each case we give conditions for these upper bounds to be attained

on the growth of the polynomial entropy integrals for measures in the szego class let be a probability borel measure on the unit circle t and f ng be the orthonormal polynomials with respect to we say trhat is a szego measure if it has an arbitrary singular part s and t log 0dm 1 where 0 is the density of the absolutely continuous part of m being the normalized lebesgue measure on t the entropy integrals for n are de ned as n z t j nj2 log j njd it is not di cult to show that n o p n in this paper we construct a measure from the szego class for which this estimate is sharp over a subsequence of n s

complex quantum group dual algebra and bicovariant differential calculus the method used to construct the bicovariant bimodule in ref csww is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group the complex quantum group fun q sl n c is defined by requiring that it contains fun q su n as a subalgebra analogously to the quantum lorentz group analyzing the properties of the fundamental bimodule we show that the dual algebra has the structure of the twisted product fun q su n fun q su n reg then the bicovariant differential calculi on the complex quantum group are constructed

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

spreading lengths of hermite polynomials the renyi shannon and fisher spreading lengths of the classical or hypergeometric orthogonal polynomials which are quantifiers of their distribution all over the orthogonality interval are defined and investigated these information theoretic measures of the associated rakhmanov probability density which are direct measures of the polynomial spreading in the sense of having the same units as the variable share interesting properties invariance under translations and reflections linear scaling and vanishing in the limit that the variable tends towards a given definite value the expressions of the renyi and fisher lengths for the hermite polynomials are computed in terms of the polynomial degree the combinatorial multivariable bell polynomials which are shown to characterize the finite power of an arbitrary polynomial play a relevant role for the computation of these information theoretic lengths indeed these polynomials allow us to design an error free computing approach for the entropic moments weighted l q norms of hermite polynomials and subsequently for the renyi and tsallis entropies as well as for the renyi spreading lengths sharp bounds for the shannon length of these polynomials are also given by means of an information theoretic based optimization procedure moreover the existence of a linear correlation between the shannon length as well as the second order renyi length and the standard deviation is computationally proved finally the application to the most popular quantum mechanical prototype system the harmonic oscillator is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed

covariant differential complexes on quantum linear groups we consider the possible covariant external algebra structures for cartan s 1 forms omega on glq n and slq n our starting point is that the omega realize an adjoint representation of quantum group and all monomials of the omega possess the unique ordering for the external algebras obtained define the differential mapping d possessing the usual nilpotence condition and the generally deformed version of the leibnitz rules the status of the known examples of glq n differential calculi in the proposed classification scheme and the problems of slq n reduction are discussed