multivariate pade approximation abstract this paper is a survey on the multivariate pade approximation two types of approximants are considered those which can approximate general meromorphic functions f h g where both h and g are holomorphic and those which are specialized to the approximation of functions of the same form where g is a polynomial algorithms are described together with the different techniques used for proving convergence

multivariate pade approximation multivariate pade approximation is considered in the framework of the theory developed by a cuyt 1982 the denominators are determined from determinants of matrices whose entries are homogeneous polynomials the main difference to the univariate case is a typical shift of the degree of the polynomials singularities atx 0 are analyzed since it is the rule rather than the exception that simultaneous zeros of the numerator and the denominator cannot be removed

generalized coherent states for associated hypergeometric type functions we introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric type operators this makes possible the definition of generalized coherent states for the associated eigenfunctions these generalized coherent states satisfy the relevant properties i e label continuity overcompleteness temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system

wave functions for a duffin kemmer petiau particle in a time dependent potential we present the solution of the duffin kemmer petiau equation of spin 0 in a time dependent linear scalar field the treatment is based on the use of the lewis jr riensfield invariant method

a direct geometrical method for bounding the error exponent for any specific family of channel codes i cutoff rate lower bound for block codes a direct general and conceptually simple geometrical method for determining lower and upper bounds on the error exponent of any specific family of channel block codes is presented it is considered that a specific family of codes is characterized by a unique distance distribution exponent the tight linear lower bound of slope 1 on the code family error exponent represents the code family cutoff rate bound it is always a minimum of a sum of three functions the intrinsic asymptotic properties of channel block codes are revealed by analyzing these functions and their relationships it is shown that the random coding technique for lower bounding the channel error exponent is a special case of this general method the requirements that a code family should meet in order to have a positive error exponent and at best attain the channel error exponent are stated in a clear way using the direct distance distribution method presented

identification of discontinuous nonlinear systems via a multivariate pade approach we present a nonlinear system identification technique based on multi dimensional rational polynomials a multi dimensional pade legendre approximation is developed to circumvent challenges in dealing with sharp shocks the purpose of this paper is to investigate the accuracy of such approximations for identification of various nonlinear systems particularly systems with a non smooth response surface this identification approach utilizes the generalized form of a pade legendre approximation for studying multivariable functions in the studied problems the nonlinearity is a function of state variables displacement and velocity which requires multi dimensional formulation furthermore a spatial filter is applied to minimize the effects of the singular points in the applicable rational function of the response surface this study presents different types of nonlinearities including smooth irregular and hysteretic functions in order to demonstrate the performance of the approach under different conditions in order to study the robustness of the method in comparison to other identification techniques based on plain polynomial representation a nonlinear system with a sharp discontinuous restoring force surface is considered the performance of both approaches is investigated for different degrees of sharpness in addition the accuracy of the identified models to represent the nonlinear system is verified by comparing the output of the system computed on the basis of the identified model from data sets corresponding to different excitations than those used for identification purposes it is shown that the proposed approach provides a robust identification technique for a broad class of highly nonlinear systems and it is particularly advantageous to use when dealing with systems incorporating discontinuous properties

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

capacity achieving ldpc codes through puncturing the performance of punctured low definition parity check ldpc codes under maximum likelihood ml decoding is studied in this correspondence via deriving and analyzing their average weight distributions awds and the corresponding asymptotic growth rate of the awds in particular it is proved that capacity achieving codes of any rate and for any memoryless binary input output symmetric mbios channel under ml decoding can be constructed by puncturing some original ldpc code with small enough rate moreover it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process these results show high potential for puncturing to be used in designing capacity achieving codes and in rate compatible coding under any mbios channel

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

a general theory of phase space quasiprobability distributions we present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are finite dimensional lie groups the family of distributions on a phase space is postulated to satisfy the stratonovich weyl correspondence with a generalized traciality condition the corresponding family of the stratonovich weyl kernels is constructed explicitly in the presented theory we use the concept of generalized coherent states that brings physical insight into the mathematical formalism

q rotations and other q transformations as unitary nonlinear automorphisms of quantum algebras for the quantum algebras with three generators suq 2 suq 1 1 q oscillator and others the unitary operator u s q is found in terms of q exponents from algebra s generators being the q analog of the linear automorphism for the lie algebras the operator u s q is shown to generate nonlinear automorphisms of the quantum algebras q rotations q shifts bogoliubov q transformations etc the matrix elements tnn of the operator u s q are explicitly expressed in terms of askey wilson polynomials the operator u s q is shown to give rise the new systems of q coherent states of the quantum algebras

likelihood approximation by numerical integration on sparse grids the calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions approaches for extending gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions we propose an extension that is very general and easily implemented and does not suffer from the curse of dimensionality monte carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques

coherent states for anharmonic oscillator hamiltonians with equidistant and quasi equidistant spectra two kinds of transformation for the time dependent schrodinger equation i e the differential and integral transformations are introduced if one considers only stationary solutions of this equation both transformations reduce to the well known darboux transformation for the stationary schrodinger equation when applied to non stationary solutions they give different results both transformations are invertible in appropriate spaces with the help of these transformations alternative systems of coherent states to those in the literature are obtained for isospectral hamiltonians with equidistant spectra these transformations are also applied to the construction of coherent states for hamiltonians whose spectrum consists of an equidistant part and one separately disposed level with an energy gap equal to the k skipped levels

the growth of bargmann functions and the completeness of sequences of coherent states the growth of bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of bargmann functions of an order not exceeding and of a type not exceeding if of order these results generalize known results on the completeness of von neumann lattices the practical significance of this formalism in the context of quantum optics is also discussed

analysis of some pade chebyshev approximants we present some rational approximations of the sign function and analyze their convergence the rate of convergence is shown to increase with the degree of the denominator of the rational approximation several numerical tests are presented

non random coding error bounds for lattices an upper bound on the error probability of specific lattices based on their distance spectrum is constructed the derivation is accomplished using a simple alternative to the minkowski hlawka mean value theorem of the geometry of numbers in many ways the new bound greatly resembles the shulman feder bound for linear codes based on the new bound error exponent and channel dispersion expressions are derived for specific lattice sequences of increasing dimension over the awgn channel measuring a sequence s gap to capacity using the new asymptotics is demonstrated additional finite dimension results encountered along the way are presented

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

identification and prediction of stochastic dynamical systems in a polynomial chaos basis non parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation in a class of such models multi dimensional chebychev polynomials in the state variables are fitted to the observed dynamical state of the system due to the approximative nature of this non parametric model as well as to various other sources of uncertainty such as measurement errors and non anticipative excitations the parameters of the model exhibit a scatter that is treated here in a probabilistic context the statistics of these coefficients are related to the physical properties of the model being analyzed and are used to endow the model predictions with a probabilistic structure they are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy the proposed methodology is quite simple and robust

the serial concatenation of rate 1 codes through uniform random interleavers until the analysis of repeat accumulate codes by divsalar et al 1998 few people would have guessed that simple rate 1 codes could play a crucial role in the construction of good binary codes we construct good binary linear block codes at any rate r 1 by serially concatenating an arbitrary outer code of rate r with a large number of rate 1 inner codes through uniform random interleavers we derive the average output weight enumerator we for this ensemble in the limit as the number of inner codes goes to infinity using a probabilistic upper bound on the minimum distance we prove that long codes from this ensemble will achieve the gilbert varshamov 1952 bound with high probability numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes in essence this construction produces codes with good distance properties which are also compatible with iterative turbo style decoding for selected codes we also present bounds on the probability of maximum likelihood decoding mld error and simulation results for the probability of iterative decoding error

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

variations on the gallager bounds with some applications by generalizing the framework of the second version of the duman salehi bound encompassing also deterministic and random codes we demonstrate its rather broad features and show that this variation provides the natural bridge between the 1961 and 1965 gallager bounds this approach entails a natural geometric interpretation encompassing also a large class of e0cient recent bounds or their cherno2 versions which are demonstrated to be special cases of the generalized second version of the duman salehi bound implications and applications of these observations are pointed out referring to known bounds as well as a novel extended version of the shulman feder bound c 2001 elsevier science b v all rights reserved

improved error exponent for time invariant and periodically time variant convolutional codes an improved upper bound on the error probability first error event of time invariant convolutional codes and the resulting error exponent is derived the improved error bound depends on both the delay of the code k and its width the number of symbols that enter the delay line in parallel b determining the error exponent of time invariant convolutional codes is an open problem while the previously known bounds on the error probability of time invariant codes led to the block coding exponent we obtain a better error exponent strictly better for b 1 in the limit b spl rarr spl infin our error exponent equals the yudkin viterbi 1967 1971 1965 exponent derived for time variant convolutional codes these results are also used to derive an improved error exponent for periodically time variant codes

pade legendre approximants for uncertainty analysis with discontinuous response surfaces a novel uncertainty propagation method for problems characterized by highly non linear or discontinuous system responses is presented the approach is based on a pade legendre pl formalism which does not require modifications to existing computational tools non intrusive approach and it is a global method the paper presents a novel pl method for problems in multiple dimensions which is non trivial in the pade literature in addition a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities the numerical examples include fluid dynamic problems characterized by shock waves a simple dual throat nozzle problem with uncertain initial state and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain results are presented in terms of statistics of both shock position and strength and are compared to monte carlo simulations

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

coherent states for transparent potentials the darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials an isometric correspondence between hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established it is shown that the darboux transformation operator being unbounded and closed cannot realize an isometric mapping between hilbert spaces a quasispectral representation of transformation operators in terms of continuous basis sets is obtained different families of coherent states of the multisoliton potential are introduced measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated these measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

dual universality of hash functions and its applications to quantum cryptography in this paper we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography we begin by establishing the one to one correspondence between a linear function family f and a code family c and thereby defining e almost dual universal2 hash functions as a generalization of the conventional universal2 hash functions then we show that this generalized and thus broader class of hash functions is in fact sufficient for the security of quantum cryptography this result can be explained in two different formalisms first by noting its relation to the biased family introduced by dodis and smith we demonstrate that renner s two universal hashing lemma is generalized to our class of hash functions next we prove that the proof technique by shor and preskill can be applied to quantum key distribution qkd systems that use our generalized class of hash functions for privacy amplification while shor preskill formalism requires an implementer of a qkd system to explicitly construct a linear code of the calderbank shor steane css type this result removes the existing difficulty of the construction of a linear code of css code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function we also show that a similar result applies to the quantum wire tap channel finally we compare our results in the two formalisms and show that in typical qkd scenarios the shor preskill type argument gives better security bounds in terms of the trace distance and holevo information than the method based on the biased family

multivariate pade approximants abstract for an operator f r n r analytic in the origin the notion of abstract multivariate pade approximant apa is introduced by making use of abstract polynomials the classical pade approximant n 1 is a special case of the multivariate theory and many interesting properties of classical pade approximants remain valid such as covariance properties and the block structure annie a m cuyt j oper theory6 2 1981 207 209 of the pade table also a projection property for multivariate pade approximants is proved

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

darboux transformations of coherent states of the time dependent singular oscillator darboux transformation of both barut girardelo and perelomov coherent states for the time dependent singular oscillator is studied in either case a measure that realizes the resolution of the identity operator in terms of coherent states is found and a corresponding holomorphic representation is constructed for the particular case of the free particle moving with the fixed value of the square of angular momentum equal to 2 it is shown that the barut giriardelo coherent states are more localized at the initial time moment while the perelomov coherent states are more stable with respect to time evolution it is also illustrated that darboux transformation may keep unchanged this different time behaviour

phase operators temporally stable phase states mutually unbiased bases and exactly solvable quantum systems we introduce a one parameter generalized oscillator algebra that covers the case of the harmonic oscillator algebra and discuss its finite and infinite dimensional representations according to the sign of the parameter we define an hamiltonian operator associated with and examine the degeneracies of its spectrum for the finite when 0 and the infinite when 0 representations of we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators to overcome the difficulties related to the phase operator in the infinite dimensional case and to avoid the degeneracy problem for the finite dimensional case we introduce a truncation procedure which generalizes the one used by pegg and barnett for the harmonic oscillator this yields a truncated generalized oscillator algebra where s denotes the truncation order we construct two types of temporally stable states for as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of two applications are considered in this paper the first concerns physical realizations of and in the context of one dimensional quantum systems with finite morse system or infinite poschl teller system discrete spectra the second deals with mutually unbiased bases used in quantum information

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

supersymmetry and supercoherent states of a nonrelativistic free particle coordinate atypical representation of the orthosymplectic superalgebra osp 2 2 in a hilbert superspace of square integrable functions constructed in a special way is given the quantum nonrelativistic free particle hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system the supercoherent states defined by means of a supergroup displacement operator are explicitly constructed these are the coordinate representation of the known atypical abstract super group osp 2 2 coherent states we interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement in the appropriate coordinate system in its odd sector