Results 11  20
of
68
Computing the Algebraic Relations of Cfinite Sequences and Multisequences
, 2006
"... We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing these generators makes it possible to use Gröbner basis methods for carrying out certain basic operatio ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing these generators makes it possible to use Gröbner basis methods for carrying out certain basic operations in the ring of such sequences effectively. In particular, one can answer the question whether a given sequence can be represented in terms of other given sequences. A collection of examples, which were done with an implementation of our algorithm, is included. 1
qBernoulli and qEuler Polynomials, an Umbral approach II
, 2009
"... We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of d ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of down corresponding to inversion of basis. We also find new qEulerMaclaurin expansions.
Symmetric Functions and
 Combinatorial Operators on Polynomials, CBMS Reg. Conf. Ser. Math. 99, AMS
, 2003
"... ..."
Invariant manifolds and their zeroviscosity limits for NavierStokes equations
 Dynamics of PDE
"... Abstract. First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis i ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Abstract. First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H ℓ (ℓ = 0, 1, 2, · · ·). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zeroviscosity limits (ν → 0 +) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the centerstable manifold of a steady state are O ( √ ν), while the sizes of the stable manifold, the center manifold, and the centerunstable manifold are O(ν), as ν → 0 +. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete estimate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other
Umbral calculus, discretization, and Quantum Mechanics on a lattice
 J. Phys. A: Math. Gen
, 1996
"... ‘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 publicati ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
‘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. 1
COMBINATORIAL MODELS OF CREATION–ANNIHILATION
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65C
, 2011
"... Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or “diagrams”, that are composed of elementary “gates”. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to qanalogues, multivariate frameworks, and urn models are also briefly discussed.
On Differences of Zeta Values
 in "Journal of Computational and Applied Mathematics
, 2008
"... ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of Nörlu ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of NörlundRice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li’s criterion for the Riemann hypothesis.
Fast Convolution
"... We present a very simple and fast algorithm to compute the convolution of an arbitrary sequence x with a sequence of a specific type, a. The sequence a is any linear combination of polynomials, exponentials and trigonometric terms. The number of steps for computing the convolution depends on a certa ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We present a very simple and fast algorithm to compute the convolution of an arbitrary sequence x with a sequence of a specific type, a. The sequence a is any linear combination of polynomials, exponentials and trigonometric terms. The number of steps for computing the convolution depends on a certain complexity of a and not on its length, thus making it feasible to convolve a sequence with very large kernels fast. Computing the convolution (correlation, filtering) of a sequence x together with a fixed sequence a is one of the ubiquitous operations in graphics, image and signal processing. Often the sequence a is a polynomial, exponential or trigonometric function sampled at discrete points or a piecewise sum of such terms, such as, splines, or else the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.
Symbolic Computation of Divided Differences
, 1999
"... Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulati ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulations. This paper provides a guide to divided difference theory and practice, with a special eye toward the needs of computer algebra systems that should be programmed to deal with these oftenmessy formulas.