Convergent Power Series Research Articles

Three-Dimensional Dynamic Problems for an Elastic Body Containing a Shallow Crack

By combining the methods of boundary integral equations and small parameter, we solve a three-dimensional problem of low-frequency harmonic loading of the surfaces of a shallow crack in an infinite elastic body. The functions of crack opening displacements and the stress intensity factors are obtained in the form of double convergent power series in the wave number and in a geometric parameter characterizing the curvature of the crack. We study the influence of inertial terms on the level of stresses in the vicinity of spheroidal cracks with various values of eccentricity under the action of uniform dynamic pressure with constant amplitude.

Approximation and convergence of formal CR-mappings

An important step in understanding the existence of analytic objectswith certain properties consists of understanding the same problem at the level of formal power series. The latter problem can be reduced to a sequence of algebraic equations for the coefficients of the unknown power series and is often simpler than the original problem, where the power series are required to be convergent. It is therefore of interest to know whether such power series are automatically convergent or can possibly be replaced by other convergent power series satisfying the same properties. A celebrated result of this kind is Artin’s approximation theorem [1] which states that a formal solution of a system of analytic equations can be replaced by a convergent solution of the same system that approximates the original solution at any prescribed order. In this paper, we study convergence and approximation properties (in the spirit of [1]) of formal (holomorphic) mappings sending real-analytic submanifolds M ⊂ C and M ′ ⊂ C ′ into each other, N,N ′ ≥ 2. In this situation, the above theorem of Artin cannot be applied directly. Moreover, without additional assumptions on the submanifolds, the analogous approximation statement is not even true. Indeed, in view of an example of Moser-Webster [23], there exist real-algebraic surfaces M,M ′ ⊂ C that are formally but not biholomorphically equivalent. However, our firstmain result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C (see Section 2.1 for notation and definitions). Theorem 1.1. Let M ⊂ C be a real-analytic minimal CR-submanifold and M ′ ⊂ C ′ a real-algebraic subset with p ∈ M and p ′ ∈ M ′. Then for any formal (holomorphic)

On roots of polynomials with power series coefficients

We give a deepened version of a lemma of Gabrielov and then use it to prove the following fact: if $h \in {{\mathbb K}}[[X]]$ (${{\mathbb K}} = {{\mathbb R}}$ or ${{\mathbb C}}$) is a root of a non-zero polynomial with convergent power series coefficients

Weighted least-square design of FIR filters using a fast iterative matrix inversion algorithm

It has been shown by some researchers that in a problem of weighted least-square (WLS) design of finite-impulse response (FIR) filters, bulk of the design computation is concerned with the evaluation of the inverse of a matrix in order to solve a system of equations. In this paper, a new algorithm for the WLS design of FIR filters is presented, in which an iterative procedure is developed for the inversion of the matrix involved in the design. By imposing a mild constraint on the updation factor of the weighting function, the inverse of a matrix is expanded as a convergent power series. By investigating the properties of some of the matrices from the design formulation, a modified version of the series that converges rapidly is then proposed to evaluate the inverse in each iteration. It is shown that due to the fast convergence of the power series, one needs to evaluate only the first two or three terms of the series except during the initial stages of the iterations, implying that the conventional operation for matrix inversion is simplified significantly.

Precision matched solution of the coupled beam envelope equations for a periodic quadrupole lattice with space charge

The coupled Kapchinskij–Vladimirskij (K–V) envelope equations for a charged particle beam transported by a periodic system of quadrupoles with self-consistent space charge force have previously been solved by various approximate methods, with accuracy ranging from 1% to 10%. A new method of solution is introduced here, which is based on a double expansion of the beam envelope functions in powers of the focal strength and either the beam’s emittance or its dimensionless perveance. This method results in accuracy better than 0.1% for typical lattice and beam parameters when carried through one consistent level of approximation higher than employed in previous work. Several useful quantities, such as the values of the undepressed tune and the beam’s perveance in the limit of vanishing emittance, are represented by very rapidly converging power series in the focal strength, with an accuracy of 0.01% or better.

On the solution of the non‐linear Korteweg–de Vries equation by the decomposition method

The Adomian decomposition method is used to implement the non‐linear Korteweg–de Vries equations. The analytic solution of the equation is calculated in the form of a convergent power series with easily computable components. The non‐homogeneous problem is quickly solved by observing the self‐cancelling “noise” terms whose sum vanishes in the limit. Comparing this methodology with some known techniques shows that the present approach is highly accurate.

A non-standard proof of the Briançon-Skoda theorem

Using a tight closure argument in characteristic p and then lifting the argument to characteristic zero with the aid of ultraproducts, I present an elementary proof of the Briancon-Skoda Theorem: for an m-generated ideal a of C[[X 1 ,...,X n ]], the m-th power of its integral closure is contained in a. It is well-known that as a corollary, one gets a solution to the following classical problem. Let f be a convergent power series in n variables over C which vanishes at the origin. Then f lies in the ideal generated by the partial derivatives of f.

Construction of orthonormal wavelets using Kampé de Fériet functions

One of the main aims of this paper is to bridge the gap between two branches of mathematics, special functions and wavelets. This is done by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one and two variables and a generalization of the latter, known as Kampé de Fériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy and fast numerical evaluation. Explicit representation of wavelets facilitates, among other things, the study of the analytic properties of wavelets.

LINDSTEDT SERIES FOR PERTURBATIONS OF ISOCHRONOUS SYSTEMS: A REVIEW OF THE GENERAL THEORY

We give a proof of the persistence of invariant tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions of the equations of motion. With respect to the case of anisochronous systems, there is the additional problem of finding the set of allowed rotation vectors, because they cannot be given a priori simply by looking at the unperturbed system. By considering the involved parameters (size of the perturbation, rotation vector and average action of a persisting invariant torus) as independent parameters we can introduce a function which is analytic in such parameters and only when the latter satisfy some constraint it becomes a solution: this can be regarded as a sort of singular implicit function problem. Therefore, although the dependence of the parameters, hence of the solution, upon the size of the perturbation is not smooth, in this way we construct explicitly the solution by using an absolutely convergent power series.

Algebraic braided model of the affine line and difference calculus on a topological space

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions f=f(x): Z →k (with an arbitrary commutative ring k) which are constant when x↦+∞ or x↦−∞ and their “differentials” considered as functions x↦ f( x+1)− f( x) (difference calculus). Thanks to this new “ differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra D ∗(X) which is similar in spirit to the algebra W ∗(X) introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of D ∗(X) . In particular, if k= Z , we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.

Analytic solutions of a second-order iterative functional differential equation

This paper is concerned with a second-order iterative functional differential equation x″( z) = ( x m ( z)) 2. Its analytic solutions are discussed by locally reducing the equation to another functional differential equation with proportional delays μ 2 y″(μ z) y′( z) = μ y′(μ z) y″( z) + [ y′( z)] 3[ y(μ m z)] 2 and by constructing a convergent power series solutions for the latter equation.

Analytic Solutions of an Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation x′(x[r](z))=c0z+c1x(z)+c2x(x(z))+⋯+cmx[m](z), where r and m are nonnegative integers, x[0](z)=z,x[1](z)=x(z),x[3](z)=x(x(x(z))), etc. are the iterates of the function x(z), and ∑j=0mcj≠0. By constructing a convergent power series solution y(z) of a companion equation of the form αy′(αr+1z)=y′(αrz)∑j=0mcjy(αjz), analytic solutions of the form y(αy−1(z)) for the original differential equation are obtained.

Convergent Perturbative Solutions of the Schrödinger Equation for Two-Level Systems with Hamiltonians Depending Periodically on Time

We study the Schrodinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2e between the free energy levels is sufficiently small with respect to the strength of the external interaction. Under suitable conditions we show that this equation has a solution in terms of converging power series expansions in e. In contrast to other expansion methods, like in the Dyson expansion, the method we present is not plagued by the presence of secular terms. Due to this feature we were able to prove uniform convergence of the Fourier series involved in the computation of the wave functions and to prove absolute convergence of the e-expansions leading to the secular frequency and to the coefficients of the Fourier expansion of the wave function.

Analytic Solutions of a Functional Equation for Invariant Curves

An iterative functional equation is deduced by C. T. Ng and W. Zhang (1997, J. Differ. Equations Appl.3, 147–168) from the problem of invariant curves. In this paper, its analytic solutions are discussed by locally reducing the equation to another functional equation without iteration and by constructing solutions in uniformly convergent power series for the latter equation.

The Fredholm determinant of an almost periodic arithmetical function

For functions f in the space of 2-limit-periodic functions on \({\Bbb Z}\), the Fredholm determinant D(f; z) is defined as an everywhere convergent power series whose coefficients are mean values of certain determinants involving the values of f. A Weierstras product representation is given for D(f; z) which shows in particular that the zeros of D(f; z) are the reciprocals of the nonvanishing Fourier coefficients of f.

Analytic Solutions of a Second-order Functional Differential Equation with a State Dependent Delay

This paper is concerned with a second-order functional differential equation of the form x″(z) = x(az + bx(z)). By constructing a convergent power series solution of an auxiliary equation of the form α2y″ (αz) y′ (z) = αy′(z)y″(z)+(y′(z))3[y(α2z)−ay(αz)], analytic solutions of the form (y(αyt - 1(z)) − az)/b for the original differential equation are obtained.

Finite element method for two-dimensional vibrational wave functions: Theory and application to van der Waals molecules

A variational formulation finite element method is developed for calculation of vibrational wave functions in a domain spanned by close-coupled, or Jacobi, coordinates R and γ. C1 tensor-product basis functions, which allow straightforward separation of kinetic and overlap integrals into products of one-dimensional integrals, are used. Furthermore, representation of the potential energy surface in terms of the same tensor-product basis functions used to represent the wave functions allows the potential energy integrals to also be written as a sum of products of one-dimensional integrals. Factorization of the integrals, together with expression of one-dimensional integrals in analytic or rapidly convergent power series form, reduces the computational effort of calculation of all matrix elements to a small, and arguably insignificant, level. It is shown that the theoretical error in eigenvalue, i.e., O(h6) for bicubic Hermite functions, is achieved for a number of rare gas van der Waals triatomics for which surfaces have been previously published. We also present illustrative calculations on NeHCl and A′2 and A″2 NeHCl+, which have not been previously studied, for surfaces calculated at the CCSD(T)/cc-pVTZ level.

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form ∑ j=j 0 ∞ z j φ j j μ∏ i=1 s (j+α i )−ν i , where μ∈R, ν i ≥0 and α i ∈C, 1≤i≤s, are known parameters, and φ j =φ(j), φ being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of φ. The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.

On the ring of constants for derivations of power series rings in two variables

Let $k[[x,y]]$ be the formal power series ring in two variables over a field $k$ of characteristic zero and let $d$ be a nonzero derivation of $k[[x,y]]$. We prove that if $\mathop{\rm Ker}\nolimits (d)\neq k$ then $\mathop{\rm Ker}\nolimits (d) =\mathop{

The constuctive solution of linear systems of partial difference and differential equations with constant coefficients

This paper gives a survey of past work in the treated subject and also contains several new results. We solve the Cauchy problem for linear systems of partial difference equations on general integral lattices by means of suitable transfer operators and show that these can be easily computed with the help of standard implementations of Grobner basis algorithms. The Borel isomorphism permits to transfer these results to systems of partial differential equations. We also solve the Cauchy problem for the function spaces of convergent power series and for entire functions of exponential type. The unique solvability of the Cauchy problem implies that the considered function spaces are large injective cogenerators for which the duality between finitely generated modules and behaviours holds. Already in the beginning of the last century C. Riquier considered and solved problems of the type discussed here.