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1.
J. L. López 《Constructive Approximation》2001,17(4):535-559
Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional
approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these
expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the
asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric
functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic
integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases
for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by
an error bound at any order of the approximation.
January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000. 相似文献
2.
In this paper, we shall give in Abel type theorem of Jacobi series and then based on it discuss asymptotic expressions near
the ellipse of convergence of Jacobi series in complex plane. 相似文献
3.
4.
《Mathematical and Computer Modelling》1999,29(8):81-95
The aim of this paper is twofold. First, we shall focus on Lyapunov functions for discrete dynamical systems. We shall propose a methodology for building Lyapunov functions. This methodology will be based upon the introduction of small random perturbations in the deterministic dynamics. Then we shall deal with concentration results for the perturbed dynamics. Our ultimate goal is to force the convergence of the perturbed process towards a set of specified attractors of the deterministic system. We shall illustrate our results on the paradigms of global minimization and associative memory. Our formalism will be illustrated on new algorithms for which the asymptotic analysis can be done rigorously. 相似文献
5.
The authors consider Gauss-Christoffel formulas with preassigned node 0 for evaluating Cauchy singular integrals with an even generalized smooth Jacobi weight. A convergence theorem is given, and some asymptotic estimates of the remainder are established. 相似文献
6.
Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small
number of critical points. We show that using values of the integrand at certain complex points close to the critical points
can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting
ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type
methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order,
but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary
points as well, through a simple change of variables. 相似文献
7.
Yukio Ogura 《Stochastic Processes and their Applications》1983,14(1):19-40
We shall study the asymptotic behavior of the particle numbers in bounded domains of a binary splitting one-dimensional branching diffusion process. We shall give a Yaglom type limit theorem in the so-called locally subcritical case, and almost sure convergence of the normalized particle number in the locally supercritical case. 相似文献
8.
This paper deals with boundary value problems whose nonlinear part involves periodic functions and such that the linear part has a one-dimensional solution space. We shall study the existence and multiplicity of solutions using various methods of Nonlinear Analysis such as the Lyapunov-Schmidt reduction and methods of critical point theory. The proofs are based on some general results on the oscillation and asymptotic behavior of certain parametric integrals. 相似文献
9.
Avram Sidi 《BIT Numerical Mathematics》1996,36(1):143-157
In an earlier paper by the author a detailed convergence and stability analysis of a generalization of the Richardson extrapolation process was given under certain conditions. In the present work these conditions are modified and relaxed considerably, and results are obtained on convergence and stability under the new conditions. As the previous ones, these new results are asymptotic in nature, and contain the former. The conditions of the present paper are naturally satisfied, e.g., by the trapezoidal rule approximations of finite range integrals of functions having algebraic and logarithmic end point singularities. 相似文献
10.
Two-Point Taylor Expansions of Analytic Functions 总被引:2,自引:0,他引:2
Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points. 相似文献
11.
The notion of asymptotically equivalent sequences was presented by Pobyvanets in 1980. Using this definition, he presented Silverman–Toeplitz-type matrix conditions that ensure that a summability matrix preserves asymptotic equivalency. This work begins with an extension of Pobyvanets’ definition of convergence in probability. This definition is also used to present Silverman–Toeplitz-type conditions for ensuring that a summability matrix preserves asymptotic probability equivalence. In addition, we shall also present a Marouf-type characterization of such a sequence space. 相似文献
12.
José L. Ló pez Nico M. Temme 《Transactions of the American Mathematical Society》2004,356(11):4323-4342
Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.
13.
We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals
with stationary points of high order. The method is based on substituting the original interval of integration by a set of
contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially
decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ
from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known
methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required
number of evaluations of the integrand. 相似文献
14.
In this paper,new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities.To avoid singularities,the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs.The solutions of one can be obtained explicitly,while the other kind of ODEs can be solved efficiently by collocation methods.The proposed methods can attain arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points.Several numerical experiments are presented to validate the efficiency of the proposed methods. 相似文献
15.
Bounded, monotonic, and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed. Some necessary and sufficient conditions for boundedness of all solutions are established. Moreover, all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals. The obtained results extend and improve many known results of various authors. 相似文献
16.
The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/sin 2(x-s) is discussed,and the main part of the asymptotic expansion of error function is obtained.Based on the main part of the asymptotic expansion,a series is constructed to approach the singular point.An extrapolation algorithm is presented and the convergence rate is proved.Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms. 相似文献
17.
Wang Zhenyuan 《Fuzzy Sets and Systems》1985,16(3):277-290
In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary. 相似文献
18.
In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results. 相似文献
19.
In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results. 相似文献
20.
Fang Hua Lin 《纯数学与应用数学通讯》1991,44(4):453-468
Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects. 相似文献