Showing papers on "Asymptotology published in 2015"
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17 Jun 2015221 citations
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TL;DR: In this article, the authors presented a new formula for approximating the gamma function, which is very fast in comparison with other classical or recently discovered asymptotic series, and some conjectures are proposed.
Abstract: It is the scope of this paper to present a new formula for approximating the gamma function. The importance of this new formula consists in the fact that the convergence of the corresponding asymptotic series is very fast in comparison with other classical or recently discovered asymptotic series. Inequalities related to this new formula and asymptotic series are established. Some conjectures are proposed.
31 citations
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04 Jun 2015
TL;DR: Asymptotic estimates for ODEs with turning points have been studied in this paper, where the authors show that the integration of nonlinear ODE's can be achieved by regular perturbation.
Abstract: Asymptotic Estimates.- Asymptotic Estimates for Integrals.- Regular Perturbation of ODE's.- Singularly Perturbed Linear ODE's.- Linear ODE's with Turning Points.- Asymptotic Integration of Nonlinear ODE's.- Bibliography.- Index.
18 citations
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TL;DR: In this article, asymptotic stability in probability and stabilization designs of discrete-time stochastic systems with state-dependent noise perturbations were investigated. And the convergence theorem of supermartingale was proved.
Abstract: Summary
This paper investigates asymptotic stability in probability and stabilization designs of discrete-time stochastic systems with state-dependent noise perturbations. Our work begins with a lemma on a special discrete-time stochastic system for which almost all of its sample paths starting from a nonzero initial value will never reach the origin subsequently. This motivates us to deal with the asymptotic stability in probability of discrete-time stochastic systems. A stochastic Lyapunov theorem on asymptotic stability in probability is proved by means of the convergence theorem of supermartingale. An example is given to show the difference between asymptotic stability in probability and almost surely asymptotic stability. Based on the stochastic Lyapunov theorem, the problem of asymptotic stabilization for discrete-time stochastic control systems is considered. Some sufficient conditions are proposed and applied for constructing asymptotically stable feedback controllers. Copyright © 2014 John Wiley & Sons, Ltd.
13 citations
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TL;DR: In this paper, the authors investigated the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction.
Abstract: The article is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second order elliptic problems by Chipot and Rougirel, where the force functions are considered on the cross section of domains, we prove the non-local counterpart of their result.
Furthermore, recently Yeressian established a weighted estimate for solutions of nonlocal Dirichlet problems which exhibit the asymptotic behavior. The case whens= 1=2 was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this article, we extend this result to each order between 0 and 1.
9 citations
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TL;DR: In this article, a general theory for finite asymptotic expansions in real powers was developed for expansions of type (*),x → x0 where the ordered n-tuple forms an asymptic scale at x 0, i.e., as x → x 0, 1 ≤ i ≤ n − 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o.
Abstract: After studying finite asymptotic expansions
in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where
the ordered n-tuple forms an asymptotic scale at x0 , ie as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended
complete Chebyshev system on a one-sided neighborhood of x o As in previous papers by the author concerning polynomial, real-power
and two-term theory, the locution “factorizational theory” refers to the
special approach based on various types of factorizations of a differential
operator associated to Moreover, the guiding thread of our theory is the property of formal
differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained
by formal applications of suitable linear differential operators of orders 1,2,…,n-1
Some considerations lead to restrict the attention to two sets of operators
naturally associated to “canonical factorizations” This gives rise to
conjectures whose proofs build an analytic theory of finite asymptotic
expansions in the real domain which, though not elementary, parallels the
familiar results about Taylor’s formula One of the results states that to each
scale of the type under consideration it remains associated an important class
of functions (namely that of generalized convex functions) enjoying the
property that the expansion(*), if valid, is automatically formally
differentiable n-1 times in two special senses
8 citations
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TL;DR: The mathematical framework of singular analysis is introduced and a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems is discussed, which corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator.
Abstract: The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to short-range correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.
7 citations
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TL;DR: In this paper, local asymptotic properties of likelihood ratios of certain Heston models are studied in three cases: subcritical, critical, and supercritical models for the drift parameters.
Abstract: We study local asymptotic properties of likelihood ratios of certain Heston models We distinguish three cases: subcritical, critical and supercritical models For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality
6 citations
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TL;DR: In this paper, the authors present a detailed factorizational theory of asymptotic expansions of type (?),,,, where the scale is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of.
Abstract: This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?) , , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.
6 citations
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TL;DR: In this paper, the authors continue the factorizational theory of asymptotic expansions of type (*),,, where the expansion is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x 0.
Abstract: Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.
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TL;DR: In this paper, an approach to construct multi-soliton asymptotic solutions for non-integrable equations is described, and the general idea is realized in the case of three waves and for the KdV-type equation with nonlinearity $u^4$.
Abstract: We describe an approach to construct multi-soliton asymptotic solutions for non-integrable equations. The general idea is realized in the case of three waves and for the KdV-type equation with nonlinearity $u^4$. A brief review of asymptotic methods as well as results of numerical simulation are included.
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TL;DR: In this article, the asymptotic solution for the displacement distribution in the Reissner plate becomes infinite for some special vertex angles of the notch, this is a paradox and the corresponding bounded solutions are explained by the Jordan form solution according to the methods of mathematical physics.
01 Jan 2015
TL;DR: This part II-C of this work completes the factorizational theory of asymptotic expansions in the real domain by presenting two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymPTotic scale at an endpoint.
Abstract: This part II-C of our work completes the factorizational theory of asymptotic expansions in the real domain. Here we present two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymptotic scale at an endpoint. These algorithms arise quite naturally in our asymptotic context and prove very simple in special cases and/or for scales with a small numbers of terms. All the results in the three Parts of this work are well illustrated by a class of asymptotic scales featuring interesting properties. Examples and counterexamples complete the exposition.
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TL;DR: In this article, the authors present two algorithms for constructing canonical factorizations of disconjugate operators starting from a basis of their kernel, which form a Chebyshev asymptotic scale at an endpoint.
Abstract: This part II-C of our work completes the factorizational theory of asymptotic expansions in the real domain. Here we present two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymptotic scale at an endpoint. These algorithms arise quite naturally in our asymptotic context and prove very simple in special cases and/or for scales with a small numbers of terms. All the results in the three Parts of this work are well illustrated by a class of asymptotic scales featuring interesting properties. Examples and counterexamples complete the exposition.
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TL;DR: In this article, the problem of asymptotic equivalence of non-homogeneous differential equations with exponentially equivalent right-hand sides was studied, and with the help of this result, the behavior of solutions to nonhomogeneous solutions to nonsmooth differential equations was described.
Abstract: This paper is devoted to the problem of asymptotic equivalence of $n$-th order differential equations with exponentially equivalent right-hand sides. With the help of this result asymptotic behavior of solutions to nonhomogeneous differential equations is described.
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TL;DR: In this article, a method to obtain asymptotic evaluations for double sums from the asymPTotic evaluations of a single sum was proposed. But this method requires the assumption that the Euler totient function is a function of the divisor.
Abstract: In this paper, we indicate a method to obtain asymptotic evaluations for double sums from the asymptotic evaluations of a single sum. As applications, we show that all results in the recent paper [New extensions of some classical theorems in number theory, J. Number Theory 133(13) (2013) 3771–3795] can be used to obtain some asymptotic evaluations for double sums. Further, we give asymptotic evaluations for double sums which involve the divisor and the Euler totient function.
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TL;DR: In this article, the authors investigated the asymptotic equilibrium of the integro-differential equations with infinite delay in a Hilbert space and showed that the problem is NP-hard.
Abstract: The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by several authors. In this paper, we investigate the asymptotic equilibrium of the integro-differential equations with infinite delay in a Hilbert space.
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TL;DR: In this paper, the existence of an asymptotic expansion is characterized by the nice property that a certain quantity F(t) has an approximate linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve, which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T.
Abstract: We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F(t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.
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01 Jan 2015TL;DR: In this article, the authors attempt to compute asymptotic expansions for fast-slow systems by more or less brute force, and they show that such a substitution seems to give a good approximation, and where are modifications required.
Abstract: In this chapter, we shall just attempt to compute asymptotic expansions for fast–slow systems by more or less brute force. It is a very instructive technique: just substitute an asymptotic expansion for the solution and see what happens. In other words, where does such a substitution seem to give a good approximation, and where are modifications required?
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01 Jan 2015TL;DR: In this paper, the authors analyze the finite-sample behavior of estimators and show that robust estimators, advertised as resistant to heavy-tailed distributions, can be themselves heavy-tail.
Abstract: The asymptotic distribution of an estimator approximates well the central part, but less accurately the tails of its true distribution. Some properties of estimators are always non-asymptotic, regardless a widely accepted view that their properties under moderate sample sizes are inherited from the asymptotic normality. Robust estimators, advertised as resistant to heavy-tailed distributions, can be themselves heavy-tailed. They are asymptotically admissible, but not finite-sample admissible for any distribution. While the asymptotic distribution of the Newton-Raphson iteration of an estimator, starting with a consistent initial estimator, coincides with that of the non-iterated estimator, its tail-behavior is determined by that of the initial estimator. Hence, before taking a recourse to the asymptotics, we should analyze the finite-sample behavior of an estimator, whenever possible. We shall try to illustrate some distinctive differences between the asymptotic and finite-sample properties of estimators, mainly of robust ones.
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01 Jan 2015TL;DR: In this article, the behavior of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries (k-gKdV) equations is studied.
Abstract: This chapter is concerned with the longtime behavior of solutions to the initial value problem (IVP) associated to the k-generalized Korteweg-de Vries (k-gKdV) equations We shall restrict ourselves to consider only real solutions of the associated IVP We will discuss global well-posedness results as well as some blow-up results
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TL;DR: In this article, a second-order linear parabolic problem with lower order coefficients rapidly oscillating in time is investigated, where the coefficient multiplying the principal stationary operator is assumed to have a simple zero eigenvalue.
Abstract: Asymptotic methods are used to investigate a second-order linear parabolic problem with lower order coefficients rapidly oscillating in time. The coefficient multiplying the principal stationary operator is assumed to be singular, i.e., it has a simple zero eigenvalue. Under certain additional conditions, the problem is proved to have a unique time-periodic solution and its complete asymptotic expansion is constructed and justified.
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TL;DR: Asymptotic I-sequences and I-grade as discussed by the authors have been shown to play a role analogous to asymptotically sequences and asymptonically grade, respectively.
01 Jan 2015
TL;DR: In this article, the authors provide a precise account of some commonly used results from asymptotic theory in probability in probability, and provide a more precise analysis of the results.
Abstract: We provide a precise account of some commonly used results from asymptotic theory in probability.