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Showing papers on "Asymptotology published in 2016"




Journal ArticleDOI
TL;DR: In this paper, the authors considered the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure and constructed an asymptotic expansion of its solution.
Abstract: In the paper, we consider the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure. Our goal is to construct an asymptotic expansion of its solution. We provide error estimates and also introduce and justify the asymptotic partial domain decomposition for this problem.

25 citations


Journal ArticleDOI
TL;DR: In this paper, the asymptotic stability and global stability of equilibria in autonomous systems of differential equations are analyzed in terms of compact invariant sets and positively invariants.
Abstract: The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.

16 citations


Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of all solutions to the fourth-order Emden-Fowler type differential equation with singular nonlinearity is investigated, where the equation is transformed into a system on the three-dimensional sphere.
Abstract: The asymptotic behavior of all solutions to the fourth-order Emden– Fowler type differential equation with singular nonlinearity is investigated. The equation is transformed into a system on the three-dimensional sphere. By investigation of the asymptotic behavior of all possible trajectories of this system an asymptotic classification of all solutions to the equation is obtained.

8 citations


Journal ArticleDOI
09 May 2016
TL;DR: In this article, an asymptotic perturbation of the limit set generated from a nitely family of conformal contraction maps endowed with a directed graph was studied, and it was shown that the Hausdor dimension of the restricted limit set behaves in the same order.
Abstract: We study an asymptotic perturbation of the limit set generated from a nitely family of conformal contraction maps endowed with a directed graph. We show that if those maps have asymptotic expansions under weak conditions, then the Hausdor dimension of the limit set behaves asymptotically by the same order. We also prove that the Gibbs measure of a suitable potential and the measure theoretic entropy of this measure have asymptotic expansions under an additional condition. In nal section, we demonstrate degeneration of graph iterated function systems. Mathematics Subject Classi cation (2010). Primary: 37B10; Secondary: 37C45, 37D35.

8 citations


Book ChapterDOI
01 Jan 2016
TL;DR: In this article, the authors present an asymptotic expansion of a martingale with a asmptotically mixed normal distribution, where the expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle.
Abstract: The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida 1997 as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) is discussed.

8 citations


Journal ArticleDOI
TL;DR: In this paper, a class of second order difference equations with three paremeters is studied and the asymptotic behavior of positive solutions with positive initial values is investigated.
Abstract: In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.

5 citations


Journal ArticleDOI
TL;DR: Some asymptotic estimates of the θ-methods with constant stepsize discretization are given and this gives an upper bound estimate of the solution for the long-time behaviour of the numerical solutions to a class of linear non-autonomous neutral delay differential equation with proportional delays.
Abstract: This paper is concerned with the long-time behaviour of the numerical solutions to a class of linear non-autonomous neutral delay differential equation with proportional delays. Our purpose is to give some asymptotic estimates of the θ-methods with constant stepsize discretization and formulate their upper bounds. Asymptotic estimate not only describes more accurate than asymptotic stability, but also gives an upper bound estimate of the solution for the long-time behaviour. We also compare the known results and show that our formulae improve and generalize these results. Some numerical examples are given in the end of this paper to confirm the theoretical results.

4 citations



Proceedings ArticleDOI
01 Dec 2016
TL;DR: A sufficient condition which ensures the asymptotic stability in probability of this class of systems is given using a Lyapunov approach.
Abstract: In this paper, we deal with asymptotic stability in probability of a class of stochastic systems, which is described by a stochastic differential equation having a square root on the diffusion part. This type of equations are usually used for the modeling of some systems such as population dynamics. A sufficient condition which ensures the asymptotic stability in probability of this class of systems is given using a Lyapunov approach.

Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of solutions of the Cauchy problem for the linearized system of magnetohydrodynamic equations with initial conditions localized near a two-dimensional surface was refined.
Abstract: The asymptotic behavior of solutions of the Cauchy problem for the linearized system of magnetohydrodynamic equations with initial conditions localized near a two-dimensional surface was obtained by the authors earlier. Here, this asymptotic behavior is refined.

Journal ArticleDOI
TL;DR: In this paper, a new systematic method to obtain discrete asymptotic numerical models for incompressible free-surface flows is discussed, which consists of first discretizing the Euler equations in the horizontal direction, keeping both the vertical and time derivatives continuous, and then performing an analysis on the resulting system.
Abstract: In this paper, we discuss a new systematic method to obtain discrete asymptotic numerical models for incompressible free-surface flows. The method consists of first discretizing the Euler equations in the horizontal direction, keeping both the vertical and time derivatives continuous, and then performing an asymptotic analysis on the resulting system. The asymptotics involve the ratios wave amplitude over depth, denoted by $\varepsilon$, and depth over wavelength, denoted by $\sigma$. For simplicity, in this paper we only consider the weakly nonlinear scaling in which both $\sigma^4$ and $\varepsilon\sigma^2$ are very small and of the same order. We investigate the properties of the fully discrete Boussinesq model obtained by neglecting terms proportional to these quantities. Our study reveals that if the interaction between terms arising from the discretization and from the PDE is properly accounted for, the resulting discrete system has improved linear dispersion and shoaling approximations w.r.t. the d...

Proceedings ArticleDOI
23 Jun 2016
TL;DR: In this article, the analysis of asymptotic behavior of the solutions of differential systems is considered using integral inequalities sufficient conditions for the existence of an equilibrium state are presented for various classes of differentials.
Abstract: In the paper the analysis of asymptotic behavior of the solutions of differential systems is considered. Using integral inequalities sufficient conditions for the existence of an asymptotic equilibrium state are presented for various classes of differential systems.

Journal ArticleDOI
27 Apr 2016
TL;DR: In this article, a generalized method of boundary functions was proposed for constructing complete asymptotic expansions of the solutions to the Dirichlet problem for bisingular perturbed linear inhomogeneous second-order elliptic equations with two independent variables in the ring.
Abstract: The Dirichlet problem for elliptic equations with a small parameter in the highest derivatives takes a unique place in mathematics. In general case it is impossible to build explicit solution to these problems, which is why the researchers apply different asymptotic methods. The aim of the research is to develop the asymptotic method of boundary functions for constructing complete asymptotic expansions of the solutions to such problems. The proposed generalized method of boundary functions differs from the matching method in the fact that the growing features of the outer expansion are actually removed from it and with the help of the auxiliary asymptotic series are fully included in the internal expansions, and differs from the classical method of boundary functions in the fact that the boundary functions decay in power-mode nature and not exponentially. Using the proposed method, a complete asymptotic expansion of the solution to the Dirichlet problem for bisingular perturbed linear inhomogeneous second-order elliptic equations with two independent variables in the ring with quadratic growth on the boundary is built. A built asymptotic series corresponds to the Puiseux series. The basic term of the asymptotic expansion of the solution has a negative fractional degree of the small parameter, which is typical for bisingular perturbed equations, or equations with turning points. The built expansion is justified by the maximum principle

01 Jan 2016
TL;DR: In this paper, it was shown that the set A = ∪3 i=0 Ag(Wi) is a minimal asymptotic basis of order four, where i = 0, 1, 2, 3, and Wi = {n ∈ N | n ≡ i (mod 4)}.
Abstract: Abstract Let N denote the set of all nonnegative integers and A be a subset of N. Let W be a nonempty subset of N. Denote by F∗(W ) the set of all finite, nonempty subsets of W . Fix integer g ≥ 2, let Ag(W ) be the set of all numbers of the form ∑ f∈F afg f where F ∈ F∗(W ) and 1 ≤ af ≤ g − 1. For i = 0, 1, 2, 3, let Wi = {n ∈ N | n ≡ i (mod 4)}. In this paper, we show that the set A = ∪3 i=0 Ag(Wi) is a minimal asymptotic basis of order four.

Journal ArticleDOI
TL;DR: In this article, it was shown that the positive momentum density of the Camassa-Holm equation is a combination of Dirac measures supported on the positive axis, and that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.
Abstract: The paper addresses the asymptotic properties of Camassa-Holm equation on the half-line. That is, using the method of asymptotic density, under the assumption that it is unique, the paper proves that the positive momentum density of the Camassa-Holm equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.

Posted Content
TL;DR: In this paper, the authors combine the method of multiple scales and the matched asymptotic expansions to construct uniformly-valid solutions to autonomous and non-autonomous difference equations in the neighbourhood of a period-doubling bifurcation.
Abstract: In this paper, we combine the method of multiple scales and the method of matched asymptotic expansions to construct uniformly-valid asymptotic solutions to autonomous and non-autonomous difference equations in the neighbourhood of a period-doubling bifurcation. In each case, we begin by constructing multiple scales approximations in which the slow time scale is treated as a continuum variable, leading to difference-differential equations. The resultant approximations fail to be asymptotic at late time, due to behaviour on the slow time scale, it is necessary to eliminate the effects of the fast time scale in order to find the late time rescaling, but there are then no difficulties with applying the method of matched asymptotic expansions. The methods that we develop lead to a general strategy for obtaining asymptotic solutions to singularly-perturbed difference equations, and we discuss clear indicators of when multiple scales, matched asymptotic expansions, or a combined approach might be appropriate.

Journal ArticleDOI
TL;DR: In this article, the wave number is taken as a small asymptotic parameter, and the Lagrangian multipliers are introduced into the governing equations to transform them into saddle point problems.

01 Jan 2016
TL;DR: The nonstandard asymptotic analysis is universally compatible with any devices to read, and is available in the book collection an online access to it is set as public so you can get it instantly.
Abstract: Thank you very much for reading nonstandard asymptotic analysis. As you may know, people have search hundreds times for their chosen books like this nonstandard asymptotic analysis, but end up in infectious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they cope with some malicious virus inside their laptop. nonstandard asymptotic analysis is available in our book collection an online access to it is set as public so you can get it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the nonstandard asymptotic analysis is universally compatible with any devices to read.

Journal ArticleDOI
TL;DR: In this article, an exact asymptotic value of the logarithm for a counting process in the max-scheme is obtained for the case where the counting process can be expressed as
Abstract: Abstract An exact asymptotic value of the logarithm for a counting process in the max-scheme is obtained.

Proceedings ArticleDOI
01 Jun 2016-Stahlbau
TL;DR: In this paper, conditions for asymptotic stability and global stability of nonlinear time-invariant systems are obtained in terms of compact invariant sets and positively invariant set.
Abstract: Conditions for asymptotic stability and global asymptotic stability of equilibrium points of nonlinear time-invariant systems are obtained in terms of compact invariant sets and positively invariant sets. To verify these conditions the functional localization method of compact invariant sets is proposed. The obtained results can be applied to investigate the asymptotic stability in critical cases, and to find the Lyapunov function if the equilibrium point is (globally) asymptotically stable.

Posted Content
TL;DR: In this article, it is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases.
Abstract: Inspired by the works of Y. Ohta and J. Yang, one constructs the lumps solutions in the Kadomtsev-Petviashvili-(I) equation using the Grammian determinants. It is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. Also, one can prove that all the locations of peaks are on a vertical line when time approaches - $\infty$, and then they will be on a horizontal line when time approaches $\infty$, i.e., there is a rotation $\frac{\pi}{2}$ after interaction.

Posted Content
TL;DR: A computational and asymptotic analysis of the solutions of Carrier's problem is presented in this paper, which reveals a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurbcations as the bifurlcation parameter tends to zero.
Abstract: A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag is incorrect.

DOI
01 Jan 2016
TL;DR: An unbiased Monte Carlo estimator with finite variance for computing expectations of the solution to random PDEs and an analytical form for the exponential decay rate of error probabilities of the generalized likelihood ratio test for testing two general families of hypotheses are derived.
Abstract: ASYMPTOTIC THEORY AND APPLICATIONS OF RANDOM FUNCTIONS XIAOOU LI Random functions is the central component in many statistical and probabilistic problems. This dissertation presents theoretical analysis and computation for random functions and its applications in statistics. This dissertation consists of two parts. The first part is on the topic of classic continuous random fields. We present asymptotic analysis and computation for three non-linear functionals of random fields. In Chapter 2, we propose an efficient Monte Carlo algorithm for computing P{supT f(t) > b} when b is large, and f is a Gaussian random field living on a compact subset T . For each pre-specified relative error e, the proposed algorithm runs in a constant time for an arbitrarily large b and computes the probability with the relative error e. In Chapter 3, we present the asymptotic analysis for the tail probability of ∫ T edt under the asymptotic regime that σ tends to zero. In Chapter 4, we consider partial differential equations (PDE) with random coefficients, and we develop an unbiased Monte Carlo estimator with finite variance for computing expectations of the solution to random PDEs. Moreover, the expected computational cost of generating one such estimator is finite. In this analysis, we employ a quadratic approximation to solve random PDEs and perform precise error analysis of this numerical solver. The second part of this dissertation focuses on topics in statistics. The random functions of interest are likelihood functions, whose maximum plays a key role in statistical inference. We present asymptotic analysis for likelihood based hypothesis tests and sequential analysis. In Chapter 5, we derive an analytical form for the exponential decay rate of error probabilities of the generalized likelihood ratio test for testing two general families of hypotheses. In Chapter 6, we study the asymptotic property of the generalized sequential probability ratio test, the stopping rule of which is the first boundary crossing time of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero. These results have important theoretical implications in hypothesis testing, model selection, and other areas where maximum likelihood is employed.