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Showing papers on "Asymptotology published in 2013"


Book
11 Jun 2013
TL;DR: In this paper, asymptotics of Green's kernels in domains with singularly perturbed boundaries and meso-scale approximations of physical fields in non-periodic domains with many inclusions are discussed.
Abstract: Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale approximations for bodies with non-periodic multiple perforations Contains illustrations and numerical examples for a range of physically realisable configurationsThere are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary Examples include perforated domains and bodies with defects of different types The accurate direct numerical treatment of such problems remains a challenge Asymptotic approximations offer an alternative, efficient solution Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems The uniformity of the asymptotic approximations is the principal point of attention We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusionsThe main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions The novel feature of these asymptotic approximations is their uniformity with respect to the independent variablesThis book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations

68 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V (x ) = κ x 2 q + ω x 2, q = 2, 3, etc.

20 citations


Journal ArticleDOI
TL;DR: In this article, the authors give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase.

13 citations


Posted Content
TL;DR: Asymptotic property C for metric spaces was introduced by Dranishnikos as generalization of finite asymptotics -asdim as discussed by the authors, and it turns out that this property can be viewed as transfinite extension of the original metric dimension.
Abstract: Asymptotic property C for metric spaces was introduced by Dranishnikos as generalization of finite asymptotic dimension - asdim. It turns out that this property can be viewed as transfinite extension of asymptotic dimension. The original definition was given by Radul. We introduce three equivalent definitions, show that asymptotic property C is closed under products (open problem stated "Open problems in topology II") and prove some other facts, i.e. by defining dimension of a family of metric spaces. Some examples of spaces enjoying countable trasfinite asymptotic dimension are given. We also formulate open problems and state "omega conjecture", which inspired most part of this paper.

11 citations


Journal ArticleDOI
TL;DR: In this paper, the Stirling series was extended to the Gamma function with shifted argument, which is the generalization of the well-known Stirling Series. But no explicit error bounds exist in the literature for this expansion.
Abstract: In this paper we reconsider the asymptotic expansion of the Gamma function with shifted argument, which is the generalization of the well-known Stirling series. To our knowledge, no explicit error bounds exist in the literature for this expansion. Therefore, the first aim of this paper is to extend the known error estimates of Stirling’s series to this general case. The second aim is to give exponentially-improved asymptotics for this asymptotic series.

10 citations


Journal ArticleDOI
TL;DR: In this paper, an extension of the method of asymptotic decompositions of vector fields with finite-time singularities was proposed, by applying the central extension technique of Poincar\'e to the dominant part of the vector field on approach to the singularity.
Abstract: We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincar\'e to the dominant part of the vector field on approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces.

9 citations


Journal ArticleDOI
TL;DR: In this article, different asymptotic representations for correlation functions of critical integrable systems were discussed, and it was shown that in the one-dimensional boson model, the correlation functions obtained by the multiple-integral method coincides with the conformal field theory predictions in the low-temperature limit.
Abstract: We discuss different asymptotic representations for correlation functions of critical integrable systems. We prove that in the one-dimensional boson model, the asymptotic series for correlation functions obtained by the multiple-integral method coincides with the conformal field theory predictions in the low-temperature limit.

4 citations


Journal ArticleDOI
TL;DR: In this paper, the authors discuss the construction of solutions to the inverse Cauchy problem by using characteristics, and propose a solution based on the characteristics of the inverse cauchy problems.
Abstract: We discuss the construction of solutions to the inverse Cauchy problem by using characteristics.

3 citations


Book ChapterDOI
01 Nov 2013

2 citations





Posted Content
TL;DR: The asymptotic, near-equilibrium neural response of the sensory periphery can be derived theoretically using information theory, asymPTotic Bayesian statistics and a theory of complex systems.
Abstract: The asymptotic, near-equilibrium neural response of the sensory periphery can be derived theoretically using information theory, asymptotic Bayesian statistics and a theory of complex systems. Almost no biological knowledge is required. The theoretical approach shows good agreement with experimental data across different sensory modalities and animal species. The theory is reminiscent of statistical physics.

Journal ArticleDOI
TL;DR: For systems of nonlinear functional equations, the asymptotic properties of their solutions continuously differentiable and bounded for t ≥ T > 0 in a neighborhood of the singular point t = + ∞ were studied in this paper.
Abstract: For systems of nonlinear functional equations, we study the asymptotic properties of their solutions continuously differentiable and bounded for t ≥ T > 0 in a neighborhood of the singular point t = + ∞.

Posted Content
TL;DR: In this article, the authors considered the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one and proved the existence of the full asymptotic expansions of these spherical integrals.
Abstract: We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion. Using asymptotic expression of the spherical integral, we derive the asymptotic freeness of Wigner matrices with (deterministic) Hermitian matrices.

01 Jan 2013
TL;DR: This work develops the classical idea of limiting step to the asymptotology of multiscale reaction networks and proposes the concept of limit simplification, which is applied to the analysis of microRNA-mediated mechanisms of translation repression.
Abstract: We develop the classical idea of limiting step to the asymptotology of multiscale reaction networks. The concept of limit simplification is proposed. For multiscale reaction networks the dynamical behavior is to be approximated by the system of simple dominant networks. The dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning. They give an answer to an important question: given a network model, which are its critical parameters? Many of the parameters of the initial model are no longer present in the dominant system: these parameters are non-critical. Parameters of dominant systems indicate putative targets to change the behavior of the large network. Following Kruskal [1], asymptotology is “the art of describing the behavior of a specified solution (or family of solutions) of a system in a limiting case.” We analyze dynamics and steady states of multiscale reaction networks. We focused mostly on the case when the elementary processes have significantly different time scales. In this case, we obtain “limit simplification” of the model: all stationary states and relaxation processes could be analyzed “to the very end”, by straightforward computations, mostly analytically. For any ordering of reaction rate constants we look for the dominant kinetic system. The dominant system is, by definition, the system that gives us the main asymptotic terms of the stationary state and relaxation in the limit for well separated rate constants. The theory of dominant systems for linear reaction networks and Markov chains is well developed [2; 3]. Complete theory for linear networks with well separated reaction rate constants allows us to elaborate algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. We found the explicit asymptotics of eigenvectors and eigenvalues. All algorithms are represented topologically by transformation of the graph of reaction (labeled by reaction rate constants). The reaction rate constants for dominant systems may not coincide with constant of original network. In general, they are monomials of the original constants. In the simplest cases, the dominant system can be represented as dominant path in the reaction network. In the general case, the hierarchy of dominant paths in the hierarchy of lumped networks is needed. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple benchmarks and on multiscale biochemical networks [4]. These methods are applied, in particular, to the analysis of microRNA-mediated mechanisms of translation repression [5; 6; 7]. Although remarkable progress has been made in deciphering the mechanisms used by miRNAs to regulate translation, many contradictory findings have been published that stimulate active debate in this field. There is a hot debate in the current literature about which mechanism and in which situations has a dominant role in living cells. The same experimental systems dealing with the same pairs of mRNA and miRNA can provide ambiguous evidences about which is the actual mechanism of translation repression observed in the experiment. We analyse dominant systems for the reaction kinetic network that includes all known mechanisms of miRNA action and demonstrate that among several coexisting miRNA mechanisms, the one that will effectively be measurable is that which acts on or changes the sensitive parameters of the translation process. This analysis of dominant systems explains the majority of existing controversies reported. For general nonlinear systems, the problem of dominant systems is still open. It is discussed in the framework of the modern theories of tropical asymptotic [8; 9]. For nonlinear reaction networks, we present a new heuristic algorithm for calculation of hierarchy of dominant paths. Our approach is based on the asymptotic analysis of fluxes on the Volpert graph [10; 11]. The results of the analysis of the dominant systems often support the observation by Kruskal [1]: “And the answer quite generally has the form of a new system (well posed problem) for the solution to satisfy, although this is sometimes obscured because the new system is so easily solved that one is led directly to the solution without noticing the intermediate step.”

Journal Article
TL;DR: In this paper, the authors investigate the asymptotic behavior of the solutions of an economic model and show that the model can be solved by a simple linear combination of the solution and the model.
Abstract: The objective of this paper is to investigate the asymptotic behavior of the solutions of an economic model


Journal ArticleDOI
TL;DR: The equations of the Kirchhoff-love and Reissner-Mindlin plate theories were derived with the use of the asymptotic homogenization method as discussed by the authors.
Abstract: The equations of the Kirchhoff-Love and Reissner-Mindlin plate theories are derived with the use of the asymptotic homogenization method.

Posted Content
TL;DR: In this article, a classification of homogenization-based Numerical Methods and (in particular) of numerical methods that are based on the Two-Scale Convergence is done, and the classification stand: direct homogenisation-based numerical methods; H-Measure-based numerical methods; two-scale numerical methods and TSAPS: two-Scale asymptotic preserving schemes.
Abstract: In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of Numerical Methods that are based on the Two-Scale Convergence is done. In this classification stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Numerical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes.