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Showing papers on "Asymptotology published in 2009"


Book ChapterDOI
01 Jan 2009
TL;DR: Asymptotic safety is a set of conditions, based on the existence of a nontrivial fixed point for the renormalization group flow, which would make a quantum field theory consistent up to arbitrarily high energies as mentioned in this paper.
Abstract: Asymptotic safety is a set of conditions, based on the existence of a nontrivial fixed point for the renormalization group flow, which would make a quantum field theory consistent up to arbitrarily high energies. After introducing the basic ideas of this approach, I review the present evidence in favor of an asymptotically safe quantum field theory of gravity. To appear in “ Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter” ed. D. Oriti, Cambridge University Press.

59 citations


Book
30 Oct 2009
TL;DR: This chapter discusses Stochastic processes: an overview, asymptotic distributions, categorical data models, and Regression models, which describe weak convergence and Gaussian processes.
Abstract: Exact statistical inference may be employed in diverse fields of science and technology. As problems become more complex and sample sizes become larger, mathematical and computational difficulties can arise that require the use of approximate statistical methods. Such methods are justified by asymptotic arguments but are still based on the concepts and principles that underlie exact statistical inference. With this in perspective, this book presents a broad view of exact statistical inference and the development of asymptotic statistical inference, providing a justification for the use of asymptotic methods for large samples. Methodological results are developed on a concrete and yet rigorous mathematical level and are applied to a variety of problems that include categorical data, regression, and survival analyses. This book is designed as a textbook for advanced undergraduate or beginning graduate students in statistics, biostatistics, or applied statistics but may also be used as a reference for academic researchers.

47 citations


Journal ArticleDOI
TL;DR: In this paper, the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from a semiconductor model was studied.
Abstract: We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

35 citations


Journal ArticleDOI
TL;DR: This work presents a simplified and improved method of computing asymptotic unfoldings that can be used in any normal form style using normal (and hypernormal) form methods.

19 citations


Journal ArticleDOI
TL;DR: This paper presents a general scheme for the design of such ''asymptotic extrapolation algorithms'' using discrete differentiation and techniques from automatic asymptotics, to approximate the coefficients in the expansion with high accuracy.

17 citations


Journal ArticleDOI
TL;DR: A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed in this paper, including the classification and methods of constructing uniform asymptotics used to describe the structure of wave fields in these domains.
Abstract: A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed The fundamentals of the wave catastrophe theory are considered, including the classification and methods of constructing uniform asymptotics used to describe the structure of wave fields in these domains, together with an analysis of the structure of the field Classes of special functions used to construct uniform asymptotic expansions of wave fields are generally described together with the properties of these classes and the methods of computation

16 citations


Journal ArticleDOI
TL;DR: In this paper, for an unknown parameter in the drift function of a diffusion process, an M-estimator based on continuously observed data, and obtain its distributional asymptotic expansion up to the third order.
Abstract: For an unknown parameter in the drift function of a diffusion process, we consider an M-estimator based on continuously observed data, and obtain its distributional asymptotic expansion up to the third order. Our setting covers the misspecified cases. To represent the coefficients in the asymptotic expansion, we derive some formulas for asymptotic cumulants of stochastic integrals, which are widely applicable to many other problems. Furthermore, asymptotic properties of cumulants of mixing processes will be also studied in a general setting.

14 citations


Journal ArticleDOI
TL;DR: In this paper, a non-linear numerical method is applied to solve the viscous-elastic-plastic material impact problem and a finite element simulation agrees with the celebrated European crash safety analysis.
Abstract: Non-linear numerical method is applied to solve the viscous-elastic-plastic material impact problem The finite element simulation agrees with the celebrated European crash safety analysis The complex material stress distribution in the large deformation has been obtained, when the impact happens Also the posteriorestimate solver and asymptotic analysis have been used for the sensitive pre-stage deformation before the impact happening This part of simulation is very interesting for the passive safety in automotive protection devices It is an important part of the mathematical modelling

12 citations


Posted Content
TL;DR: In this article, an asymptotic expansion approach to numerical problems on valuation of financial assets and securities is described, where the authors describe an expansion approach for numerical problems in financial asset valuation.
Abstract: This paper describes an asymptotic expansion approach to numerical problems on valuation of financial assets and securities.

11 citations


Journal ArticleDOI
TL;DR: First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models, and give sufficient and in a sense necessary conditions for asymPTotic normality of test statistics under alternative hypotheses.

9 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the oscillatory and asymptotic behavior of solutions of the nonlinear difference equation of the form (1) has been considered by Thandapani and Selvaraj.
Abstract: where ∆ is the forward difference operator defined by ∆xn = xn+1 −xn, α and β are positive constants, {pn} and {qn} are positive real sequences defined for all n ∈ N(n0) = {n0, n0 + 1, ...}, and n0 a nonnegative integer. By a solution of equation (1), we mean a real sequence {xn} that satisfies equation (1) for all n ∈ N(n0). If any four consecutive values of {xn} are given, then a solution {xn} of equation (1) can be defined recursively. A nontrival solution of equation (1) is said to be nonoscillatory if it is either eventually positive or eventually negative, and it is oscillatory otherwise. The oscillatory and asymptotic behavior of solutions of the nonlinear difference equation of the form (1) has been considered by Thandapani and Selvaraj

Journal ArticleDOI
TL;DR: In this paper, a numerical algorithm for solving the asymptotic stabilization problem by the initial data to a fixed hyperbolic point with a given rate is proposed and justified, which makes it possible to apply the results to a wide class of semidynamical systems including those corresponding to partial differential equations.
Abstract: A numerical algorithm for solving the asymptotic stabilization problem by the initial data to a fixed hyperbolic point with a given rate is proposed and justified. The stabilization problem is reduced to projecting the resolving operator of the given evolution process on a strongly stable manifold. This approach makes it possible to apply the results to a wide class of semidynamical systems including those corresponding to partial differential equations. By way of example, a numerical solution of the problem of the asymptotic stabilization of unstable trajectories of the two-dimensional Chafee-Infante equation in a circular domain by the boundary conditions is given.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the non-local parabolic problem with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is nonincreasing.
Abstract: In this paper, we consider the asymptotic behaviour for the non-local parabolic problem \\begin{eqnarray} \\[ u_{t}=\\Delta u+\\displaystyle\\frac{\\lambda f(u)}{(\\int_{\\Omega}f(u)dx)^{p}},\\quad x\\in \\Omega,\\ t>0, \\\\end{eqnarray} with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is non-increasing. It is found that (a) for 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 < p < 2, u(x, t) is globally bounded for any λ > 0; (c) for p = 2, if 0 < λ < 2|∂Ω|2, then u(x, t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution and u(x, t) is a global solution and u(x, t) → ∞ as t → ∞ for all x ∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution and u(x, t) blows up in finite time for all x ∈ Ω; (d) for p > 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* and u0(x) sufficiently large, u(x, t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour of u(x, t) as it blows up are obtained for p ≥ 2.

Journal ArticleDOI
TL;DR: In this article, asymptotic expansions of solutions of the first initial boundary value problem for strong Schrodinger systems near a conical point of the boundary of a domain are considered.
Abstract: We consider asymptotic expansions of solutions of the first initial boundary-value problem for strong Schrodinger systems near a conical point of the boundary of a domain

Journal ArticleDOI
TL;DR: In this article, the authors investigated the large-time asymptotic behavior of solutions of the Cauchy problem for a non-linear Sobolev-type equation with dissipation.
Abstract: The large-time asymptotic behaviour of solutions of the Cauchy problem is investigated for a non-linear Sobolev-type equation with dissipation. For small initial data the approach taken is based on a detailed analysis of the Green's function of the linear problem and the use of the contraction mapping method. The case of large initial data is also closely considered. In the supercritical case the asymptotic formulae are quasi-linear. The asymptotic behaviour of solutions of a non-linear Sobolev-type equation with a critical non-linearity of the non-convective kind differs by a logarithmic correction term from the behaviour of solutions of the corresponding linear equation. For a critical convective non-linearity, as well as for a subcritical non-convective non-linearity it is proved that the leading term of the asymptotic expression for large times is a self-similar solution. For Sobolev equations with convective non-linearity the asymptotic behaviour of solutions in the subcritical case is the product of a rarefaction wave and a shock wave. Bibliography: 84 titles.


Posted Content
TL;DR: In this paper, asymptotic expansion for solution of singularly perturbed equation for Markov random evolution in R d is studied. The views of regular and singular parts of solution are discussed.
Abstract: Is studied asymptotic expansion for solution of singularly perturbed equation for Markov random evolution in R d . The views of regular and singular parts of solution

Journal ArticleDOI
TL;DR: In this article, an asymptotic expansion in integer nonnegative powers of small parameter for a solution of a discrete optimal control problem for one class of weakly controllable systems is constructed by substituting an assumed expansion into the problem conditions and obtaining a series of problems in the coefficients of the coefficients.
Abstract: An asymptotic expansion in integer nonnegative powers of small parameter for a solution of a discrete optimal control problem for one class of weakly controllable systems is constructed by substituting an assumed asymptotic expansion into the problem conditions and obtaining a series of problems in the coefficients of the asymptotics. Conditions of existence of a solution to the perturbed problem for sufficiently small values of the parameter are found. Estimates of closeness of the approximate and exact solutions in terms of trajectory, control, and functional are obtained. The values of the minimized functional are proven not to increase when higher-order asymptotic approximations of the optimal control are used. The discussion is illustrated by examples.

01 Jan 2009
TL;DR: In this paper, the mean-square asymptotic stability of the zero solution of a general class of stochastic pantograph differential equations was studied and sufficient conditions for the zero-solution stability were derived.
Abstract: This paper studies mean-square asymptotic stability of the zero solution of initial problems of a general class of stochastic pantograph differential equation.The sufficient conditions of mean-square asymptotic stability of the zero solution are derived.In the third section,numerical methods based on θ-methods are suggested and mean-square asymptotic stability conditions for the presented methods are derived.

Journal ArticleDOI
TL;DR: In this article, the authors obtained an asymptotic expansion of solutions of Problem (1.1, 1.3) using the Vishik-Lyusternik method.
Abstract: The author obtains an asymptotic expansion of solutions of Problem (1.1), (1.3) using the Vishik-Lyusternik method.

Journal ArticleDOI
TL;DR: In this paper, one-dimensional generalized Riccati-type inequalities can be employed to analyse asymptotic behaviour of solutions of elliptic problems, and Liouville-type theorems as well as necessary conditions for the existence of solutions are given.
Abstract: We show how one-dimensional generalized Riccati-type inequalities can be employed to analyse asymptotic behaviour of solutions of elliptic problems. We give Liouville-type theorems as well as necessary conditions for the existence of solutions of specified asymptotic behaviour of nonlinear elliptic problems.

Journal ArticleDOI
TL;DR: In this paper, asymptotic properties of solutions of a class of linear $q$-difference equations were investigated and connections with the continuous case were made with the corresponding continuous case.
Abstract: In this paper we investigate asymptotic properties of solutions of a class of linear $q$-difference equations. We relate their behaviour to the asymptotics of some $q$-exponential functions and mention possible connections with the corresponding continuous case.

01 Jan 2009
TL;DR: In this article, Lu et al. build an asymptotic solution in the cases of simple and multiple characteristics, provided that the Cauchy data has an exponential expansion, and observe, like other authors, that the phase function is a solution of the classical eikonal equation.
Abstract: The term of wave is used in a general way to name any solution of a hyperbolic problem () 0 Lu = , and the method used is a generalization of a geometrical process of optics (W.K.B.) consisting in seeking the solution in the form g ( attenuating factors). We thus build such an asymptotic solution in the cases of simple and multiple characteristics, provided that the Cauchy data has an asymptotic expansion, and observe, like other authors, that the phase function is a solution of the classical eikonal equation, and the terms of the series are determined by a recursive system of differential equations. We then deduce a condition for genuine non linearity of the problem which generalizes that of Lax (7) and John (3), and highlight from it the singular behaviour of the first term of the formal solution when the characteristics are distinct. We note that, for sufficiently small frequency, the asymptotic solution is almost global.


Book ChapterDOI
TL;DR: In this paper, an explicit criterion for positive linear Volterra-stieltjes differential systems is given, and new explicit criteria for uniform and exponential asymptotic stability of positive linear VLSD systems are presented.
Abstract: An explicit criterion for positive linear Volterra-Stieltjes differential systems is given. Then new explicit criteria for uniform asymptotic stability and exponential asymptotic stability of positive linear Volterra-Stieltjes differential systems are presented. Finally, a crucial difference between the uniform asymptotic stability and the exponential asymptotic stability of linear Volterra-Stieltjes differential systems is shown.