Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.     

A Variational Principle for the Ackerberg-O'Malley Resonance Problem

A new variational principle is proposed for determining the asymptotic expansion of the solution of the Ackerberg-O'Malley resonance problem [Stud. Appl. Math. 49:277–295 (1970)] to any order in ε. The method used yields new higher-order results not permitted by the technique of Grasman and Matkowsky [SIAM J. Appl. Math. 32:588–597 (1977)]. Explicit results using the method are reported to O(ε) and confirmed with asymptotic expansions of the exact solution; the O(1) results agree with those reported in the literature. In the case where the coefficient functions are analytic, an exact solution is presented. It is not difficult to perform the higher-order calculations using the proposed variational approach, in contrast to the current methods in use.     

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [[5.]]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].     

New versions of the Colombeau algebras

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half‐space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions . The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one‐dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and [50]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

Polynomials of the best uniform approximation to sgn(<Emphasis Type="Italic">x</Emphasis>) on two intervals

We describe polynomials of best uniform approximation to sgn(x) on the union of two intervals [−A,−1] ⊂ [1, B] in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error in this approximation.     

Localization results for densities associated with stable small-noise diffusions

Summary This paper concerns asymptotic properties of the stationary density associated with small-noise diffusion processes, such as considered in the well-known work of Ventcel and Freidlin [12]. We assume that the origin is a globally attracting asymptotically stable equilibrium point of the underlying deterministic flow. For a bounded domain D, containing the origin, we derive estimates which establish the asymptotic independence, as the size of the noise vanishes, of the equilibrium density in D from the coefficients of the process outside D. These results are applied to generalize a result of Sheu [10] on an asymptotic representation of the equilibrium density.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

Parameter studies of modal vibrations associated with impurity pinned dislocations in a metal single crystal

Koehler's model [1–2] of motion for edge‐type dislocations in a metal single crystal that are pinned down by impurity atoms is studied. An exact solution can be found, which is composed of a rapidly decaying transient and a steady time‐oscillating, steady state vibration. This solution is used to improve Koehler's [1] approximation to the steady time‐oscillating steady state vibration. General parameter studies of the modes of oscillation are then performed. The present result is of some significance, because it allows insight into the behavior of crystalline solids over a wide parameter range, whereas Koehler's asymptotic approach is valid only for materials that exhibit order‐of‐magnitude variation in system parameters. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 427–439, 2001.     

Martingale expansion in mixed normal limit

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 by Yoshida [28] 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”) are discussed.     

On the distribution of square-full integers

Let A(x) A(x) be the number of square-full integers \leqq x \leqq x and let D(x) \Delta(x) be the error term in the asymptotic formula for A(x) A(x) . Under the Riemann hypothesis, we show that D(x) << x^[12/85]+e \Delta(x)\ll x^{{12\over 85}+\varepsilon} . This improves the earlier results of Zhu and Yu [17], Cao [4, II], Liu [9] and Wu [16], which requires [ 1/7 ] 1\over 7 in place of [ 12/85 ] 12\over 85 .     

Modified Wave Operators for the Hartree Equation with Data,Image and Convergence in the Same Space,II

We study modified wave operators for the Hartree equation with a long-range potential |x|^-n |x|^{-\nu} , extending the result in [12] to the whole range of the Dollard type 1/2 < n \nu < 1. We construct the modified wave operators in the whole space of (1 + |x|)^-sL² (1 + |x|)^{-s}L^2 . We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound $ s > 1 - \nu / 2 $ s > 1 - \nu / 2 of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.     

Approximation in eigenvalue problems for holomorphic fredholm operator functions I

The approximation of a holomorphic eigenvalue problem is considered. The main purpose is to present a construction by which the derivation of the asymptotic error estimations for the approximate eigenvalues of Fredholm operator functions can be reduced to the derivation of these estimations for the case of matrix functions. (Some estimations for the latter problem can be derived, in one's turn, from the error estimations for the zeros of the corresponding determinants.) The asymptotic error estimations are considered in part II of this paper, in [10]. By the presented construction also the stability of the algebraic multiplicity of eigenvalues by regular approximation is proved in Section 3

The presented construction, in essence, reproduces the constructions in [7] for the case of the compact approximation in subspaces and in [9] for the case of projection—like methods. It is simpler to use than similiar construction in [8], and allows unified consideration of the general case and the case of projection—like methods, what in [8, 9] was not achieved     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

On a new stopping rule for stochastic approximation

Summary A new stopping rule for the Robbins-Monro process, based on an F-statistic criterion is proposed and its asymptotic behavior established. On the basis of evidence obtained through experimental sampling, the procedure seems to work well over a wide variety of situations. A two-stage procedure, coupling the new rule with an earlier one proposed by Sielken [1973] is recommended for practical use.     

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Polynomial Approximation of Conformal Maps

Let G be a finite domain, bounded by a Jordan curve Γ , and let f ₀ be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f ₀ , in the case that Γ is piecewise-analytic without cusps. In particular, we consider the problem of approximating f ₀ by the Bieberbach polynomials π _n and derive results better than those in [5] and [6] for the case that the corners of Γ have interior angles of the form π/N . In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms. December 6, 1995. Date revised: August 5, 1996.