Asymptotic normality of Powell’s kernel estimator

We establish asymptotic normality of Powell’s kernel estimator for the asymptotic covariance matrix of the quantile regression estimator for both i.i.d. and weakly dependent data. As an application, we derive the optimal bandwidth that minimizes the approximate mean squared error of the kernel estimator. We also derive the corresponding results to censored quantile regression.     

Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt, Proc Natl Acad Sci USA 42:43–47, 1956 and Parzen, Ann Math Stat 33:1965–1976, 1962) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg’s method rather than on Bernstein’s small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the strong mixing coefficients which do not depend on the bandwidth.     

Asymptotic behavior of an eigenvalue of the two-particle discrete Schr?dinger operator

We consider a two-particle discrete Schr?dinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential. We show that there is a unique eigenvalue below the bottom of the essential spectrum for all values of the coupling constant and two-particle quasimomentum. We obtain a convergent expansion for the eigenvalue.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

On an Asymptotic Behavior of Exponential Functional Equation

The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.     

On the existence of ɛ-fixed points

In this paper we prove some approximate fixed point theorems which extend, in a broad sense, analogous results obtained by Brânzei, Morgan, Scalzo and Tijs in 2003. By assuming also the weak demiclosedness property we state two fixed point theorems. Moreover, we study the existence of ?-Nash equilibria.     

Asymptotic behavior as t→∞ of the solutions of initial-boundary value problems for the equations of motions of linear viscoelastic fluids

It is shown that for the nonstationary equations of motion of the linear viscoelastic fluids, whose defining equation has the form the stationary system is the Navier-Stokes stationary system with viscosity coefficient v: It is proved that for small Reynolds numbers the solutions of the initial-boundary value problems for the equations of motion of the Oldroyd fluids (M=L=1, 2, ...) and Kelvin-Voight fluids (M=L + 1, L=0, 1, 2, ...) converge for t to the solution of the first boundary value problem for the stationary Navier-Stokes system (*).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 171, pp. 174–181, 1989.     

Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

We study the asymptotic behaviours for the trajectory fitting estimator in the Ornstein–Uhlenbeck process with linear drift. Deviation inequality, moderate deviations, Berry–Esseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry–Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals [P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966–988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448–505].     

Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type”

We derive a new integral equation that linearizes the Cauchy problem for the Korteweg—de Vries equation for the initial condition of the threshold type, where the initial function vanishes as x- and tends to some periodic function as x+. We also expand the solution of the Cauchy problem into a radiation component determined by the reflection coefficient and a component determined by the nonvanishing initial condition. For the second component, we derive an approximate determinant formula that is valid for any t0 and x(-,X _N), where X _N with the unboundedly increasing parameter N that determines the finite-dimensional approximation to the integral equation. We prove that as t, the solution of the Cauchy problem in the neighborhood of the trailing edge decays into asymptotic solitons, whose phases can be explicitly evaluated in terms of the reflection coefficient and other parameters of the problem.     

Asymptotic behavior of the Landau—Lifshitz model of ferromagnetism

According to the classical theory of Weiss, Landau, and Lifshitz, on a microscopic scale a ferromagnetic body is magnetically saturated (i.e., |M| =: constant) and consists of regions in which the magnetization is uniform, separated by thin transition layers. Any stationary configuration corresponds to a minimum point of an energy functional in which a small parameter is present. The asymptotic behavior as 0 is studied here. It is easy to see that any sequence of minimizers contains a subsequenceM _j that converges to a fieldM. By means of a -limit procedure it is shown that this fieldM is a minimizer of a new functional containing a term proportional to the area of the surfaces separating different domains of uniform magnetization. TheC ^1, -regularity of these surfaces, for < 1/2, is also proved under suitable assumptions for the external magnetic field.     

Convergence acceleration of some sequences by the ɛ-algorithm

Summary It is proved that the sequences formed by the columns and the diagonals of the array of the -algorithm converge faster than the initial sequence when it is either a non logarithmic totally monotonic sequence or a converging totally oscillating sequence.     

Robust empirical optimization is almost the same as mean–variance optimization

We formulate a distributionally robust optimization problem where the deviation of the alternative distribution is controlled by a $?$-divergence penalty in the objective, and show that a large class of these problems are essentially equivalent to a mean–variance problem. We also show that while a “small amount of robustness” always reduces the in-sample expected reward, the reduction in the variance, which is a measure of sensitivity to model misspecification, is an order of magnitude larger.     

Asymptotic behavior of reaction–advection–diffusion population models with Allee effect

In this work, a qualitative analysis is carried out for reaction–advection–diffusion (RAD) systems modeling the interactions between two species with Allee effect. In particular, we study different scenarios: mutualism, competition, and a predator–prey relationship in order to investigate the survival or extinction of both populations. Global existence and uniqueness of positive solutions of the proposed RAD problems are demonstrated. Equilibrium states and asymptotic behavior of solutions are obtained using the monotone method and the upper and lower solutions technique. Numerical simulations by a Crank–Nicolson monotone iterative method of the different asymptotic solution dynamics are shown to illustrate our theoretical results.     

On the Relation between the Nonconvexity of a Set and the Nonconvexity of Its ɛ-Neighborhoods

To each closed subset P of a Banach space, a real function characterizing the nonconvexity of this set is associated. Inequalities of the type _P(.),)< \nomathbreak an extensor, etc. In this paper, examples of sets whose nonconvexity functions substantially differ from the nonconvexity functions of arbitrarily small neighborhoods of these sets are constructed. On the other hand, it is shown that, in uniformly convex Banach spaces, conditions of the type the function of nonconvexity is less than one are stable with respect to taking -neighborhoods of sets.     

Asymptotic bubble evolutions of the Rayleigh–Taylor instability

We examine the multiple harmonic model for the single-mode Rayleigh–Taylor instability, and present a new class of the asymptotic solution for the bubble evolution. Previously reported solutions for the bubble curvature and velocity from the model were quantitatively different from other theoretical models and numerical results, for small density jumps. The discrepancy between the theoretical models is resolved by our new approach to the model. Our solution agrees with the Layzer–Goncharov model, and gives the independence of the bubble curvature on the density ratio.     

Asymptotic Analysis of Multilump Solutions of the Kadomtsev–Petviashvili-I Equation

We construct lump solutions of the Kadomtsev–Petviashvili-I equation using Grammian determinants in the spirit of the works by Ohta and Yang. We show that the peak locations depend on the real roots of the Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. We also prove that if the time goes to ?∞, then all the peak locations are on a vertical line, while if the time goes to ∞, then they are all on a horizontal line, i.e., a π/2 rotation is observed after interaction.