The convex hull of a Banach–Saks set

A subset A of a Banach space is called Banach–Saks when every sequence in A has a Cesàro convergent subsequence. Our interest here focuses on the following problem: is the convex hull of a Banach–Saks set again Banach–Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.     

Asymptotic behavior of the solutions of the p-Laplacian equation

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

Closed convex hull of the family of multivalently close-to-convex functions of order β

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ₀ ^k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

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 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.     

Closed convex hull of multivalent symmetric close-to-convex functions of order β with the Montel normalization

We obtain a one-to-one tranformation of the family of multivalent symmetric close-to-convex functions of order β, onto the family of multivalent symmetric close-to-convex functions of order β with the Montel normalization. Using this transformation we determine the closed convex hull and extreme points of the latter class for β≥1. The present investigation is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India under Grant No. 48/2/2003-R & D II.     

Closed convex hull of multivalent symmetric close-to-convex functions of order β with the Montel normalization

We obtain a one-to-one tranformation of the family of multivalent symmetric close-to-convex functions of order β, onto the family of multivalent symmetric close-to-convex functions of order β with the Montel normalization. Using this transformation we determine the closed convex hull and extreme points of the latter class for β≥1.     

Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.     

Asymptotic behavior on (−∞, 0) of the spectral measures of a family of singular Sturm-Liouville operators

We find the asymptotics as λ/? → ?∞ of the density of the spectral measure of the Sturm-Liouville operator in L ²(0,+∞) generated by the expression ?y″ + ?q(x)y, ? > 0, with the boundary condition y(0) cos α+y′(0) sinα = 0. The potential q(x) tends to ?∞ as x → +∞ and is assumed to satisfy the Sears condition and some additional regularity conditions.     

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice ? ³ (here k is the total quasimomentum of a two-particle system, $k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$ . We show that for any $k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$ , there is a potential $\hat v$ such that the two-particle operator H(k) has infinitely many eigenvalues z_n(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ? S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k ₂ , k ₃ ) ∈ (?π,π) ² , and $\hat v \in W(3)$ , then the operator H(k) has infinitely many eigenvalues z_n(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $\hat v \in W(ij)$ , then the eigenvalues z_nm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.     

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Mathematical Programming - The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer...     

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Lévy Brownian motion.     

Asymptotic behavior of the loss rate for Markov-modulated fluid queue with a finite buffer

We consider a Markov-modulated fluid queue with a finite buffer. It is assumed that the fluid flow is modulated by a background Markov chain which may have different transitions when the buffer content is empty or full. In Sakuma and Miyazawa (Asymptotic Behavior of Loss Rate for Feedback Finite Fluid Queue with Downward Jumps. Advances in Queueing Theory and Network Applications, pp. 195–211, Springer, Cambridge, 2009), we have studied asymptotic loss rate for this type of fluid queue when the mean drift of the fluid flow is negative. However, the null drift case is not studied. Our major interest is in asymptotic loss rate of the fluid queue with a finite buffer including the null drift case. We consider the density of the stationary buffer content distribution and derive it in matrix exponential forms from an occupation measure. This result is not only useful to get the asymptotic loss rate especially for the null drift case, but also it is interesting in its own light.     

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.     

On the location of the maximum of a process: Lévy,Gaussian and Random field cases

ABSTRACT

We show how the techniques presented in Pimentel [On the location of the maximum of a continuous stochastic process, J. Appl. Prob. 51 (2014), pp. 152–161] can be extended to a variety of non-continuous processes and random fields. For the Gaussian case, we prove new covariance formulae between the maximum and the maximizer of the process. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein–Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among other processes.     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

On using an automatic scheme for obtaining the convex hull defining inequalities of a Weismantel 0–1 knapsack constraint

In this short note we obtain the full set of inequalities that define the convex hull of a 0–1 knapsack constraint presented in Weismantel (1997). For that purpose we use our O(n) procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates, as well as our schemes for coefficient increase based tightening cover induced inequalities and coefficient reduction based tightening general 0–1 knapsack constraints.