Asymptotic Behavior Research Articles

Weak Convergence of Probability Measures on Hyperspaces with the Upper Fell-Topology

Let E be a locally compact second countable Hausdorff space and F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} the pertaining family of all closed sets. We endow F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak convergence of probability measures on the corresponding hyperspaces with a focus on the upper Fell topology. The results can be transferred to distributional convergence of random closed sets in E with applications to the asymptotic behavior of measurable selections.

On the Oscillatory Behavior of Solutions of Second-Order Damped Differential Equations with Several Sub-Linear Neutral Terms

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results.

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

A consistent estimator for confounding strength

Regression on observational data can fail to capture a causal relationship in the presence of unobserved confounding. Confounding strength measures this mismatch, but estimating it requires itself additional assumptions. A common assumption is the independence of causal mechanisms, which relies on concentration phenomena in high dimensions. While high dimensions enable the estimation of confounding strength, they also necessitate adapted estimators. In this paper, we derive the asymptotic behavior of the confounding strength estimator by Janzing and Schölkopf (2018) and show that it is generally not consistent. We then use tools from random matrix theory to derive an adapted, consistent estimator.

Existence and asymptotic behavior of normalized solutions for Choquard equations with Kirchhoff perturbation

Existence and asymptotic behavior of normalized solutions for Choquard equations with Kirchhoff perturbation

On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty <x<+\infty , t>0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi <\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi>\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.

Asymptotic analysis of sign-changing transmission problems with rapidly oscillating interface

We study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmission problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems. For more information see https://ejde.math.txstate.edu/Volumes/2024/60/abstr.html

Asymptotic behavior of stochastic p-Laplacian equations with dynamic boundary conditions

In this paper, we investigate the asymptotic behavior for the non-autonomous stochastic [Formula: see text]-Laplacian equations with dynamic boundary conditions via the stationary Wong–Zakai approximations given by the Wiener shift. We first establish the existence and uniqueness of pullback random attractor for the Wong–Zakai approximation equations in [Formula: see text]. Then we prove the upper semi-continuity of random attractors in [Formula: see text] and [Formula: see text] when the step size of approximations approaches zero.

Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

For bounded domains Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} with Lipschitz boundary Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type A=B∗B0-NbBγ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {A}}=\\begin{pmatrix} B^*B & 0 \\\\ -{\\mathscr {N}}^{\\mathfrak {b}}B & \\gamma \\end{pmatrix}, \\end{aligned}$$\\end{document}where B is associated to a quadratic form b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {b}}$$\\end{document} and Nb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^{{\\mathfrak {b}}}$$\\end{document} an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space H:=L2(Ω)×L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}:=L^2(\\Omega ) \ imes L^2(\\Gamma )$$\\end{document}. Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form B=-divQ∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B=-\\mathop {{div} }Q \ abla $$\\end{document} with Q∈C1,1(Ω¯,Rd×d),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q \\in C^{1,1}({\\overline{\\Omega }},{\\mathbb {R}}^{d\ imes d}),$$\\end{document} also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.

Asymptotic behavior of SMOTE-generated samples using order statistics

Asymptotic behavior of SMOTE-generated samples using order statistics

New decay rates for a weakly damped magneto-thermo-elastic model in [formula omitted]

We consider the Cauchy problem for a linear magneto-thermo-elastic model in R3 without a damping term in the equation of motion. This system has the decay property of regularity-loss type. Working with the system in Fourier space, the dissipative structure becomes very weak in the high frequency region. To circumvent this difficulty, we imposed more regularity on the initial data and adapted ideas introduced in the article of J. E. M. Rivera and R. Racke (2001) [23], using appropriate multipliers and working with the system in terms of its components. We obtain new decay rates for the total energy of the system. The approach developed in this work can be applied in the study of the asymptotic behavior of other linear, weakly dissipative, systems in Rn.

Multivariate asymptotic normality determined by high moments

We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines asymptotically normality, from the one dimensional to the multidimensional setting. This approach differs from the usual moment method which requires that the moments of each fixed order converge. We illustrate our results by considering a joint distribution of the numbers of bins (having the same, finite, capacity) containing a prescribed number of balls in a classical allocation scheme.

Limiting Behavior of Mixed Coherent Systems With Lévy‐Frailty Marshall–Olkin Failure Times

ABSTRACTIn this article, we show a limit result for the reliability function of a system—that is, the probability that the whole system is still operational after a certain given time—when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non‐repairable, with failure‐times governed by a Lévy‐Frailty Marshall–Olkin (LFMO) distribution—a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first‐passage time of a Lévy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

Abelian- and Tauberian-type results for the generalized Stockwell and wavelet transforms

Tauberian-type results characterizing the quasiasymptotic behavior of polynomial multiplication of Lizorkin distributions in terms of their Stockwell transforms are obtained. Some Abelian-type results relating the quasiasymptotic behavior and quasiasymptotic boundedness of Lizorkin distributions to the asymptotic behavior of their Stockwell transforms are given. Several Abelian-type results for the generalized wavelet transform are also presented.

EXISTENCE OF THE GLOBAL ATTRACTOR FOR A HYPERBOLIC PHASE FIELD SYSTEM OF CAGINALP TYPE WITH RELAXATION, GOVERNED BY A POLYNOMIAL GROWTH POTENTIAL OF DEGREE

Phase field systems have attracted the attention of many researchers in physics and mathematics for several years. They have a wide range of applications, including materials science, thermal welding, crystal formation, and others. In such phenomena, perturbations can occur. The industrial world seeks to understand the behavior of the perturbed system while seeking better control of the perturbation. Our goal in this paper is to study the asymptotic behavior of the solution, proving the existence of the global attractor for perturbed phase field systems with a regular potential and polynomial growth typically of degree .

Convergence of solutions of the porous medium equation with reactions

Consider the Cauchy problem of one-dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as t\to \infty to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behavior of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Estimation of Expectile‐Based Marginal Expected Shortfall Under Asymptotic Independence

ABSTRACTThe expectile‐based marginal expected shortfall (XMES) is an analogue of the quantile‐based marginal expected shortfall (QMES). We study the asymptotic behaviour of XMES when the underlying random variables are asymptotically independent. We consider two different methods for the estimation of XMES: one based on least asymmetrically weighted squares and the other making use of the QMES. We establish the asymptotic theories for both estimators. Additionally, we investigate the finite sample performance of our methods through a simulation study and present a concrete application to financial data.

Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth

We are concerned with a class of generalized Chern-Simons-Schrödinger systems{−Δu+λV(x)u+A0u+∑j=12Aj2u=f(u),∂1A2−∂2A1=−12|u|2,∂1A1+∂2A2=0,∂1A0=A2|u|2,∂2A0=−A1|u|2, where λ>0 denotes a sufficiently large parameter, V:R2→R admits a potential well Ω≜intV−1(0) and the nonlinearity f fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Under some suitable assumptions on V and f, based on variational method together with some new technical analysis, we are able to get the existence of positive solutions for some large λ>0, and the asymptotic behavior of the obtained solutions is also investigated when λ→+∞.

Stochastic Optimal Control Analysis for HBV Epidemic Model with Vaccination

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV.

Secrecy Rate Bounds in Spatial Modulation-Based Visible Light Communications under Signal-Dependent Noise Conditions

This study examines the physical-layer security of an indoor visible light communication (VLC) system using spatial modulation (SM), which consists of several transmitters, an authorized receiver, and a passive adversary. The SM technique is applied at the transmitters so that only one transmitter is operational at any given time. A uniform selection (US) strategy is employed to choose the active transmitter. The two scenarios under examination encompass the conditions of non-negativity and average optical intensity, as well as the conditions of non-negativity, average optical intensity, and peak optical intensity. The secrecy rate is then obtained for these two scenarios while accounting for both signal-independent noise and signal-dependent noise. Additionally, the high signal-to-noise ratio (SNR) asymptotic behavior of the derived secrecy rate constraints is investigated. A channel-adaptive selection (CAS) strategy and a greedy selection (GS) scheme are utilized to select the active transmitter, aiming to enhance the secrecy performance. The current numerical findings affirm a pronounced convergence between the lower and upper bounds characterizing the secrecy rate. Notably, marginal asymptotic differentials in performance emerge at elevated SNRs. Furthermore, the GS system outperforms the CAS scheme and the US method, in that order. Additionally, the impact of friendly optical jamming on the secrecy rate is investigated. The results show that optical jamming significantly enhances the secrecy rate, particularly at higher power levels.