Asymptotic Behavior Research Articles

Staircasing effect for minimizers of the one-dimensional discrete Perona-Malik functional

We consider the one-dimensional Perona-Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.

Global Stability Analysis of Onchocerciasis Transmission Dynamics with Vigilant Compartment in Two Interacting Populations

A deterministic compartmental model for the transmission dynamics of onchocerciasis with vigilant compartment in two interacting populations is studied. The model is qualitatively analyzed to investigate its global asymptotic behavior with respect to disease-free and endemic equilibria. It is shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptotically stable when the associated basic reproduction number, R0<1. When the basic reproduction number R0>1, under some certain conditions on the model parameters, we prove that the endemic equilibrium is globally asymptotically stable with the aid of a suitable nonlinear Lyapunov function.

Asymptotic Behaviour of Grade of Some Modules via Linkage

Asymptotic Behaviour of Grade of Some Modules via Linkage

Randomness Test of Thinning Parameters for the NBRCINAR(1) Process

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications.

Experimental investigation of Lee-Yang criticality using non-Hermitian quantum system

Abstract Lee-Yang theory clearly demonstrate where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee-Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter, we successfully obtained the partition function under different imaginary magnetic fields and observed the Lee-Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee-Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee-Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.

Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model

The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences { E n + } n = 0 ∞ \lbrace E_n^+\rbrace _{n=0}^{\infty } , { E n − } n = 0 ∞ \lbrace E_n^-\rbrace _{n=0}^{\infty } , and their large  n n asymptotic behavior with error term O ⁡ ( n − 1 / 2 ) \operatorname {O}(n^{-1/2}) is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).

A note on the 𝑉-invariant

Let R R be a finitely generated N \mathbb N -graded algebra domain over a Noetherian ring and let I I be a homogeneous ideal of R R . Given P ∈ A s s ( R / I ) P\in Ass(R/I) one defines the v v -invariant v P ( I ) v_P(I) of I I at P P as the least c ∈ N c\in \mathbb N such that P = I : f P=I:f for some f ∈ R c f\in R_c . A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that A s s ( R / I n ) Ass(R/I^n) is constant for large n n . So it makes sense to consider a prime ideal P ∈ A s s ( R / I n ) P\in Ass(R/I^n) for all the large n n and investigate how v P ( I n ) v_P(I^n) depends on n n . We prove that v P ( I n ) v_P(I^n) is eventually a linear function of n n . When R R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the v \text {v} -number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023].

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We study four-dimensional \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 2 superconformal circular, cyclic symmetric quiver theories which are planar equivalent to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 super Yang-Mills. We use localization to compute nonplanar corrections to the free energy and the circular half-BPS Wilson loop in these theories for an arbitrary number of nodes, and examine their behaviour in the limit of long quivers. Exploiting the relationship between the localization quiver matrix integrals and an integrable Bessel operator, we find a closed-form expression for the leading nonplanar correction to both observables in the limit when the number of nodes and ’t Hooft coupling become large. We demonstrate that it has different asymptotic behaviour depending on how the two parameters are compared, and interpret this behaviour in terms of properties of a lattice model defined on the quiver diagram.

Geometric and analytic results for Einstein solitons

AbstractWe compute a lower bound for the scalar curvature of a gradient Einstein soliton under a certain assumption on its potential function. We establish an asymptotic behavior of the potential function on a noncompact gradient shrinking Einstein soliton. As a result, we obtain the finiteness of its fundamental group and its weighted volume. We also prove some geometric and analytic results for constructing gradient Einstein solitons that are realized as warped metrics, and we give a few explicit examples.

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

Topological dyonic black holes of massive gravity with generalized quasitopological electromagnetism

In this paper we investigate new dyonic black holes of massive gravity sourced by generalized quasitopological electromagnetism in arbitrary dimensions. We begin by deriving the exact solution to the field equations defining these black holes and look at how graviton’s mass, dimensionality parameter, and quasitopological electromagnetic field affect the horizon structure of anti-de Sitter dyonic black holes. We also explore the asymptotic behaviour of the curvature invariants at both the origin and infinity to analyze the geometric structure of the resultant black holes. We also compute the conserved and thermodynamic quantities of these dyonic black holes with the help of established techniques and known formulas. After investigating the relevancy of first law, we look at how various parameters influence the local thermodynamic stability of resultant black hole solution. We also examine how thermal fluctuations affect the local stability of dyonic black holes in massive gravity. Finally, we study the shadow cast of the black hole.

Forest fire spreading: A nonlinear stochastic model continuous in space and time

AbstractForest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed‐point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

On stability of a reaction diffusion system described by difference equations

This work investigates the dynamics of discrete reaction-diffusion Gierer-Meinhardt system as mathematical models of biological pattern formation. We study the system's local asymptotic behaviour with and without the diffusion once developing the discrete integer variant of the well-known Gierer-Meinhardt model and proving that the model has a unique equilibrium. The requirements for the steady-state solution's local and global stability are found with the help of relevant approaches and the Lyapunov technique. Two large biological models and simulations are used throughout the work to validate the utility of the suggested technique.

Mathematical analysis of a modified Volterra-Leslie chemostat Model.

In this paper, we investigate the asymptotic behavior of a modified chemostat model. We first demonstrate the existence of equilibria. Then, we present a mathematical analysis for the model, the invariance, the positivity, the persistence of the solutions, and the asymptotic global stability of the interior equilibrium. Some numerical simulations are carried out to illustrate the main results.

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

Exact exponent for atypicality of random quantum states

We study the properties of the random quantum states induced from the uniformly random pure states on a bipartite quantum system by taking the partial trace over the larger subsystem. Most of the previous studies have adopted a viewpoint of ‘concentration of measure’ and have focused on the behavior of the states close to the average. In contrast, we investigate the large deviation regime, where the states may be far from the average. We prove the following results: first, the probability that the induced random state is within a given set decreases no slower or faster than exponential in the dimension of the subsystem traced out. Second, the exponent is equal to the quantum relative entropy of the maximally mixed state and the given set, multiplied by the dimension of the remaining subsystem. Third, the total probability of a given set strongly concentrates around the element closest to the maximally mixed state, a property that we call conditional concentration. Along the same line, we also investigate an asymptotic behavior of coherence of random pure states in a single system with large dimensions.

Global boundedness and asymptotic behavior of the chemotaxis system for alopecia areata with singular sensitivity

This paper is concerned with a three-component chemotaxis system for alopecia areata with singular sensitivity ut=Δu−χ1∇⋅uw∇w+w−μ1u2,x∈Ω,t>0,vt=Δv−χ2∇⋅vw∇w+w+ruv−μ2v2,x∈Ω,t>0,wt=Δw+u+v−w,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ωunder the homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂R2, where the parameters χi, μi(i=1,2) and r are positive. It is showed that if χ1,χ2<52, this system admits a globally bounded classical solution. Furthermore, under the particular conditions of μ1<μ2<3μ1 and r=μ2−μ1, the global bounded solution converges to the steady state (2μ1,2μ1,4μ1) as t→∞.

On Asymptotic Behavior of Mean Values of Some Functionals on Branching Random Walk

On Asymptotic Behavior of Mean Values of Some Functionals on Branching Random Walk

Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay

In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results.

An extreme worst-case risk measure by expectile

Abstract Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$ . With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.