Second-order Evolution Research Articles

Multiple Topological Phases with Electronic Correlation in Intrinsic Ferromagnetic Semimetal VI3 Monolayer.

2D topological materials with magnetic ordering have become hot topics due to their nontrivial band topology and quantum states. In this work, the second-order topological states and evolution of linear band crossing are successfully predicted utilizing the effective k· p and tight binding models in the intrinsic ferromagnetic VI3 monolayer under various effective Hubble interaction Ueff. Upon inclusion of spin orbit coupling, a small bandgap (Eg-1) of 12.7 meV is opened with a Chern invariant C = -1 at Ueff = 0 eV. The Eg-1 undergoes a transition from the non-trivial state to trivial state at Ueff = 0.80 eV, accompanied by the appearance of Dirac cone. Remarkably, the increase of Ueff causes the band inversion and adjustment of crystal symmetry, resulting in two unreported coexistingtopological bandgaps (Eg-2 and Eg-3). Furthermore, a gapless node-loop appears at Ueff = 1.06 eV and disappears at Ueff = 1.09 eV around Γ point. Moreover, for the first time, the existence of second-order topological states with quantized corner fractional charges (e/3) is also observed in the VI3 monolayer at Ueff ≥0.96 eV. These results make the VI3 monolayer a compelling candidate for exploring topological devices.

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Second-order McKean–Vlasov stochastic evolution equation driven by Poisson jumps: Existence, uniqueness and averaging principle

In this paper, a class of second-order McKean–Vlasov stochastic evolution equations driven by Poisson jumps with non-Lipschitz conditions is considered. The existence and uniqueness of the mild solution are established by means of the Carathéodory approximation technique. Furthermore, an averaging principle is obtained between the solution of the second-order McKean–Vlasov stochastic evolution equation and that of the simplified equation in the mean-square sense.

Approximate controllability of non-autonomous evolution system with infinite delay

This article deals with the existence and approximate controllability results for a class of nonautonomous second-order evolution systems with infinite delay. To establish sufficient conditions for the proposed control problem the theory of evolution operator with Schauder’s fixed point theorem is used. Further, we extend the approximate controllability results to the integro-impulsive differential system. Finally, to emphasize our theoretical concepts, an example is provided.

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

Micro- and Macroevolution: A Continuum or Two Distinct Types of Change?

How microevolution and macroevolution are related is one of the major unanswered questions in evolutionary biology. The most prevalent view is that microevolution and macroevolution are part of a continuum of one type of change and that macroevolution is the cumulative result of microevolution. Mathematics, however, distinguishes two fundamentally different, singular types of change: change of a vector in its parameters versus its dimensions. This mathematical distinction may help to articulate the concept of evolution by distinction of two fundamentally different types of evolution: the change of the state vector of an organism in 1) its parameters (= ‘first-order evolution’) and 2) its dimensions (= ‘second-order evolution’). This distinction can be operationalized by identifying genes and regulatory elements in the nucleotide code of an organism as dimensions and the level of expression as parameters of its state vector. This operationalization allows us to substitute the phenotype-based analysis of evolution with a genotype-based analysis and draws attention to the molecular mechanisms that change the parameters or the dimensions of the state vector, respectively. We illustrate the distinction between first- and second-order evolution with a simulation of the adaptive dynamics of a population of digital amoebae. Our genotype-based systems approach reveals that micro- and macroevolution are largely similar to first- and second-order evolution respectively, and are not a continuum of change.

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

BV Solutions to Evolution Inclusion with a Time and Space Dependent Maximal Monotone Operator

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel.

Stock portfolio optimization based on factor analysis and second-order memetic differential evolution algorithm

Stock portfolio optimization based on factor analysis and second-order memetic differential evolution algorithm

Second-order neutral impulsive stochastic evolution equations with infinite delay: existence, uniqueness and averaging principle

In this paper, a class of second-order neutral impulsive stochastic evolution equations with infinite delay driven by fractional Brownian motion and Poisson jumps is considered. We utilise the Carathéodory approximation technique, the Burkhölder–Davies–Gundy inequality and the Gronwall inequality to present the existence and uniqueness of mild solutions for the addressed system under Carathéodory conditions. Furthermore, we derive an averaging principle for the proposed system which is approximated by the simplified equation in the sense of mean square. Finally, an application is used to illustrate the obtained theoretical results.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Numerical treatment for some abstract degenerate second-order evolutionary problem

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay

<p>In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.</p>

Approximate controllability for second-order stochastic neutral evolution equations with infinite delay

Approximate controllability for second-order stochastic neutral evolution equations with infinite delay

Stability for a Coupled System of Second-Order Evolution Equations with Indirect Memory-Damping

Stability for a Coupled System of Second-Order Evolution Equations with Indirect Memory-Damping

Three-Level Schemes with Double Change in the Time Step

Nonstationary problems are solved numerically by applying multilevel (with more than two levels) time approximations. They are easy to construct and relatively easy to study in the case of uniform grids. However, the numerical study of application-oriented problems often involves approximations with a variable time step. The construction of multilevel schemes on nonuniform grids is associated with maintaining the prescribed accuracy and ensuring the stability of the approximate solution. In this paper, three-level schemes for the approximate solution of the Cauchy problem for a second-order evolution equation are constructed in the special case of a doubled or halved step size. The focus is on the approximation features in the transition between different step sizes. The study is based on general results of the stability (well-posedness) theory of operator-difference schemes in a finite-dimensional Hilbert space. Estimates for stability with respect to initial data and the right-hand side are obtained in the case of a doubled or halved time step.

Approximate Controllability Outcomes of Impulsive Second-Order Stochastic Neutral Differential Evolution Systems

In this manuscript, we present a collection of suitable requirements for the approximate controllability outcomes of impulsive second-order stochastic neutral differential evolution systems. We use the ideas from the sine and cosine functions and the fixed point strategy to demonstrate the primary findings. The analysis then moves to second-order nonlocal stochastic neutral differential systems. Eventually, in order to make our topic more useful, we will provide an epistemological implementation.

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

A Lie group treatment on a generalized evolution Fisher type equation with variable coefficients

In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are studied by a non-traditional Lie symmetry method (LSM). Prolongation of the model is an essential part of the Lie symmetry method, which in the current work, we analyze by the heir-equations. Finally, different types of solutions depending on the variable coefficients, such as the exponential solutions, trigonometric solutions, Bessel solutions, and Mathieu solutions, are extracted.