In this paper, we study the symmetry on a class of multiparameter Markov Processes in Khoshnevisan (2002), and obtain some important and interesting results.

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition of multivariate polynomials, and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algorithm, the multiplicities of zeros could be directly read out, and the real solution isolation with multiplicity can also be easily produced. As a main theoretical result of this paper, we analyze the structure of dual space of the saturated ideal generated by a simple set as well as a regular set. Experiments with a preliminary implementation show the efficiency of our method.

It is comprehended that the systems without any limitation on their Zeno action are enthralled in a vast class of hybrid systems. This article is influenced by a new category of non-autonomous second order measure differential problems with state-dependent delay (SDD) and non-instantaneous impulse (NII). Some new sufficient postulates are created that guarantee solvability and approximate controllability. We employ the fixed point strategy and theory of Lebesgue–Stieltjes integral in the space of piecewise regulated functions. The measure of non-compactness is applied to establish the existence of a solution. Moreover, the measured differential equations generalize the ordinary impulsive differential equations. Thus, our findings are more prevalent than that encountered in the literature. At last, an example is comprised that exhibits the significance of the developed theory.

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well-posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system (E ,S( t)) associated with the problem in the space H 0 2 ( Ω ) × L 2 ( Ω ) and D ( A 3 4 ) × H 0 1 ( Ω ) , respectively. After that, the asymptotic smoothness of the dynamical system (E ,S( t)) and the further quasi-stability are demonstrated by the energy reconstruction method. Finally, different from [21] we show not only existence of the finite global attractor but also existence of the generalized exponential attractor.

Let $H_c$ be a $(2n)\times(2n)$ symmetric tridiagonal matrix with diagonal elements $c \in \mathbb{R}$ and off-diagonal elements one, and $S$ be a $(2n)\times(2n)$ diagonal matrix with the first $n$ diagonal elements being plus ones and the last $n$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil $\mathcal{A}_c=H_c-\lambda S$ as $2n$ approaches to infinity. It was conjectured by DL that for any $n \in \mathbb{N}$ the non-real eigenvalues $\lambda$ of $\mathcal{A}_c$ satisfy both $|\lambda + c|<2$ and $|\lambda - c|<2$. The conjecture has been verified numerically for a wide range of $n$ and $c$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non-real eigenvalues of $\mathcal{A}_c$. We provide a proof of the conjecture for $n \leq 3$, and also in the case where $|\lambda + c|=|\lambda - c|$. In addition, numerics indicate that some phenomena may occur for more general linear pencils.

In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure-valued solutions. We prove that the dissipative measure-valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well-prepared initial data when the Mach number tends to zero.

This article proposes an adaptive neural output feedback control scheme in combination with state and disturbance observers for uncertain fractional-order nonlinear systems containing unknown external disturbance, input saturation and immeasurable state. The radial basis function neural network (RBFNN) approximation is used to estimate unknown nonlinear function, and a state observer as well as a fractional-order disturbance observer is developed simultaneously by using the approximation output of the RBFNN to estimate immeasurable states and unknown compounded disturbances, respectively. Then, a fractional-order auxiliary system is constructed to compensate the effects caused by the saturated input. In addition, by introducing a dynamic surface control strategy, the tedious analytic computation of time derivatives of virtual control laws in the conventional backstepping method is avoided. The proposed method guarantees that the boundness of all signals in the closed loop system and the tracking errors converge to a small neighbourhood around the origin. Finally, two examples are provided to verify the effectiveness of the proposed control method.

In this paper, we study the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes/Allen-Cahn system, which describes the phase transitions in two-component patterns interacting with a compressible fluid. We establish the existence and space-time pointwise behaviors of global solutions to this non-conserved system. In order to control the source term consisting of the phase variable, we make use of the Green’s function and space-time weighted estimates to prove that the phase variable only contains the diffusion wave whose amplitude decays exponentially in time, so as to show that the density and momentum of the fluid obey the generalized Huygens’ principle.

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.

Abstract: In this paper, the nonlinear Schrödinger-type equation −(∇ + iA) ^2 u + u + λ[I_α*(K|u|^2)]Ku=af(|u|)u/|u| in R ^3 is considered in the presence of magnetic field, where A ∈ C ^1 (R ^3 ,R^ 3 ), α ∈ (0,3), I_α denotes the Riesz potential, K ∈ L^ p (R ^3 ,(0,∞)) for some p ∈ (6/(1 + α),∞], a ∈ L^ q (R 3 ,[0,∞)) \ {0} for some q ∈ (3/2,∞], and f ∈ C(R,[0,∞)) is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A, K, a, and f, variational methods are used to establish the existence of ground-state solutions of the above equation for sufficiently small values of the parameter λ.

In the present paper, we study a class of quasilinear Choquard equations involving N-Laplacian and the nonlinearity with the critical exponential growth. We discuss the existence of positive solutions of such equations.

By constructing various Pohozaev identities, we study the uniqueness of minimizers for L 2 -subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain x=0 as a singular point.

In this paper, we consider the Ricker model with delay and constant or periodic stocking. The impact of delay and stocking on stability is known to reflect opposing effect, which motivates investigating the interaction between these factors and their influence on overall stability. We found that the high stocking density tends to neutralize the delay effect. Conditions are established on the parameters in order to guarantee the global stability of the equilibrium solution in the case of constant stocking, as well as the global stability of the 2-periodic solution in the case of 2-periodic stocking. Our approach extensively relies on the utilization of the embedding technique. Whether constant stocking or periodic stocking, the mode has the potential to undergo a Neimark-Sacker bifurcation in both cases. However, the Neimark-Sacker bifurcation in the 2-periodic case results in the emergence of two invariant curves that collectively function as a single attractor. Finally, we pose open questions in the form of conjectures about global stability for certain choices of the parameters.

The generalized MHD-Boussinesq equations are studied in this paper. The well-posedness of the strong solutions for the generalized MHD-Boussinesq equations is proved. Asymptotic regularity for the generalized MHD-Boussinesq equations is proved in $H^{\frac{9}{2}}\times H^{\frac{9}{2}}\times H^{\frac{9}{2}}$. The higher regularity for the generalized MHD-Boussinesq equations is proved in $H^{a}\times H^{a}\times H^{a}$ for $a\geq\frac{9}{2}$.

This paper settles event-triggered synchronization and H∞ synchronization matters for two types of coupled delayed reaction-diffusion memristive neural networks (CDRDMNNs). First of all, several synchronization and H∞ synchronization conditions are acquired for CDRDMNNs with state coupling in virtue of exploiting Lyapunov stability theory in combination with proper controllers of the triggering event. Then, for CDRDMNNs with spatial diffusion coupling, event-triggered synchronization and H∞ synchronization are investigated as well. Finally, the correctness of the deduced synchronization and H∞synchronization results is verified by two given numerical examples.

The inappropriate use of insecticides may lead to disastrous pest outbreaks. Aiming at avoiding the outbreak of pests by optimizing the control strategies, we propose a novel mathematical model base on the idea of pulse in this study, namely a discrete-time model of pest-natural enemies, which considers the implementation of spraying insecticides within the time interval between two generations of pests. We first investigates the existence and stability of fixed points, and then using the central manifold theory, we proved that the system can exist period-doubling bifurcation. The main theoretical results are verified by numerical simulations. The experiments show that if the insecticidal time is within a certain range, the lower insecticidal rate stimulates the growth of pests and natural enemies, while the larger insecticidal rate can inhibit the growth of pests and natural enemies. In addition, the effects of pest growth rate, timing of pesticide spraying, and pesticide intensity on the system are comprehensively discussed. The main research provides good reference significance to choose an appropriate time to prevent the pest outbreaks.

The aim of this paper is to give a precise proof of the completeness of Lamb modes and associated modes. This proof is relatively simple and short but relies on two powerful mathematical theorems. The first one is a theorem on elliptic systems with a parameter due to Agranovich and Vishik. The second one is a theorem due to Locker which gives a criterion to show the completeness of the set of generalized eigenvectors of a Hilbert-Schmidt discrete operator.

We study the Helmholtz equation with periodic coefficients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.