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.

In this paper, we are concerned with an integral system $$ \left\{ \begin{aligned} &u(x)= W_{\beta,\gamma}(u^{p-1}v)(x), \ u>0 \ \text{in} \ R^{n},\\ &v(x)=I_{\alpha}(u^{p})(x), \ v>0 \ \text{in} \ R^{n}, \end{aligned} \right. $$ where $p>0,$ $0<\alpha, \beta\gamma1$. Base on the integrability of positive solutions, we obtain some Liouville theorems and the decay rates of positive solutions at infinity. In addition, we use the properties of the contraction map and the shrinking map to prove that $u$ is Lipschitz continuous. In particular, the Serrin type condition is established, which plays an important role to classify the positive solutions.

Motivated by an important geophysical application, we analyze the nonlinear dynamics of the number of earthquakes per unit time in a given Earth’s surface area. At first, we consider a dynamical model of earthquakes describing their rhythmic behaviour with time delays. This model comprises different earthquake scenarios divided into three types (A, B, and C) accordingly to various system dynamics. We show that the deterministic system contains stable equilibria and a limit cycle whose size drastically depends on the production rate $\alpha$ of earthquakes and their time-delay effect. As this takes place, the frequency of earthquakes possesses an oscillatory behaviour dependent on $\alpha$. To study the role of $\alpha$ in more detail, we have introduced a white Gaussian noise in the governing equation. First of all, we have shown that the dynamical system is stochastically excitable, i.e. it excites larger-amplitude noise-induced fluctuations in the frequency of earthquakes. In addition, these large-amplitude stochastic fluctuations can alternate with small-amplitude fluctuations over time. In other words, the frequency of earthquakes can change its amplitude in an irregular manner under the influence of white noise. Another important effect is how close the current value of $\alpha$ is to its bifurcation point. The closer this value is, the less noise generates large-amplitude fluctuations in the earthquake frequency.

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.

We provide a generalization of Beurling's theorem for the Clifford- Fourier transform and we give some of its applications. Indeed, analogues of Hardy, Cowling-Price and Gelfand-Shilov theorems are obtained in the Clifford analysis setting.

In this paper, we propose an algorithm for solving a two-point boundary-value problem for a linear differential equation with constant delay subject to a nonlinear boundary condition. We derive sufficient conditions for the convergence of the algorithm and for the existence of an isolated solution to the problem under study. A numerical example is provided.

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, i.e. attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux’s theorem when the derivative is continuous, states that a continuous real-valued function $f$ defined on $[-1,1]$ satisfying $f(-1)<0$ and $f(1)>0$ 0, has a zero, i.e. $f(x) = 0$ for at least one number $-1

In this paper, being investigated an initial - boundary value problem for a one - dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow - up of weak solutions in a finite time interval.

This article is devoted to the study of the existence of an exponential attractor for a family of problems, in which diffusion $d_{\lambda}$ blows up in localized regions inside the domain \begin{equation*} \begin{cases} u_t^\lambda-\mathrm{div}(d_\lambda(x)(|\nabla u^\lambda|^{p(x)-2}+\eta ) \nabla u^\lambda)+ |u^\lambda|^{p(x)-2}u^\lambda=B(u^\lambda), & \mbox{ in } \Omega \\ u^\lambda = 0, & \mbox{ on } \partial\Omega\\ u^\lambda(0)=u^\lambda_0 \in L^2(\Omega),& \end{cases} \end{equation*} and their limit problem via the $l$-trajectory method.

A method for practical realization of the inverse scattering transform method for the Korteweg-de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently by the authors. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.

Two dimensional (2D) images of light beams reflected off the objects in space impinge on the retinal photoreceptors of our two laterally separated eyes. Nevertheless, we experience our visual percept as a single 3D entity—our visual world that we tend to identify with physical world. However, experiments point to different geometries in these two worlds. Using the binocular system with the asymmetric eyes (AEs), this article studies the global geometric aspects of visual space in the Riemannian geometry framework. The constant-depth curves in the horizontal field of binocular fixations consist of families of arcs of ellipses or hyperbolas depending on the AE parameters and the eyes’ fixation point. For a single set of AE’s parameters, there is a unique symmetric fixation at the abathic distance such that the constant-depth conics are straight frontal lines. Critically, the distribution of the constant-depth lines is independent of such fixations. In these cases, a two-parameter family of the Riemann metrics is proposed based on the retinal topography and simulated constant-depth lines. The obtained geodesics for a subset of metric parameters include the incomplete geodesics that give finite distances to the horizon. The Gaussian curvature of the phenomenal horizontal field is analyzed for all the metric parameters. The sign of the curvature can be inferred from the global behavior of the constant-depth ellipses and hyperbolas only when for the metric parameters for which the constant-depth frontal lines at the abathic distance fixations are geodesics.

Nanocomposites are heterogeneous media with two or more microstructural levels. For instance, a nano-level characterized by isolated nano-inclusions and a micro-level represented by the clusters resulting from aggregation processes. Our goal is to present a procedure to study the influence of this aggregation process and interfacial thermal resistance on the effective thermal conductivity. The procedure is based on the Reiterated Homogenization Method and consists of two stages. First, an effective intermediate thermal property is obtained by taking into account only the influence of the individual nano-inclusions in the matrix. Second, the final effective thermal coefficient ($\hat{k}_{RH}$) is calculated considering the clusters immersed in the intermediate effective medium derived in the first step. The conductivity gain ($k_{gain}$) is defined as the quotient ($\hat{k}_{RH}/\hat{k}_{CH}$) where $\hat{k}_{CH}$ is the effective thermal coefficient computed considering only one micro-structural level. The scheme is exemplified for 2-D square arrays of circular cylinders. The analytical formulas of the effective coefficient used in the calculations generalize other well known formulas reported in the literature. Finally, the effect of thermal conductivity gain is illustrated as a function of the Biot number, the quotient of the thermal conductivity, the fibers volume fraction and an aggregation parameter. The present contribution could be useful for nano-reinforced fibers applications and nanofluids. Furthermore, the present formulas can be used to assess numerical computations. An appendix is included showing similarities and differences between the obtained analytical formulas of effective coefficients, for different truncation orders, and those derived from the trifasic model.

The subject of this research is the time-domain scattering problem for a three-dimensional layered elastic shell submerged in a two-layered fluid separated by an unbounded rough surface. The essence of the problem is to model the scattering interaction between the given elastic shell and some incident wave in a two-layered medium. Using the exact transparent boundary condition (TBC), we reformulate the unbounded scattering problem into an equivalent initial-boundary value problem. The well-posedness is proved for the problem by the Laplace transform and variational method in the s-domain. Moreover, we show that the reduced problem has a unique weak solution by the energy method. The priori estimates with explicit dependence on the time are derived for the acoustic pressure and the elastic displacement in the time domain.

We shall investigate the asymptotic behavior of solutions to the Cauchy problem for the one-dimensional bipolar quantum Euler-Poisson system with time-dependent damping effects $\frac{J_i}{(1+t)^{\lambda}}(i=1,2)$ for $-1<\lambda<1$. Applying the technical time-weighted energy method, we prove that the classical solutions to the Cauchy problem exist uniquely and globally, and time-algebraically converge to the nonlinear diffusion waves. This study generalizes the results in [Y.-P. Li, Nonlinear Anal., 74(2011), 1501-1512] which considered the bipolar quantum Euler-Poisson system with constant coefficient damping.

In this paper, a class of stochastic fractional differential equations (SFDEs) with impulses is considered. By virtue of the Monch’s fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen’s inequality, Holder inequality, Burkholder-Davis-Gundy inequality, Gronwall-Bellman inequality and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.

In this study, we give the structure of ruled surfaces parameterized on a product time scale. Mean and Gaussian curvatures of these surfaces are constructed on distinct time scales. Then, we obtain principal curvatures for ruled surfaces. Finally, we will clarify the issue with some examples.

We consider the least energy solutions of Hatree-Fock system with the coupling term Φu,v in the two equations, and the nonlinearity are general subcritical with a small perturbation. By Nehari’s manifold approach, the existence of non-trivial ground state solutions is obtained. The asymptotic behaviors with respect to parameters λ and β are also studied.

We are interested in problems of diffraction of an acoustic or electromagnetic wave by a perfectly conducting planar obstacle coated with thin multilayers of dielectric materials. The aim is to obtain boundary condition that replaces the effect of dielectric thin layers. This condition is constructed from an approximation of the Dirichlet-to-Neumann operator. In this paper, we analyze the construction and the approximation of this operator.