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The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

Some errors exist in the above paper. [ABSTRACT FROM AUTHOR]

In this work, a system of non‐linear difference equations is employed to model the opinion dynamics between a small group of agents (the target group) and a very persuasive agent (the indoctrinator). Two scenarios are investigated: the indoctrination of a homogeneous target group, in which each agent grants the same weight to his (or her) partner's opinion and the indoctrination of a heterogenous target group, in which each agent may grant a different weight to his or her partner's opinion. Simulations are performed to study the required times by the indoctrinator to convince a group. Initially, these groups are in a consensus about a doctrine different to that of the ideologist. The interactions between the agents are pairwise. [ABSTRACT FROM AUTHOR]

The main work of this paper is to investigate two kinds of generalized focal surfaces and two kinds of evolutes generated by spacelike curve γ lying in lightlike surfaces in Minkowski three‐space. Applying the method of unfolding theory in singularity theory to our study, it is shown that there exist the cuspidal edge and the swallowtail types of singularities in each of two classes of generalized focal surfaces under certain conditions; the only cusps will appear in each of evolutes. Two new geometric invariants are presented to classify the singularities of generalized focal surfaces and evolutes. Much more importantly, we reveal the correspondence among the geometric invariants, the types of singularities on generalized focal surfaces and evolutes, the singularities of two kinds of evolutes, and the contact of γ with the osculating spheres. Finally, several examples are presented to demonstrate the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

The equations studied in the above paper are dimensionally incorrect. [ABSTRACT FROM AUTHOR]

The present note presents some errors in the aforementioned paper published in Mathematical Methods in the Applied Sciences. Two errors are found in the definition of the non-dimensional parameters and correct results are presented for temperature profiles included in figure 10 of the previous paper. [ABSTRACT FROM AUTHOR]

There is no correct form of the term HT ht in the literature. In Physics it is not allowed to add quantities with different units and the term HT ht is wrong. [Extracted from the article]

No abstract is available for this article. [ABSTRACT FROM AUTHOR]

This paper focuses on the three‐dimensional (3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space Hk(ℝ3)(k≥3)$$ {H}^k\left({\mathrm{\mathbb{R}}}^3\right)\left(k\ge 3\right) $$ of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena. [ABSTRACT FROM AUTHOR]

This paper deals with compatibility of space‐time kernels with (either) full, spatially dynamical, or space‐time compact support. We deal with the dilemma of statistical accuracy versus computational scalability, which are in a notorious trade‐off. Apparently, models with full support ensure maximal information but are computationally expensive, while compactly supported models achieve computational scalability at the expense of loss of information. Hence, an inspection of whether these models might be compatible is necessary. The criterion we use for such an inspection is based on equivalence of Gaussian measures. We provide sufficient conditions for space‐time compatibility. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics. Some results of independent interest relate about the space‐time spectrum associated with the classes of kernels proposed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the jerk vector that is the rate of change of the acceleration vector over time. In three‐dimensional space, the decomposition of the jerk vector is a new concept in the field. This decomposition expresses the jerk vector as the sum of three unique components in specific directions: the tangential direction, the radial direction in the osculating plane, and the radial direction in the rectifying plane. The snap vector is the rate of change of the jerk vector over time. In this paper, the authors examine non‐relativistic particles moving along non‐lightlike Frenet curves at low speeds compared to the speed of light in Minkowski 3‐space. They resolve the jerk and snap vectors using Frenet–Serret frames. Additionally, the cases for motion along non‐lightlike Frenet planar curves in the Minkowski 3‐space are given as corollaries. To help understand these results, the paper provides some illustrative examples [ABSTRACT FROM AUTHOR]

In this paper, a Hopfield‐type neural network system with Caputo–Hadamard fractional derivative is discussed. The importance of the existence of the equilibrium point in the analysis of artificial neural networks is well known. Another important investigation is the stability properties. So, the stability of a neural network system is dealt with in the present paper. First, a theorem that asserts the existence and uniqueness of the equilibrium point of the system is proven. Later, the conditions that ensure the Mittag‐Leffler stability of the equilibrium point is obtained by using the Lyapunov's direct method. In addition, an example is given with numerical simulations to show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with transmission‐blocking drugs (TBDs). The paper's main aim is to demonstrate the impact the chosen model for demographic growth has on the disease's transmission and the effect of its treatment with TBDs. We calculate the model's control reproduction number and equilibria and perform a global stability analysis of the disease‐free equilibrium point. The mathematical analysis reveals that, depending on the model's demography, the model can exhibit forward, backward, and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short‐time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy. [ABSTRACT FROM AUTHOR]

This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references. [ABSTRACT FROM AUTHOR]

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function ψα$$ {\psi}_{\alpha } $$ and investigate their approximation properties. By choosing an appropriate function ψα$$ {\psi}_{\alpha } $$, the order of approximation of our operators to a function f$$ f $$ is at least as good as the classical Bernstein–Kantorovich operators on the interval [0,1]$$ \left[0,1\right] $$. We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided. [ABSTRACT FROM AUTHOR]

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study a higher order Kirchhoff problem with variable exponent of type M∫Ω|Dru|m(x)m(x)dxΔm(x)ru=f(x,u)inΩ,Dαu=0,on∂Ω,for eachα∈ℝNwith|α|≤r−1,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left|{\mathcal{D}}_ru\right|}^{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}^ru=f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}^{\alpha }u=0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}^N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}^N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1

In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using the Cauchy data u(x0,t),ux(x0,t)$$ u\left({x}_0,t\right),{u}_x\left({x}_0,t\right) $$ measured at one interior point x=x0∈(0,L)$$ x={x}_0\in \left(0,L\right) $$ or using an interior data u(x0,t)$$ u\left({x}_0,t\right) $$ and the assumption limx→∞u(x,t)=0$$ {\lim}_{x\to \infty }u\left(x,t\right)=0 $$. However, the flux ux(x0,t)$$ {u}_x\left({x}_0,t\right) $$ is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x=1$$ x=1 $$ and x=2$$ x=2 $$, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill‐posed and further constructs an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

Geometrically, a complex dynamical network (CDN) can be regarded as the interconnected system composed of the node subsystem (NS) and the link subsystem (LS) coupled with each other. Guided by this idea, in order to achieve the goal of each node following asymptotically its own reference target in a CDN, this paper investigates the model following adaptive control (MFAC) problem of NS via the dynamics of links, which implies that the LS plays the important dynamic auxiliary role in the MFAC realization of nodes. Meanwhile, we focus on the condition that the links state information is unavailable, due to sensor practical application and measurement cost constraints. To obviate this restriction, we construct the asymptotical state observer for the LS. Next, to achieve the control goal of this paper, an appropriate Lyapunov candidate function is selected, by which the MFAC scheme for NS is synthesized based on the state observer of LS. Finally, the simulation example is performed to demonstrate the theoretical results in this paper. [ABSTRACT FROM AUTHOR]

This paper discusses the issue of passivity analysis of the neutral‐type Cohen–Grossberg neural networks (NTCGNNs) with distributed, transmission, and neutral time‐varying delays. A new method based on the system solutions was proposed to analyze passivity. It is important to note that this paper focuses solely on introducing the proposed method and does not provide a comprehensive analysis of NTCGNNs. This method does not require for the construction of Lyapunov–Krasovskii functional (LKF) or model transformation, thus reducing computational effort and complexity. By using this new way and the inequality technique, a passivity criterion for NTCGNNs with multiple time‐varying delays is derived. The applicability of the derived criterion was demonstrated through a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we find a new regularity criterion of the 3D tropical climate model in terms of one component of the barotropic velocity. Our criteria of the velocity field improve the previous result by Berti et al. Moreover, we extend the regularity criteria of the Lebesgue spaces to that of larger Lorentz spaces. The key of this paper is by using interaction between partial and usual H1$$ {H}^1 $$ estimate to get upper bound of velocities. [ABSTRACT FROM AUTHOR]

This paper introduces and establishes new Hermite‐Hadamard type inequalities specifically tailored for m$$ m $$‐preinvex functions within the (p,q)$$ \left(p,q\right) $$‐calculus framework. These newly developed inequalities come with accompanying left‐right estimates, which enhance their practical utility. The primary objective of this research is to investigate the properties of (p,q)$$ \left(p,q\right) $$‐differentiable m$$ m $$‐preinvex functions and derive inequalities that extend and generalize existing results in the domain of integral inequalities. The techniques employed in this study hold broader implications, finding relevance in various fields where symmetry is paramount. The findings presented in this paper make a significant contribution to the field of analytic inequalities, offering valuable insights into the behavior and characteristics of m$$ m $$‐preinvex functions. Moreover, the established results demonstrate the wider applicability and generalization of analogous findings from prior literature. The techniques and inequalities introduced herein pave the way for further exploration and research in the realm of integral inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping ‖ut‖2qut$$ {\left\Vert {u}_t\right\Vert}_2^q{u}_t $$ and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical ( E(0)

This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has i(i=1,2)$$ i\left(i=1,2\right) $$ small limit cycle around the positive equilibrium. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐calculus in the quaternionic context, which is proposed via two kinds of (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators, with the theory of slice regular functions. Specifically, we shall work in suitable subclasses of slice regular functions in which the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators can be related with the slice derivative. The paper presents some results such as an integral formula and series expansion. [ABSTRACT FROM AUTHOR]

A recent paper (Zhang R, Yu X. Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence. Math Meth Appl Sci. 2022; 45: 113‐136) presented a four‐compartment lattice epidemic system with exposed class and standard incidence. The authors studied the existence and non‐existence of traveling wave solutions of this system. However, the limit behavior of R$$ R $$‐component of traveling wave solutions is still open. In this paper, we solve this open problem and establish the boundedness of traveling wave solutions by analysis technique. Meanwhile, we study the non‐existence of traveling wave solutions with non‐positive wave velocity. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of multiple high‐energy solutions for a p$$ p $$‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in ℝN$$ {\mathrm{\mathbb{R}}}^N $$. Considering that the "double" lack of compactness in the system is caused by the unboundedness of ℝN$$ {\mathrm{\mathbb{R}}}^N $$ and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to ℝN$$ {\mathrm{\mathbb{R}}}^N $$ of Struwe's classical global compactness result for double p$$ p $$‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work. [ABSTRACT FROM AUTHOR]

This paper discusses pseudospectra and stability radii for structured nonlinear matrix functions, such as Hermitian, skew‐Hermitian, H‐even, H‐odd, complex symmetric, and complex skew‐symmetric. To compute pseudospectra and stability radii, eigenvalue backward error is required. Hence, we initially present the structured eigenvalue backward error. Subsequently, we compute the structured pseudospectra using the obtained results for the eigenvalue backward error of a class of structured nonlinear matrix functions. Finally, we discuss the stability radii of the above‐structured problems arising in different applications. The paper also generalizes the results on the eigenvalue backward error of matrix polynomials in the literature for the above structures. [ABSTRACT FROM AUTHOR]

This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

This paper explores the finite‐time stability of reaction–diffusion genetic regulatory networks with time‐varying mixed delays. These delays encompass discrete delays and distributed delays. Unlike many existing studies, this paper assumes that both types of delays are bounded and continuous, without requiring differentiability. Additionally, the analysis combines the inequality approach and comparison method, while neither a Lyapunov–Krasovskii functional is constructed nor the finite‐time stability theorem is involved. The finite‐time stability of the addressed networks is then ensured by several suitable algebraic criteria. Some extended results are deduced for networks without considering reaction–diffusion and distributed delays effects as special cases. Finally, the validness of the derived results is proved by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation. [ABSTRACT FROM AUTHOR]

The stability analysis of dynamic continuous structural system (DCSS) has often been investigated by discretizing it into several low‐dimensional elements. The integrated results of all elements are employed to describe the whole dynamic behavior of DCSS. In this paper, DCSS is regarded as the complex dynamic network with the discretized elements as the dynamic nodes and the time‐varying stiffness as the dynamic link relations between them, by which the DCSS can be regarded to be the large‐scale system composed of the node subsystem (NS) and link subsystem (LS). Therefore, the dynamic model of DCSS is proposed as the combination of dynamic equations of NS and LS, in which their state variables are coupled mutually. By using the model, this paper investigates the stability of DCSS. The research results show that the state variables of NS and LS are uniformly ultimately bounded (UUB) associated with the synthesized coupling terms in LS. Finally, the simulation example is utilized to demonstrate the validity of method in this paper. [ABSTRACT FROM AUTHOR]

Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral drawings in hyperbolic space. In this paper, we consider a modified hyperbolic geometry by removing the condition that a geodesic is orthogonal to the unit circle in the Poincaré model. We show that spiral symmetry and the similarity property exist in this modified geometry so that the creation of uncommon hyperbolic spiral drawings is possible. To this end, we first establish the theoretical foundation for the proposed method by deriving a contraction mapping and a rotation for constructing modified hyperbolic spiral tilings (MHSTs) and introduce symmetry groups to analyze the structure of MHSTs. Then, to embed a pre‐designed wallpaper template into the tiles, we derive a one‐to‐one mapping between a tile of MHST and a rectangle. Finally, we specify some technical implementation details and give a gallery of the resulting MHST drawings. Using existing wallpaper templates, the proposed method is able to generate a great variety of exotic Escher‐like drawings. [ABSTRACT FROM AUTHOR]

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti‐) η$$ \eta $$‐Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti‐) η$$ \eta $$‐Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the scattering function and discrete spectrum of the impulsive Sturm‐Liouville type differential operator with a turning point and nonlinear eigenparameter‐dependent boundary condition. Using hyperbolic type representations of the fundamental solutions, the operator's discrete spectrum was constructed. We presented asymptotic equations for the fundamental solutions and the Jost function. Finally, we stated an example to demonstrate the paper's major points. [ABSTRACT FROM AUTHOR]

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space Lp(풞) for 2≤p<∞$$ 2\le p<\infty $$. First, we investigate the existence and uniqueness of Lp$$ {L}&#x0005E;p $$‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of Lp$$ {L}&#x0005E;p $$‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact delay range making input-delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete-time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well-known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems. [ABSTRACT FROM AUTHOR]

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier-Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and Córdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation. [ABSTRACT FROM AUTHOR]

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems. [ABSTRACT FROM AUTHOR]

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of octonion‐valued stochastic shunting inhibitory cellular neural networks with delays. First, we give an estimate of the distance between two different moments of finite‐dimensional distributions of a stochastic process. Then, based on this and by using fixed point theorems and inequality techniques, we establish the existence and global exponential stability of Besicovitch almost periodic (B$$ \mathcal{B} $$‐almost periodic for short) solutions in finite‐dimensional distributions for this kind of networks. Our results are new even if the networks we consider in the paper are real‐valued ones. At the same time, the method proposed in this paper can be applied to study the existence of Besicovitch almost periodic solutions in finite‐dimensional distributions for other types of stochastic neural networks. Finally, an example is given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we would like to consider the Cauchy problem for semilinear σ$$ \sigma $$‐evolution equations with time‐dependent damping for any σ≥1$$ \sigma \ge 1 $$. Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from σ=1$$ \sigma &#x0003D;1 $$ to σ>1$$ \sigma >1 $$ and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ$$ \sigma $$ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results. [ABSTRACT FROM AUTHOR]