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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]

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 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]

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]

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

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

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 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$$ \sigma =1 $$ 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]

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]

We consider general classes of Euler type linear and half‐linear difference equations, which are conditionally oscillatory. Applying the adapted Riccati technique, we improve known oscillation criteria for these equations. More precisely, our presented main criterion is the full oscillatory counterpart of a non‐oscillation criterion. Thus, in this paper, we enlarge the set of conditionally oscillatory Euler type difference equations. We highlight that our results are new even for linear equations with periodic coefficients. This fact is documented by simple examples of such equations at the end of this paper. [ABSTRACT FROM AUTHOR]

Each stage of the construction of the rail transit network has unique dynamic characteristics. It can provide recommendations for rail transit network planning and phased construction by evaluating the degree of evolution of the urban rail transit network and dividing the evolution stages precisely. Starting from the allometric growth relationship between transfer nodes and ordinary nodes in the urban rail transit network, this paper defines the evolution level of the rail transit network and the growth rate difference between transfer nodes and ordinary nodes and deduces the dynamic logistic equation of URT network evolution level using the mathematical derivative and logarithmic relationship. This paper reveals the dynamic law of the evolution process of the urban rail transit network at a theoretical level, analyzes the dynamic evolution process on this basis, determines the threshold for phasing the evolution stage, and divides it into four evolution stages. Finally, the results of the theoretical derivation are validated by the evolution data of the Beijing Rail Transit Network (1984–2020) over 45 years. The verification results are substantially congruent with the conclusions of the theoretical derivation, demonstrating the accuracy of the theoretical model and its practical usefulness in directing the building of the rail transit network. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to construct a common solution of the split null point problem associated with the maximal monotone operators and the fixed point problem associated with a finite family of k‐demicontractive operators in Hilbert spaces. We compute the optimal common solution via inertial parallel hybrid algorithm under a suitable set of control conditions. The viability of parallel implementation of the algorithm is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space 피14. We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in 피14. Moreover, the importance of this paper is the definition of these surfaces by using the octonions in 피14. Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space 피14. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain the results on the degree of convergence of an H$$ H $$‐periodic continuous function in generalized Hölder spaces using Hausdorff operator with monotonically non‐decreasing and monotonically non‐increasing rows of its hexagonal Fourier series. Some important corollaries are also deduced from our main theorems. Applications of main theorems are also obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, two mathematical methods are used for solving a complex multicriteria optimization problem as the considered convex differentiable vector optimization problem with vanishing constraints. First of them is the linearized approach in which, for the original vector optimization problem with vanishing constraints, its associated multiobjective programming problem is constructed at the given feasible solution. Since the aforesaid multiobjective programming problem constructed in the linearized method is linear, one of the existing methods for solving linear vector optimization problems is applied for solving it. Thus, the procedure for solving the considered differentiable vector optimization problems with vanishing constraints is presented in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Choquard equation with weight (−Δ)su=1|x|N−α∗h(x)|u|ph(x)|u|p−2uinℝN,$$ {\left(-\Delta \right)}&#x0005E;su&#x0003D;\left(\frac{1}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{N-\alpha }}\ast h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;p\right)h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where 02s,p>2,α>0$$ 02s,p>2,\alpha >0 $$ and h$$ h $$ is a positive weight function satisfying h(x)≥C|x|a$$ h(x)\ge C{\left&#x0007C;x\right&#x0007C;}&#x0005E;a $$ at infinity, for some a≥0$$ a\ge 0 $$. We establish, in this paper, a Liouville type theorem saying that if maxN−4s−2a,0<α

The primary aim of this paper is the derivation and a proof of a simple and tractable formula for the stray field energy in micromagnetic problems. The formula is based on an expansion in terms of a special family of recently discovered functions. It remains valid even if the magnetization is not of constant magnitude or if the sample is not geometrically bounded. The paper continues with a direct and important application which consists in a fast summation technique of the stray field energy. The convergence of this method is established, and its efficiency is proved by various numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

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

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)$$ x&#x0003D;{x}_0\in \left(0,L\right) $$ or using an interior data u(x0,t)$$ u\left({x}_0,t\right) $$ and the assumption limx→∞u(x,t)=0$$ {\lim}_{x\to \infty }u\left(x,t\right)&#x0003D;0 $$. 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$$ x&#x0003D;1 $$ and x=2$$ x&#x0003D;2 $$, 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]

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]

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 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]

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]

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]

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]

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,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left&#x0007C;{\mathcal{D}}_ru\right&#x0007C;}&#x0005E;{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}&#x0005E;ru&#x0003D;f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}&#x0005E;{\alpha }u&#x0003D;0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}&#x0005E;N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$ where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1

This paper discusses the existence of multiple high‐energy solutions for a p$$ p $$‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$. Considering that the “double” lack of compactness in the system is caused by the unboundedness of ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$ and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$ 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]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [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]

This document is an editorial summarizing the Applied Geometric Algebras in Computer Science and Engineering (AGACSE) conference held in Brno, Czech Republic in September 2021. The conference aimed to promote the use of geometric algebra in fields such as image processing, robotics, and quantum computing. The conference proceedings were published in the journal Mathematical Methods in the Applied Sciences. The editorial provides a list of accepted papers, covering topics such as applied geometry, technological applications, algebra, and quantum phenomena. One specific paper explores the use of geometric algebra in teaching rotations through neural networks. The document is a compilation of research papers showcasing the applications of geometric algebra in various fields, including robotics, control systems, image processing, cryptography, and physics. Each paper presents a specific problem or application and proposes a unique approach or solution using geometric algebra. The authors compare their methods with existing techniques and provide mathematical analysis to support their claims. Overall, the papers demonstrate the versatility and effectiveness of geometric algebra in different domains. [Extracted from the article]

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the p$$ p $$ order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

This paper provides a new robust control approach to uncertain mechanical manipulators in the computed torque framework. With respect to the fact that model uncertainties occurring from the framework could make the overall system unstable although the resulting nominal plant is stabilized, this paper develops a readily applicable method for estimating such uncertain elements. More precisely, we propose a time‐delay compensation method, by which prior values relevant to the model uncertainties in a sufficiently small time instant are used for such an estimation. The theoretical effectiveness of the proposed method is validated by establishing the two different arguments on operator‐theoretic exponential stability and Lypaunov‐based disturbance input‐to‐state stability. Finally, some simulation and experiment results are provided to demonstrate the overall arguments developed in this paper for both the theoretical and practical aspects. [ABSTRACT FROM AUTHOR]

In the paper mentioned in the title, we studied the semi‐Markovian random walk processes with jumps and delaying screen in zero. More precisely, we obtained a mathematical modeling of the semi‐Markov random walk processes with a delaying screen in zero, given in general form by means of fractional differential equation. In this note, we provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with the multiplicity results to a class of p$$ p $$‐Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by the abstract linking lemma due to Brézis and Nirenberg. We obtain the twofold results in subcritical and critical cases, which is a meaningful addition and completeness to the known results about Kirchhoff equation. At the same time, this paper also gives a method to deal with p$$ p $$‐Laplacian, Kirchhoff equation, and some Kirchhoff type equation in a unified variational framework. [ABSTRACT FROM AUTHOR]

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: i∂tu−(−Δ)su+|x|−b|u|p−1u=0,(t,x)∈ℝ×ℝN,$$ i{\partial}_tu-{\left(-\Delta \right)}&#x0005E;su&#x0002B;{\left&#x0007C;x\right&#x0007C;}&#x0005E;{-b}{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-1}u&#x0003D;0,\kern0.30em \left(t,x\right)\in \mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}&#x0005E;N, $$where N≥2$$ N\ge 2 $$, 1/2

We consider the incremental subgradient algorithm employing dynamic step sizes for minimizing the sum of a large number of component convex functions. The dynamic step size rule was firstly introduced by Goffin and Kiwiel [Math. Program., 1999, 85(1): 207‐211] for the subgradient algorithm, soon later, for the incremental subgradient algorithm by Nedic and Bertsekas in [SIAM J. Optim., 2001, 12(1): 109‐138]. It was observed experimentally that the incremental approach has been very successful in solving large separable optimizations and that the dynamic step sizes generally have better computational performance than others in the literature. In the present paper, we propose two modified dynamic step size rules for the incremental subgradient algorithm and analyse the convergence and complexity properties of them. At last, the assignment problem is considered and the incremental subgradient algorithms employing different kinds of dynamic step sizes are applied to solve the problem. The computational experiments show that the two modified ones converges dramatically faster and more stable than the corresponding one in [SIAM J. Optim., 2001, 12(1): 109‐138]. Particularly, for solving large separable convex optimizations, we strongly recommend the second one (see Algorithm 3.3 in the paper) since it has interesting computational performance and is the simplest one. [ABSTRACT FROM AUTHOR]

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch'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 inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–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. [ABSTRACT FROM AUTHOR]

This paper numerically investigates the effect of pulsating flow on the settling dynamics of rigid circular particles. This is an interdisciplinary subject and spans several areas ranging from mathematical and numerical modeling to fluid mechanics. For this purpose, pulsatile flow characteristics are embedded in the combination of the direct‐forcing immersed boundary method and the split‐forcing lattice Boltzmann method. Inter‐collision forces between the solid boundaries (particles and boundaries) and the added mass force due to acceleration are considered. Adequate verification tests are done to ensure the credibility of the findings. The critical parameters of pulsating flow, such as amplitude and frequency of pulsation, are investigated in detail. The paper especially puts emphasis on the interaction between particles and studies the well‐known drafting, kissing, and tumbling (DKT) phenomena. Two different scenarios are taken into account and also compared with the stationary flow. The first case is when the pulsating flow is in the direction of gravity (co‐flow), while in the latter, there is an opposing flow (counterflow). The sedimentation manners of 12 particles in a vertical channel are also presented. The findings shed light on the importance of pulsating flow and the extension of the proposed computational method for such problems. It is also revealed that pulsation and its variables can alter DKT by either postponing or speeding up the process. Also, in some cases, the cycle of DKT can be maintained incompletely, and particles would just stick together. The results can be useful for various engineering problems like filtration and particle sorting. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞

In this paper, we study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems. [ABSTRACT FROM AUTHOR]

This paper proposed a new class of stochastic cellular neural networks (SCNNs) with distributed time delays. It is driven by time‐varying Brownian motion. By constructing the Lyapunov functions with a more concise form and using the time‐varying Itô formula, combining some inequality techniques, this paper proved the almost surely exponential stability and the p$$ p $$‐th moment exponential stability of the solution of the SCNNs. The numerical example verifies the theoretical results of this paper. [ABSTRACT FROM AUTHOR]

The paper focuses on applying the octonions to explore the influence of the external torque on the angular momentum of fluid elements, revealing the interconnection of the external torque and the vortices of vortex streets. J. C. Maxwell was the first to introduce the quaternions to study the physical properties of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to investigate the electromagnetic theory, gravitational theory, quantum mechanics, special relativity, general relativity, curved spaces, and so forth. The paper adopts the octonions to describe the electromagnetic and gravitational theories, including the octonionic field potential, field strength, linear momentum, angular momentum, torque, and force. In case the octonion force is equal to zero, it is able to deduce eight independent equations, including the fluid continuity equation, current continuity equation, and force equilibrium equation. Especially, one of the eight independent equations will uncover the interrelation of the external torque and angular momentums of fluid elements. One of its inferences is that the direction, magnitude, and frequency of the external torque must impact the direction and curl of the angular momentum of fluid elements, altering the frequencies of Karman vortex streets within the fluids. It means that the external torque is interrelated with the velocity circulation, by means of the liquid viscosity. The external torque is able to exert an influence on the direction of downwash flows, improving the lift and drag characteristics generated by the fluids. [ABSTRACT FROM AUTHOR]

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation y′′+λ/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm. [ABSTRACT FROM AUTHOR]

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well. [ABSTRACT FROM AUTHOR]

Acupuncture is an important part of traditional Chinese medicine (TCM). Although the efficacy of acupuncture has been widely accepted, the scientific basis behind it is still lack of exploration. Some experimental studies have shown that acupuncture can lead to changes in cell membrane potential, thus affecting the transmission of nerve signals to a certain extent. The change of cell membrane potential mainly depends on a large number of voltage dependent ion channels on the cell membrane. Relevant studies have proved that such voltage dependent ion channels have certain mechanical sensitivity, but the principle is not clear. Based on the theoretical model of cell action potential excitability proposed by Hodgkin–Huxley, taking rat skeletal muscle cell fibers as an example, this paper makes a rough stability analysis of the nonlinear system, explores the conditions of repeated discharge of cell membrane, and simulates the law of cell membrane potential frequency variation under different stimulating currents by numerical calculation. Based on the results of numerical simulation, we propose a hypothesis: Mechanical stimulation will produce a certain amount of stimulating current on the cell membrane, making the cell membrane potential in the state of repeated discharge, so as to block or interfere with the transmission of nerve signals and achieve the analgesic effect and so forth. [ABSTRACT FROM AUTHOR]