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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]

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]

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]

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 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 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$$ {R}_0 $$. It is proved that the disease‐free equilibrium is asymptotically stable if R0<1$$ {R}_0<1 $$, and it is unstable if R0>1$$ {R}_0>1 $$. 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]

In this paper, we consider a predator–prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag‐Leffler function and the generalized singular Gronwall inequality, we get a priori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. The stability estimate is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study several problems of the fifth‐order Camassa‐Holm type (FOCHT) equation. The local well‐posedness and blow‐up scenario are established at first. Then we prove the global existence under some conditions and analyze the large‐time behavior of the support of momentum density. Finally, we discuss the persistence property in Sobolev space. [ABSTRACT FROM AUTHOR]

We consider the asymptotic nonlinear stability of viscous shock profiles for an initial‐boundary value problem of the scalar conservation laws with an artificial heat flux satisfying Cattaneo's law in the negative half line ℝ−=(−∞,0)$$ {\mathrm{\mathbb{R}}}_{-}&#x0003D;\left(-\infty, 0\right) $$ with Dirichlet boundary condition. When the nonlinear flux function is assumed to be strictly convex and the unique global entropy solution of the corresponding Riemann problem of the resulting scalar conservation laws consists of shock wave with negative speed, it is shown in this paper that the large time behavior of its global smooth solutions can be precisely described by the suitably shifted viscous shock profiles, where the time‐dependent shift function is uniquely determined by both the boundary value and the initial data. We also show that the shift function converge to a constant time asymptotically. Our analysis is based on weighted L2−$$ {L}&#x0005E;2- $$energy method. [ABSTRACT FROM AUTHOR]

In this paper, bipartite consensus of linear multi‐agent systems with structually balanced signed network is investigated. In particularly, control protocols for consensus are considered to work under an intermittent framework due to some unavoidable factors. Meanwhile, the event‐triggered mechanism is introduced in order to reduce the update frequency of control input. Based on distributed intermittent event‐triggered control, some sufficient conditions are proposed to ensure leaderless and leader‐following bipartite consensus. Moreover, minimum event‐triggered intervals are set to be positive in advance to exclude Zeno behavior. Finally, two numerical examples are provided to verify the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional reaction–diffusion equation using Jacobi spectral method. Reaction–diffusion equation is used as model for spatial effects in ecology. In this method, the reaction–diffusion equation is changed to the systems of equations. Convergence analysis for the proposed spectral method is established from theoretical as well as numerical point of view. Some numerical examples are solved using proposed numerical method showing the feasibility of the method. The efficiency of the spectral method is shown by listing CPU time of computation. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm for both u$$ u $$ and ρ$$ \rho $$. Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data. [ABSTRACT FROM AUTHOR]

In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conjecture posed in a recent paper, concerning the power‐series solution in a Lebesgue sense. By using the finiteness of the moment‐generating function of the equation's coefficient, mean‐square convergence is derived. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of a ground state solution for a logarithmic weighted biharmonic equation under Dirichlet boundary conditions in the unit ball B$$ B $$ of ℝ4$$ {\mathrm{\mathbb{R}}}&#x0005E;4 $$. The reaction term of the equation is assumed to have exponential growth, in view of Adams' type inequalities. It is proved that there is a ground state solution using min‐max techniques and the Nehari method. The associated energy functional loses compactness at a certain level. An appropriate asymptotic condition allows us to bypass the non‐compactness levels of the functional. [ABSTRACT FROM AUTHOR]

This paper considers the exponential stabilization of genetic regulatory networks (GRNs) with time‐varying delays by means of event‐triggered impulsive control (ETIC). Firstly, an ETIC strategy is designed, and impulse instants are determined by the state of the system. Then, some sufficient conditions for exponential stability of the GRNs with time‐varying delays under the proposed ETIC are established by Lyapunov functions together with Razumikhin technique. Also, it's shown that Zeno behavior can be eliminated in the designed strategy. Finally, two numerical examples are given to illustrate the effectiveness of the presented results. [ABSTRACT FROM AUTHOR]

This paper studies two nonlocal diffusion problems with a free boundary and a fixed boundary in heterogeneous environment. The main goal is to understand how the evolution of the two species is affected by the heterogeneous environment. We first prove the existence and uniqueness of a global solution for such systems. Then, for models with Lotka–Volterra type competition or predator–prey growth terms, we establish the spreading‐vanishing dichotomy. Sharp criteria of spreading and vanishing are also obtained. Furthermore, we show that accelerated spreading occurs if and only if the kernel function violates the threshold condition. [ABSTRACT FROM AUTHOR]

In this paper, a free boundary model of tumor growth with drug application including the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) is studied. Heterogeneity is included in the model by a variable −1≤y≤1$$ -1\le y\le 1 $$. It is assumed that converting from mutation y1$$ {y}_1 $$ to mutation y2$$ {y}_2 $$ happens with probability P(y1,y2)$$ P\left({y}_1,{y}_2\right) $$. A Caputo time fractional‐order hyperbolic equation describes the evolution of tumor cells depending on y$$ y $$. It also includes two Caputo time fractional‐order parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. Instead of integer‐order time derivatives, the fractional ones are considered. In this study, it is aimed to employ the least squares curve fitting method to fit the order of fractional derivatives, coefficient, and rates of the model. For this purpose, using the mathematical model, we have considered the boundary densities of the tumor cells of different types and near‐boundary concentrations of drug and nutrient as the functions of the unknown orders, coefficients and rates (unknown variables). Due to the complexity of the problem, we have obtained the functions numerically. For this, using a change of variable, we have changed the free boundary problem to a problem with fixed domain. Thus, Riemann–Liouville fractional‐order integrals are added to the problem. In the spatial domain, the problem is discretized using the collocation method. In the temporal domain, the fractional derivatives and integrals (with order α$$ \alpha $$) are approximated in mesh points (with step size t∗$$ {t}&#x0005E;{\ast } $$) applying a method with error O((t∗)2−α)$$ O\left({\left({t}&#x0005E;{\ast}\right)}&#x0005E;{2-\alpha}\right) $$. Then, the unknown variables are obtained by fitting the functions for unknown variables to the data. In order to obtain the variables, a quadratic objective function is considered. The discretized function is substituted into the objective function, then the objective function is minimized using the trust‐region reflective algorithm. Finally, some numerical examples are presented to verify the efficiency of the method. In the examples, we have added noises generated from Gaussian distributions to the data and the effects of noise on the fitted coefficients, and rates are illustrated using some figures and tables. The noisy data are also plotted to have a clear vision of the effects of noises on the data. It is shown that despite the noise, the prediction of the radius of the tumor is acceptable. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta &#x0002B;{\left(-\Delta \right)}&#x0005E;s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}&#x0005E;s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N&#x0002B;4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]