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The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with traveling waves in a class of reaction–diffusion equations with nonlocal delays. By introducing new variables, we transform equations with nonlocal delays to non‐delayed system; the existence of traveling waves is obtained by means of Routh–Hurwitz criterion, and by considering the regularity of the Cauchy problem and using spectral theory, we investigate the global stability of traveling waves and then the uniqueness of wave speeds with the help of upper and lower solution method. Finally, our results are applied to two nonlocal delayed reaction–diffusion equations. [ABSTRACT FROM AUTHOR]

This paper presents the computational analysis of fractional differential equations of variable‐order delay systems. To the proposed problem, the existence of solutions is derived using Arzela‐Ascoli theorem, and the Banach fixed point theorem is used for uniqueness results. To investigate and address the computational solutions, Adams‐Bashforth‐Moulton technique is established. To demonstrate the method's efficiency, computational simulations of chaotic behaviors in several one‐dimensional delayed systems with distinct variable orders are employed. The numerical solution of the proposed problem gives high precision approximations. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear model for analyzing the behavior of functionally graded material Euler‐Bernoulli thin‐walled beams with varying cross‐sections, while considering periodic perturbations in the rotating speed. The study focuses on an analytical exploration of the natural frequency and amplitude frequency response associated with the nonlinear parametric vibrations within this system. Initially, the governing differential equations for the transverse vibrations of the beams are derived through Hamiltonian principle. Subsequently, the perturbation method is employed to solve the established equations, and Galerkin method is adopted to determine the mode shapes and frequencies. Furthermore, this research extensively examines the influence of significant parameters on the vibration characteristics of the system. [ABSTRACT FROM AUTHOR]

In this paper, we study stability of hybrid stochastic differential equations (SDEs) with multiple delays by intermittent control based on time‐varying delay observations. By constructing an auxiliary non‐delayed hybrid stochastic differential equation and using M$$ M $$‐matrix theory, Lyapunov method, and the comparison principle, sufficient criterions to guarantee the q$$ q $$th moment exponential stability and almost sure exponential stability of hybrid SDEs are derived. Finally, a simulation example is shown to illustrate the validity of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a second‐order accurate scheme is considered for the temporal discretization of the abstract Volterra integrodifferential equation with a weakly singular kernel. The time discrete scheme is constructed by the Crank–Nicolson method for approximating the time derivative and product integration (PI) rule for approximating the integral term. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at t=0$$ t=0 $$. Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order k2$$ {k}^2 $$, where k$$ k $$ is the parameter for the time grids. Numerical experiments validate the theoretical estimate. [ABSTRACT FROM AUTHOR]

Precision matrix (inverse covariance matrix) estimation is a rising challenge in contemporary applications while dealing with high‐dimensional data. This paper focuses on large‐scale precision matrix of the random vector that only has lower polynomial moments. We mainly investigate upper bounds of the proposed estimator under the spectral norm in terms of the probability and mean estimation respectively. It is shown that the data‐driven estimator is fully adaptive and achieves the same optimal convergence order as under Gaussian assumption on the data. Simulation studies further support our theoretical claims. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the invariance of the microsupport under the influence of the evolution of the solution in L2(ℝn)$$ {L}^2\left({\mathrm{\mathbb{R}}}^n\right) $$ of the following Cauchy problem: 1(hDt+P)u=0,$$ \left(h{D}_t+P\right)u=0, $$ 2u(0,x)=u0(x),$$ u\left(0,x\right)={u}_0(x), $$where h$$ h $$ is a semiclassical parameter, Dt=1i∂∂t$$ {D}_t=\frac{1}{i}\frac{\partial }{\partial_t} $$ and P=Opth(p)$$ P=O{p}_t^h(p) $$ (t∈[0,1]$$ t\in \left[0,1\right] $$) is a pseudo‐differential operator of symbol p(x,ξ,h)=p0(x,ξ)+ε(h)r(x,ξ,h)∈S2nhol(1),$$ p\left(x,\xi, h\right)={p}_0\left(x,\xi \right)+\varepsilon (h)r\left(x,\xi, h\right)\in {S}_{2n}^{hol}(1), $$where ε(h)→0$$ \varepsilon (h)\to 0 $$ as h→0$$ h\to 0 $$ and p0(x,ξ)$$ {p}_0\left(x,\xi \right) $$ is real valued, that is, p0(x,ξ)∈ℝfor all(x,ξ)∈ℝ2n.$$ {p}_0\left(x,\xi \right)\in \mathrm{\mathbb{R}}\kern0.30em \mathrm{for}\ \mathrm{all}\kern0.1em \left(x,\xi \right)\in {\mathrm{\mathbb{R}}}^{2n}. $$ [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniform exponential stability of the system dx(t)dt=Ax(t)−ρBx(t),(ρ>0),$$ \frac{dx(t)}{dt}= Ax(t)-\rho Bx(t),\kern3.0235pt \left(\rho >0\right), $$ where the unbounded operator A$$ A $$ is the infinitesimal generator of a linear C0−$$ {C}_0- $$semigroup of contractions S(t)$$ S(t) $$ in a Hilbert space X$$ X $$ and B$$ B $$ is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior of nonclassical diffusion equations with time‐dependent perturbed parameter and lacking instantaneous damping is considered on whole space ℝn$$ {\mathrm{\mathbb{R}}}^n $$. The existence and regularity of time‐dependent global attractors are proved by using a new analytical technique. It is remarkable that there is not any restriction on the upper growth of the nonlinearity f$$ f $$. The result obtained essentially extends and improves earlier ones. [ABSTRACT FROM AUTHOR]

Based on the relevant theories of M‐matrix and the Lyapunov stability technique, this paper investigates the multistability of equilibrium points and periodic solutions for Clifford‐valued memristive Cohen–Grossberg neural networks. With the help of Cauchy convergence criterion, the exponential stability inequality is derived. The system has ∏A(KA+1)n$$ {\left[\prod \limits_A\left({K}_A+1\right)\right]}^n $$ locally exponentially stable equilibrium points and periodic solutions, which greatly increases the number of solutions compared with the existing Cohen–Grossberg neural networks, where the KA$$ {K}_A $$ is a Clifford‐valued and there is no limitation of linearity and monotonicity for activation functions. Furthermore, the attraction basins of stable periodic solutions are obtained, and it is proved that the basins can be enlarged. It is worth mentioning that the results can also be used to discuss the multistability of equilibrium points, periodic solutions, and almost periodic solutions for real‐valued, complex‐valued, and quaternion‐valued memristive Cohen–Grossberg neural networks. Finally, two numerical examples with simulations are taken to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with the design problem of networked cascade control systems (NCCSs) with two different event‐triggered mechanisms (ETMs) for the feedback loops. A combination of stochastic denial‐of‐service (DoS) attack and deception attack has been considered for the communication network. To comply with the practical point of view, the primary and the secondary systems have been considered with distinct disturbances. First, the closed‐loop system with state feedback controllers is provided in a unified stochastic delayed system. Then, the Lyapunov–Krasovskii stability theory and stochastic analysis techniques are employed to derive and formulate H∞$$ {\mathcal{H}}_{\infty } $$ control design conditions in terms of linear matrix inequalities (LMIs). Finally, a superheater steam temperature control is utilized to assess the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of strong solutions to the two‐dimensional Boussinesq‐MHD equations without the temperature diffusion for general initial data in Hm$$ {H}^m $$ with m>2$$ m>2 $$. As an application, we shall prove the propagation of the Hm+1$$ {H}^{m+1} $$ regularity of the interface between fluids with different temperatures. [ABSTRACT FROM AUTHOR]

The Levin‐Nohel integro‐differential equation is one of the famous equations in various fields of science and engineering. It was first studied by Volterra in connection with biological applications. This paper deals with the nonlinear neutral Levin‐Nohel integro‐differential equation with Caputo fractional derivative and variable delays. The Ulam‐Hyers‐Rassias stability, Ulam‐Hyers stability, semi‐Ulam‐Hyers‐Rassias stability are studied. The existence and uniqueness of solutions are established by using Krasnoselskii's fixed point theorem and contraction mapping principle. An illustrative example is also given and that is studied by the help of the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem for the time‐fractional two‐dimensional Cahn–Hilliard equation. The ill‐posedness and a conditional stability of the inverse problem are proved. We present a new iterative variational regularization method to solve it. The convergence rates of the regularized solutions under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we systematically present some main results of the fractional calculus, almost periodic functions, fuzzy functions, fuzzy fractional calculus, and almost periodic generalized fuzzy dynamic equations on timescales. Moreover, the potential future research of almost periodic fractional fuzzy dynamic equations on timescales is discussed. The results presented in this survey can be applied to study the qualitative theory of almost periodic fractional fuzzy dynamic equations and fuzzy fractional calculus on timescales. [ABSTRACT FROM AUTHOR]

The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam–Hyers and Ulam–Hyers–JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag‐Leffler–Ulam–Hyers and Mittag‐Leffler–Ulam–Hyers–JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field‐controlled DC servo motor with position control. [ABSTRACT FROM AUTHOR]

This paper considers the Galerkin spectral method for solving linear second‐kind Volterra integral equations with weakly singular kernels on large intervals. By using some variable substitutions, we transform the mentioned equation into an equivalent semi‐infinite integral equation with nonsingular kernel, so that the inner products from the Galerkin procedure could be evaluated by means of Gaussian quadrature based on scaled Laguerre polynomials. Furthermore, the error analysis is based on the Gamma function and provided in the weighted L2$$ {L}^2 $$‐norm, which shows the spectral rate of convergence is attained. Moreover, several numerical experiments are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness. [ABSTRACT FROM AUTHOR]

This paper is devoted to study six novel inequalities associated with the Riesz‐type fractional integral operator for synchronous function and bounded function. [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetry on a class of multiparameter Markov processes and obtain some interesting results. It is very important for us to research and apply some multiparameter stochastic processes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional Schrödinger equation ε2s(−Δ)su+V(x)u=P(x)f(u)+Q(x)|u|2s∗−2uinℝN,$$ {\varepsilon}^{2s}{\left(-\Delta \right)}^su+V(x)u=P(x)f(u)+Q(x){\left|u\right|}^{2_s^{\ast }-2}u\kern0.30em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^N, $$where ε>0$$ \varepsilon >0 $$ is a parameter, s∈(0,1),2s∗=2NN−2s,N>2s,(−Δ)s$$ s\in \left(0,1\right),{2}_s^{\ast }=\frac{2N}{N-2s},N>2s,{\left(-\Delta \right)}^s $$ is the fractional Laplacian and f$$ f $$ is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration‐compactness principle, we prove the existence of a positive solution for the above equation. [ABSTRACT FROM AUTHOR]

This study uses a delayed matrix exponential to illustrate the solution of both linear and nonlinear delay problems using permutable matrices. We outline several prerequisites that are necessary to ensure the exponential stability of the trivial solution of delay systems. The conditions that are necessary and sufficient for the linear system to possess relative controllability are established by introducing a delayed Grammian matrix‐like criterion. A sufficient condition is obtained by Krasnoselskii's fixed point theorem for the relative controllability of nonlinear systems. Numerical examples provided at the end of the paper demonstrate the validity of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive L∞0,T;H01(Ω)$$ {L}^{\infty}\left(0,T;{H}_0^1\left(\Omega \right)\right) $$ bound as well as L2(0,T;H2(Ω))$$ {L}^2\left(0,T;{H}^2\left(\Omega \right)\right) $$ bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of O(h+k2−α)$$ O\left(h+{k}^{2-\alpha}\right) $$, where α(0<α<1)$$ \alpha \kern0.1em \left(0<\alpha <1\right) $$ is the fractional derivative exponent and h$$ h $$ and k$$ k $$ are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 σ$$ {}_{\sigma } $$ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims. [ABSTRACT FROM AUTHOR]

In this paper, we consider the strong solutions under three different types of boundary problems to a compressible nematic liquid crystals system in three dimensions. We establish the blow‐up criterion in terms of the density and the gradient of direction field, provided the shear viscosity and bulk viscosity coefficients of the fluid satisfy an additional relation (i.e., μ>λ$$ \mu >\lambda $$). In particular, the initial vacuum is allowed. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve an inverse source problem of time fractional diffusion‐wave equation in a cylinder. The ill‐posedness, existence, and uniqueness of the inverse source problem are proved. We propose an iterative generalized quasi‐boundary value regularization method to deal with the inverse source problem. The convergence rates of the regularized solution are obtained under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule. Numerical examples are presented to show the validity and stability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a coupled disease‐economic model to study the impact of poverty on cholera outbreaks. We consider the stock of capital per effective units of labor, reflecting the effect of the stock of capital on cholera outbreak and transmission. The system has positive bounded solutions. We show that the disease‐free equilibrium is globally asymptotically stable whenever R01≤1$$ {\mathcal{R}}_{01}\le 1 $$, while when R01>1$$ {\mathcal{R}}_{01}>1 $$, the disease‐free equilibrium is unstable. The sensitivity analysis of the model has been performed in order to determine the impact of related parameters on outbreak severity. We further study various cases for the investment rate upon critical value of the investment in order to describe the effect of poverty on the infected individuals over time. The results we obtained show that increasing of the stock of capital per effective units of labor and the implementation of strategies combining awareness programs and therapeutic treatments will reduce the proportion of the infected population. To verify our analytical results, the numerical simulations on the model are performed. [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions on the incompressible viscous resistive MHD equations and the viscous non‐resistive MHD equations to ensure the energy equality. Part of our results also holds for the ideal MHD equations. [ABSTRACT FROM AUTHOR]

In this paper, we study nontrivial periodic solutions for seasonal epidemiological models with information‐dependent vaccination. Using Leray–Schauder degree theory, we establish new results concerning the existence of at least one nontrivial periodic solution for the SIS$$ SIS $$ and SIR$$ SIR $$ models with seasonality, state‐dependent vaccination coverage and delayed information. Some numerical simulations are presented to illustrate the effect of such vaccination coverage. [ABSTRACT FROM AUTHOR]

In this paper, we analyze rigorously the asymptotic behaviors for perturbation of eigenvalues and eigenfunctions associated to the Stokes eigenvalue problem in the presence of a rapidly oscillating boundary. The aim is to construct asymptotic approximations, as δ⟶0$$ \delta \longrightarrow 0 $$, of the eigenvalues and corresponding eigenfunctions for the case where the eigenvalue of the reference problem is simple or multiple. Taking advantage of small oscillations, we use the method of matching of asymptotic expansions to construct the associated leading terms. We believe that our results are ambitious tools for determining shape and/or size of the small perturbed part of the domain by taking eigenvalue measurements. [ABSTRACT FROM AUTHOR]

Variance swaps is a kind of financial instrument that plays an important role in volatility risk management. In this paper, we study the pricing problem of log‐return variance swaps under the double mean reversion DMR (Heston‐CIR) model. Compared with Kim's work, we introduce the square‐root process into the diffusion term of the long‐term mean and present a stochastic approach that greatly simplify the solution of the problem without solving PDEs. An analytical solution and approximate solution are obtained. Some numerical examples show that the exact solution and MC simulation fit well. It is worth mentioning that the difference between the approximate solution and the exact solution is small when the parameters are selected appropriately. By the mean time, the parameter of the long‐term mean has an important impact on the solution, which implies that the introduction of a multi‐factor model is necessary. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence, uniqueness, and the Mittag–Leffler–Ulam stability of solutions for fractional Duffing equations involving three fractional derivatives. Uniqueness result for solution of the underlying Duffing problem is given with the help of Banach's fixed point theorem, where the existence result is computed using Leray–Schauder's alternative. Also, the Mittag–Leffler–Ulam stability results are computed by employing generalized singular Gronwall's inequality. An illustrative example is also given. [ABSTRACT FROM AUTHOR]

The issue of event‐triggered H∞$$ {H}_{\infty } $$ performance state estimation for neural networks with time‐varying delays is addressed in this paper. An innovative event‐triggered approach is presented, designed to strike a harmonious equilibrium between the state estimator's performance and the communication bandwidth of the network. The proposed approach captures the relationship between the time‐varying delay and system states by employing a delay derivative‐dependent integral inequality with matrices that account for delay derivatives. This novel formulation ensures the asymptotic stability of the estimation error system, thereby fulfilling the H∞$$ {H}_{\infty } $$ performance requirement. The desired event‐triggered estimators are then obtained through the use of linear matrix inequalities. The effectiveness of this approach is demonstrated through a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we devoted to investigate a hyperelliptic estimate of the ultimate bound and positively invariant set for the new system via the Lyapunov function method. Furthermore, we obtained the global exponential attractive set Cλ,k$$ {C}_{\lambda, k} $$, which is an exponential decay rate. The rate of the trajectories of the system going from the exterior to the interior of the set Cλ,k$$ {C}_{\lambda, k} $$. Numerical simulations are presented to show the effectiveness and the feasibility. The theoretical results obtained will find wide applications in cascade control and chaos synchronization. [ABSTRACT FROM AUTHOR]

In this paper, we take a fresh look at the determination of the transition matrix for a homogeneous, biisotropic particle employing the Null‐field approach. The condition for passivity of the material is discussed. In particular, we focus on some previously unproved properties of the Beltrami spherical vector waves, such as completeness, orthogonality, and linear independence, which are all instrumental in the analysis of the scattering problem with the Null‐field approach. Some numerical examples illustrate the approach. [ABSTRACT FROM AUTHOR]

Understanding and predicting novel diseases have become very important owing to the huge global health burden. The organization and study of mathematical models are critical in predicting disease behavior of the disease. In this paper, a new stochastic Susceptible‐Infected‐Recovered‐Death (SIRD) model for spreading epidemic disease is investigated. First, the deterministic SIRD model is considered, and then, by allowing randomness in the recovery and death rates that are not deterministic, the system of nonlinear stochastic differential equations is derived. For the suggested model, the existence and uniqueness of a positive global solution are demonstrated. The parameter estimation is done with the conditional least square estimator for deterministic models and the maximum likelihood estimator for stochastic ones. After that, we investigate a nonadditive state‐space model for spreading epidemic disease by considering infected as the hidden process variable. The problem of the hidden process variable from noisy observations is filtered, predicted, and smoothed using a recursive Bayesian technique. For estimating the hidden number of infected variables, closed‐form solutions are obtained. Finally, numerical simulations with both simulated and real data are performed to demonstrate the efficiency and accuracy of the current work. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce the notion of polyconvolution for Fourier‐cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms are investigated in detail. The Watson‐type theorem is given, to establish necessary and sufficient conditions for this operator to be isometric isomorphism (unitary) on L2(ℝ+)$$ {L}_2\left({\mathrm{\mathbb{R}}}_{+}\right) $$, and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young's type theorem, as well as the norm‐inequalities in weighted space, is also obtained. As applications, we investigate the solvability of a class of Toeplitz plus Hankel type integral equations and linear Barbashin's equations with the help of factorization identities of such polyconvolution. Several examples are provided to illustrate the obtained results to ensure their validity and applicability. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a precise illustration for the structure of the nonconstant steady states for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis. We treat the nonlinear prey‐taxis as a bifurcation parameter to analyze the bifurcation structure of the system. Furthermore, the exported global bifurcation theorem, under a rather natural condition, offers the existence of nonconstant steady states. In the proof, a priori estimates of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given. Finally, some conclusions including biological meanings are performed to summarize our main analytic results and future investigations. [ABSTRACT FROM AUTHOR]

In this paper, the improved Fan subequation method is employed to construct exact traveling wave solutions for a generalized fifth‐order dispersive equation. By making use of bifurcation theory of dynamical systems, we have succeeded to obtain the phase portraits of the subequations involving all possible parameter conditions, and many different types of traveling wave solutions as well, which include more general soliton solutions, kink solutions, triangular function solutions, rational solutions, and Jacobian elliptic function solutions with double periods and so on. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors and could be applicable to explain diversity in qualitative features of wave phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

When the sparse regularizer is convex and its proximal operator has a closed‐form, first‐order iterative algorithms based on proximal operators can effectively solve the sparse optimization problems. Recently, plug‐and‐play (PnP) algorithms have achieved significant success by incorporating advanced denoisers to replace the proximal operators in iterative algorithms. However, convex sparse regularizers such as the ℓ1$$ {\ell}_1 $$‐norm tend to underestimate the large values within the sparse solutions. In contrast, the convex non‐convex (CNC) sparse regularization enables the non‐convex regularizer while preserving the convexity of the objective function. In this paper, we propose several PnP algorithms for solving the CNC sparse regularization model and discuss their convergence properties. Specifically, we first derive the proximal operator for CNC sparse regularization in iterative form and subsequently integrate it with several first‐order algorithms to yield different PnP algorithms. Then, based on the monotone operator theory, we prove that the proposed PnP algorithms are convergent under the condition that the data‐fidelity term is strongly convex and the residuals of the denoisers are contractive. We also emphasized the significance of strong convexity in the data‐fidelity term for the CNC sparse regularization model. Additionally, we demonstrate the superiority of the proposed PnP algorithms through extensive image restoration tasks. [ABSTRACT FROM AUTHOR]

In this paper, we present an analysis of stability of solutions corresponding to a variable coefficient's wave equation subject to a locally Kelvin–Voigt damping and distributed effect driven by a nonnegative function b(x)≥0$$ b(x)\ge 0 $$ with dynamic Wentzell boundary conditions and delay term. By using frequency domain approach method, we show that under a suitable assumption between the internal damping function c$$ c $$ and the boundary delay feedback, considering some geometrical assumptions on the boundary of Ω$$ \Omega $$, supposing that the relaxation function h$$ h $$ decay exponentially to zero, that the energies of the problem decay exponentially to zero. [ABSTRACT FROM AUTHOR]

In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only "authorized" sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness ("unauthorized" sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras. [ABSTRACT FROM AUTHOR]

If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points. In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a 'best fit' rotor between these sets and compare our results with other methods. [ABSTRACT FROM AUTHOR]

Different ways of representing the group SU(3)$$ SU(3) $$ within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for SU(3)$$ SU(3) $$ and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of SU(3)$$ SU(3) $$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing SU(3)$$ SU(3) $$, such as G2$$ {G}_2 $$ and SU(8)$$ SU(8) $$, are also considered. [ABSTRACT FROM AUTHOR]

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras. [ABSTRACT FROM AUTHOR]

This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of restoring the orthogonality of a numerically noisy 4D rotation matrix by finding its nearest (in Frobenius norm) correct rotation matrix. This problem can be straightforwardly solved using the Singular Value Decomposition (SVD). Nevertheless, to avoid numerical methods, we present two new closed‐form methods. One relies on the direct minimization of the mentioned Frobenius norm, and the other on the passage to double quaternion representation. A comparison of these two methods with respect to the SVD reveals that the method based on a double quaternion representation is superior in all aspects. [ABSTRACT FROM AUTHOR]

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]