This manuscript investigated the stability of Hopfield neural networks with time-varying transmission delays and leakage delays, and specially discussed the impact of the time-varying delays’ dynamic characteristics on network stability. First, to characterize the dynamic features of time-varying delays transitioning continuously between short and large delays, two key parameters were introduced: a critical threshold for distinguishing whether the time-varying delays are short or large delays, and the ratio of the measure of the union of time periods, in which the time-varying delays appear as short delays, to the measure of the whole time interval. Then, by utilizing the Lyapunov stability method, some sufficient conditions for the global exponential stability of the neural networks were derived. It is pointed out that when the measure of time periods in which the time-varying delays appear as short delays is large enough, the above two parameters related to the time-varying delays’ dynamic characteristics will have an important impact on the system stability, and the upper bound that the time-varying delays can achieve in the whole time interval will no longer be the dominant factor influencing stability. Additionally, the relationship between the leakage delays and the stabilization ability of the negative feedback terms was explored. Two admissible upper bounds were presented, below which the leakage delays do not completely undermine the capacity of negative feedback terms to stabilize the neural networks. Finally, some simulation experiments were conducted to validate our theoretical findings.

Medical image denoising is particularly important in brain image processing. Noise in acquisition and transmission degrades image quality and affects the reliability of diagnosis and research. Due to the complexity of the brain’s structure and minor density differences, noise can increase diagnosis difficulty, so high-quality images are essential for disease detection, prognosis assessment, and treatment plan development. This paper proposes a multi-convolutional neural network based on feature distillation learning and dense residual attention to enhance the quality of brain images and improve denoising performance. The overall network structure contains four parts: a global sparse network (GSN), a dense residual attention network (DRAN), a feature distiller network (FDN), and a feature processing block (FPB). Before feeding the brain images into the denoising network model, they are preprocessed using a modified watershed algorithm based on a combination of a morphological gradient, Sobel’s operator, and Canny’s operator. The GSN is used to extract global features and increase the sensory field, and the DRAN efficiently extracts key features by combining improved channel attention and spatial attention mechanisms. The FDN extracts useful features through two feature distillation blocks, suppresses redundant information, and reduces computational complexity. The FPB performs feature fusion. Experimental results on brain image datasets and ground-based open datasets show that the proposed model outperforms existing methods in several metrics, and helps to improve the accuracy of brain disease diagnosis and treatment.

This paper is devoted to dealing with the dynamic event-triggered H_∞ quantized control for neural networks with sensor saturations and stochastic deception attacks. To save the limited network resources, a dynamic event-triggered scheme is offered, which includes the general one. And a lower trigger frequency can be obtained by appropriately adjusting the triggering error. Then, a new closed- loop quantized control model is established under a dynamic event-triggered scheme, sensor saturations, and stochastic deception attacks, which is described by two independent Bernoulli-distributed variables. Moreover, by Lyapunov-Krasovskii functional theory, a new H_∞ performance criterion is given, and based on the criterion, the controller design approach is derived. Finally, simulations are listed to verify the validity of derived methods.

In this paper, we have constructed two muscular blood vessel systems influenced by external disturbances by using the non-autonomous delayed differential equation (DDE). As an important physiological structure of the human body, muscle blood vessels participate in many activities such as blood flow. Many diseases are associated with abnormal dynamics of muscle blood vessels. From a mathematical point of view, vasospasm is caused by a chaotic state of blood vessels. Vasospasm is the manifestation of this disease in the vascular system, which can cause blood vessel blockage and even harm human health when it is serious. We conducted the dynamical analysis of the S- type and N-type muscular blood vessel systems by utilizing bifurcation diagrams, time histories, and pseudo-phase portraits, and investigated the effects of different parameters on these systems. Specifically, when parameters change, rich dynamical phenomena occur, such as equilibria, periodic solutions, quasi-periodic solutions, Hopf bifurcation, and chaos, as well as the route of period-doubling bifurcation to chaos. Meanwhile, we analyzed approximate analytical solutions by using the method of multiple scales (MMS) and determined the stability of steady-state solutions through the Routh- Hurwitz criterion. The results indicate that the MMS can deduce better analytical results for some non-autonomous DDEs.

This paper is devoted to the study of the Liouville-type theorem of the stationary fractional compressible MHD systems in anisotropic Lebesgue spaces in R³. We showed that the solution is trivial when certain anisotropic integrability conditions are satisfied in terms of the velocity and the magnetic field components.

Let $M_{n, m}:=\operatorname{Mat}_{n}(\mathbb{Z} / m \mathbb{Z})$ be the ring of matrices of $n \times n \text { over } \mathbb{Z} / m \mathbb{Z} \text { and } G_{n, m}:=G l_{n}(\mathbb{Z} / m \mathbb{Z})$ be the multiplicative group of units of M_n,m with n ⩾ 2, m ⩾ 2. In this paper, we obtain an exact formula for the number of representations of any element of M_2,m as the sum of k units in M_2,m. Furthermore, by using the technique of Fourier transformation, we also give a formula for the case n ≥ 3 and m = p is a prime.

It is not well-known that for each separable real locally convex space, its Bohr compactification is isomorphic as a topological group to the Bohr compactification of the topological group R of all real numbers. This is the case for each separable real Banach space. In this expository research note, we state the results explicitly and provide accessible proofs.

Although scholars have proven the existence of a pair of homoclinic orbits to the origin, or a pair of heteroclinic orbits to the origin along with a pair of nontrivial equilibria in symmetric Lorenz, Chen, and Lü systems, they have rarely dealt with asymmetric ones of the corresponding asymmetric analogues, to the best of our knowledge. To clarify this subject, this work revisited an asymmetric Chen system and reveals a single/a pair of asymmetric heteroclinic/homoclinic orbits, which are justified with numerical experiments.

In this study, we derive new criteria that ensure the oscillation of solutions to noncanonical dynamic equations that are half-linear sublinear functional. These results not only resolve an open issue in numerous works in the literature but also emulate Ohriskatype and Hille-type criteria for canonical dynamic equations. We provide examples to demonstrate the accuracy, usefulness, and flexibility of the main results.

The purpose of this paper is to study nonabelian embedding tensors on 3-Lie algebras, and to explore the fundamental algebraic structures, cohomology and deformations associated with them. First, we introduce the concept of nonabelian embedding tensors on 3-Lie algebras. Then, we present the concept of a 3-Leibniz-Lie algebra, which constitutes the fundamental algebraic framework for a nonabelian embedding tensor on a 3-Lie algebra. Additionally, we examine the 3-Leibniz- Lie algebras that are derived from Leibniz-Lie algebras. Finally, we develop the cohomology of nonabelian embedding tensors on 3-Lie algebras and utilize the first cohomology group to characterize infinitesimal deformations.

Several binomial sums are established for the Pell polynomials and the Pell-Lucas polynomials, as well as two types of the Chebyshev polynomials and the Fibonacci-Lucas numbers, which include two special cases proposed by Hideyuki Othsuka in 2024.

In this paper, the local convergence of a high-order Chebyshev-type method without the second derivative is studied. We study the convergence under ω-continuity conditions based on the first derivative. The uniqueness of the solution and the radii of convergence domains are obtained. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. In addition, the attractive basins of the family with different parameters are studied, which can show the different stability of the family. Finally, in numerical experiments, the iterative method is used to solve different nonlinear models, including vertical stresses, civil engineering problem, blood rheology model, and so on. Theoretical results of convergence criteria are verified.

Let ϕ(z) = z^d + c be a polynomial over a field K. We study the inverse stability of ϕ(z) over K. In this paper, we establish some sufficient conditions for the inverse stability of ϕ(z) over the field of rational numbers and a function field. Furthermore, we also provide necessary and sufficient conditions for the inverse stability of ϕ(z) over a finite field.

In this paper, we study the privacy-preserving distributed optimization problem on directed graphs, aiming to minimize the sum of all agents’ cost functions and protect the sensitive information. In the distributed optimization problem of directed graphs, agents need to exchange information with their neighbors to obtain the optimal solution, and this situation may lead to the leakage of privacy information. By using the state decomposition method, the algorithm ensures that the sensitive infor- mation of the agent will not be obtained by attackers. Before each iteration, each agent decomposes their initial state into two sub-states, one sub-state for normal information exchange with other agents, and the other sub-state is only known to itself and invisible to the outside world. Unlike traditional optimization algorithms applied to directed graphs, instead of using the push-sum algorithm, we in- troduce the external input, which can reduce the number of communications between agents and save communication resources. We prove that in this case, the algorithm can converge to the optimal solu- tion of the distributed optimization problem. Finally, a numerical simulation is conducted to illustrate the effectiveness of the proposed method.

Recently, correlation filter-based tracking methods have been widely adopted in UAV target tracking due to their outstanding performance and excellent tracking efficiency. However, existing correlation filter-based tracking methods still face issues such as redundant visual features with weak discriminative ability, inadequate spatio-temporal information mining, and filter degradation. In order to overcome these challenges, this paper proposes a spatial saliency-aware strategy that reduces redundant information in spatial and channel dimensions, thus improving the discriminative ability between the target and background. Also, this paper proposes a position estimation mechanism under spatio-temporal joint constraints to fully mine spatio-temporal information and enhance the robustness of the model in complex scenarios. Furthermore, this paper establishes a positive expert group using historical positive samples to assess the reliability of candidate samples, thereby effectively mitigating the filter degradation issue. Ultimately, the effectiveness of the proposed method is demonstrated through the evaluation of multiple public datasets. The experimental results reveal that this method outperforms others in tracking performance under various challenging conditions.

In this paper, we aim to study the stability and convergence of a finite difference scheme for solving the two-dimensional nonlinear multi-term time fractional subdiffusion equation with weakly singular solutions. We apply the L1 scheme to discretize the multi-term temporal Caputo derivatives, a standard central difference method in space, and a backward formula to approximate the nonlinear term on the uniform mesh, respectively. Stability and pointwise-in-time error estimates are obtained for the fully discrete scheme. The global convergence order is α₁, and the local convergence order is 1 in the temporal direction. The theoretical analysis is verified by some numerical results.

In this study, the numerical solution of a two-dimensional coupled Burgers’ equation is investigated by the barycentric interpolation collocation method. The spatial-temporal domain of the equation is made discrete by employing barycentric interpolation, and a corresponding differentiation matrix based on this interpolation technique is constructed. The effectiveness and high accuracy of the proposed method are demonstrated for solving the two-dimensional coupled Burgers’ equation.

Let q ≥ 3 be a positive integer. For any integers m, n, k, h, the two-term exponential sums $C(m, n, k, h ; q)=\sum_{a=1}^{q} e\left(\frac{m a^{k}+n a^{h}}{q}\right)$, where k > h ≥ 2. The main purpose of this paper is to use analytic methods and the properties of classical Gauss sums to study the mean value involving two-term exponential sums and fourth Gauss sums, and to provide some asymptotic formulas and identities. Previously, only the case of h = 1 had been studied.

This paper presents a simplified weak Galerkin (WG) finite element method for solving biharmonic equations avoiding the use of traditional stabilizers. The proposed WG method supports both convex and non-convex polytopal elements in finite element partitions, utilizing bubble functions as a critical analytical tool. The simplified WG method is symmetric and positive definite. Optimal- order error estimates are established for WG approximations in both the discrete H² norm and the L² norm.

In this study, we investigate the existence of at least one solution and the existence of an infinite number of solutions for a discrete fractional boundary value problem. Requiring an algebraic condition on the nonlinear term for small values of the parameter, and requiring an additional asymptotical behavior of the potential at zero, we investigate the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior of the nonlinearity at infinity, for exact collections of the parameter, we discuss the existence of a sequence of solutions for the problem. We also present some examples that illustrate the applicability of the main results.

In this paper, Kou’s method is extended to solve nonlinear systems. The convergence order of the iterative method is proved. Using fractal theory, we study the theoretical operators related to the iterative method, and analyze the stability of the iterative method. Properties related to strange fixed points and critical points are explored. The fractal results indicate that the iterative method is most stable when the parameter γ equals zero. The extended iterative method is applied to solve the Hammerstein equation and some nonlinear systems. The dynamic plane and numerical experiments show that the extended iterative method can solve the nonlinear system of equations with good convergence and stability.

A nonlinear correction finite volume scheme preserving the discrete maximum principle (DMP) was presented to solve diffusion equations with anisotropic and discontinuous coefficients. It is well-known that existing cell-centered finite volume schemes for the diffusion problem with the general discontinuous coefficient often impose severe restrictions on the mesh-cell geometry to maintain the DMP. We proposed a nonlinear method for modifying the flux to obtain a new scheme which eliminated the requirement for nonnegative interpolation coefficients at the midpoint of cell-edge unknowns while still preserving the DMP. That is, our new scheme satisfied the DMP unconditionally and can be applied to the diffusion problem with the discontinuous coefficient on arbitrary distorted meshes. We then provided a priori estimation under a coercivity assumption and proved that the scheme satisfied the DMP. Numerical results were presented to demonstrate that our scheme can handle diffusion equations with anisotropic and discontinuous coefficients, satisfied the DMP, and, in some cases, outperformed existing schemes which preserved the DMP in terms of accuracy.

The mathematical analysis of spatiotemporal distributions in many species exhibiting different predation mechanisms has attracted considerable attention in biology and ecology. In this article, we investigated a prey-taxis model involving hunting cooperation, which has more strong coupling structures. Utilizing energy estimates and semigroup theory, the global boundness of its classical solution was established when the hunting cooperation is weak in two dimensions. By means of Lyapunov functionals, the global asymptotically stability of the non-negative constant steady-state solution for the discussed model was established under certain assumptions on parameters. These results enrich the related researches on the prey-taxis model with Lotka-Volterra functional response, which has been studied by Jin and Wang.

Numerous adaptations of traditional entropy concepts and their residual counterparts have emerged in statistical research. While some methodologies incorporate supplementary variables or reshape foundational assumptions, many ultimately align with conventional formulations. This study introduces a novel extension termed residual cumulative generalized exponential entropy to broaden the scope of residual cumulative entropy for continuous distributions. Key attributes of the proposed measure include non-negativity, bounds, its relationship to the continuous entropy measure, and stochastic comparisons. Practical implementations are demonstrated through case studies involving established probability models. Additionally, insights into order statistics are derived to characterize the measure’s theoretical underpinnings. The residual cumulative generalized exponential entropy framework bridges concepts such as Bayesian risk assessment and excess wealth ordering. For empirical implementation, non-parametric estimation strategies are devised using data-driven approximations of residual cumulative generalized exponential entropy, with two distinct estimators of the cumulative distribution function evaluated. A practical application is showcased, using clinical diabetes data. The study further explores the role of generalized exponential entropy in identifying distributional symmetry, mainly through its application to uniform distributions to pinpoint symmetry thresholds in ordered data. Finally, the utility of generalized exponential entropy is examined in pattern analysis, with a diabetes dataset serving as a benchmark for evaluating its classification performance.

In this paper, a multi-time scale stochastic eco-epidemic model where the prey population was infected with disease was proposed. The stochastic factors in the ecological environment and the fact that the growth and loss rates of predators were much smaller than those of prey were considered. First, the dynamical behavior of the deterministic model was analyzed, including the existence and the stability of the equilibrium points and the bifurcation phenomena. Second, the existence and uniqueness of global positive solutions and the ergodic property of stochastic model were discussed. Meanwhile, the solution trajectory which was perturbed was also analyzed by using random center-manifold and random averaging method. Finally, the stochastic P-bifurcation is shown by applying singular boundary theory and invariant measure theory. Numerical simulation also verified the correctness of the theoretical analysis.

Feature selection is a crucial data processing method used to reduce dataset dimensionality while preserving key information. In this paper, we proposed a multi-strategy enhanced dung beetle optimization algorithm (mDBO) that integrates multiple strategies to effectively address the feature selection problem. First, a novel population initialization strategy based on a hybrid tent-sine map and random opposition-based learning was proposed to generate initial population. This strategy yielded a more uniform distribution of the initial population, significantly improving the quality of the population distribution within the search space. Second, a new differential evolution mutation strategy with a periodic retrospective adaptive mutation factor was proposed. This strategy effectively improved the algorithm’s ability to jump out of the local optimal and explore potential candidate solutions. Third, based on Padé approximation technology and the novel adaptive evolutionary boundary constraint method, an innovative approximation strategy was proposed. The strategy was integrated into the framework of the dung beetle optimizer, significantly improving the solution accuracy and population quality of the algorithm. Finally, the binary version of the mDBO algorithm (bmDBO) was applied to feature selection tasks. Experiments entailing CEC2017 benchmark functions and 17 datasets showed that both mDBO and bmDBO outperformed other algorithms. The mDBO method outperformed other algorithms in 11 of the 29 benchmark functions, ranked second in 8 functions, and achieved an average rank of 1.62 in the Friedman ranking, securing the overall first place; the bmDBO method outperformed in 12 of 17 datasets, achieving an average ranking of 1.35 in the Friedman ranking, securing the first position.

In this paper, we analyzed and tested a nonlinear implicit compact finite difference scheme for the pseudo-parabolic Burgers’ equation. The discrete conservation laws and boundedness of the scheme were rigorously established. We then proved the unique solvability of the numerical scheme by reformulating it as an equivalent system. Furthermore, using the energy method, we derived an error estimate for the proposed scheme, achieving a convergence order of O(τ² + h⁴) under the discrete L^∞-norm. The stability of the compact finite difference scheme was subsequently proven using a similar approach. Finally, a series of numerical experiments were performed to validate the theoretical findings.

We explored the phenomenon of finite-time synchronization for coupled complex-valued neural networks that were subject to mixed-variable delays. To address this challenge, an impulsive pinning control strategy was proposed. The method presented entailed the dynamic adjustment of specific nodes at distinct impulsive intervals, contingent upon the magnitude of the errors observed in those nodes. Furthermore, an enhanced technique utilizing the sign function was employed to simultaneously manage the real and imaginary components of the complex-valued neural networks. By applying finite-time stability theorems and utilizing complex-valued inequalities, sufficient conditions for achieving finite-time synchronization and determining stability time under the influence of delayed impulsive effects were established. A comprehensive discussion on the interaction between impulsive effects and pinning strategies was also included. It was noted that integrating impulsive effects with pinning ratios enabled precise control over nodes exhibiting significant errors, thereby promoting rapid convergence within finite time frames. Our findings highlight the effectiveness of impulsive pinning control in enhancing synchronization stability, providing significant insights into the practical applications of complex-valued neural networks, particularly in image processing.

This study examined a single-compartment pharmacokinetic model with intravenous and oral administration. It investigated the trends in steady-state drug exposure and average steady-state plasma drug concentration under multiple dosing regimens. First, the combined drug administration model was a non-autonomous system, and we approximated the solution of the model by estimating the exact upper and lower bounds in conjunction with the comparison theorem for differential equations. We also demonstrated the existence, uniqueness, and stability of the solution. Second, we derived the steady-state drug exposure and compared it with the solution of the intravenous drug delivery model alone. The results indicate that the combined drug delivery scheme offers superior performance. Finally, we theoretically proved the change rule of average steady-state blood drug concentration under different dosing regimens, and verified its feasibility and rationality by combining the numerical simulation results of Phenytoin sodium.

An optimized potential formula for the m × n apple surface network has been introduced in this paper. Compared with the original potential formula, this method significantly enhances the efficiency required for rapid and large-scale numerical simulations. Based on the optimized potential function, we proposed a metaheuristic algorithm suitable for apple surface environment path planning. Chebyshev polynomials of the first class were employed to represent the potential function. Subsequently, a fast algorithm for calculating the potential utilizing the first kind of discrete sine transform (DST-I) was devised. We proposed potential formulas for several cases to visually present the distribution of the potential and illustrated them using three-dimensional graphs. We also conducted simulation experiments on the computational efficiency of the original and optimized formulas at different data scales, verifying the superiority of the optimized formulas. These findings provided new perspectives and tools for the computation of resistor networks and the design of path planning algorithms. Experiments were conducted to analyze the efficiency and availability of various techniques for computing potential.

We proved that on every Stiefel manifold $V_{2} \mathbb{R}^{n} \cong \operatorname{SO}(n) / \operatorname{SO}(n-2)$ with n ≥ 3 the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems, we proved that for every invariant set Σ of the normalized Ricci flow on $V_{2} \mathbb{R}^{n}$ defined as $x_{1}^{n-2} x_{2}^{n-2} x_{3}=c, c>0$, there exists a smaller invariant set $\Sigma \cap \mathscr{R}_{+}$ for every n ≥ 3, where $\mathscr{R}_{+}$ is the domain in $\mathbb{R}_{+}^{3}$ responsible for parameters x₁, x₂, x₃ > 0 of invariant Riemannian metrics on $V_{2} \mathbb{R}^{n}$ admitting positive Ricci curvature.