In this study, we establish new oscillation criteria for the fourth-order delay differential equation. Our main objective is to build upon recent advancements in the study of the oscillatory behavior of second-order equations and extend these findings to higher-order equations. Although fourth-order equations have numerous applications, their study presents significant analytical challenges due to the complex nature of their solutions, which we will discuss in this study. We use the comparison technique with first-order equations in several approaches. Our results show an improvement in the oscillation test due to the development of some monotonic and asymptotic properties of positive solutions. We present a comparison of the new criteria to test their effectiveness, as well as a comparison with previous studies to illustrate the novelty.

This paper introduces a matrix-free variant of the Davidon-Fletcher-Powell (DFP) method for unconstrained optimization problems with applications in compressive sensing and image restoration. The main contribution lies in the new search direction incorporating a scaling parameter that ensures the satisfaction of the sufficient descent condition, independent of the line search conditions. A rigorous convergence analysis guarantees the boundedness and theoretical validity of the proposed method. Comprehensive numerical experiments on benchmark unconstrained optimization test problems and compressive sensing problems demonstrate the efficiency and robustness of the algorithm. Specifically, in image restoration tasks, our method outperforms CG-DESCENT, MDL, and NSMA, achieving a 100% success rate compared to 95.8%, 84.5%, and 53.5%, respectively. Additionally, results on computational time, relative error, and PSNR confirm the superior performance of the proposed approach. These findings establish the proposed method as a competitive alternative for large-scale optimization problems.

In this article, a Cauchy problem for a coupled system of the generalized Korteweg-de Vries equations (gKdV) is considered. In the periodic case, it is shown that the system is locally well-posed in a large class of analytic functions and conditions for which weak solutions extend holomorphically in a symmetric strip of the complex plane around the x-axis at large times. In addition, the uniform analyticity radius of the solution does not change as time progresses. Also, information about the regularity of the solution in the time variable is obtained.

Bird sound recognition has important applications in bird monitoring and ecological protection. However, in complicated environments, noise and insufficient sample data are the major factors affecting recognition accuracy. We proposed a bird sound recognition method based on a developed transfer deep residual shrinkage network. First, a deep residual shrinkage network with noise resistance was constructed based on the structural characteristics of the residual shrinkage module, multi-scale operations, and the characteristics of bird sound Mel spectrograms. Then, the deep residual shrinkage network was pre-trained using a bird sound dataset, applying an unfreezing fine-tuning strategy, to mitigate the impact of insufficient training data. A transfer learning alleviated the problem of data scarcity by utilizing pre-trained models, while the deep residual shrinkage network enhanced the performance of the model in a noisy environment by optimizing the network structure. Experimental results showed that this method achieves high recognition accuracy under noise and small data sets. It has advantages over the compared methods and is suitable for ecological monitoring fields such as bird population monitoring. The method has good application prospects.

Hashing methods have been widely used in large-scale content-based image retrieval systems, which have extensive and practical applications in search engines and social networks. However, existing deep supervised hashing methods focus on enforcing the supervision information to learn the discrete hash codes, ignoring the latent clustering in the Hamming space. In addition, the discrete constraints in the above optimization problem are usually hard to solve. To address these issues, we propose a new deep supervised hashing method for image retrieval tasks. Specifically, we utilized a metric learning strategy to minimize the intra-class Hamming distance and maximize the inter-class Hamming distance as much as possible. Then, a novel label fitting approach was developed to generate discrete proxies and hash codes simultaneously. Finally, an effective alternating optimization method was designed to solve the discrete optimization problem. Experimental results on various datasets demonstrate that the proposed method can achieve promising retrieval performance.

Compared with the recently reported hidden two-scroll Lorenz-like attractors in symmetric quadratic and sub-quadratic Lorenz-like dynamical systems, little seems to be concerned with the generation of hidden n-scroll (n ∈ N) attractors as far as one knows, especially the one whose number of scrolls equals the one of equilibria. To achieve this target, we first constructed a new asymmetric quadratic Lorenz-like analogue and seized hidden single-scroll Lorenz-like attractors, which were also created through the collapse of pseudo singularly degenerate heteroclinic cycles. Then, utilizing the fractal process or rotation symmetry, the proposed system may exhibit hidden n-scroll Lorenz-like attractors coexisting with n stable equilibria, and two examples of hidden two/three-scroll Lorenz-like attractors coexisting with two/three stable equilibria were given. In addition, the existence of a single heteroclinic orbit was proved with the help of suitable Lyapunov functions. The obtained results may not only generalize the second part of Hilbert’s 16th problem (i.e., the degree may determine the geometrical structure of strange attractors), but also provide reference for practical application.

This paper investigates the space-time decay properties of solutions to the three- dimensional compressible quantum magnetohydrodynamic (QMHD) model. By employing weighted Sobolev space techniques, we establish the optimal decay rate for the k-th order spatial derivatives of solutions with k ∈ [0, 4], which concides with the heat equation. Specifically, we prove that the decay rate in the weighted space $H_{\gamma}^{2}\left(\mathbb{R}^{3}\right)$ is given by $t^{-\frac{3}{4}-\frac{k}{2}+\gamma}$ for the spatial derivatives of order k. The key contribution lies in developing a unified framework that connects the weighted energy estimates with time decay analysis, which enables us to simultaneously capture both the spatial regularity and temporal decay characteristics of the solution. This result generalizes the previous decay estimates and provides a new description of the solution’s asymptotic behavior in quantum magnetohydrodynamic systems.

This paper investigated a class of nonlinear ψ−Caputo fractional parabolic equations with impulsive. We reformulated the fractional parabolic equations into abstract evolution equations. By using the nonlinear analysis method and fixed point theorems, we obtained the existence and uniqueness of the global of mild solutions for the problem.

In this paper, we consider a cross-diffusive model for pursuit-evasion processes which was described as

in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$ (N ≥ 1) under homogeneous Neumann boundary conditions. For nonnegative initial data with appropriate regularity, the unique globally bounded classical solution is obtained provided that min $\left\{\left(r_{1}-1\right)\left(r_{2}-1\right),\left(r_{1}-r\right)\left(r_{2}-1\right)\right\}>\frac{(N-2)_{+}}{N}$ , where $r, \lambda_{i}, \mu_{i}, \chi, \xi>0$ and r_i > 1(i = 1, 2). This generalizes the results of Li et al. [1], Zheng et al. [2], and Zheng and Zhang [3].

The dynamic parallel magnetic resonance imaging (MRI) reconstruction presents additional challenges due to the multi-coil acquisition process and an increase in data dimensions, which lead to a more ill-posed inverse problem compared to single coil dynamic MRI. Relevant optimization strategies usually involve solving the model after convex relaxation, which often leads to suboptimal solutions that either overshrink important signals or fail to adequately promote low-rankness and sparsity. In this paper, we propose a dynamic parallel MRI reconstruction model based on weighted nuclear norm and L₁ norm regularization in the framework of low-rank plus sparse (L+S) decomposition. The introduced weighted nuclear norm imposes nonuniform penalties on singular values and has a more flexible approximation to a low-rank property. To solve the resulting non-convex optimization problem, we employ the alternating direction method of multipliers (ADMM) based on variable splitting. The experimental results on multi-coil dynamic datasets show that this method provides a higher reconstruction quality than the traditional convex method.

This paper investigated the dynamical behaviors of the SIQS (susceptible, infected, isolated, and again susceptible) infectious disease model with nonlinear incidence rate and degenerate diffusion in a stochastic environment. By introducing nonlinear contagion rate, the model was able to more realistically reflect the complexity of real-world disease transmission, including the effects of social behavior, medical resource constraints, and public health interventions. It was proved that the infectious disease will be extinct when R^s₀ < 1. Furthermore, by utilizing Markov semigroup theory, we obtained that there existed stationary distribution for the system when R^s₀ > 1. Numerical simulations were conducted by introducing three different forms of nonlinear incidence rates (standard incidence, non-monotonic incidence, Beddington-DeAngelis incidence) to verify our results.

In this paper, we consider the Cauchy problem for the two-dimensional anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation, where the density at infinity is in the state $\widetilde{\rho}>0$. We obtain the global well-posedness of strong solutions and the large-time behavior of the velocity field under the condition that $\left\|\sqrt{\rho_{0}} \mathbf{u}_{0}\right\|_{L^{2}}$ is sufficiently small. It is worth noting that our results include the case of vacuum.

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation

on metric measure spaces with m-Bakry-Emery Ricci curvature bounded from below. Here τ > 0 and l were constants, and λ(x) was allowed to change sign. As applications, we also obtained a Liouville- type result and Harnack’s inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of τ and l.

Nonlinear tuned mass dampers (TMDs) are widely recognized for their superior damping performance, which stems from their broad damping bandwidth. In this study, we propose a single-pendulum TMD for application in vibration absorption within train compressors to protect critical components. However, the inherent nonlinear vibration characteristics of the TMD may undermine its damping effectiveness. To address this challenge, we develop a three-degree-of-freedom mechanical model of a train compressor integrated with a single-pendulum TMD. The equations of motion for the system are derived using the second kind of Lagrange formula. By employing the fourth-order Runge-Kutta numerical integration method and using the excitation frequency as the bifurcation parameter, we analyze the bifurcation behavior and the transition to chaos via the Poincaré cross-section method. Our results demonstrate that the system can transition to chaos through multiple routes, including multiplicative bifurcation, Hopf bifurcation, and residual-dimension bifurcation, depending on the excitation frequency. These findings provide critical insights for the dynamic design and chaotic motion prediction of single-pendulum TMD systems, offering practical guidance to enhance their reliability and performance in real-world applications.

By employing trigonometric sums and solving congruence equations, this paper explores the power mean of a specific class of high–order three–term exponential sums, and derives intriguing identities in the process. The findings presented in this paper offer insights into establishing the lower bound for these sums.

The two-region migration model is an important tool for analyzing population movements and evaluating their economic, social, and environmental impacts. This model mathematically examines the reasons and consequences of individuals that migrate from one region to another, thereby shedding light on issues such as labor force distribution, income differences, urbanization dynamics, and regional development. This migration model, which has a significant impact on classical analyses, is defined in bigeometric calculus, and solved by the bigeometric Laplace transform. In addition, the numerical patterns and curves presented by mathematical modeling in these two migration analyses are interpreted within the framework of sociological theories, thus demonstrating that the phenomenon of migration is not only quantitative but is also too complex to be limited to socially constructed numerical data and is closely linked to the historical process.

This paper is an extension of our earlier research [1]. In this article, we discussed the fractionally damped plate equation with infinite memory. We proved that solutions are global under different initial and boundary conditions. Subsequently, we investigated the exponential stability of the solutions through Lyapunov functionals in order to establish stability conditions sufficient for stability analysis.

The purpose of this paper was to discuss the existence of a random exponential attractor for non-autonomous coupled Klein-Gordon-Schrödinger (KGS) lattice equations with multiplicative noise. We employed the method of estimation on the tails of solutions to prove the existence of a random attractor for a continuous cocycle generated by the random KGS lattice equations on an infinite-dimensional sequence space, and used this abstract result to prove the Lipschitz continuity of the continuous cocycle. Then, we verified the existence of a random exponential attractor for the investigated system according to a known criterion.

This study focuses on the efficient and accurate solution of time-fractional nonlinear partial differential equations (PDEs), which arise in various scientific and engineering applications but are often challenging to solve due to their complexity. We propose a novel method that integrates the Laplace transform with the accelerated Adomian decomposition method (AADM), forming the Laplace transform accelerated Adomian decomposition method (LAADM). The key innovation of this approach lies in its ability to handle the fractional time derivative expressed in the Caputo sense, while enhancing convergence speed and reducing computational effort. The methodology is systematically formulated, and several numerical experiments are conducted to validate its performance. The results demonstrate that LAADM provides highly accurate solutions and exhibits superior efficiency when compared to traditional solution techniques for fractional PDEs.

In this paper, we simultaneously study reaction identification and the optimal control problem for a reaction-diffusion system modeling carrier-borne epidemics with a general transmission function and vaccination. The state equations are given by a susceptible-infected-recovered reaction-diffusion system with zero-flux boundary conditions and initial conditions. The reaction is modeled by three terms: a general transmission function modeling the force of the infection or the effective contact between susceptible and infected individuals, a linear function for transition between susceptible and infected individuals, and a function for control of vaccination of susceptible individuals. The cost function consists of two parts: two terms related to parameter identification, comprising a regularized least squares cost function, and five terms related to the control of the population through vaccination. The optimal control problem is analyzed by applying the Dubovitskii and Milyutin formalism. In the main results, we deduce the well-posedness of the state equation, the existence of the optimal control problem, the existence of solutions of the adjoint state, and a first-order optimality condition. We develop a numerical approximation for the optimal control problem by employing an IMEX method to approximate the state equations. In this approach, the coefficients of the reaction terms and the control functions depend on a finite set of parameters. We provide two numerical examples to demonstrate the agreement of our numerical solution with the measurement observations.