This research was concerned with a linear theory of thermoelasticity with microtempera- tures where the second thermal displacement gradient and the second gradient of microtemperatures are included in the classical set of independent constitutive variables. The master balance laws of micromorphic continua, the theory of the strain gradient of elasticity, and Green-Naghdi thermome- chanics were used to derive a second gradient theory. The semigroup theory of linear operators al- lowed us to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed. For the equations of isotropic rigids, we presented a natural extension of the Cauchy- Kovalevski-Somigliana solution of isothermal theory. In the case of stationary vibrations, the funda- mental solutions of the basic equations were obtained. Uniqueness and instability of the solutions were obtained in the case of antiplane shear deformations.

This paper was based on a kernel-free boundary integral (KFBI) method for solving the reaction-diffusion equation. The KFBI method serves as a general elliptic solvers for boundary value problems in an irregular problem domain. Unlike traditional boundary integral methods, the KFBI method avoids complicated direct integral calculations. Instead, a Cartesian grid-based five-point compact difference scheme was used to discretize the equivalent simple interface problem, whose solution is the integral involved in the corresponding boundary integral equations (BIEs). The resulting linear system was treated with a fast Fourier transform (FFT)-based elliptic solver, and the BIEs were iteratively solved by the generalized minimal residual (GMRES) method. The first step in solving the reaction-diffusion equation was to discretize the time variable with a two-stage second-order semi-implicit Runge-Kutta (SIRK) method, which transforms the problem into a spatial modified Helmholtz equation in each time step and can be solved by the KFBI method later. The proposed algorithm had second-order accuracy in both time and space even for small diffusion problems, and the computational work was roughly proportional to the number of grid nodes in the Cartesian grid due to the fast elliptic solver used. Numerical results verified the stability, efficiency, and accuracy of the method.

The payload, coupler slack, and buffer device performance in heavy-haul trains substantially affect their longitudinal dynamic systems under operational conditions. These factors result in the bifurcation of the system, consequently leading to chaos. To study this phenomenon in depth, a two-degrees-of-freedom longitudinal dynamics model of the train is established. The system is analyzed using the fourth-order Runge–Kutta (R-K₄) numerical integration method, incorporating bifurcation diagrams, phase planes, Poincaré mapping, and time-domain analysis to elucidate the trajectory of the system as it transitions into a chaotic state of motion via period-doubling bifurcations and quasi-periodic motions. A comprehensive analysis of the complex nonlinear dynamics of the train’s longitudinal system can establish a theoretical foundation for forecasting and regulating the chaotic motion of the train system.

In this paper, we investigate the existence of a normalized solution for the following Kirchhoff system in the entire space $\mathbb{R}^{N}(N \geq 3)$:

under the constraints $\int_{\mathbb{R}^{N}}|u|^{2} d x=m_{1}$ and $\int_{\mathbb{R}^{N}}|v|^{2} d x=m_{2}$, where m₁, m₂ > 0 are prescribed. The parameters $\mu_{1}, \mu_{2}, \beta>0,2 \leq p, q<2+\frac{8}{N}, r_{1}, r_{2}>$, and satisfy $r_{1}+r_{2}=2^{*}=\frac{2 N}{N-2}$. The frequencies λ₁, λ₂ appear as Lagrange multipliers. With the help of the Pohožaev manifold and the minimization of the energy functional over a combination of the mass constraints and the closed balls, we obtain a positive ground state solution to (P). We mainly extend the results of Yang (Normalized ground state solutions for Kirchhoff-type systems) concerning the above problem from a single critical to a coupled critical nonlinearity.

The modeling of discrete space-time stochastic heterogeneous complex networks with unknown factors was achieved through the utilization of differencing techniques with respect to the time and space variables of the nodes’ states. Via the space-time discrete Lyapunov-Krasovskii functional and the approach of linear matrix inequality, this paper derived the mean-squared asymptotic anti-synchronization of the aforementioned discrete networks. This was achieved by defining an updated law for the hitherto unknown parameters and incorporating boundary values within the feedback controller. The theoretical and experimental findings indicated that the feedback controller at the boundary represented a more effective and cost-effective control technique for the networks. Furthermore, an adaptive rule has been designed to identify the uncertainties that occur in the networks with a high degree of accuracy. In particular, this rule enabled the response networks to identify unknown information in the drive networks with a high level of precision by incorporating an adaptive updating mechanism. Finally, a numerical example was provided to elucidate the viability of the ongoing investigation.

In this work, we aimed at a kind of multi-term variable-order time fractional mobile- immobile diffusion (TF-MID) equation satisfying the Neumann boundary condition, with fractional orders α^m(t) for m = 1, 2, ...,P, and introduced a QSC-L1⁺ scheme by applying the quadratic spline collocation (QSC) method along the spatial direction and using the L1⁺ formula for the temporal direction. This new scheme was shown to be unconditionally stable and convergent with the accuracy $\mathcal{O}\left(\tau^{\min \left\{3-\alpha^{*}-\alpha(0), 2\right\}}+\Delta x^{2}+\Delta y^{2}\right)$, where ∆x, ∆y, and τ denoted the space-time mesh sizes. α^* was the maximum of α^m(t) over the time interval, and α(0) was the maximum of α^m(0) in all values of m. The QSC-L1⁺ scheme, under certain appropriate conditions on α^m(t), is capable of attaining a second order convergence in time, even on a uniform space-time grid. Additionally, we also implemented a fast computation approach which leveraged the exponential-sum-approximation technique to increase the computational efficiency. A numerical example with different fractional orders was attached to confirm the theoretical findings.

In this paper, we study space-time decay rates of a nonconservative compressible two-phase flow model with common pressure in the whole space R³. Based on previous temporal decay results, we establish the space-time decay rate of the strong solution. The main analytical techniques involve delicate weighted energy estimates and interpolation.

Tidal time series are affected by a combination of astronomical, geological, meteorological, and anthropogenic factors, revealing non-stationary and multi-period features. The statistical features of non-stationary data vary over time, making it challenging for typical time series forecasting models to capture their dynamism. To solve this challenge, we designed memory factors, leveraging the fusion of statistical data at the channel dimension to enhance the model’s prediction capacity for non- stationary data. On the other hand, traditional approaches have limitations in trend and cycle decomposition, making it difficult to detect complicated multi-period patterns and accurately separate the components. We combined integrated frequency domain optimization and multi-level, multi-scale convolutional kernel technologies. By employing Fourier-based methods and iterative recursive decomposition strategies, we effectively separated periodic and trend components. Then, the periodic multi-level wavelet block was applied to extract the periodic interaction features, aiming to deeply mine the latent information of periodic components and enhance the model’s long-term prediction capabilities. In this paper, we used the Informer model as the foundational framework for further research and development. In comparative experiments, our proposed model outperformed LSTM, Informer, and MICN by 61.4%, 51.7%, and 23.8%, respectively. In multi-time-span prediction, the model’s error remained stable as the prediction span increased from 48 to 96 steps (from 0.059 to 0.067). Under multi-site conditions, the model achieved varying degrees of improvement over the baseline in three key evaluation metrics, with average increases of 35.2%, 35.6%, and 61.2%, respectively. In this study, we focused on the extraction of short-period features from tidal data, providing an innovative and reliable solution for tidal height prediction. The results are significant for tidal assessments and protective engineering construction.

Diabetic retinopathy (DR) is a major cause of vision loss. Accurate grading of DR is critical to ensure timely and appropriate intervention. DR progression is primarily characterized by the presence of biomarkers including microaneurysms, hemorrhages, and exudates. These markers are small, scattered, and challenging to detect. To improve DR grading accuracy, we propose FF-ResNet-DR, a deep learning model that leverages frequency domain attention. Traditional attention mechanisms excel at capturing spatial-domain features but neglect valuable frequency domain information. Our model incorporates frequency channel attention modules (FCAM) and frequency spatial attention modules (FSAM). FCAM refines feature representation by fusing frequency and channel information. FSAM enhances the model’s sensitivity to fine-grained texture details. Extensive experiments on multiple public datasets demonstrate the superior performance of FF-ResNet-DR compared to state-of-the-art models. It achieves an AUC of 98.1% on the Messidor binary classification task and a joint accuracy of 64.1% on the IDRiD grading task. These results highlight the potential of FF-ResNet-DR as a valuable tool for the clinical diagnosis and management of DR.

In natural ecosystems, the external environment is constantly changing, and is affected by various factors, thus presenting a certain degree of randomness and uncertainty. Therefore, having a suitable habitat is essential for the reproductive success of many species. Understanding the impact of habitat selection provides valuable insights into how species locate and adapt to suitable living environments based on their specific needs. For this purpose, a prey-predator system model with prey habitat selection in an environment subject to stochastic disturbances is formulated. The properties of the proposed model without and with stochastic disturbances are investigated, including the existence of a unique ergodic stationary distribution, the stochastically ultimate bounded-ness of the solutions, and the extinction and persistence of the populations. The study demonstrates that prey can persist at a low intensity noise, whereas stronger stochastic disturbances may lead to the extinction of both the prey and predator species. To illustrate the theoretical results, numerical simulations are presented step by step. This work provides a theoretical reference for further studies on populations with habitat selection in an environment subject to stochastic disturbances.

Low-frequency vibration isolation is an attractive research topic in vibration control. In this study, a novel quasi-zero stiffness isolator utilizing shape memory alloy (SMA) springs is proposed. Leveraging the inherent stress plateau characteristics caused by the super-elastic effect of SMA, this design significantly improves the isolation performance at low-frequency excitations. We began by reformulating the static constitutive equation of the SMA spring, and the torsional strain of the spring was taken into account into the static constitutive equation. Subsequently, the dynamics of the SMA spring was modeled as an ordinary differential equation using the Euler-Lagrange equation. The SMA spring was fabricated and tensile tests were performed to validate the model given by dynamic differential function. Building on the validated spring model, a dynamic model of the quasi-zero stiffness isolator using SMA springs was proposed and its response under sinusoidal excitation was analyzed. The amplitude-frequency response of the system was determined using the harmonic balance method (HBM), and superior performance of the isolator in attenuating low-frequency vibrations was confirmed. Finally, an experimental platform was constructed to evaluate the isolator’s performance under low-frequency excitations at 0.5, 1.0, 1.5, and 2.5 Hz. Our results demonstrated the effectiveness of the proposed quasi-zero stiffness vibration isolator system in isolating low-frequency vibrations, and the simulation results were verified by the experimental counterparts.

This study provided a significant contribution to developing an adaptable trigonometric extension of the power half-logistic distribution. To be more specific, we created an innovative two- parameter lifetime model called the sine power half-logistic distribution (SPHLD) by using features from the sine-generated family of distributions. The novel distribution could be more effective in modeling lifetime phenomena when asymmetric data was presented, which was the primary motivating factor. The SPHLD’s density function plots showed that the distribution adopted several asymmetric shape configurations. Furthermore, the SPHLD’s hazard rate plots displayed both monotonic increases and decreases. The quantile function, moments, incomplete moment, and stress-strength reliability were among the statistical characteristics of the SPHLD that were computed. Statistical inference using sixteen distinct classical estimating techniques was utilized to estimate the SPHLD parameters. A simulation study was done to evaluate the consistency of the different estimates and determine the best estimating approach based on some accuracy measures. Analyses of real data revealed that the SPHLD performed better than a number of alternative distributions.

In this work, a novel approach based on a single-layer machine learning Legendre spectral neural network (LSNN) method is used to solve an elliptic partial differential equation. A Legendre polynomial based approach is utilized to generate neurons that fulfill the boundary conditions. The loss function is computed by using the error back-propagation principles and a feed-forward neural network model combined with automatic differentiation. The main advantage of using this methodology is that it does not need to solve a system of nonlinear and nonsparse equations compared with other traditional numerical schemes, which makes this algorithm more convenient for solving higher-dimensional equations. Further, the hidden layer is eliminated with the help of a Legendre polynomial to enlarge the input pattern. The neural network’s training accuracy and efficiency were significantly enhanced by the innovative sampling technique and neuron architecture. Moreover, the Legendre spectral approach can handle equations on more complex domains because of numerous networks. Several test problems were used to validate the proposed scheme, and a comparison was made with other neural network schemes consisting of the physics-informed neural network (PINN) scheme. We found that our proposed scheme has a very good agreement with PINN, which further enhances the reliability and efficiency of our proposed method. The absolute and relative error in both L₂ and L_∞ between exact and numerical solutions are provided, which shows that our numerical method converges exponentially.

We clarify a relation between the arithmetic autocorrelation and pattern distribution of binary sequences, then we apply the relation to study the upper bound of arithmetic autocorrelation for two binary sequences constructed by Fermat quotient and the generalized cyclotomic class of order 2, respectively. Our results indicate that the sequences with large “long term” correlations may have small “short term” pattern distribution; and thus have rather small arithmetic autocorrelations.

This article studies a distributed data-driven iterative learning control (ILC) strategy based on the identified input–output coupled parameters (IOCPs) to address the consensus trajectory tracking problem of discrete time-varying multi-agent systems (MASs). First, by leveraging the repeatability of the control system, a special learning scheme is designed by using system input and output data to identify the unknown IOCPs. Then the reciprocal of the identified IOCPs is selected as the learning gain to construct the ILC law of the MASs. Second, the case of measurement noise in the MASs is considered, where the maximum allowable control deviation is incorporated into the learning mechanism for identification of the IOCPs, thereby minimizing adverse effects of the noise on the learning scheme’s performance and bolstering robustness. Finally, three numerical simulations are employed to validate the effectiveness of the designed IOCP identification method and iterative learning control strategy.

This paper adopted the alternating direction method of multipliers (ADMM) which aims to delve into data-driven differential equations. ADMM is an optimization method designed to solve convex optimization problems. This paper attempted to illustrate the conceptual ideas and parameter discovery of the linear coupled first-order ODE. The estimation of the coefficients of the underlying equation utilized a combination of algorithms between physics-informed neural networks (PINNs) and sparse optimization. Both methods underwent a sufficient amount of looping during the search for the best combinations of coefficients. The PINNs method took charge of updating weights and bi- ases. The updated trainable variables were then fetched to the sparse optimization method. During the sparse optimization process, ADMM was used to restructure the constrained optimization problems into unconstrained optimization problems. The unconstrained optimization problem usually consists of smooth (differentiable) and non-smooth (non-differentiable) components. By using the augmented Lagrangian method, both smooth and non-smooth components of the equations can be optimized to suggest the best combinations of coefficients. ADMM has found applications in various fields, such as signal processing, machine learning, and image reconstruction, which involve decomposable struc- tures. The proposed algorithm provides a way to discover sparse approximations of differential equa- tions from data. This data-driven approach provides insights and a step-by-step algorithm guide to allow more research opportunities to explore the possibility of representing any physical phenomenon with differential equations.

We investigated an observer-based fault diagnosis method for the high altitude airship (HAA). The nonlinear kinematics and dynamics model, which considers model uncertainties and external disturbances, were obtained. The nonliear portion of the model was re-expressed to the linear parameter varying frame mathematically with an auxiliary design. Subsequently, in order to realize fault detection, isolation, and reconstruction, diverse types of robust were observers designed, respectively. Furthermore, all these mentioned observers were achieved via the linear matrix inequation framework. The effectiveness of the proposed method was verified by different fault cases in an HAA simulation system.

In this article, we refine the modification theorem for a compact solvmanifold given in 2006 and completely solve the problem of finding the cohomology ring on compact solvmanifolds.

In this paper, we investigate an initial boundary value problem of a planar magnetohydrodynamics system with temperature-dependent viscosity, heat conductivity, and resistivity. When all of the relative coefficients mentioned above are power functions of temperature, the existence and uniqueness of a global-in-time non-vacuum strong solutions are proved under some special assumptions. At the same time, we obtain the nonlinear exponential stability of the solution. In fact, the initial data could be large if the power of viscosity is small enough.

Dynamic modeling, analysis, and control of fish ecosystems are important for promoting the sustainable development of fish stocks. The objective of this study is to analyze the dynamic behavior of prey-predator systems with discontinuous prey refuge effect and different types of harvesting activities in an uncertain environment. Initially, a Filippov-type prey-predator model with fuzzy parameters is formulated and the positivity and bounded-ness of the solutions and the dynamic properties of Filippov prey-predator system are discussed. Next, from the perspective of effective exploitation and utilization of fish resources, a state linearly dependent fishing strategy is adopted into the system and a fishing model based on threshold feedback is established, as well as an analysis on the complex dynamics of the control system. Finally, to illustrate the theoretical results, computer simulations are presented step by step with an explanation on the practical significance. This study provides a reference for in-depth understanding of the development dynamics of fish resources and scientific planning of fishery resources exploitation.

In this paper, a multiple delays stage-structure predator-prey model with refuge and coop- eration is established. First, the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed by analyzing the characteristic equations of the system. Second, taking time delays as the bifurcation parameters, the existence of Hopf bifurcation at the positive equi- librium is given. Next, the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed based on the center manifold theorem and normal form theory. Moreover, the optimal harvesting policy of the system is showed by using Pontryagin’s maximum principle. Finally, we give the global sensitivity analysis of some parameters by calculating the partial rank correlation coeffi- cients, and some numerical simulations are performed to verify the correctness and feasibility of the theoretical results by using the MATLAB software.

During a first or second course in number theory, students soon encounter several sets of “number theoretic interest”. These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent m, among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.

Binder interphases inside the highly filled polymer bonded explosives (PBXs) are irregularly distributed and extremely thin, but play an essential role in affecting the overall moduli and explosive performance of such heterogeneous media. In the present paper, a spring-type interface model, which was physically equivalent to these practical layers within a fixed error bound, was briefly derived, at first taking account of the fact that the stiffness of the binder material was much lower than that of the explosive crystals. Hereafter, a simplified PBX model consisting of a spherical explosive particle bonded to an infinite explosive matrix by the spring-type interface is designed, and its effective isotropic moduli were analytically determined via the generalized self-consistent scheme. The upper and lower bounds of these moduli were also derived based on the elasticity extremum principles of minimum potential and minimum complementary energies. These explicit expressions can be applied to predict the preliminary elastic properties of highly filled PBXs as benchmarks to validate numerical evaluations and so forth. Eventually, some discussions were made on the size-dependent effect of PBXs with the aid of the simplified model.

The Kronecker product is widely utilized to construct higher-dimensional spaces from lower-dimensional ones, making it an indispensable tool for efficiently analyzing multi-dimensional systems across various fields. This paper investigates the representation of analytic functions within hyper-elliptical regions through infinite series expansions involving sequences of Kronecker product bases of polynomials. Additionally, we examine the growth order and type and T_ρ-property of series composed of Kronecker product bases that represent entire functions. We also delve into the convergence properties of Kronecker product bases associated with special functions, including Bessel, Chebyshev, Bernoulli, Euler, and Gontcharoff polynomials. The obtained results extend and enhance the existing findings of such representations in hyper-spherical regions.

As an emerging architecture, graph Transformers (GTs) have demonstrated significant potential in various graph-related tasks. Existing GTs are mainly oriented to graph-level tasks and have proved their advantages, but they do not perform well in node classification tasks. This mainly comes from two aspects: (1) The global attention mechanism causes the computational complexity to grow quadratically with the number of nodes, resulting in substantial resource demands, especially on large-scale graphs; (2) a large number of long-distance irrelevant nodes disperse the attention weights and weaken the focus on local neighborhoods. To address these issues, we proposed a new model, dual-branch graph Transformer (DCAFormer). The model divided the graph into clusters with the same number of nodes by a graph partitioning algorithm to reduce the number of input nodes. Subsequently, the original graph was processed by graph neural network (GNN) to obtain outputs containing structural information. Next, we adopted a dual-branch architecture: The local branch (intracluster Transformer) captured local information within each cluster, reducing the impact of long- distance irrelevant nodes on attention; the global branch (intercluster Transformer) captured global interactions across clusters. Meanwhile, we designed a hybrid feature mechanism that integrated original features with GNN outputs and separately optimized the construction of the query (Q), key (K), and value (V) matrices of the intracluster and intercluster Transformers in order to adapt to the different modeling requirements of two branches. We conducted extensive experiments on 8 benchmark node classification datasets, and the results showed that DCAFormer outperformed existing GTs and mainstream GNNs.

In this study, we developed a mathematical framework, based on the SIR model, to study the dynamics of two competing virus variants with different characteristics of transmissibility, immune escape, and cross-immunity. The model includes variant-specific transmission and recovery rates and enables flexible parameterization of partial and waning cross-immunity. We conducted stability and bifurcation analyses and numerical simulations to explore the conditions of coexistence, dominance, and extinction of the variants, studying variations in epidemiological parameters that affect endemic prevalence and infection ratios. Our results indicated that transmission rates, levels of cross-immunity, and immunity waning rates are critical in determining disease outcomes, which influence variant prevalence and competitive dynamics. The sensitivity analysis provided the relative importance of these parameters and provided valuable insight into designing intervention strategies. This work contributes to furthering our understanding of multi-variant epidemic dynamics and lays the bedrock for tackling complex interactions involving arising virus variants, finding applications in real-world public health planning.

In this article, we studied the spatial property for a coupled system of wave-plate type in a two-dimensional cylindrical domain. Using an integral differential inequality, we obtained the spatial decay estimates result that the solution can decay exponentially as the distance from the entry section tended to infinity. The result can be viewed as a version of Saint-Venant principle.

This paper is devoted to the study of tractability of the L₂-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class Λ^all, including all continuous linear functionals, and from the class Λ^std, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and (σ, τ)-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.

Traditional recommendation algorithms usually rely on the user’s existing data and historical behavioral records to make recommendations, which often leads to low recommendation accuracy and insufficient personalized experience. To solve these problems, this paper proposes an innovative recommendation algorithm model, neural collaborative filtering with multiple attention mechanism (NCF-MAH). The goal of this model is to enhance the effectiveness of the recommender system. The specific implementation includes constructing a negative sample set and applying matrix decomposition techniques to map user and item IDs to a low-dimensional embedding vector space. In addition, the model processes these embedding vectors using a multi-head attention mechanism to transform them into query vectors, key vectors, and value vectors, and further computes the attention scores and the corresponding weighted sums. Finally, the score prediction is accomplished by fusing the output of the multi-head attention mechanism with the results of the multilayer perceptual machine. The experimental results show that the NCF-MAH model exhibits significant advantages over the baseline model in two key evaluation metrics, hit rate and normalized discount cumulative gain (NDCG), on the MOOC platform and other datasets. Specifically, hit rate and NDCG improved by 13% vs. 9.8% and 15.7% vs. 12.8% when Top-k was set to 10 and 20, respectively.