We investigate Mahler’s real T-numbers through Cantor-type constructions inside $\mathcal{T} \cap \mathbb{G}$, where G is the middle-third Cantor set and $\mathcal{T}$ is the set of real Mahler T-numbers. We build explicit families $\mathcal{T}$^(t) which are homeomorphic to Cantor space and use them to analyze structural, combinatorial, and additive properties of $\mathcal{T}$ . Our results include the existence of descending chains of Cantor subsets of $\mathcal{T}$ of length c; a characterization of ternary expansions in $\mathcal{T}$^(t), showing non- normality but maximal block complexity under sparse forcing, linked to the Adamczewski–Bugeaud criterion; and a sumset theorem proving that for suitable parameters t₁, t₂ one has the interval identity $\mathcal{T}$^(t1) + $\mathcal{T}$^(t2) = [0, 1], which yields the global corollary $\mathcal{T}$ + $\mathcal{T}$ = $\mathbb{R}$ (Erd˝os property) by integer translation invariance. We also discuss implications for cardinal invariants and entropy of the shift map, highlighting the interplay between thin Diophantine sets and large additive structure. To address a natural concern about existence, we include a non-emptiness lemma which shows that our scheduled deletion-and-witness procedure always leaves a non-empty perfect set $\mathcal{T}$ (t).

This paper constructs a Susceptible-Exposed-Infectious-Recovered (SEIR) model that takes into account the dynamic changes in the birth population and includes both vaccinated and unvaccinated populations. The basic reproduction number R₀ is derived, and its global stability is analyzed. It is proven that when R₀ < 1, the disease-free equilibrium point is globally stable. When R₀ > 1, there exist an unstable disease-free equilibrium and a unique endemic equilibrium. Subsequently, we conduct the numerical simulations to validate theoretical results and plot data visualizations for various scenarios. In addition, the parameter sensitivity analysis and the forward bifurcation of the model are performed to examine how each system parameter specifically affects disease transmission. Our results offer practical insights for public health planning, particularly in optimizing vaccination strategies in populations with high birth rates or waning immunity.

Distributed delay is a core concept in time-delay systems and it has been incorporated into the networked respiratory disease model that elucidates the occurrence of respiratory diseases induced by air pollution. Next we studied a respiratory disease model with distributed delay and discrete delay. By analyzing the linearized system, we showed that if the disease-free equilibrium E₀ exists, exhibits global asymptotic stability without any constraint on the variable space. In addition, we proved the global stability of the endemic equilibrium E_* by constructing the Liapunov functional. Our findings contributed networked reaction-diffusion models with distributed delay and discrete delay to the existing body of knowledge. Our research found that distributed delay altered the transmission rhythm of respiratory diseases, which weakened local stability and disrupted global stability, which leads to disease recurrence. Discrete delay could disrupt the ”synchrony” of respiratory disease transmission, thereby inducing Hopf branches that lead to periodic disease outbreaks and undermined global stability, making it impossible to completely eradicate the disease.

This paper investigates the numerical solution for Allen-Cahn equations with perturbation parameters and strong nonlinear terms in general computational domains. To get a global second-order accuracy, we use the second-order semi-implicit Runge-Kutta (SIRK) method and some implicit- explicit (IMEX) methods for time discretization. Then the original problem is transformed into boundary value problems (BVPs) of a modified Helmholtz equation at each time step, which can be solved by a Cartesian grid-based kernel-free boundary integral (KFBI) method. In the KFBI method, the BVPs are reformulated into a corresponding boundary integral equation and then solved iteratively by a class of subspace methods such as the matrix-free generalized minimal residual(GMRES) method, while integrals involved are regarded as solutions to their equivalent interface problems. Unlike traditional boundary integral methods, this method avoids numerical integration of the singular or nearly singular integrals. Instead, it utilizes grid-based operations as an alternative to direct evaluation. Therefore, integral evaluation only requires solving equivalent but much simpler interface problems in a bounding box so that fast elliptic solvers such as fast fourier transforms(FFTs) and geometric multigrid methods are applicable. This makes the KFBI method accurate and efficient when solving constant coefficient elliptic problems in general irregular domains. It can be seen that the accuracy of the present method is verified by numerical examples.

Graph neural networks (GNNs) have been widely studied to handle graph-structured data due to their superior learning capability. Despite the successful applications of GNNs in many areas, their performance suffers heavily from the imbalanced degree distribution (long-tail issue). Most prior studies tackle this issue by graph augmentation, which explicitly increases the communication among nodes by optimizing original topology. In this paper, we employed the perspective of taylor interaction to explore the long-tail issue, and analyzed that there is insufficient interaction between low-degree nodes and their neighbors. In detail, we proposed a novel GNN framework named with degree-aware feature interaction (GraphDAFI), in order to bridge the gap of neighborhood aggregation between head-node embeddings and tail-node embeddings. GraphDAFI comprises two collaborative modules: adaptive feature interaction and degree-aware neighborhood transfer. Adaptive feature interaction leverages node embeddings of the current layer and interactions of the historical layer to perceive potential local information. Then, a unified feature encoder was designed that enhances the interaction to increase the model’s generalization ability. To inject relevant information into low-degree nodes, a degree-aware neighborhood transfer was developed, which updates the node-edge adjacency matrix through a degree-aware strategy to achieve knowledge transfer. Experimental results demonstrate that GraphDAFI achieves excellent performance in semi-supervised node classification compared with the state-of-the-art models.

We study a susceptible–exposed–infected–recovered (SEIR) reaction–diffusion epidemic model that includes susceptible individuals with underlying diseases, focusing on how these comorbidities, diffusion coefficients, and spatial heterogeneity affect disease’s spread. The basic reproduction number R₀ is central to understanding and controlling infectious diseases’ spread. We define R₀, analyze its behavior under low diffusion rates, and investigate the persistence of infection in relation to R₀. Our results show that underlying health conditions increase the value of R₀, enhancing the disease’s transmission potential and persistence. In a homogeneous environment, if R₀ > 1, the system admits a constant endemic equilibrium that is globally asymptotically stable; if R₀ < 1, the disease-free equilibrium is globally attractive, implying eventual disease eradication. Furthermore, we analyze the asymptotic behavior of the endemic equilibrium as the diffusion rates approach zero. Our results indicate that limiting the mobility of susceptible, exposed, and infectious individuals alone is insufficient to eliminate the disease. By examining the influence of diffusion coefficients on the spatial dynamics and disease persistence, we conclude that effective control strategies must extend beyond diffusion control and incorporate interventions targeting additional transmission factors.

This paper has studied a predator-prey model that incorporates a nonlocal fear effect and nonlinear cross-diffusion under homogeneous Neumann boundary conditions. We have derived necessary and sufficient conditions for Turing instability in the presence of both nonlocal fear and nonlinear cross-diffusion by means of linear stability analysis. Moreover, we have investigated steady-state bifurcations induced by the nonlocal fear effect using the Lyapunov-Schmidt reduction.

In this work, a relaxation modulus-based matrix splitting iteration method for large sparse horizontal nonlinear complementarity problems was established. The convergence analysis was presented, where the proposed conditions were shown to be weaker than the existing result. Furthermore, a practical selection strategy of the relaxation parameter was provided by analyzing the error function in each iteration. Numerical examples were given to verify the theoretical improvement and show the effectiveness of the proposed method with the suggested relaxation parameters.

We consider the existence of solutions for some Schrödinger systems on locally finite graphs. Using variational method and Nehari manifold method, the existence and multiplicity of nontrivial solutions are proved. More explicitly, we first give a functional setting of problem and prove the compactness of Sobolev embedding. Then for the Schrödinger system with quadratical nonlinear terms, the existence of nontrivial solutions is proved by using the Nehari manifold method. For the Schrödinger system with cubic type nonlinearity, we first prove the existence of ground state solution and then prove the multiplicity of solutions by combining the Nehari manifold method as well as the Lusternik-Schnirelmann theory.

In this paper, we use the elementary and analytic methods to investigate the computational problem of quadratic character sums related to quaternary and quintuple symmetric polynomials modulo a prime p, respectively. By applying properties of Gauss sums and those of third-order characters modulo p, we establish some identities for such quadratic character sums in the cases of (3, p − 1) = 1 and (3, p − 1) = 3.

In this paper, we study initial-boundary value problem of a predator-prey model with taxis strategies and a nonlinear growth rate for the predator. We establish that, for any spatial dimension (N ≥ 1), the model admits positive classical solutions that are globally existent and uniformly bounded. Our results demonstrate that the nonlinear growth rate can effectively restrain the aggregation of predators. Furthermore, by constructing a suitable energy functional and combining it with the previously established uniform boundedness result, we analyze the global asymptotic stability of the coexistence steady state.

Amid the global challenge of mosquito-borne disease transmission, sterile insect technology (SIT) has emerged as a promising biological control strategy. Based on the classical Beverton–Holt model and the assumption of complete cytoplasmic incompatibility (CI), we develop a discrete-time model with overlapping generations and a corresponding integro-difference equation (IDE) to investigate the population dynamics under different release strategies for infected male mosquitoes. For the discrete-time model, we apply stability and bifurcation theory to determine the existence and stability of equilibria and derive the release threshold r^* above which the wild mosquito population is successfully suppressed. Analysis of the IDE model yields a lower release threshold r^** and a critical patch-size L^*. Comparison of the two models demonstrates that spatial diffusion reduces the required release threshold for achieving population suppression.

This work provides rigorous verification of the super-smoothing effect of higher-order fractional Laplacian dissipation. Shang and Zhao (2017) have proved the global regularity of classical solutions of the 2D incompressible magneto-micropolar equations with linear velocity damping u, microrotational dissipation (−∆)ω, and magnetic diffusion $(-\Delta)^{\beta} b, \beta>1$. This paper is devoted to further investigating the large-time behavior of global smooth solutions of the system with $1<\beta \leq \frac{3}{2}$. We apply the negative Sobolev space to overcome the difficulty caused by fractional-order dissipation and establish $\int_{0}^{t}\|\nabla b(\tau)\|_{L^{2}} d \tau \leq C$. Furthermore, by fully exploiting the special structure of the system and combining the properties of a heat operator with the generalized Fourier splitting methods, we obtain the decay rates of the solutions and their first-order derivatives for $1 \leq p \leq \frac{2}{\beta}$.

Every ring containing a unity has a maximal right ring of quotients. In this paper, we investigate a skew n-derivation ϕ on a class of triangular rings and use the theory of maximal right ring of quotients to demonstrate that ϕ is an extremal skew n-derivation. This result not only significantly generalizes previous findings on derivations, but also forges a strong connection between the theory of derivations and that of rings of quotients, providing a tool for characterizing functional identities in various specific operator algebras.

This paper addresses the numerical method for the generalized Allen–Cahn equation featuring nonlinear mobility and a convection term. We propose a linear second–order finite difference scheme that adheres to the discrete maximum bound principle (MBP). The scheme is discretized using the leapfrog finite difference approach, incorporating a stabilized term in time, an upwind scheme for the convection term, and a central–difference scheme for the diffusion term. It is demonstrated that the discrete MBP holds under reasonable constraints on both the time step size and the coefficient of the stabilized term. Additionally, we provide an L^∞–error estimate for our proposed scheme. Several numerical experiments are conducted to validate our theoretical findings.

In this paper, we consider the Dirichlet problem of three-dimensional inhomogeneous incompressible micropolar equations with density-dependent viscosity. Under the assumption that the coefficients are power functions of the density, we establish the global existence of strong solutions as long as the initial density is linearly equivalent to a large constant state. There is no restriction on the size of the initial velocity and micro-rotational velocity. As a byproduct, we prove the exponential decay for the solution.

In this paper, we study the SIR model and give a strategy to control the epidemic. Based on discrete-time observation, the linear control term is added and the upper bound of time delay is given in order to make the epidemic disappear. Both the deterministic and stochastic cases are considered. The novelty of this paper lies in the fact that it only controls for the infected population. Numerical examples verify the theory results.

Electrical impedance tomography (EIT) is a noninvasive imaging technique that reconstructs the internal conductivity distribution of an object by measuring electric potentials on its surface. Mathematically, EIT is twofold: the forward problem solves a generalized Laplace equation with conductivity coefficients to determine the electric potential, while the inverse problem requires estimating conductivity from noisy and incomplete boundary measurements-an ill-posed task highly sensitive to noise. This paper proposed a hybrid deep learning–evolutionary framework for EIT image reconstruction. The fully-connected feedforward neural networks (FNNs) and convolutional neural networks (CNNs) were trained to approximate the forward map from conductivity to electric potential, eliminating the need for repeated finite element simulations during inversion. For the inverse problem, we formulate the recovery of conductivity distributions as a global optimization task using differential evolution and its variants. Among them, success-history based adaptive differential evolution (SHADE) achieved the most accurate and robust results. While the proposed CNN-SHADE algorithm demonstrated competitive reconstruction performance, the FNN-SHADE approach provided a favorable trade-off between accuracy and computational efficiency. Furthermore, the FNN-SHADE framework outperformed the traditional finite element-based method in computational speed, while maintaining accuracy. By embedding a neural forward operator into the differential evolution loop, the framework offered a scalable, data-driven alternative to traditional EIT reconstruction methods.

In this paper, we defined and classified the algebraic Schouten solitons that are associated with the perturbed Bott connection on three-dimensional Lorentzian Lie groups possessing three distinct distributions. By transforming equations of perturbed Bott algebraic Schouten solitons into algebraic equations, we found that G₃,G₅,G₆, and G₇ have perturbed Bott algebraic Schouten solitons under the first distribution.G₃,G₆, and G₇ have such solitons under the second distribution. Additionally,G₂,G₃,G₄,G₅,G₆, and G₇ have such solitons under the third distribution.

In this paper, the problem of finite-time control for a group of quadrotor aircraft with an unknown input-saturation structure is investigated. Based on backstepping design, finite-time formation control algorithm is designed. Within the multi-agent systems, leaders are tasked with achieving formation consensus with an external leader. Concurrently, for the followers, a leader-follower consensus controller incorporating the dynamic leader’s input is developed utilizing adaptive strategies and low-gain techniques. Thus, a hierarchical formation is achieved where leaders track the exogenous leader, followers maintain consensus with their leaders, accomplishing complex multi-unmanned aerial vehicle formation tasks.

This paper proposes a novel numerical scheme based on an improved charge simulation method for computing conformal mappings of bounded multiply connected domains. The core of the method reformulates the mapping problem into a constraint system constructed via charge simulation. This system is then solved efficiently using the symmetric successive over-relaxation incomplete Cholesky conjugate gradient method, which is particularly well-suited for handling the ill-conditioned systems inherent to such problems. Numerical experiments show that our method achieves significantly higher accuracy and improved convergence rates compared to the conventional Gauss-Seidel iteration. The results confirm the robustness and practical potential of the proposed framework, establishing it as an efficient and reliable tool for computing conformal mappings of domains with high connectivity and complex geometry.

Dental segmentation is a critical step in computer-aided orthodontic treatment planning, but accurate segmentation still faces numerous challenges due to complex tooth morphology, ambiguous gingival boundaries, and clinical issues such as malformed teeth, crowding, and malocclusion. This paper proposes GFACNet, a network that integrates geometric features and anatomical constraints for 3D dental segmentation from intraoral scan data. Our method comprises three key innovations: 1) a morphology-aware graph construction (MAGC) mechanism that adaptively constructs graph structures based on dental geometric characteristics, 2) a multi scale transformer (MST) feature integration module that processes features at different scales while capturing both local and global context, and 3) a hierarchical anatomical constraint loss (HACL) that incorporates multi level anatomical features to guide anatomically consistent segmentation. Experiments on real intraoral scanning datasets demonstrate that GFACNet significantly outperforms existing methods in handling complex dental morphologies, particularly in cases of malformed and missing teeth. Additionally, our method requires reduced computational resources while providing a more practical solution for clinical applications.

Different subthreshold resonance patterns characterized by different amplitudes and frequencies are evoked from the resting potential of the medial superior olive neurons, playing important roles in sound localization. In the present paper, the conditions and nonlinear cooperation mechanisms between two ionic currents for the different patterns are obtained in a neuronal model. As the resting potential is close to the half-activation voltage of low threshold potassium current (I_KLT), simulation results of resonances match the experimental observations. For weak stimulation with small amplitude, slow hyperpolarization-activated cation current (I_H) is not activated, whereas fast I_KLT is activated to mediate a common single resonance, exhibiting small amplitude with symmetry to the resting potential and high frequency for sound localization. For strong stimulation with large amplitude,I_H is also activated to mediate a low frequency resonance and enhance the frequency of resonance mediated by I_KLT for precise sound localization. Then, an uncommon phenomenon of double resonances appears, which exhibits large and non-symmetrical amplitude. The different resonances are well explained with the fast frequency response of I_KLT and slow response of I_H. Interestingly and paradoxically, an oscillation not around its steady value appears at a high frequency of stimulation, since I_H is too slow to recover. If the time scale of two currents becomes approximate, the paradoxical oscillation disappears. Finally, for the double resonances, the phase trajectory is wide enough to cover a large part of nullclines with nonlinearity, resulting in large and asymmetrical amplitudes. For the single resonance with small and symmetrical amplitude, the trajectory is narrow to cover only a local linear part of nullcline. These results present feasible measures to modulate the high frequency response of the medial superior olive neurons for sound localization.

The gamma distribution is an essential distribution for modeling data in different areas such as insurance, finance, reliability, and many fields of engineering. This study proposes a new sophisticated distribution as an alternative to the gamma distribution with tractable properties. Mathematical characteristics and the parameter estimation process of the newly defined model are studied. Two datasets from two different disciplines, education and finance, are used to demonstrate the importance of the new model. Moreover, the WMLdist cloud-based application is developed to simplify and spread the use of the proposed distribution.

In this work, the analytical pricing of maximum and minimum options under the integrated scenario of interdependent stochastic volatility and jump, and the stochastic interest rate, were investigated by means of the composite Mellin transform approach. The analytic expressions of Mellin transform functions of the price of the maximum put option, minimum call option, and the exchange option were derived by different partial differential-integral equations (PDIEs). Meanwhile, the explicit price of other maximum and minimum options were obtained by means of the payoff decomposition technique and the parity relation of max-min and exchange options. In addition, the convergence of solutions of PDIEs was further demonstrated by transform techniques and decomposition skills. The simulation of the price process of two underlying assets was given to present the effectiveness and uniqueness of the proposed model. Finally, numerical analysis was implemented to examine the accuracy of the PDIE method and the validity of key parameters.

Convolutive non-negative matrix factorization has been a dominant analytical technique for deriving interpretable insights from data in speech processing, image analysis, data mining, biomedicine, and other fields. In this paper, a sparse convolutive non-negative matrix factorization model was introduced by incorporating an ℓ₁ regularization on representation matrices. This enhancement not only preserved the inherent characteristics of convolutive non-negative matrix factorization, but also promoted sparse data representation, thereby facilitating more efficient data storage and analysis. An alternating minimization algorithm for the presented model was proposed by integrating the alternating direction method of multipliers with the accelerated iterative shrinkage- thresholding algorithm. In addition, a convergence result was presented that the convergence point of the algorithm necessarily constitutes a stable point of the problem. Experimental results showed that the proposed algorithm yielded sparser solutions for synthetic data designed to simulate sparse representation scenarios, and achieved practical applicability in speech dataset, validating its potential for real-world signal processing tasks.

In this paper, a predator-prey model with cross-diffusion and two delays is investigated. First, the conditions for local stability and Turing instability of positive steady-state solution are studied separately when the system was without and with diffusion. Second, the existence and the stability of Hopf bifurcation were investigated by computing stability switching curves in the parameter plane with two delays. Moreover, explicit formulas for determining the stability and the direction of the bifurcation periodic solutions were derived using the normal form theory and the center manifold theorem. Finally, the theoretical results were verified by numerical simulations.

Let p₁,p₂, ... ,p₇ be primes. In this paper, we first show that when k₁ = 23, every sufficiently large odd integer can be represented as the sum of one prime square, five prime cubes, one prime biquadrate and at most k₁ powers of 2. We further prove that for k₂ = 45, every pair of sufficiently large odd integers satisfying certain necessary conditions can be represented as a pair of equations involving one prime square, five prime cubes, one prime biquadrate and at most k₂ powers of 2.

This paper proposes and studies a predator-prey model incorporating distributed memory and gestation delay to more accurately describe animal movement. First, the stability conditions of the positive equilibrium in the absence of delays are analyzed. Second, the conditions for the occurrence of Turing and Hopf bifurcation without gestation delay are derived. Subsequently, the combined effects of memory delay and gestation delay on the stability of the positive equilibrium are investigated, revealing that their interaction can generate more complex spatiotemporal patterns. Furthermore, normal form theory is employed to determine the direction and stability of the Hopf bifurcation induced solely by memory delay in the absence of gestation delay. Finally, numerical simulations are conducted to validate the theoretical results. In addition, variations in the memory- based diffusion coefficient, memory delay, and gestation delay are shown to trigger transitions among spatially homogeneous/nonhomogeneous steady states and spatially homogeneous/nonhomogeneous periodic patterns.

This expository paper recounts the development and application of the concept of a diffeological groupoid, from its introduction in 1985 to its use in current research. We demonstrate how this single concept has served as a powerful and unifying tool for defining fundamental structures, analyzing the stratification of complex spaces like orbifolds, building a bridge to noncommutative geometry, and, most recently, forging new approaches to geometric quantization. The paper aims to provide a cohesive narrative of this journey, making explicit certain concepts like the “Klein groupoid” and showcasing the enduring vitality of the diffeological groupoid in modern geometry and physics.

We establish the well-posedness of linear elliptic equations with critical-order drifts in L^d and positive zero-order coefficients in L¹ or $L^{\frac{2 d}{d+2}}$, where classical methods are often too restrictive. Our approach relies on a divergence-free transformation and a structural condition on the drift vector field, which admits a decomposition into a regular component and another whose weak divergence belongs to L^$\tilde{q}$ for some $\tilde{q}$ > $\frac{d}{2}$. This condition is essential for constructing a suitable weight function ρ via the weak maximum principle and the Harnack inequality. Within this framework, we prove the existence and uniqueness of weak solutions, significantly relaxing the regularity assumptions on the zero-order coefficients in $L^{\frac{d}{2}}$.

The differentiation of neural activity in the primary visual cortex (V1) for the edges and surface regions of stimuli is a key feature in representing visual information. This differentiation is closely related to the phenomenon of surface inhibition, where neural activity in central surface regions is suppressed relative to edges of square stimuli. However, due to the challenges in manipulating biological experiments and the limitations of simplified models, the detailed synaptic-level analysis of the regulatory mechanisms underlying this differentiation has not been revealed. In this study, a visual information transmission pathway from visual stimuli to the lateral geniculate nucleus (LGN) of the thalamus, and further to the input and output layers of V1, is constructed by using real biological anatomical data to investigate the regulatory mechanisms underlying the aforementioned neural activity differentiation. The model successfully replicates the surface inhibition characteristics observed in biological experiments: the input layer exhibits relatively uniform responses to both surface and edge stimuli, while the output layer shows strong activity in edge regions and suppressed activity in central surface regions, creating a “hole” effect. Through control experiments-specifically, by eliminating long-range connections in L2/3-we find that long-range connections in layer 2/3 of V1 are the necessary conditions for generating the surface inhibition phenomenon within this layer. Furthermore, modulating either the proportion or the spatial distribution of long-range connections in L2/3 can exert regulatory effects on the surface inhibition phenomenon. This not only facilitates our understanding of the neural mechanisms underlying visual processing but also demonstrates the advantages of computational modeling in elucidating internal cortical mechanisms that are difficult to manipulate in biological experiments.