Let H($\mathbb{D}$) be the class of all analytic functions on $\mathbb{D}$, ϕ be an analytic self-map of $\mathbb{D}$, u₀,u₁,...,u_n ∈ H($\mathbb{D}$), and $\vec{u}=\left(u_{0}, u_{1}, \ldots, u_{n}\right)$. In this paper, we studied the operators

where $D_{u_{j}, \varphi}^{j}$ is the generalized weighted composition operator To be precise, we characterized the boundedness and compactness of the operators $P_{\vec{u}, \varphi}^{n}$ acting from the weighted Dirichlet spaces $\mathscr{D}_{\alpha}^{p}$ to the weighted Bloch spaces for p ≥ 1, and we also calculated the essential norm of such operators for p > 1.

This article studies the Coleman-Gurtin thermoelastic model with second-gradient effects on the temperature from both analytical and numerical perspectives. Semigroup theory is used to obtain the existence, uniqueness, and continuous dependence of the solutions on the initial parameters. In the one-dimensional case, the polynomial decay of the solutions is obtained. The numerical analysis of the problem is then considered. Employing the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations of a weak form of the thermoelastic problem are introduced. A discrete stability property and an a priori error analysis are also provided. Finally, some numerical simulations are presented to show the numerical behavior of the approximations and the discrete energy.

Conventional 3D image reconstruction algorithms suffer from fundamental limitations in disentangling 3D scene geometry from rendering processes, which results in poor generalization to novel viewpoints and object poses. To address this, we propose adversarial neural radiance fields (NeRF) for efficient 3D-aware synthesis. By reformulating NeRF as the generator within an adversarial paradigm, our approach bypasses traditional encoding-decoding pipelines through a direct coordinate-to-pixel transformation, thus significantly reducing the computational overhead. This bidirectional enhancement enables a generative adversarial network (GAN) to leverage NeRF’s efficient 3D-aware rendering, while NeRF benefits from adversarial regularization to overcome scene-specific overfitting. Extensive evaluations demonstrate that the proposed adversarial NeRF (ANeRF) performs well on benchmark datasets. ANeRF achieves a 3 dB improvement over traditional NeRF in terms of the peak signal-to- noise ratio (PSNR) upon convergence, while accelerating convergence to 6 × 10⁻³ loss in 34.4% fewer iterations. The framework maintains exceptional efficiency with 31.42 minutes per epoch and a memory footprint of 5.25 GB. It represents a 43% memory reduction compared to the baseline methods. This explicit 3D representation enables robust multiview consistency and precise scene manipulation, thus establishing a high-efficiency architectural foundation for potential Augmented Reality/Virtual Reality (AR/VR) content generation and digital twin simulations.

Let G be a group with identity e. The proper trivial intersection (TI) power graph $\mathcal{N}(G)^{*}$ defined on G is the undirected graph whose vertices are the elements of $G \backslash\{e\}$, and two distinct vertices a,b are adjacent if $\langle a\rangle \cap\langle b\rangle=\{e\}$. The purpose of this paper is to explore how the graph theoretical properties of $\mathcal{N}(G)^{*}$ can affect the group theoretical properties of G. In particular, this paper determines the domination number of a proper TI-power graph and classifies finite groups whose proper TI-power graph is planar.

This paper investigated the sphericity test for high-dimensional covariance matrices, which is a fundamental problem in probability and statistics courses. While several existing powerful tests in textbooks and literature are available, they often rely on either the normality assumption or an unknown fourth moment. This work aimed to relax these restrictions, serving as a complement and extension of traditional course content on this subject. Specifically, we constructed a consistent estimator related to the fourth moment and proposed a new test procedure. The proposed test statistic depends on the second, third, and fourth arithmetic means of the eigenvalues of the sample covariance matrix, but does not require the normality assumption. Numerical simulations illustrated that the proposed test maintains satisfactory empirical size, and achieves power comparable to, or in some cases superior to, the existing tests.

In this paper, we investigate qualitative properties of solutions to second-order q-difference inequalities within the framework of quantum calculus. We establish a strong maximum principle on a nonuniform q-grid and show that an interior maximum forces the solution to be constant. As a consequence, we derive a comparison principle and a uniqueness result for linear boundary value problems. In addition, we construct explicit barriers in the constant-coefficient case and obtain pointwise bounds along the grid. The resulting bounds are used to prove a right-hand Hermite–Hadamard-type inequality.

In this paper, we proposed and studied a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods. First, the existence of a stationary distribution for the model was verified using the Lyapunov method. Then, by solving the corresponding Fokker-Planck equation, the explicit expression of the probability density function around the quasi-stable equilibrium point was obtained. Furthermore, the global approximation of the invariant probability density function for the model was explored. In addition, sufficient conditions for disease extinction were derived by constructing a suitable Lyapunov function. Finally, numerical simulations are presented to illustrate the analytical results and to reveal the impact of stochastic perturbations on disease transmission.

This paper investigates the problem of event-triggered stabilization for memristive neural networks (MNNs). To reduce the consumption of limited network communication resources, a memory-based switched event-triggered rule is proposed, in which an integral-type time-memory function is incorporated into the triggering threshold to characterize the average historical state information. The proposed event-triggered rule divides the entire time interval into a variable-length waiting interval and a continuous, event-triggered interval. To enlarge the maximum allowable variable-length waiting interval, a switched controller with different feedback gains is designed for these two intervals. A sufficient condition ensuring asymptotic stability of the closed-loop MNN is derived by constructing a piecewise Lyapunov functional, employing several integral inequalities, and introducing appropriate slack matrices. Subsequently, by decoupling the nonlinearities, a convex matrix-inequalities-based condition is established, enabling the explicit computation of the feedback gains. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed memory-based switched event-triggered stabilization scheme.

Many special functions are defined by differential equations, but useful arithmetic continued fractions often come from their discrete recurrences. This paper studied Riccati-type recurrences and their continued-fraction expansions in the setting of machine-discovered patterns from the enumerated signed-continued-fraction Massey approve algorithm (ESMA). The main contribution was a block method for Riccati recurrences, which was used to prove identities for signed interlaced continued fractions with affine partial denominators. We showed that several ESMA-type conjectures can be reduced to finite matrix identities in the block index. Three representative identities were proved in detail. They involved modified Bessel functions, half-integer Bessel functions, and ratios of integer- order Bessel functions. We also gave positivity checks and computable tail estimates. These estimates justified the passage to the infinite limit and gave explicit truncation bounds. The results showed that the representative formulas studied here were not isolated numerical coincidences.

We mainly investigated the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph G = (V,E). We first presented an existence result of mountain pass type nontrivial solutions when the nonlinear term F satisfies the super-(p,q) linear growth condition, which is a generalization of those results in [6]. Then, we mainly showed that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also presented the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the local minimum type nontrivial solutions. Subsequently, we presented an example on optimal control as an application of our results. Finally, we gave a nonexistence result and some results for the corresponding scalar equation.

In this paper, we proposed an orthogonal series nonparametric modal regression (OS-NMR) method for estimating the conditional mode under a unimodal setting. The unknown modal regression function was approximated by a truncated expansion of orthogonal basis functions, which represents the nonparametric component through a finite-dimensional coefficient vector. These coefficients were estimated using a kernel-density-based modal regression criterion and computed by the modal expectation-maximization (MEM) algorithm. Compared with local polynomial modal regression, the proposed method provides a global approximation and avoids repeated pointwise estimation. The orthogonal structure further separates coefficient estimation error from truncation approximation error, which makes the theoretical arguments clearer and more tractable. We established the convergence rate and asymptotic normality of the proposed estimator, and evaluated its finite-sample performance through simulations and a real data application.

In this paper, we present a modified generalized Newton method (MGNM) for solving absolute value equation (AVEs) Ax −|x| =b. Some known iterative schemes are special cases of our method. We prove that the generated sequence of iterates is well defined and converges linearly under condition $\left\|A^{-1}\right\|<\frac{1}{3\left(2 t_{0}+t_{1}\right)+1}$ (t₀ and t₁ are constants greater than or equal to 0, and not both zero). Compared to Newton’s method, the MGNM can guarantee the convergence of the iterative method under weaker conditions. Numerical experiments on various types of problems are conducted to illustrate the superior efficiency and accuracy of the proposed method, showing its superiority over some existing methods.

In this paper, we investigated the geometric and variational properties of the four-dimensional Thurston geometry $N i l^{3} \times \mathbb{R}$. First, we provided a complete classification of Ricci soliton structures by solving the associated system of partial differential equations. We proved that all Ricci solitons on this manifold were strictly shrinking with λ = −3/2 and explicitly showed that none of them admitted a gradient formulation. In contrast, by examining the Yamabe flow counterpart, we demonstrated that $N i l^{3} \times \mathbb{R}$ did not admit any nontrivial Yamabe solitons; the only solutions were trivial and generated by Killing vector fields, revealing a stark geometric duality between volume-preserving and conformal deformations. Finally, we shifted to the variational theory of maps and characterized the space of harmonic vector fields. While left-invariant fields were trivially harmonic, we utilized the tension field framework to construct a rich, nontrivial family of harmonic vector fields depending solely on the x-coordinate, providing their precise analytical description.

We established a new Gronwall inequality for the discrete generalized proportional fractional derivative. The result extended known discrete fractional inequalities and provided explicit bounds, uniqueness criteria, and Ulam–Hyers stability for nonlinear delayed fractional systems. An example was provided to illustrate the theory.

This paper investigates calibration of the jackknife empirical likelihood for U-statistics and U-shaped estimating equations. By adjusting the mean and variance of the asymptotic distribution, we propose two calibration methods that control the Type I error rate in high-dimensional settings. To overcome their limitations, we develop an average calibration method for the high-dimensional jackknife empirical likelihood. Numerical results show that this average calibration significantly outperforms standard approaches.

Let E be an extension of a centerless finite group by another finite group. Under some conditions, it is proved that E has the normalizer property. As an application, we prove that the holomorphs of all the alternating groups and the sporadic simple groups have the normalizer property.

We studied a predator–prey reaction–diffusion system with cross-diffusion and a saturated herd-type functional response. The saturation effect introduces an extra ecological parameter that shifts both the Hopf and Turing–Hopf bifurcation thresholds away from those of the classical square-root model. The positive equilibrium was obtained in closed form, and linearization at this equilibrium yields characteristic equations for every spatial mode. Sharp conditions for local stability, Turing instability, and the codimension-two Turing–Hopf bifurcation were derived from these equations. A complete spectral description was provided so that the center spectrum at the bifurcation point consists of one pair of simple purely imaginary eigenvalues together with one simple zero eigenvalue. Using center manifold reduction and normal form theory for reaction–diffusion systems with cross-diffusion, we derived the third-order normal form on the center manifold. Numerical simulations verified the normal-form classification and displayed the stable spatially homogeneous and inhomogeneous steady states and periodic solutions that emerge near the bifurcation point.

We investigate the well-posedness, regularity, and numerical approximation of a viscoelastic membrane equation involving a Caputo fractional derivative of order α ∈ (1, 2). The fractional term describes an acceleration history memory effect and is therefore different from the velocity memory damping terms commonly used in fractional viscoelastic wave models. By introducing the velocity variable $z=\partial_{t} u$ and setting $\beta=\alpha-1$, the original problem is reformulated as an order-reduced first-order system in which a Caputo derivative of order β ∈ (0, 1) acts on the velocity. On the basis of this reformulation, we establish the well-posedness of the initial boundary value problem under homogeneous Neumann or Robin boundary conditions. For the Neumann case, a shifted spectral scale is introduced to include the zero eigenvalue and to obtain regularity estimates uniformly for both the constant and nonconstant modes. We then develop a fully discrete finite element method by combining continuous piecewise linear finite elements in space with the L1 approximation in time. Under suitable regularity and compatibility assumptions, we prove the error estimate

Numerical experiments are presented to confirm the predicted first-order temporal convergence and optimal second-order spatial convergence in the L² norm. In addition to the Neumann case, a Robin boundary manufactured solution test is included to demonstrate the applicability of the proposed framework under homogeneous Robin boundary conditions. Additional simulations illustrate the influence of the fractional order on viscoelastic membrane vibration and demonstrate the applicability of the proposed method to modal response analysis.

A class of delayed predator-prey models with the Crowley-Martin functional response was investigated in this study. First, the conditions for the stability of the coexistence equilibrium were obtained. Second, by applying the normal form method and the center manifold theorem, the conditions for the occurrence of a Hopf bifurcation induced by the time delay were established, and the direction of the Hopf bifurcation was determined. Finally, numerical simulations were conducted to illustrate the theoretical results. The findings provided insights into the effects of time delay on stability and oscillatory behavior in predator-prey interactions, which are significant for understanding population dynamics in ecology.

In this paper, we investigate multiplayer nonzero-sum differential graphical games for continuous-time nonlinear systems with unknown dynamics. Conventional reinforcement-learning- based optimal control methods for such problems usually suffer from the curse of dimensionality in parameter estimation and high online computational cost. To address these issues, an indirect adaptive control algorithm with distributed sparse identification is developed within an actor–critic–identifier architecture. Specifically, graphical game structural priors are incorporated to reduce the dimension of the unknown parameter space, and a gradient-based integral concurrent learning identifier using a historical data stack is designed for online system identification. In contrast to conventional least- squares-based concurrent learning methods, the proposed identifier avoids covariance matrix inversion and reduces the per-step computational complexity from O(q²) to O(q). By combining the identifier with integral reinforcement learning, the proposed method achieves online approximation of the Nash equilibrium solution to the coupled Hamilton–Jacobi–Bellman equations without requiring an exact system model. A Lyapunov-based analysis is further conducted to establish exponential convergence of the identifier parameters under finite excitation and uniformly ultimately bounded stability of the closed-loop system under a persistence-of-excitation condition on the critic regressor. Comparative simulation results for the considered benchmark example show that, while maintaining a weight approximation error on the order of 10⁻³, the proposed algorithm reduces the convergence time by 75%, thereby improving learning efficiency and alleviating the online computational burden.

An improved single-component Metropolis–Hastings (ISC–MH) algorithm for estimating a kind of time fractional delay diffusion equation (FDDE) is provided. We introduce the symmetric proposal distribution for generating the candidate point of each parameter to obtain the ISC–MH algorithm. To estimate the unknown parameter of the FDDE, firstly, we construct the compact difference scheme to numerically discretize the FDDE. Then the ISC–MH algorithm is used to estimate the fractional order, the diffusion coefficient, and the time delay in the system of the FDDE simultaneously. Finally, some simulation tests of two examples are provided to validate the performance of the algorithms.

Large-scale low earth orbit (LEO) satellite constellations face three fundamental communication challenges: severe channel dynamics caused by orbital velocities exceeding 7.5 km s⁻¹, resource scarcity under heterogeneous quality-of-service demands, and strategy incoordination induced by stochastic multi-hop inter-satellite delays. This paper proposes the intelligent perception communication protocol (IPCP), a unified three-layer perception–decision–execution architecture driven by multi-agent deep reinforcement learning (MARL). We develop the information-exchange deep Q-Network (IEDQN), in which each satellite compresses its local observation–Q-value history into a low-dimensional structured token via a shared Info-Block, aggregates global context through a receiver-specific coordination network (Mess-Net), and selects actions via a shared Q-Net on near- fully-observable augmented states. We prove that IEDQN converges almost surely to an ϵ-approximate Nash equilibrium, with the gap bounded in closed form by the Lipschitz constant of the coordination network. To handle multi-hop delay, we propose the dual-attention delay-aware communication network (DADC), integrating a Transformer-based delay-aware information integration module (DAII) with a dual-attention filtering module (DAIF) that combines Gumbel-softmax hard attention and scaled dot-product soft attention. A formal delay-compensation bound shows that the performance gap to the oracle zero-delay policy decreases exponentially with reception window T_r. Experiments on an NS3-based LEO simulator (5, 7, and 10 agents) show that DADC consistently outperforms the best prior delay-aware method (DACOM) and all delay-agnostic baselines across constellation scales, with advantages that widen as the number of agents increases, while the full IPCP stack doubles throughput (18.4→36.8 Mbps) and reduces end-to-end latency by 39.9% over a static protocol baseline.

Unmanned aerial vehicle (UAV) visual tracking is an important task in low-altitude perception and aerial monitoring, where trackers must handle fast camera motion, scale variation, occlusion, background clutter, and limited onboard computational resources. These challenges can easily lead to tracking drift or failure. To address these issues, we propose a correlation filter tracker named correlation filter learning with dual-modal motion fusion and total variation for UAV tracking (CFLDT). First, the search region is updated using the dual-modal motion fusion (DMMF), and features are then extracted from the calibrated area. This process integrates local motion cues and correlation response information to improve target localization under complex UAV motion. In addition, a multi-response spatial penalty matrix (MRSP) is introduced to dynamically constrain the filter, while the environment-aware regularization term is optimized through the total variation constraint module (TVCM). Finally, both the MRSP and the environment-aware regularization term are incorporated into the filter training process, yielding a more discriminative model for distinguishing the target from the background. Extensive comparative experiments on multiple UAV tracking datasets demonstrate that CFLDT achieves competitive tracking accuracy, robustness, and real-time performance.

We investigate a stochastic shallow water wave equation in Itô form with its deterministic counterpart containing the Degasperis-Procesi and Camassa-Holm equations. The existence of a short time solution is established by transforming the equation into a stochastic transport equation endowed with the initial data in the space $H^{s}(\mathbb{R})(s>3 / 2)$. Moreover, assuming that the initial value meets with the sign condition and belongs to $H^{s}(\mathbb{R}) \cap L^{1}(\mathbb{R})(s>3 / 2)$, we verify the global existence of the solution for the stochastic equation.

While fully coupled fluid-structure interaction models are often used to capture cardiovascular hemodynamics, establishing the well-posedness of the fluid equations on a moving domain constitutes a crucial mathematical foundation. Therefore, this paper investigates an initial- boundary value problem for a blood flow model where the vessel wall displacement, denoted η(t,x), is explicitly prescribed as an a priori given function, rather than being treated as a structural unknown. The inherent moving boundary introduces significant mathematical challenges, compounded by operator singularities in cylindrical coordinates and complex nonhomogeneous boundary conditions. To overcome the singularity at the polar axis, we truncate the domain by introducing a sufficiently small positive constant. We then apply the arbitrary Lagrangian-Eulerian mapping to transform the moving boundary problem into a fixed reference domain, enabling the use of standard mathematical analysis methods. The superposition principle is subsequently used to decouple the problem. Specifically, we use BogovskiĬ’s theorem to construct auxiliary functions that resolve the nonhomogeneous boundary conditions, and apply the Galerkin method alongside energy estimates and compactness arguments to solve the resulting homogeneous parabolic problem. Finally, by applying inverse mapping, we establish the existence and uniqueness of weak solutions in the original time-dependent fluid domain. This rigorous mathematical analysis provides a theoretical complement to the numerical simulations of vascular hemodynamics.

In this paper, we explored the following p-Kirchhoff equation with the L^p-mass constraint

where $a>0, b>0, \frac{3}{2}<p<3, p<q<p^{*}:=\frac{3 p}{3-p}, \Delta_{p} u=\operatorname{div}\left(|\nabla u|^{p-2} \nabla u\right)$, and $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. We considered L^p-mass subcritical, L^p-mass critical, and L^p-mass supercritical cases. Making full use of the Gagliardo-Nirenberg inequality, we established the existence of the minimizers in L^p-mass subcritical and L^p-mass critical cases and the nonexistence in the L^p-mass critical case. For the L^p-mass supercritical case, a radially symmetric ground state was obtained using the Nehari-Pohozaev manifold method. On the other hand, employing a σ-homotopy stable argument of fountain theorem type, we obtained infinitely many radial normalized solutions. In addition, the asymptotic behavior of normalized solutions as $b \rightarrow 0^{+}$ was also explored. Finally, we proved the uniqueness of normalized solutions and provided precise descriptions when $\frac{3}{2}<p \leq 2$.

A subgroup H of a finite group G is called s^∗-normal in G if some normal subgroup T of G satisfies H_G ≤ T, G = HT, and the quotient (H ∩ T)/H_G has square-free order, where H_G is the core of H in G. This paper gave several new criteria for finite groups to be solvable by considering the s^∗-normality of certain n-maximal invariant subgroups under coprime actions, where n ∈{1, 2, 3}.

The objective of this work is to address Sturm-Liouville boundary value problems associated with fractional p-Laplacian equations, where instantaneous as well as non-instantaneous impulses are taken into account. Using variational methods, we establish the existence and multiplicity of solutions for the stated problem. The main novelty of this paper is the combination of a weakened super- p-linear growth condition (which relaxes the classical Ambrosetti-Rabinowitz (AR) condition) with the concave-convex splitting technique in the mixed impulsive Sturm-Liouville p-Laplacian setting. Lastly, the correctness of the key results obtained in this paper is demonstrated through an example.

To address the performance degradation of fully connected tensor network decomposition under high missing rates, where relying solely on global low-rank priors is insufficient, we proposed a novel FCTN decomposition method based on factor double sparsity regularization, termed FCTN-FDS. The proposed method jointly incorporated gradient factor regularization and structural sparsity regularization into the FCTN framework, constraining the model at the same time from two levels: Factor representation ability and network structure, to enhance the robustness of rank selection and reduce redundant parameters. Furthermore, an efficient algorithm based on proximal alternating minimization was designed to guarantee convergence and computational efficiency. Experimental results on color image, multispectral image, and MRI recovery tasks demonstrated that the proposed method achieves superior performance in recovering fine details and complex textures compared to existing approaches, validating its effectiveness and strong representational capability.

This paper presents a stochastic human immunodeficiency vrius transmission model driven by a log-normal Ornstein–Uhlenbeck process, designed to capture the stochastic fluctuations in the transmission rates across multiple stages of infection while incorporating co-infection characteristics. First, by constructing suitable Lyapunov functions, the existence and uniqueness of the global posi- tive solution to the model are proved. Subsequently, it is shown that the system admits a stationary distribution when R^S 0 > 1; under this condition, an analytical expression for the probability density function around the quasi-equilibrium state is derived. Furthermore, a sufficient condition for disease extinction is established, namely R^E 0 < 1. Finally, the theoretical results are validated through numer- ical simulations, and the impacts of the stochastic process’s parameters on disease transmission are investigated. The results indicate that a smaller reversion speed and a larger fluctuation intensity of the Ornstein–Uhlenbeck process tend to destabilize the epidemic system.

Based on Morse theory and the mountain pass lemma, we investigate the existence and multiplicity of nontrivial periodic solutions for a class of second-order partial difference equations. If the nonlinearity is resonant or sublinear at infinity, a series of results under reasonable assumptions are established. Also, two examples are provided to illustrate our main results. Our obtained results generalize and improve some existing ones.

This work proposed a stochastic SEIR epidemic model incorporating psychological effects coupled with an Ornstein-Uhlenbeck (OU) process. The model focuses on the fact that the incidence rate of diseases is disturbed by environmental noise, and it is characterized by a mean-reverting OU process. Unlike traditional white noise, OU noise is consistent with the continuity and stability characteristics of real-world environmental disturbances. First, the existence and stability of equilibria for the corresponding deterministic model were discussed, laying the foundation for the subsequent dynamical analysis. Second, utilizing Itô’s formula, the existence and uniqueness of a globally nonnegative solution for the stochastic model were proven, and the sufficient condition for disease extinction was discussed. Subsequently, by constructing appropriate Lyapunov functions and employing ergodic theory, the sufficient condition for the existence of a stationary distribution was established, namely,R^s 0 > 1, which characterizes the stochastic persistence of the disease. Furthermore, by linearizing the stochastic system around the endemic equilibrium and solving the resulting covariance matrix equation, we derived an explicit expression for the probability density function characterizing the stationary distribution in the vicinity of the endemic equilibrium. Finally, we employed numerical simulations to validate the theoretical findings and to investigate the influence of fluctuation intensity on the system’s dynamics.