This brief survey is devoted to the famous Brezis-Nirenberg problem, firstly studied in the celebrated paper [1]. Since then, critical equations have been widely studied from many perspectives and in various contexts, including, among others, general local operators, nonlocal operators, mixed local and nonlocal operators, higher-order operators, and operators in non-Euclidean contexts. It is interesting to note that, regardless of the context, most results concerning critical Dirichlet problems are obtained through suitable adaptations of the original argument of Brezis and Nirenberg. The purpose of this paper is to consider the nonlocal counterpart of the Brezis-Nirenberg problem (in which the classical Laplace operator is replaced by its fractional version). Here, we group and summarize the results obtained in [2–6], focusing on the similarities and differences between the local and nonlocal cases, highlighting the innovations and adaptations to consider in the treatment of the fractional case.

In this paper, a new type of fractional-order multi-agent systems (FOMASs) with parameter uncertainties and Takagi-Sugeno (T-S) membership functions is developed on a directed communication topology graph. Unlike existing results that either adopt fixed-gain impulsive control or deal with fractional-order consensus without simultaneously considering communication efficiency and parameter uncertainties, we propose a codesigned T-S fuzzy (TSF) impulsive control framework. By collecting instantaneous local information, leader-following and leaderless distributed TSF impulsive controllers are designed to realize the exponential consensus for the proposed FOMASs, where the impulsive gain is dynamically adjusted by the same set of fuzzy rules describing the system. In addition, some sufficient conditions with regard to the exponential consensus of FOMASs are addressed in terms of linear matrix inequalities (LMIs). Compared with the traditional impulsive controllers with the fixed gain, the designed TSF impulsive controller has faster convergence speed to achieve the exponential consensus of FOMASs with uncertainty, and it has lower energy consumption. We then present an application example in a single-link manipulator to validate the practical efficacy for the proposed theoretical findings.

In this work, we consider an inverse problem for a stationary kinetic equation. Our aim is to determine the source function from boundary measurements together with additional information provided at an interior point of the domain. Unlike existing works, the boundary information comprises not the direct problem solution itself, but the gradients the gradients of the solution and the source function are prescribed on the boundary. We develop a numerical algorithm based on a hybrid strategy that combines the finite-difference method with a bilinear interpolation polynomial approximation. The performance of the proposed approach is demonstrated through several numerical experiments, and the results are reported comparatively via graphs and tables. The numerical tests indicate that the reconstruction errors for the unknown functions remain sufficiently small.

We are concerned with the zero dissipation limit to rarefaction waves with vacuum for a non-conservative viscous compressible two-fluid system in one dimension. In this paper, given a rarefaction wave with one-side vacuum state, we establish the existence of a solution sequence for the system that approaches this rarefaction wave with vacuum as viscosity vanishes and we derive a uniform convergence rate for this approximation. The result is proved by a scaling argument and an energy method.

We establish the existence and multiplicity of normalized solutions to the coupled nonlinear Schrödinger system with linear and nonlinear couplings

satisfying the total mass constraint where the nonlinear coupling parameter β = µ > 0. The system comes from the research on standing waves of coupled Gross–Pitaevskii equations for describing Bose–Einstein condensates. First, we show that, up to translations and sign symmetries, there exists exactly one class of normalized solutions when the linear coupling parameter κ = 0. The least energy level is also explicitly determined. Second, we prove that at least three nontrivial normalized solutions exist under suitable conditions on the linear coupling parameter κ , 0. We present a new approach based on transforming the Schrödinger system with both linear and nonlinear couplings into a Schrödinger system with purely nonlinear coupling. This seems to be the first result concerning the multiplicity of normalized solutions to the Schrödinger system with both linear and nonlinear couplings.

Heating, ventilation, and air conditioning (HVAC) systems are a major contributor to global building energy consumption; however, their control is complicated by inherent parametric uncertainties and time-varying disturbances. To address the limitations of conventional methods, a novel two-stage ”learning-and-reusing” framework was proposed, which fundamentally differs from existing methods by: (i) decoupling parameter learning from disturbance rejection to avoid the single-stage trade-off; (ii) using concurrent learning and an estimator for parameter identification under relaxed excitation conditions and input saturation. In the first learning stage, a concurrent-learning-based adaptive controller accurately identifies key thermodynamic parameters, such as the heat transfer coefficient and the cross-sectional areas of the wall, while simultaneously maintaining precise temperature regulation, thereby building a reliable knowledge base. In the second reusing stage, the identified model is used within a disturbance observer-based robust controller to precisely compensate for time-varying disturbances, such as fluctuating solar radiation and internal heat loads. Simulations on multi-zone building models validated the framework, demonstrating successful parameter convergence and superior robust tracking performance compared to conventional methods. This work offers an efficient, bio- inspired solution for intelligent building thermal management.

This paper investigated a non-autonomous allelopathic phytoplankton model with threshold inhibition and feedback controls. We applied the comparison theorem of differential equations and constructed some suitable Lyapunov functions to obtain sufficient conditions for the permanence and global attractivity of the system. By constructing some suitable Lyapunov-type extinction functions, we further derived sufficient conditions for the extinction of one of the species. The analysis reveals that inter-specific inhibition and toxic substances critically influence the permanence, extinction, and global stability of the two-species system. Some known results are extended.

The Cameron-Praeger conjecture stands as a central problem at the intersection of group theory and combinatorial design, and has inspired sustained research by mathematicians worldwide for decades. In this paper, we took a step forward in the proof of the Cameron-Praeger conjecture. We studied a special case of the famous Cameron-Praeger conjecture in design theory and proved that there are no block-transitive 6-(v,k, 2) designs with k dividing v.

Accurately estimating kinetic parameters of stochastic gene expression is a key challenge in quantitative single-cell transcriptomics. Building on the strengths of traditional methods, such as the method of moments and maximum likelihood estimation (MLE), we propose a novel framework: moment-reduced MLE (MR-MLE). We combined these three parameter estimation methods with four steady-state mRNA distribution computation strategies (exact distribution expression, Laplace approximation, finite state projection (FSP), and Gauss–Jacobi quadrature) to form nine distinct inference methods. These methods were systematically compared on both synthetic and real experimental data across two core dimensions: computational efficiency and parameter estimation accuracy. Results show that the Laplace approximation yields unreliable parameter estimates; the method of moments is computationally efficient but suffers from small-sample bias; the exact distribution solution is the least computationally efficient; and the FSP method exhibits fluctuating efficiency with varying gene expression levels. Notably, MR-MLE integrated with Gauss–Jacobi quadrature achieves the optimal balance between computational speed and inference robustness. Thus, we recommend this method as the preferred choice for fitting experimental data, providing a reliable and efficient computational solution for large-scale single-cell gene expression data analysis.

Brucellosis is a zoonotic disease caused by bacteria of the genus Brucella, posing a significant threats to human health and the development of the livestock industry. Experimental studies have demonstrated significant differences in infection rates among sheep of different ages and sexes. Based on this, this paper develops a time-delayed dynamical model for brucellosis transmission with age-sex structure, coupling the sheep population with environmental contamination, and incorporates a latency time delay to characterize the temporal delay in disease transmission. First, the basic reproduction number is calculated, and the global stability of both disease-free and endemic equilibria is established. Second, for three control measures (i.e., vaccination, isolation of infected sheep, and environmental disinfection) the target reproduction numbers are computed to quantitatively evaluate the relative effectiveness of each strategy in mitigating brucellosis transmission. Numerical simulations reveal that increasing the time delay reduces the number of infected sheep and the concentration of Brucella in the environment. When the target reproduction number exceeds 1, vaccination achieves optimal control efficacy, with required coverage decreasing as the time delay increases. In addition, by adjusting the input rates of female and male lambs and the time delay, the cost-effectiveness of age-based and sex-based vaccination strategies is compared, identifying the parameter regions where each strategy is economically dominant.

This study addresses the consensus problem of multi-agent systems (MAS) subject to external disturbance. Considering the presence of external disturbance factors, a theoretical analysis of consensus for distributed MAS is conducted. First, a state observer and an adaptive disturbance observer are designed to estimate both the node states and disturbance states of each agent in the system. Subsequently, a dynamic event-triggered control strategy is formulated based on the estimated values from the observers. Notably, the dynamic event-triggered strategy incorporates dynamically adjustable parameters, offering enhanced flexibility compared to conventional static event-triggered strategies. Using Lyapunov stability theory and inequality techniques, sufficient conditions for achieving consensus in disturbed MAS are derived. Finally, a numerical example is provided to demonstrate the effectiveness of the observers, control strategies, and theoretical results. This work provides new solutions for the consensus control of heterogeneous MAS with external disturbance, where both the obsever and the dynamic event-triggered controller are equipped with adjustable parameters, rendering the approach more practical for real-world applications.

This paper investigates the existence and stability of a fractional-order labor dynamics model formulated using the fractional Nabla difference operator, specifically the Atangana-Baleanu fractional derivative in the Caputo sense. The model incorporates fractional dynamics to capture memory effects and the complex interactions associated with workforce layoffs. First, we present the mathematical formulation of the system and discuss its relevance to labor dynamics. Using the fixed-point theory, we establish the existence and uniqueness of solutions, demonstrating that the system is well posed. Furthermore, we examine the stability of the model in the Mittag-Leffler-Hyers-Ulam sense, providing insight into its long-term qualitative behavior. Numerical simulations support the theoretical findings and demonstrate that the fractional-order parameter significantly influences the system’s dynamics. Overall, this study offers a more general and flexible framework for modeling layoff processes using fractional calculus.

Let R be an Artin algebra and e ∈ R an idempotent. We study when the recollement ideal ReR induces (left) recollement of Gorenstein defect categories. As a consequence of this phenomenon, concrete triangle-equivalences between Gorenstein defect categories (resp. stable categories of Gorenstein projective modules) of R and R/ReR (resp. eRe) are obtained, and reduction conditions for Gorensteinness (resp. CM-freeness) of R and Gorenstein projective conjecture over R are provided. Finally, some applications in the context of triangular matrix algebras are given.

To understand the significant roles that seasonal variations and diffusion play in population growth, we proposed and investigated an almost periodic reaction-diffusion model incorporating age structure for a single species. The model was formulated as a reaction-diffusion system with a nonlocal term. Using the principal Lyapunov exponent λ^* as a threshold, we analyzed the global dynamics of this model. The results showed that the population will go extinct if λ^* < 0, but persist if λ^* > 0. We further established the existence of a globally attractive almost periodic solution when the population was uniformly persistent in a monotone case. The Nicholson blowflies model was investigated by means of numerical simulations, and the dependence of population development on maturation period, diffusion rate, and spatial environment was discussed.

This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $O\left(k^{2}+h_{x}^{2}+h_{y}^{2}\right)$, where k is a parameter of the maximal time steps and h^x,h^y are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.

Parkinson’s disease (PD) is characterized by increased β-band (13–30 Hz) synchronization. Although previous studies have indicated that the striatum plays a crucial part in generating and transmitting the β-band oscillations, the underlying biophysical basis remains unclear. An extended thalamic–basal ganglia (BGTC) model including the striatum (BGSTC) is used to examine the striatal influence on β-band synchronous oscillations in the globus pallidus pars interna (GPi). This influence is implemented by modulating striatal inhibitory input. Phase-locking value (PLV) and phase consistency (PC) show that the basal ganglia (BG) network exhibits pronounced β-band synchrony in the Parkinsonian state. Moreover, enhancing striatal synaptic inputs may induce β-band desynchronization in GPi via the direct pathway. In contrast, the modulation via the indirect pathway has a much weaker influence and may even exert an opposite effect. These conclusions not only reveal the striatum’s regulatory capacity, but also provide a new avenue for PD therapy.

This paper investigates time-inconsistent linear-quadratic control problems within an open-loop framework, where the objective functional incorporates a non-exponential discounting structure. By introducing a two-point boundary value problem and a Riccati-type equation with an error function, it is established, under appropriate conditions, that the existence of an open-loop equilibrium strategy for the time-inconsistent control problem is equivalent to the existence of solutions to both the two-point boundary value problem and the Riccati-type equation. Furthermore, through an illustrative example, it is demonstrated that there exists an essential distinction between open-loop and closed-loop equilibria in time-inconsistent control problems.

This paper investigated impulsive insulin therapy within an extended glucose-insulin regulatory system that included insulin-degrading enzyme dynamics. Two treatment modes were examined: periodic delivery and state-dependent feedback. For the periodic model, we established solution positivity and boundedness, proved the global asymptotic stability of a unique positive periodic solution for type I diabetes, and derived permanence bounds along with a safe dosing region for type II diabetes. For the state-dependent model, under mild conditions, we demonstrated the existence and global orbital asymptotic stability of a unique order-1 periodic orbit within a physiologically relevant glucose range. Numerical simulations supported the theoretical results and illustrated how the choice of dosing period, bolus size, and threshold level influenced long-term glucose control. The results provided a unified theoretical framework for comparing and optimizing impulsive insulin therapies.

Let B denote a unital prime *-algebra over the complex field $\mathbb{C}$, and let $\xi \in \mathbb{C} \backslash\{0, \pm 1\}$. This paper characterized a family $\Psi=\left\{\varphi_{m}\right\}_{m \in \mathbb{N}}$ of maps (not necessarily additive) from B into itself that satisfy the functional identity

for all $A, B, C \in \mathfrak{B}$, where $A \diamond_{\xi} B=A B+\xi B A^{*}$. It was proved that Ψ satisfies the above identity if and only if Ψ is an additive higher *-derivation and φ_m(ξA) = ξφ_m(A) holds for every A ∈ $\mathcal{B}$. As an application, the result not only generalizes the structure of nonlinear ξ-skew-Jordan triple derivations on prime *-algebras, but also yields a description of nonlinear ξ-skew-Jordan triple higher derivations on several important operator algebras, including standard operator algebras and factor von Neumann algebras.

The main purpose of this paper is to study behaviors of solutions to the Cauchy problem for coupled systems of semilinear damped wave equations with power nonlinearities on the Heisenberg group, where the Fujita critical exponent is determined. By means of the Bihari inequality as well as the Gagliardo-Nirenberg-type inequality and the contraction mapping principle, the local well-posedness of the Cauchy problem is successfully proved. Combining energy estimates and decay estimates together with a contradiction argument, the global existence of solutions is also demonstrated. At the same time, the blow-up results and upper bound estimation of the solutions is derived by using the test function technique. Our main new contribution is the derivation of a fundamental inequality utilized in weighted energy estimates, as well as the selection of exponents in the scaling of two convex functions for constructing test functions.

The vehicle routing problem with time windows (VRPTW) is a classical NP-hard combinatorial optimization problem, where NP denotes nondeterministic polynomial time, and it plays a critical role in modern logistics and transportation systems. Although competitive swarm optimization (CSO) algorithms have demonstrated strong performance in continuous optimization, their effective application to discrete combinatorial problems such as VRPTW remains challenging. In this paper, a hybrid competitive swarm optimization and tabu search algorithm (CSO-TS) is proposed to solve the VRPTW. To enhance the search capability of the CSO framework, a tabu search mechanism with an adaptive neighborhood operation strategy is integrated. Moreover, to achieve a better balance between exploration and exploitation, a path diversity index is introduced to quantitatively evaluate solution diversity based on four distinct indices. The proposed CSO-TS algorithm is tested on 56 Solomon benchmark instances and obtains 23 optimal solutions. Extensive computational experiments and comparative analyses demonstrate that CSO-TS outperforms or is competitive with nine state-of-the-art algorithms in terms of solution quality.

Let $\mathbb{U}$ be the open unit disk, $\Omega=\left\{\omega \in \mathbb{C}: \operatorname{Re} \omega>a,-\infty<a<a_{0}<+\infty\right\}$ with $a, a_{0} \in \mathbb{R}$, and K ≥ 1 and K^′ ≥ 0 as two given constants. In this paper, we study some properties of the class $\mathcal{H}(\mathbb{U}, \Omega)$ of mappings consisting of those sense-preserving Euclidean harmonic mappings from $\mathbb{U}$ onto Ω with the real, normalized conditions of $f(0)=a_{0}, f_{z}(0)>0$ and $f_{\bar{z}}(0)$. We first show that a (K,K^′)- quasiconformal mapping $f \in \mathcal{H}(\mathbb{U}, \Omega)$ need not be Euclidean Lipschitz continuous. Subsequently, we give a sufficient and necessary condition for $f \in \mathcal{H}(\mathbb{U}, \Omega)$ to be (K,K^′)-quasiconformal. Additionally, coefficient estimates, distortion theorems, and area theorems are obtained.

This paper investigates a class of elliptic hemivariational inequalities (HVIs) arising from steady-state heat conduction problems that involve linear and nonlinear mixed boundary effects and explores their applications to optimal control problems. First, we use nonsmooth analysis and pseudomonotone operator theory to establish the existence of solutions of the HVIs for the model. Under a relaxed monotonicity condition, we obtain uniqueness of the solution. Second, we prove that the solutions of the HVIs converge strongly to the solution of the corresponding Dirichlet boundary problem as the coefficient λ tends to infinity. Finally, a convergence analysis is provided for the solutions of the optimal control problem governed by HVIs. The results provide a theoretical framework for the study of elliptic problems with nonmonotone and multivalued boundary conditions, and they contribute to the mathematical foundation of related control and engineering applications.

Transformed gamma distributions are widely used in various fields such as survival analysis, reliability engineering, and environmental studies. Statistical inference for the quantities of the transformed gamma distributions is of great importance in these applications. However, traditional inference methods often rely on large sample approximations, which is not accurate for small sample sizes. To address this issue, we introduced an exact statistical inference framework for quantities of the transformed gamma distributions and applied the framework to loggamma distribution. The performance of the proposed framework was extensively evaluated via comprehensive simulation studies. The results showed that the proposed framework outperforms the parametric bootstrapping method in terms of coverage probability and type 1 error rate. Two real data applications are provided for illustrative purposes.

This paper studied the trajectory tracking performance of electrohydraulic position servo systems subject to impulsive disturbance via sliding mode control. Different from traditional sliding mode control, which in applied in continuous state space, our proposed sliding mode control could restrain the negative effect of impulsive disturbance. The proposed method is based on the linear matrix inequality method; some sufficient conditions were presented to ensure the reachability of the sliding surface and the stability of the resulting sliding mode dynamics, where a relationship between continuous dynamics, impulsive strength, and impulsive frequency was established. It showed that this relationship can fully estimate the effect of impulsive disturbance. Moreover, we provided the estimation of the reaching time, which was related to the impulse actions and initial state. Finally, the simulations demonstrated the validity of the obtained results.

To address the synchronization control problem of heterogeneous neural networks, this paper proposed a novel adaptive impulsive hybrid control strategy incorporating state feedback and event-triggered mechanisms. First, based on Lyapunov stability theory, the effectiveness of the designed controller was established under a static event-triggered scheme, achieving synchronization of heterogeneous neural networks while avoiding Zeno behavior. Then, an auxiliary parameter was introduced to dynamically adjust the triggering threshold, thereby constructing a dynamic event-triggered mechanism. On the basis of the static triggering principle, the validity of the controller under the dynamic event-triggered condition was further verified. Finally, numerical simulations were conducted to demonstrate the effectiveness and feasibility of the proposed method. The results show that, compared with the traditional static triggering strategy, the dynamic event-triggered scheme can guarantee synchronization performance while significantly reducing the number of triggering events and lowering system resource consumption, thereby validating the advantages of the proposed approach.

An SIRS (Susceptible-Infectious-Recovered-Susceptible) model with sexual structure and dual incubation delays was proposed to characterize the effects of homosexual and heterosexual behaviors on the transmission dynamics and optimal control of gonorrhea. First, the nonnegativity and boundedness of solutions were obtained, and the basic reproduction number R₀ was calculated as well. Second, local and global asymptotical stability of the disease-free equilibrium was established if R₀ < 1. If R₀ > 1 the disease was uniformly persistent, and there existed at least one endemic equilibrium. An optimal control strategy was derived based on sensitivity analysis and practical intervention policy. Finally, the theoretical findings were illustrated through numerical simulations, revealing that targeted management of male infected individuals can markedly diminish gonorrhea prevalence, while disregarding the incubation periods tended to substantially overestimate the epidemic scale.

Abnormal respiratory rhythms, a hallmark of many respiratory diseases, largely arise from transitional firing dynamics in the pre-Bötzinger complex (pre-BötC) neurons. Mixed-mode bursting (MMB), characterized by alternating depolarization block (DB) and square-wave (SW) bursts, represents a particularly complex firing pattern of significant research interest. Based on Diekman’s closed-loop respiratory control model, this study applies a fast-slow dynamical analysis and the bifurcation theory to investigate how variations in hemoglobin concentration ([Hb]) govern the emergence and disappearance of MMB. Numerical simulations show that [Hb] variations induce clear transitions between MMB and single SW bursting. A dynamical analysis reveals that the intrinsic slow variable h determines the type of individual bursting, while the slow feedback variable g_tonic, which is regulated by the arterial blood oxygen partial pressure, evolves on a comparable timescale and influences the global trajectories by shifting the critical bifurcation structures. The interaction between these slow variables enables the emergence of MMB. When the slow feedback pathway is fixed such that g_tonic remains constant, MMB can no longer be sustained. These results indicate that MMB in closed-loop systems relies on the dynamic coupling of multiple slow variables rather than on the static value of a single parameter, thus providing a dynamical mechanism for complex rhythm generation.

In recent years, with the development of science and technology, deep learning technology has also been continuously advancing. As an important application field of deep learning, computer vision has increasingly broad application prospects. Among them, object detection is an extremely important branch of computer vision. This technology has been widely applied in many fields such as environmental monitoring, traffic management, and agricultural evaluation. This paper focused on introducing deep learning-based computer vision object detection and small object detection technologies. In general, object detection methods can be divided into two major categories, namely one-stage and two-stage object detection algorithms. In further subdivision, we roughly classified object detection technologies into three categories: 1) object detection frameworks based on the you only look once (YOLO) series; 2) object detection frameworks based on the region-based convolutional neural network (R-CNN) series; 3) object detection frameworks based on the SSD (single shot multibox detector) series. In addition, we also introduced a series of real application scenarios of small object detection algorithms, such as small object detection based on unmanned aerial vehicles (UAVs) and remote sensing images. Finally, we summarized object detection and small object detection, and we look forward to some possible future research or improvement directions of this technology.