This paper investigates an optimal control problem governed by variable-order Caputo fractional dynamics. The main difficulty arises from the fact that, for a type-I variable-order Caputo derivative, the corresponding adjoint operator is not obtained by a direct replacement of the constant fractional order in the standard right-sided Riemann–Liouville derivative. Instead, it must be derived from the dual kernel induced by the variable-order Caputo operator. Based on this observation, we formulate a right-sided variable-order adjoint operator and establish the associated integration-by-parts formula. Under suitable regularity assumptions, first-order necessary optimality conditions are derived for a class of quadratic optimal control problems with nonlinear variable-order fractional dynamics. The resulting optimality system consists of a forward variable-order Caputo state equation, a backward adjoint equation involving the dual-kernel operator, a stationarity condition, and a natural transversality condition. A predictor–corrector forward–backward sweep method is then developed for the numerical approximation of the coupled system. The adjoint equation is treated by a backward Volterra-type approximation motivated by the fractional memory structure. Numerical examples, including linear-quadratic, nonlinear scalar, and coupled nonlinear systems, illustrate the feasibility and mesh-refinement behavior of the proposed approach.

This paper investigated the asymptotic behavior of a generalized Nicholson’s blowflies model characterized by nonautonomous coefficients and three distinct time-varying delays. The dynamics of the population were governed by the following delay differential equation:

Using a rigorous analytical approach, we established new sufficient conditions that guaranteed the global exponential stability of the zero equilibrium. These conditions provided valuable insights into scenarios of population extinction or suppression under fluctuating environmental influences. Our results extended and improved upon existing literature by incorporating three independent delays h₁(t), h₂(t), and h₃(t), offering a more comprehensive framework for understanding the complex feedback mechanisms in biological systems. To illustrate the practical applicability of our theoretical findings, a numerical example demonstrated the convergence of solutions toward the equilibrium, confirming the robustness of the stability criteria under various conditions.

This work investigated the decentralized load frequency control (LFC) problem for multi-area power systems with renewable energy sources under aperiodic denial-of-service (DoS) attacks. A decentralized resilient event-triggered mechanism (DRETM) was proposed, in which a standard output-based ETM governs transmissions during DoS-free intervals and a forced recovery transmission is introduced after each DoS-active interval, enabling prompt closed-loop recovery while explicitly accounting for network transmission delays. A decentralized dynamic output-feedback (DOF) switched controller was developed without centralized state collection. Sufficient conditions are derived for local exponential stability and weighted H_∞ attenuation under aperiodic DoS attacks. The neighboring-area frequency terms were recognised as internal physical interconnection inputs rather than independent disturbances. After the decentralized controller gains were recovered, the original physical feedback was explicitly restored and the complete two-area closed loop was verified by a fixed-gain joint stability condition with two independent equivalent-delay channels.

Accurate prediction of drug-target interactions (DTIs) is a fundamental task in computational drug discovery, as it can effectively reduce the cost and time required for experimental screening. However, although most existing methods consider both sides of an interaction, they still focus mainly on the intrinsic features of drugs and proteins and are therefore insufficient to capture higher-order behavioral context and complementary interaction patterns in DTIs. To address this issue, we propose a novel DTI prediction framework that integrates drug molecular structure, protein sequence information, and behavior-aware contextual representations within a unified architecture. Specifically, drug compounds are encoded by a graph neural network (GNN) to capture molecular topology, protein sequences are modeled to learn semantic features, and a protein-centric behavioral sequence is introduced to characterize pharmacological interaction patterns. In addition, an attention-based fusion mechanism is introduced to model dependencies within the behavioral sequence and selectively aggregate informative behavioral contextual signals. Extensive experiments on three benchmark datasets demonstrate that the proposed method consistently outperforms representative baseline models. These results indicate that the proposed framework provides an effective computational solution for DTI prediction.

In this paper, a new general age-structured human immunodeficiency virus (HIV) model with two routes of infection, antiretroviral therapy, and humoral immunity was proposed, different from common age-structured models due to the consideration of dominant activation of latently infected cells into infected cells rather than in the boundary condition. First, the well-posedness of the model solutions was given. Second, considering the dominant activation, we discussed the existence of immune inactivation equilibrium E₁ and immune activation equilibrium E^* in two special cases. Then, the global stability of the disease-free equilibrium E₀ was analyzed by constructing appropriate Lyapunov functions. Finally, the global behavior of the model with a saturated functional response was further investigated through numerical simulations, and the role of antiretroviral therapy in slowing down HIV transmission in the body was analyzed.

Wind speed is a key factor for assessing the wind energy potential of a region. Therefore, determining the best statistical distribution for modeling wind speed data is crucial for effective resource assessment and optimized energy production. The Weibull distribution is widely used for this purpose; however, it may not always provide the most adequate fit. In this study, Kumaraswamy–Weibull (KumW) distribution is used to model wind speed data as a more flexible alternative to Weibull distribution. In estimating the parameters of the KumW distribution, maximum likelihood (ML) methodology is employed. The resulting likelihood equations are not solvable explicitly because they involve nonlinear functions of unknown parameters. In order to obtain the ML estimates of the KumW parameters, this study utilizes three popular metaheuristic algorithms, the genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE), which have emerged as effective alternatives to the traditional numerical methods for solving computationally challenging optimization problems. One of the key challenges in using metaheuristic algorithms is the proper determination of an effective search space. In the literature, search spaces are often chosen arbitrarily, which may lead to reduced accuracy. As a novel contribution, we propose a new search space based on the bootstrap confidence intervals of the unknown KumW parameters for GA, PSO, and DE algorithms. The performances of the estimators obtained using GA, PSO, and DE based on proposed search space are compared with the corresponding estimators based on fixed search space in terms of bias, mean squares error (MSE), and deficiency (Def) criteria via an extensive Monte Carlo simulation study. Finally, the annual and seasonal wind speed data obtained from two stations located in the northern part of the Edremit Gulf, Turkey are modeled using the KumW distribution to demonstrate the practical utility of the proposed search space. The fitting performance of KumW is compared with that of the well-known and widely used Weibull distribution with respect to the root mean square error (RMSE), the mean absolute error (MAE), the coefficient of determination (R²), the Akaike information (AIC) and Kolmogorov–Smirnov (KS) criteria. Moreover, the power density error (PDE) criterion is utilized to evaluate the capability of KumW distribution in estimating wind power.

In this paper, we propose an iterative method for optimal control problems governed by elliptic PDEs based on the generalized multiscale finite element method (GMsFEM). In the iterative method, we first reformulate the original optimal control system into a new system with parameter- independent coefficients and parameter-dependent right-hand sides. Subsequently, we adopt the fixed- point iteration method to solve the reformulated system to get the optimal solutions. To quantify the statistics of the optimal control problems, we need to solve a coupled optimality system for a large number of samples in the stochastic space. Therefore, the computational cost is extremely large, and it may even lead to the “curse-of-dimensionality”. When the diffusion coefficient exhibits multiscale features, the iterative process can be performed in the reduced multiscale space via GMsFEM to en- hance computational efficiency. The proposed iterative method can be split into two stages: an offline stage and an online stage. In the offline stage, local multiscale basis functions are constructed from the deterministic optimality system to build the reduced space. In the online stage, the reformulated model is projected onto the reduced space and solved via fixed-point iteration. Finally, we will present two numerical examples to verify the effectiveness of the proposed iterative method.

To address vegetation degradation in arid and semi-arid ecosystems, existing vegetation– water models neglect the directional transport of soil water induced by root uptake (cross-diffusion) and human regulation at boundaries. This paper formulates a vegetation–water reaction–diffusion model incorporating a cross-diffusion term and a Robin-type boundary condition. First, the existence and uniqueness of weak solutions are proved with a priori estimates. Second, under bilateral pointwise constraints and an integral constraint on the cumulative boundary-regulation effort, the existence of an optimal boundary control is established via a penalty method. The adjoint system is derived using the Lagrange multiplier method, yielding first-order necessary and second-order sufficient optimality conditions, and the twice continuous Fréchet differentiability of the control-to-state mapping is demon- strated. Numerical simulations verify the effectiveness of the proposed control strategy. The framework provides theoretical support for ecological restoration in water-limited regions.

Lai et al. [1] recently studied a non-autonomous predator–prey system with an additive Allee effect, prey refuge, antipredator behavior, and indirect feedback control. Their Theorem 5.1 gives a sufficient condition for predator extinction and prey persistence. We show that its hypotheses involve unknown solution components and therefore cannot be checked a priori; additionally, we identify a sign error in the proof. Then, we replace that result with conditions stated entirely in terms of the prescribed coefficients and parameters. The first condition yields predator extinction while the prey remains uniformly persistent. Its main ingredient is an optimization bound for the predator’s net benefit from the prey, which avoids the sign issue in the original argument. Additionally, we give three sufficient mechanisms for total extinction: strong intraspecific competition, uniformly bounded intrinsic prey growth combined with strong competition, and a quantitative integral condition involving the prey dissipativity bound. These conditions are not intended to be necessary, but they are directly checkable. For the autonomous model, we derive an explicit critical refuge level for predator invasion and discuss invasion numbers in autonomous and periodic settings. Numerical examples illustrate both the autonomous and non-autonomous cases.

The evaluation of the Riemann zeta function at odd integers remains an important analytical problem, partly because many of its classical series representations converge slowly. In this paper, we develop a continuous filtering framework for analyzing weighted series representations associated with ζ(5). Rather than relying on closed-form algebraic transformations, the proposed approach introduces analytically designed weighting sequences (filters) within a continuous integral formulation. We show that this construction naturally interprets discrete filtering as a continuous transformation of the underlying integral kernel. Furthermore, the proposed framework reveals a fundamental trade-off between convergence enhancement and preservation of the kernel structure. By establishing a continuous–discrete connection, the framework provides a unified perspective for studying filtered series representations and offers new insight into the analytical structure of zeta-related series.

This manuscript presents novel spectral techniques based on shifted orthonormal normalized Jacobi polynomials (SONJPs) for solving two types of fractional differential equations. The first method uses the tau technique to solve the nonlinear boundary value problem (NBVP), and the second method uses the collocation approach to solve the time-fractional generalized Fisher problem (TFGFP) that has importance in mathematical biology, population dynamics, and economic modeling. We derive new relations that form the foundation for the proposed numerical techniques, thereby reducing the problem and its governing conditions to a nonlinear algebraic system that can be solved efficiently using Newton’s method or another suitable iterative technique. We present numerical results with comparisons to illustrate the suggested algorithm’s high accuracy and applicability.

This paper investigates practical finite-time synchronization and approximate controllability of neutral fractional stochastic differential systems driven by the Rosenblatt process. The considered systems involve multiple time-varying delays, non-instantaneous impulses, and superlinear growth conditions. First, the existence and uniqueness of mild solutions are established by applying the Krasnoselskii fixed-point theorem and the Banach contraction principle. Second, a nonlinear finite-time feedback controller is designed for the drive-response systems, and sufficient conditions for practical finite-time synchronization under non-instantaneous impulses are derived. A key technical contribution is the development of a generalized Bihari-type inequality for fractional integral inequalities with weakly singular kernels and multiple time-varying delays, which provides explicit estimates for the synchronization settling time. Furthermore, based on the synchronization analysis, the approximate controllability of the response systems is investigated by employing the controllability Gramian and the resolvent operator approach. Finally, numerical simulations are presented to illustrate the effectiveness of the theoretical results.

This work investigates a coupled generalized Schrödinger-Boussinesq system with a p-power nonlinearity. These equations appear in several physical applications. To begin, we present a complete classification of infinitesimal point symmetries for the system. Then, low-order local conservation laws are derived using the multiplier method, providing conserved quantities such as momentum, mass, and charge. By introducing a potential variable, we derive a Lagrangian formulation for the potential system and find nonlocal conservation laws via Noether’s theorem. Finally, we explore travelling wave solutions by reducing the governing equations to a coupled autonomous system of ordinary differential equations. Several families of solutions are constructed, and their physical relevance is discussed.

We studied the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation. We first established its continuous mass, energy, and momentum conservation laws, and then proposed an alternating direction implicit (ADI) compact finite difference scheme combined with Strang splitting. The splitting technique isolates the nonlinear term for an exact analytical solution via phase rotation, while a 3-point Padé approximation resolves the linear term to fourth-order spatial accuracy without requiring boundary ghost points. We rigorously proved the scheme is unconditionally stable with a convergence rate of O(τ² + h⁴_x + h⁴_y). Numerical experiments validated the theoretical analysis and demonstrated the method’s high-order accuracy and efficiency.

This paper investigates the dynamical behavior of a Euler-generated discrete predator-prey map incorporating a Holling type-IV functional response and Leslie-Gower predator dynamics. Since the Leslie-Gower term is singular at zero prey density, the biologically feasible state space is restricted to x > 0 and y ≥ 0. Within this domain, the existence and local stability of the boundary and positive equilibria are analyzed, and sufficient conditions for the possible configurations of positive equilibria are derived from the discriminant of the associated cubic equation. The positivity of the uncontrolled and controlled maps is also examined through analytical one-step conditions and direct numerical feasibility checks. By employing center manifold reduction and normal form theory, analytical conditions for Flip and Neimark-Sacker bifurcations at the positive equilibrium are obtained. Two stabilization approaches are then investigated. The state-feedback scheme yields an explicit triangular stability region in the gain-parameter plane, whereas the weighted hybrid scheme is equivalent to replacing the Euler step size h by an effective step size h_eff = ωh. Numerical experiments, including convergence tests, bifurcation diagrams, maximum Lyapunov exponent (MLE) spectra, time series, and phase portraits, support the theoretical analysis and illustrate the transition among stable, periodic, quasi-periodic, and chaotic dynamics of the Euler-generated map.

Graphs associated to algebraic objects have been widely used to encode algebraic information through pairwise relations. Many algebraic relations, however, naturally involve finite collections rather than pairs. Simplicial complexes therefore provide a higher-dimensional framework for studying algebraic objects: after encoding algebraic data as a simplicial complex, one investigates the homology and homotopy type of its geometric realization. In this paper, we present several such constructions arising from commutative rings, finite groups, and cyclotomic polynomials. These examples illustrate how simplicial complexes can retain higher-order algebraic information and how the resulting topological invariants can reflect structural properties of the underlying algebraic object.

This work developed a modified Gompertz-type pest-natural enemy population model that accounts for infected natural enemies and state-dependent searching behavior. The model was then extended by incorporating two further mechanisms, namely, memory-mediated disease recovery and periodic intervention. Systematic theoretical analyses were conducted on solution boundedness, the existence and local asymptotic stability of equilibria, and transcritical bifurcation behaviors. These analyses clarify the suppression mechanism of natural-enemy predation on pest proliferation and reveal the regulatory effects of the variable searching rate and memory parameters on system dynamics. Considering that natural-enemy predation alone achieves sustainable biological control but responds slowly and cannot rapidly suppress pest outbreaks, the present work adopted an integrated strategy that combines periodic natural-enemy release with pesticide spraying to enhance pest control efficacy. The global asymptotic stability of pest-free periodic solutions was established, and parametric thresholds ensuring system persistence within a specific impulsive-period range were derived. Furthermore, sufficient criteria were provided for the global asymptotic stability of the system under pest-extinction and infected-predator-extinction scenarios, together with conditions for population persistence and the coexistence of periodic solutions. All theoretical results were validated via MATLAB numerical simulations. This study extends the theoretical framework of population dynamics and provides reliable references for integrated pest prevention and ecological regulation.

Targeting data integrity security threats, this paper investigates the anomaly detection and performance evaluation problems of data integrity attacks for finite impulse response systems with binary quantized observations. Breaking through the limitations of existing detection methods that highly depend on precise prior models and struggle with quantization non-linearity, we start from the inherent mechanisms of parameter identification and fully exploit the persistent excitation property of the system input. By cleverly designing and partitioning the observation dataset into two independent full-rank subsets, a novel detection residual is constructed. Without relying on prior knowledge of attack types, this method can provide unified and effective anomaly detection and early warning for whether the system is suffering from denial of service attacks, tampering attacks, or replay attacks. Furthermore, to measure the detection performance of the algorithm, we strictly prove the asymptotic normality of the detection residuals under both attack-free and various attack environments and derive explicit calculation expressions for the false alarm rate and missed detection rate. Finally, the impacts of factors on the algorithm’s performance are analyzed. Numerical simulations verify the effectiveness of the proposed detection algorithm and the accuracy of the theoretical derivations.

This paper primarily focuses on deriving an exact computational formula for a specific sixth-power mean of generalized two-term exponential sums. To this end, we exploit the properties of trigonometric sums and character sums, together with the solution counts of certain congruence equations modulo p.

We prove the existence of infinitely many sign-changing solutions for a variable-exponent modified quasilinear Schrödinger equation on a smooth bounded domain in three dimensions. The exponent is allowed to attain the quartic threshold on a nonempty null set. Consequently, the transformed nonlinearity is not uniformly superquadratic. The classical dual change of variables v = h(u), where $h^{\prime}(s)=\left(1+s^{2}\right)^{1 / 2}$, converts the equation into a semilinear problem on $H_{0}^{1}(\Omega)$ with subcritical growth. An exact identity for h⁻¹ provides the scaling monotonicity used in the Cerami boundedness argument and a quantitative logarithmic nonquadraticity remainder at the threshold. We establish the Cerami condition, prove the strict invariance of small neighborhoods of the positive and negative cones, and verify the hypotheses of an invariant-set symmetric minimax theorem. These steps yield an unbounded sequence of nodal critical values. A bootstrap argument and an admissible transformed test function then convert the critical points into weak solutions of the original equation. The main contribution is a nonperturbative treatment of the variable-exponent regime in which the quartic threshold is attained on a null set. The dual substitution and the invariant-set theorem are used as established tools.

This paper investigates the large deviation of two-time-scale fractional stochastic systems perturbed by fractional Brownian motion. The stochastic integrals with respect to fractional Brownian motion are understood in the sense of the generalized Riemann–Stieltjes integral, and Itô integration is employed for the standard Brownian motion component. The slow variable is governed by a Caputo fractional derivative, whereas the fast variable is described by an integer-order derivative. By combining the classical weak convergence approach with the averaging principle, a large deviation principle is established for this class of systems. Finally, the theoretical findings are illustrated through a numerical example.

We studied a curvature-dependent adapted frame, denoted for brevity by the adaptive Bishop–Frenet (ABF) frame, on Frenet-regular intervals where the curvature is strictly positive. The name refers only to the fact that the defining vector is generated explicitly from the Frenet and Bishop normals; it does not denote an endpoint-to-endpoint interpolation. Exact normalization and a uniform nondegeneracy margin ensure that the resulting adapted frame is orthonormal on this domain. The construction neither reaches the Bishop frame as an endpoint nor extends through inflection points. The contribution is local and structural: we establish the exact normalization and its uniform lower bound, derive the two curvature coefficients and the normal-plane rotation coefficient, record the associated adapted-frame structural system, and compute the exact squared-derivative energy density. The reconstruction result included in the paper concerns general adapted orthonormal frames and is not an ABF realization theorem. No universal full-energy ordering or numerical validation is claimed. Analytical examples illustrate the resulting formulas and clarify the restricted domain of validity.

The question of how algebra can be used to solve dynamical systems and characterize chaos was first posed in a fertile mathematical context by Ziglin, Morales, Ramis, and Simó using differential Galois theory. Their study was aimed at first-order, and later higher-order, variational equations of Hamiltonian systems. Recent work by this author formalized a compact yet comprehensive expression of higher-order variationals as one infinite linear system, thereby simplifying the approach. More importantly, the dual of this linear system contains all information relevant to formal first integrals, regardless of whether the original system is Hamiltonian. This applicability to formal calculation of conserved quantities is the centerpiece of this paper, following an introduction to the requisite context. Three important examples, namely, particular cases of Dixon’s system, the susceptible- infected-recovered (SIR) epidemiological model with vital dynamics, and the Van der Pol oscillator, were tackled, and explicit convergent first integrals were provided for the first two.

In this paper, we prove that both the extended minimal backward errors and the true minimal backward errors of the scaled total least squares (STLS) problem are identical to those of its core problem. This conclusion also holds for the asymptotic estimates of the extended minimal backward errors. Benefiting from its lower dimensionality, the core problem can effectively reduce the computational cost of backward error evaluation. We further derive practical, low-cost computable backward error estimates for a STLS problem using the Lanczos bidiagonalization process, and we demonstrate that our results can be readily extended to standard least squares and data least squares problems. To improve the efficiency of the Lanczos bidiagonalization-based TLS algorithm, we propose practical stopping criteria for the iterative numerical solution of STLS problems. Numerical experiments fully validate the computational advantages of our proposed approaches.

In this paper, a stochastic human immunodeficiency virus (HIV) infection model incorporating quiescent cells, cytotoxic C lymphocytes (CTLs), and antibody immune responses is formulated. First, the existence and uniqueness of the global positive solutions to the model are proven, which guarantees the well-posedness and validity of the constructed dynamic system. Second, the existence of the ergodic stationary distribution is investigated by constructing suitable stochastic Lyapunov functions. Third, the sufficient conditions for virus elimination are derived. Furthermore, optimal control strategies consistent with clinical antiviral treatment regimens are proposed to achieve a trade-off between therapeutic efficacy and economic costs. To validate the theoretical results and further compare the practical performance of different intervention schemes, numerical simulations are implemented, including theoretical verification, parameter sensitivity analysis, and comprehensive cost-effectiveness assessment of various therapeutic strategies. This study systematically clarifies the dynamical evolutionary behaviors of stochastic HIV infection and the efficacy of different control measures. The findings not only enrich the theoretical research on stochastic HIV infection model but also provide solid theoretical support and quantitative guidance for the design, assessment, and optimal selection of clinical HIV treatment strategies.

This paper addresses the well-posedness of the two-dimensional Boussinesq boundary layer system without monotonicity constraints. Compared with the classical Prandtl equations, mutual velocity-temperature nonlinear coupling generates double derivative loss in both the momentum and thermal equations, which introduces new analytical obstacles beyond those appearing in the Prandtl setting and degenerates the solution regularity. To overcome this obstacle, we adopt analytic phase modulation and Gaussian-type decaying weights, construct anisotropic weighted Sobolev spaces, and combine refined nonlinear energy estimates and fractional damping absorption techniques. Under small initial data, we establish the long-time existence and uniqueness of analytic solutions, and derive the explicit solution lifespan scaling T_ε ∼ ε^−4/3.

This paper investigates the semi-global practical finite-time adaptive neural network (NN) control problem for a class of fractional-order nonlinear systems with unknown nonlinearities. First, a novel practical finite-time stability criterion is established for fractional-order nonlinear systems. In contrast to analyses based on specific derivative properties of composite Lyapunov functions, the proposed criterion is rigorously developed through direct estimation of the fractional-order derivative of the Lyapunov function. This direct estimation framework provides a general stability analysis approach with independently adjustable convergence and residual-set parameters, thereby enhancing the flexibility of controller design. Second, an adaptive NN controller is developed based on this criterion, where NNs are employed to approximate the unknown nonlinearities. It is proven that the proposed control scheme ensures semi-global practical finite-time stability of the closed-loop system and boundedness of all closed-loop signals. Finally, numerical simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

This paper develops a unified framework for infinitesimal geometric symmetries and Ricci- type structures on two-dimensional Riemannian manifolds represented locally in orthogonal coordi- nates by a diagonal metric. Using a local orthonormal frame adapted to the metric coefficients, we derive explicit expressions for the Riemannian curvature tensor and the Ricci tensor in terms of the associated structure functions. Since, in dimension two, the Ricci tensor satisfies the pointwise iden- tity Ric =Kg, the symmetry conditions considered can be reduced to systems of differential equa- tions involving the Gaussian curvature, the metric coefficients, and the components of a vector field V =V¹ E₁ +V² E₂. We obtain necessary and sufficient conditions for conformal vector fields, ρ–Ricci vector fields, Ricci collineations, conformal Ricci collineations, and Ricci solitons and analyze the resulting systems under natural separability and cross-dependence assumptions. This leads to explicit solution families, rigidity results, and curvature restrictions for the corresponding symmetry classes. In the flat cases arising from the separability assumptions, the underlying metric is locally Euclidean, whereas, on connected open regions where the Gaussian curvature is nowhere zero, the corresponding symmetry equations retain an essential curvature dependence. In particular, we characterize the sep- arable case, determine the conditions under which the orthonormal frame fields generate Ricci-type symmetries, obtain explicit formulas for conformal Ricci collineations, and derive necessary and suffi- cient conditions for Ricci solitons. For ρ–Ricci vector fields, the classification of the diagonal metrics for which the coordinate vector fields are ρ–Ricci is carried out under the additional assumption that ρ is a prescribed nonzero constant, yielding explicit differential constraints on the metric coefficients. Several examples are provided to illustrate the existence of non-Killing conformal fields and Ricci soliton structures. The results are local in nature and apply within the orthogonal coordinate setting and the coordinate-dependence assumptions considered in this paper.

In this study, we developed and analyzed a deterministic two-strain host–vector model for dengue transmission incorporating temporary cross-immunity, antibody-dependent enhancement (ADE), disease-induced mortality during secondary infections, and explicit DENV–DENV co-infection in mosquitoes. The model distinguishes primary and secondary human infections and allows co-infected mosquitoes to contribute to strain-specific transmission. We derived the basic reproduction number R₀ and established the local stability of the disease-free equilibrium, while heterologous invasion of a one- strain endemic state was characterized through the corresponding invasion dynamics. Center-manifold analysis and numerical continuation revealed backward bifurcation, bistability between disease-free and endemic states, and Hopf-induced oscillations. Numerical simulations further illustrated transitions between disease-free, endemic, and periodic regimes. These results show how ADE, waning cross- immunity, and mosquito co-infection can jointly generate complex multi-strain dengue dynamics and provide a mechanistic framework for examining how the serotype composition of the vector population influences subsequent human infections.

This paper investigates the problem of short-interval multiplicative representations modulo primes. For a given positive integer C, an explicit lower bound p₀(C) exists such that for every prime p ≥ p₀(C), every element of the reduced residue system modulo p can be expressed as the product modulo p of two distinct integers in the interval [1,p/C). The overall argument proceeds by contradiction combined with strong induction on prime powers. For C = 4, we construct the inductive base via case analysis and derive the explicit lower bound p ≥ 199 by combining the divisibility properties of arithmetic progressions with a lower bound of the Chebyshev ψ-function. Building on this, we formulate an offset-screening and inductive framework for a specified positive integer C, in which a successful finite screening supplies the inductive base required by the subsequent theoretical argument. We carry out the complete implementation of this framework for C = 6 and obtain the explicit lower bound p ≥ 601.

This paper investigates mean-square exponential ultimate boundedness (MSEUB) for bandwidth-limited networked control systems subject to Bernoulli packet losses and finite-bit communication. A packet-loss-adaptive, event-triggered mechanism updates the triggering threshold according to the number of consecutive packet losses. It also enforces a positive minimum inter-event time. A scale-normalized encoding–decoding method is then constructed for finite-bit output transmission. Before quantization, the sampled output is normalized using a scale retained locally at the field-side node. The finite-bit index vector representing the normalized output is transmitted through the sensor-to-controller channel subject to packet losses. A stochastic closed-loop model is established by decomposing the Bernoulli packet-reception variable into its mean and zero-mean fluctuation. Tractable conditions are then derived to ensure MSEUB under Bernoulli packet losses and finite-bit reconstruction errors. Finally, numerical simulations illustrate the closed-loop and communication performance of the proposed design.