This paper presents a numerical method for the advection-diffusion process in fractured porous media described by a kind of hybrid–dimensional fracture models, in which fractures are represented as (n − 1)–dimensional interfaces embedded within the n–dimensional porous matrix. We develop a numerical approach that combines the discontinuous Galerkin (DG) method for spatial discretization with the backward Euler method for temporal discretization. Leveraging the advantage of the DG method in flexibly handling boundary conditions, the DG–backward Euler method is extended from the model with a single fracture to one with intersecting fractures. Theoretical analysis shows that the optimal order error estimate for the concentration in the discrete H¹–norm is obtained. Numerical experiments with a single fracture and intersecting fractures are all carried out to verify the accuracy of the theoretical results.

This study provides an efficient, adaptive computational framework for Bayesian model selection when likelihood functions are difficult to handle. To address model selection challenges in scenarios where complex model likelihood functions are difficult to handle, this paper proposes an adaptive approximate Bayesian computation with probability-minimizing computation (RL(Softmax)-ABC-PMC) algorithm that integrates a reinforcement learning (RL) Softmax decision mechanism. This method embeds Softmax agents into the standard ABC-PMC framework, dynamically evaluating the historical performance of candidate models to adaptively focus computational resources on potentially optimal models, thereby significantly enhancing selection efficiency. A numerical experiment is provided to select the best model from three Lotka-Volterra (LV) competition models, including a discretized LV model, a randomized LV model, and a fractional-order LV model, where the algorithm effectively distinguishes subtle differences in their fitting ability and yields reliable parameter estimates. Additionally, we further validate the efficiency of the provided algorithm by selecting the best model from three S-shaped growth curve models. Finally, an empirical analysis using Chinese automotive market sales data further validates the practicality and effectiveness of the proposed method in real-world applications, demonstrating that the RL(Softmax)-ABC-PMC algorithm can automatically identify model structures with stronger explanatory power while maintaining high computational efficiency.

Infrared imaging technology plays an indispensable role in critical applications such as military surveillance, autonomous driving, and medical diagnostics. However, its inherent low-resolution and low-contrast characteristics often limit operational performance. While deep learning-based super-resolution (SR) techniques offer a software-driven solution, models face severe feature redundancy caused by simply stacking deep layers, and a lack of discriminative power in distinguishing critical textures from thermal noise. To address these issues, we proposed a novel Convolutional and Attention-based Super-Resolution Network (CASRNet). The novelty of our model lies in the synergistic fusion of a channel splitting (CS) strategy and a dual attention mechanism. First, the CS strategy decomposes feature maps into parallel streams, extracting diverse and less redundant representations. Second, a novel channel and spatial attention residual block (CSA_ResBlock) was designed to adaptively focus on informative feature channels and critical spatial boundaries. Quantitatively, CASRNet achieved superior performance on public benchmarks. Specifically, for the FLIR dataset (× 2), our model achieved a peak signal-to-noise ratio (PSNR) of 39.73 dB and structural similarity (SSIM) of 0.9639, outperforming the state-of-the-art infrared-specific model TherISuRNet by 0.48 dB and standard models like VDSR by 0.15 dB. Similar robust improvements (e.g., an exceptional 40.45 dB PSNR on the ThermalTau2 dataset) demonstrated the general applicability and high fidelity of CASRNet for real-world infrared enhancement tasks.

Antibody-dependent enhancement (ADE) is a major safety challenge in multiserotype disease control, involving complex trade-offs among infection burden, cost, and vaccine safety. This study develops a two-strain influenza model that integrates both host immunity (endogenous) and viral cross-reactivity (exogenous) as dual mechanisms underlying ADE. Threshold analysis shows that exogenous ADE lowers transmission thresholds (eR₁, eR_A), promoting strain coexistence, whereas endogenous ADE reverses this trend. We further establish a multiobjective optimal control framework to balance outbreak control, ADE risk, and intervention costs. The resulting strategy combines early targeted measures with sustained behavioral precautions (e.g., 51–79% mask-wearing compliance), effectively reducing the peak infection rate. Crucially, we identify an optimal risk-aversion level (α ≈ 2), which halves the ADE risk with only a 2.3% increase in infections and minimal extra cost- demonstrating a clear Pareto-optimal trade-off. This quantitative framework can be applied to other contexts, such as multivalent vaccine design for dengue fever, and provides a decision-making basis for risk-benefit assessment of vaccines against emerging infectious diseases.

In this paper, drawing on the classical theory of skew n-derivations on prime and semiprime rings, we generalize the concept of (σ, σ)-derivations to the framework of 3-Lie algebras and introduce the notion of (σ, σ)- n-derivations. Subsequently, several fundamental properties of (σ, σ)-n-derivations on 3-Lie algebras are established. Furthermore, we prove that the (σ, σ)-derivation space of any non-abelian 3-dimensional 3-Lie algebra is 6-dimensional, providing a precise dimensional characterization for the low-dimensional case.

This paper studies the formulation for an optimal attack strategy, which aims to maximize the identification error of Hammerstein systems with binary measurements subject to data tampering attacks. First, the convergence of the parameter estimates under attack is analyzed, and the absolute error between the estimated value and the true value is used as the objective function. Second, an optimization model with constraints on the maximum data tampering rate and the average data tampering rate is established to maximize the objective function. Then, the sine-cosine optimization algorithm is used to search for the optimal solution that meets the constraints, and its performance is compared with other algorithms. Finally, the effectiveness of the proposed method is verified by a numerical simulation.

In this paper, we investigated a predator–prey model driven by an Ornstein–Uhlenbeck process and featuring Holling-II and Beddington–DeAngelis functional responses. To begin, the biological significance of the Ornstein-Uhlenbeck process in ecological modeling was illustrated, along with its validity in characterizing random environmental fluctuations. Subsequently, the existence and uniqueness of the global solution for this model were strictly proven and its ultimate boundedness was analyzed. By constructing Lyapunov functions and applying Itô’s formula, the existence of the stationary distribution of the system was demonstrated, while sufficient conditions for population extinction were provided. Finally, numerical simulations were conducted to confirm the validity of the conclusions. This work provides a theoretical framework for understanding complex mutualistic-predatory communities under persistent environmental fluctuations.

We develop a deterministic framework that links epidemic propagation, mortality risk, and life-insurance solvency by modeling the population through an age-structured SEIRD system in which disease-induced deaths drive an insurer’s surplus process governed by an ordinary differential equation with a continuous premium inflow and death-benefit outflow. We characterize the disease-free equilibrium and show that its local stability depends on the basic reproduction number R₀, and for any finite horizon [0,T], we derive an explicit solvency threshold in the form of a critical premiump_crit(T) that guarantees a nonnegative surplus and depends on the age-weighted infection burden. For large horizons, we obtain an analytically tractable approximation of this threshold using multi-group final-size relations and show that it monotonically increases with epidemic severity. To illustrate practical implications, we formulate an age-specific premium selection as a one-shot continuous-action reinforcement-learning problem in which a Proximal Policy Optimization (PPO) agent is trained on simulated age-structured epidemic scenarios to choose premiums that avoid ruin while remaining near actuarially fair levels; out-of-sample tests confirm that the learned premiums satisfy the analytical solvency constraints, deliver low ruin probabilities, and achieve tight fairness calibration. The combined analytical and numerical results provide a transparent basis for epidemic-sensitive premium design and stress testing of life-insurance portfolios.

We studied finite-time stabilization in probability for stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed delays and multiplicative noise under Neumann boundary conditions. By developing a Lyapunov-Krasovskii functional and using stochastic theory with the Neumann-Poincaré inequality, we obtained several verifiable stabilization criteria. A componentwise controller with a nonlinear finite-time term was further designed to induce a constant drift effect in the Lyapunov evolution, leading to an explicit stochastic settling time bound that quantifies the roles of diffusion, delays, couplings, noise, and control gains. Simulations confirmed the effectiveness of the proposed method.

This work presents a comprehensive oscillation theory for a class of second-order nonlinear neutral differential equations with a variable damping term and variable delay functions. The primary contribution consists of a set of new oscillation criteria whose validity depends on the interplay between the key parameters α and β. To address the limitations of existing approaches, a novel framework is introduced based on the construction of parameterized auxiliary equations and the use of a generalized Riccati transformation. This approach reduces the problem to the analysis of an associated integro- differential inequality, which is then analyzed via refined inequality techniques and iterative integration. The main results reveal that oscillatory behavior is dictated by a threshold condition comparing α with the rate of change of the delay, quantified by β. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.

Chaotic systems confer distinct advantages for image encryption due to their remarkable responsiveness to initial parameters, inherent unpredictability, and characteristics of pseudo-randomness. However, many existing chaos-based image encryption schemes still suffer from limitations such as narrow chaotic parameter intervals, insufficient trajectory uniformity, and inadequate diffusion depth, which may weaken the randomness of generated sequences and reduce encryption robustness. To address these issues, a square-term enhanced four-dimensional chaotic system was constructed, and a color image encryption scheme integrating hierarchical scrambling and symmetric bidirectional diffusion was developed. Dynamic analyses, including phase portraits, bifurcation diagrams, maximum Lyapunov exponents, time-series distribution, and complexity evaluation, indicate that the proposed system exhibits a broad chaotic interval, strong initial-value sensitivity, and improved trajectory uniformity. It can well-adapt to the image encryption requirements of different categories. Simulation experiments conducted using a variety of test images indicate that the encryption scheme offers an adequately large key space, with pixel correlation measured at less than 0.03, information entropy exceeding 7.9974, and robust resistance to differential attacks. The computed mean values for number of pixels change rate (NPCR) and unified average changing intensity (UACI) were found to be 99.6073% and 33.4626%, respectively, demonstrating a strong alignment with the optimal benchmarks. The algorithm achieves competitive performance in terms of efficiency and security, can effectively resist common attacks, and is suitable for the secure transmission and storage of sensitive images.

This study aimed to investigate the traveling solutions of the Camassa-Holm and Degasperis-Procesi (CH-DP) equation. By using the qualitative theory of dynamical systems and integrating along special phase orbits, we present exact explicit expressions of compacton and generalized kink wave solutions to the CH-DP equation. Our results will enrich the previous literature and help to understand the propagation of nonlinear waves.

Unsupervised domain adaptation (UDA) aims to leverage labeled source domain knowledge to improve the target domain’s performance. Source-free domain adaptation (SFDA), a recent research focus, addresses challenges such as data privacy by relying solely on a pretrained source model and unlabeled target data, thus eliminating the need for direct access to source domain data. Although many studies have proposed methods such as generating a source domain, using a proxy source domain, or using pseudo-label training, these approaches directly fine-tune the source model, overlooking the excessive bias towards the source domain data in SFDA. The source domain model contains numerous source domain-specific features, and directly updating it to shift towards the target domain is hindered by these domain-specific features. To address this issue, we propose an inverse distillation-based SFDA method. By constructing an initial target domain model, the method extracts pure target domain features and distills them back into the source model, facilitating a smoother transition towards the target domain. Additionally, it identifies stable and active target samples from both structural and scoring perspectives, applying distinct matching strategies for pseudo-label selection. Extensive experiments and ablation studies on public datasets (Digits, Office-31, Office-Home and VisDA-2017) demonstrate the superior performance of our approach in SFDA tasks.

Let R be an Artin algebra, modR be the category of finitely generated rightR- modules, and letF be a sub-bifunctor of $\operatorname{Ext}_{R}^{1}(-,-)$. In this paper, we introduce the notion of the F-extension dimension for modR. We further establish the relationships for the finiteness among the F-extension dimension,F-finitistic dimension andF-Igusa-Todorov dimension.

This study presents the bivariate Kumaraswamy discrete Lindley distribution, developed via a trivariate minimization framework. The model features closed-form formulas for its joint survival, cumulative distribution, and probability mass functions, enabling efficient computer implementation. We analyze the identifiability of the suggested model and the positive dependence structure, deriving the joint probability generating function and conditional expectations. The joint hazard rate function demonstrates considerable distributional flexibility, allowing for monotonic, bathtub, and unimodal shapes. Subsequent to the derivation of maximum likelihood estimators and the Fisher information matrix, we conduct simulation investigations across several sample sizes. The suggested model, when applied to three authentic datasets from the healthcare and manufacturing sectors, demonstrates a better fit than competitive models based on standard statistical selection criteria, confirming its efficacy for modeling count data.

As a typical form of symbolic data, interval-valued data provides an effective framework to analyze large-scale datasets. Most existing interval regression studies focus on classical methods, while research that incorporates heteroscedasticity within the Bayesian framework remains limited. This paper extends the existing parametric method for interval-valued data to a Bayesian heteroscedastic framework, and further develops the Bayesian Heteroscedastic Parametric Method (BHPM). By explicitly modeling heteroscedasticity in the regression structure, we conduct Bayesian inference using Gibbs sampling and the Metropolis–Hastings algorithm, thus enhancing the model’s interpretability and generalization performance. Both simulation studies and real-data applications demonstrate that the extended BHPM achieves superior performance over traditional methods.

In recent years, the competing failure processes gained significant attention as a way of modeling based on degradation and shock processes. A traditional independent competing failure process model assumes that soft failure and hard failure affect the life of the system independently. This modeling method is being improved by a new modeling method, that is, the interaction between soft failure and hard failure affects the life of the system. Therefore, a novel reliability modeling of systems that suffer from mutually dependent competing failure processes under the generalized mixed shock model is proposed. In reliability modeling, the external random shocks can immediately increase the degradation level and degradation rate to accelerate the system degradation. Meanwhile, the internal degradation can continuously change the hard failure threshold to weaken the system strength. Under the above assumptions, analytical expressions for the reliability function for systems under the generalized Pólya process have been derived using the stochastic process theory and probabilistic techniques based on the proposed mixed shock model. Additionally, simulation algorithms to compute the reliability based on the Monte Carlo method are provided. Finally, a case study is verified to illustrate the proposed model and the derived results.

In this article, our main aim is to explore the existence and uniqueness of solutions for a fractional-order q-equation under a q-integral boundary condition. Employing well-known fixed point theorems, we derive theoretical findings that advance the theory of q-fractional calculus. An example is provided to show the effectiveness of the results, and the stability of the solutions is discussed.

This paper establishes interior Morrey regularity for stable solutions of the p-Laplace equation $-\Delta_{p} u=h(u) \text { in } B_{1} \subset \mathbb{R}^{d}$, where h belongs to $C^{1}(\mathbb{R})$. The results cover the endpoint case left unresolved by Cabré et al. Moreover, for positive nonlinearities h, a global Morrey estimate was obtained for stable solutions of a Dirichlet problem on any smooth, bounded, strictly convex domain.

In this paper, we study nonlinear partial differential systems that describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of Camassa-Holm-type systems of the form

withm,n ≥ 2 andF,G smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of these results provide new examples of such systems, including the Song-Qu-Qiao system, the Xia-Qiao-Zhou system, and the two-component modified Camassa-Holm system. Furthermore, we construct the nonlocal symmetry for the Xia-Qiao-Zhou system from the gradients of the spectral parameter. By introducing an appropriate pseudo-potential, we prolong the nonlocal symmetry to an enlarged system and calculate the corresponding finite symmetry transformation. On this basis, we derive nontrivial solutions to the Xia-Qiao-Zhou system.

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation

where ∆_t and ∆_k denote the standard forward difference operators in the variables t and k, respectively, ∇_t denotes the standard backward difference operator in t, and is the discrete Laplacian operator in the spatial variable k. Throughout, we will assume the parameters β and ε are positive real numbers, the parameter γ is a nonzero real number, and the potential function g : $\mathbb{Z} \times \mathbb{C} \rightarrow \mathbb{C}$ is continuous.

The central objective of this paper was to investigate the structure of n-Lie (m₁, ... ,m_n)- derivations on triangular rings from two distinct perspectives: the faithful bimodule property and the maximal left ideal quotient ring. Under certain assumptions, we established that every 2-Lie (m₁,m₂)- derivation decomposes into the sum of an inner derivation, an extremal biderivation, and a bilinear central mapping. Furthermore, using mathematical induction, we proved that for n ≥ 3, each n-Lie (m₁, ... ,m_n)-derivation can be expressed as the sum of an extremal n-derivation and an n-linear central mapping. Notably, our analysis revealed that these structural forms remain invariant regardless of whether we adopt the faithful bimodule approach or the maximal left ideal quotient ring framework. As significant corollaries, we obtained explicit structural descriptions of n-Lie (m₁, ... ,m_n)-derivations on upper triangular rings and nest algebras. Additionally, our investigation yielded several interesting extensions and generalizations of existing results in this domain.

This work investigated predefined-time stability for discontinuous nonautonomous systems and its application to cluster synchronization in community networks with discontinuous intra- community interactions. First, a new predefined-time stability criterion for nonautonomous systems was established via differential inclusions and a generalized Lyapunov method, allowing indefinite time derivatives almost everywhere. Second, a community network model was introduced incorporating discontinuous intra-community interactions, time-varying parameters, and sign-dependent coupling to represent cooperative and competitive relationships. For such networks, a state-feedback control protocol was designed that, using the proposed stability result, ensures cluster synchronization within any prescribed time. A numerical example with sign-based coupling validated the theoretical analysis.

In this paper, we investigate the periodic flows of a class of time-delay dynamical systems with impulses at fixed times through an implicit mapping method. As the impulsive points coincide with the time-delay points, discrete implicit mappings at impulsive points are obtained based on the impulsive mappings and interpolation techniques according to the given accuracy. As the impulsive points coincide with the regular points, the integral limits of the impulsive points are constructed, and discrete implicit mappings at impulsive points and time-delay points are obtained respectively by using the impulsive functions and integral techniques. The approximate solutions of the periodic flows determined by regular and delay nodes in one period are presented. A second-order delay dynamical system with an impulse at a fixed time is presented as an example. The implicit mapping method provides a plan for the periodic flows of switching delay dynamical systems.

Image attribute editing based on generative adversarial networks (GANs) typically begins by mapping real images to the latent space of a pretrained StyleGAN, followed by manipulating the corresponding latent codes. However, low-rate latent codes suffer from an information bottleneck, making it challenging to faithfully reconstruct complex real images. Recent encoder-based methods enhance reconstruction by injecting high-rate features into intermediate generator layers to better preserve fine details, but they often yield misaligned details in the edited images. The primary reason is that these methods still rely on global linear transformations of high-rate features, which overlook the nonlinear and spatially localized nature of real edits. To this end, we build on a high-fidelity encoder- based GAN inversion backbone and introduce an additional adaptive feature editor that is specifically trained to convert high-rate features during editing so that fine details are correctly aligned with the edited image. The backbone refines the feature through a cross-attention mechanism and residual enhancement. Building on this, the feature editor employs a window-based cross-attention mechanism to extract a discrepancy signal between the original and edited generator features, which specifies both where to modify and what content to change. This signal is then fused into the feature through spatially adaptive modulation techniques, enabling region-selective attribute changes while preserving irrelevant details. Experiments on face and car benchmarks demonstrate that our method improves both reconstruction fidelity and editing quality compared to existing GAN inversion methods.

This paper proposes an adaptive maneuvering control for a quadrotor unmanned aerial vehicle with position constrained by geometric equations and unknown parameters. Based on the structural characteristics of the quadrotors, the inner–outer-loop controller is designed for two cascaded subsystems to solve the maneuvering problem of the position subsystem and the Euler angle tracking problem of the attitude subsystem. Meanwhile, double dynamic filters are incorporated into the the inner-loop control to eliminate the coefficient explosion problem of virtual signal differentiation. By treating the coupling term as a perturbation, the semi-global practical stability of the closed-loop system is established. Finally, the effectiveness of the obtained results are demonstrated by numerical simulations.

In this paper, we introduced and studied the (t,s)-Oresme and (t,s)-Oresme-Lucas sequences, which were generalizations of the classical Oresme and Oresme-Lucas numbers. We presented the recurrence relations, generating functions, and Binet formulas for these sequences. Furthermore, we derived various algebraic identities, including those of Tagiuri, d’Ocagne, Catalan, Cassini, Ruggles, and Honsberger. A matrix representation for these sequences was also established, and we defined the associated (t,s)-Oresme matrix sequence, exploring its fundamental properties and relations. Additionally, we provided the summation formulas for the first n terms of these sequences. This work extended the existing theory of Oresme numbers and offered a unified framework for their further investigation.

In this paper, we obtained the existence of nontrivial solutions of the nonlocal elliptic equation involving the square root of the Laplacian with asymptotic linear resonance at infinity perturbed by a bounded nonlinearity. We made use of a penalized method, local linking and Morse theory.

Let q be a 2n-th primitive root of unity and $\mathfrak{u}_{q}\left(\mathfrak{s} \mathfrak{I}_{2}\right)$ a small quantum group in the usual way. Under reasonable simplifying assumptions, we provided a complete classification of all $\mathfrak{u}_{q}\left(\mathfrak{s} \mathfrak{I}_{2}\right)$-module algebras on the truncated quantum plane $A_{n, q}[x, y]$, which was generated by x,y subject to the relations: yx = qxy, $x^{n}=y^{n}=0$.

Temperature-induced drift in the winding resistance of long-stator linear synchronous motors (LSLSMs) critically affects the levitation control performance of high-speed magnetic levitation (maglev) trains. To address this issue, this paper proposes a collaborative optimization framework that integrates multiphysics coupling analysis with active disturbance rejection control (ADRC). First, an electromagnetic-thermal-fluid coupling model of the LSLSM is developed through finite-element co-simulation. Second, a fast temperature prediction model is established by combining the parameter fitting with a lumped parameter thermal network (LPTN), enabling dynamic estimation of the winding temperature and the associated resistance drift. Third, an extended state observer (ESO) is employed to estimate and compensate for thermally induced parameter drift in real time. Numerical simulations under multiple operating scenarios show that the proposed method achieves improved tracking performance, stronger disturbance rejection capability, and better robustness against thermal-induced parameter drift than the benchmark proportional-integral-derivative (PID) controller. In addition, considering thermal-induced parameter drift in the controller design process leads to more consistent closed-loop performance in the levitation plant.

In this paper, an SEIR epidemic dynamical model incorporating a fear effect, a nonlinear incidence rate, and a time-delayed behavioral response was established. The basic reproduction number R₀ was derived using the next-generation matrix method. Theoretical analysis demonstrates that if R₀ < 1, the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, there exists a unique endemic equilibrium, which is also globally asymptotically stable. By performing stability analysis, the critical time delay threshold τ₀ for triggering a Hopf bifurcation was derived. Moreover theoretical analysis and numerical simulations consistently showed that when the time delay exceeds this threshold, the endemic equilibrium loses its stability and a stable limit cycle is generated, leading to periodic epidemic outbreaks. Furthermore, sensitivity analysis revealed an inverse relationship between fear coefficient α and the critical time delay threshold τ₀.

In Parkinson’s disease (PD), abnormal oscillations pervade the cortex–basal ganglia– thalamus (CTX-BG-Th) loop. In recent years, more and more evidence has shown the physiological phenomena of exaggerated phase–amplitude coupling (PAC) in PD, suggesting that such coupling contributes to the generation and propagation of pathological oscillations. However, current understanding of how cortical low-frequency rhythms shape downstream pathological activity is still incomplete. In this study, we aim to investigate how the theta-band (3–9 Hz) phase of layer 5 pyramidal tract (E5P) cortical neurons modulates the beta-band (13–30Hz) amplitude of globus pallidus internus (GPi) neurons. The modulation index (MI), time-lag MI analysis, and coherence analysis are introduced to quantify the strength, temporal characteristics, and pathway dependence of PAC. The results indicate that significant cross-regional θ–β PAC occurs between E5P and GPi in the PD state, and this coupling has time delay characteristics and mainly relies on indirect pathways. In addition, blocking indirect pathways or reducing the θ power of E5P will decrease the β oscillation of GPi. These findings elucidate the dynamic mechanism of cortical regulation of GPi related oscillations, providing a theoretical basis for targeted interventions in PD patients through cortical modulation.

This paper investigates the bifurcation curves and the multiplicity of positive solutions for a one-dimensional Minkowski curvature problem with constant-yield harvesting

where $\lambda, L, \mu>0$ and there exist constants $\sigma>u_{0}>0$ such that $g \in C[0, \sigma] \cap C^{2}(0, \sigma), g(0)=g(\sigma)=0$, $g^{\prime}(u)>0 \text { on }\left(0, u_{0}\right), g^{\prime}\left(u_{0}\right)=0, \text { and } g^{\prime}(u)<0 \text { on }\left(u_{0}, \sigma\right)$. We first show that the bifurcation curve has a $\subset$-like shape and then provide additional sufficient conditions under which the curve is exactly $\subset$-shaped. These results yield the exact multiplicity of positive solutions. As an application to population and ecological models, we further consider the nonlinearity $g(u)=u^{p}\left(1-u^{q} / K\right)^{r}$, where $p, q, r, K>0$.

In this paper, we adopted the steady compressible Euler system with a gravitational term as the governing equations, and focused on the effect of gravity on the transonic shock position in a three-dimensional spherically symmetric divergent nozzle. For a fixed supersonic inflow at the nozzle inlet, we proved that if the outlet pressure fell within an appropriate interval and the gravitational parameter K was sufficiently small, the shock position was uniquely determined by K and the outlet pressure, and the shock position was a strictly decreasing and continuously differentiable function of K.

This paper investigates the link exponential synchronization (LES) problem of a class of discrete-time complex dynamic networks (DTCDNs) which consist of infinite distributed delay, node complex dynamic subsystems (NCDS), and link complex dynamic subsystems (LCDS) with mutual coupling relationships. A mathematical induction method is proposed to derive the global exponential stability criterion of error systems. The obtained LES is described using linear scalar inequalities, which are very simple and easy to verify based on standard software tools such as YALMIP. Finally, it was validated through an illustrative example.