This study addresses the problem of fast fixed-time synchronization (FDTS) in multi-layer networks (MLNs), focusing on both intra-layer and inter-layer dynamic synchronization. First, we propose a more representative MLN framework in which each layer has a distinct topological structure, and node-to-node connections are established between corresponding nodes across layers. Second, we design a fast fixed-time controller to achieve synchronization of MLNs and estimate the synchronization time. Third, by applying the Lyapunov method and comparison theorem, we derive several novel synchronization conditions that are independent of initial states. Finally, numerical simulations are conducted to validate the theoretical findings.

The paper is devoted to the study of the two-phase free boundary problem for nonlinear partial differential equations describing the evolution of a foam drainage in the one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of a liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. In a series of papers, the authors have already solved the same problems without free boundary and with free boundary situated at the lower and the upper parts in the foam column, respectively. In this paper it is shown that the free boundary problem for the foam drainage equations with a sharp interface between dry and wet foams admits a unique global-in-time classical solution; this is done by a standard classical mathematical method, the maximum principle, and the comparison theorem. Moreover, the existence of the steady solution and its stability are shown.

Abstract Based on a suggestion by Katz, we determine the Tannakian monodromy group of certain $\ell $-adic hypergeometric sheaves to be the exceptional group $G_{2}$. Using the smoothing properties of the Fourier transform over the integers, known as uniformity theorems, we prove that the fourth moment is constant on an open locus of the family of hypergeometric sheaves. In our example, this implies a comparison theorem for the Tannakian monodromy groups that determines these groups if the characteristic is large.

Malaria is a catastrophic global disease transmitted to humans by bites from infected female Anopheles mosquitoes. Public awareness is a strong influencer on how individuals and groups of people perceive, respond to preventive interventions and other control measures. This work formulates mathematical model to study the impact of education on malaria control. In this work, the threshold parameter called control reproduction number was derived using next generation matrix method and used to study the global stability analysis of both the malaria-free equilibrium and malaria-endemic equilibrium. While the global stability of malaria-free equilibrium was assessed using comparison theorem, the global stability of malaria endemic equilibrium was done using non-linear Lyapunov function of Goh-Volterra type. Forward sensitivity analysis was carried out using the control reproduction number to identify the role of some critical parameters in malaria control. In the optimal control section, Pontryagin’s minimum principle was used to establish that, campaign to improve public usage of treated nets shows to be a very important optimal control parameter. The results of the sensitivity analysis and numerical simulation revealed that comprehensive healthcare measures targeted towards improved access to treatment, enhanced insecticide treated nets (ITN) compliance level, use of insecticide spray and reducing contact between human and mosquito are vital for controlling malaria prevalence.

In this paper, we prove some generalizations of the Ambrose (or Myers) theorem for complete Riemannian manifolds. We observe that the problem of finding the Ricci curvature conditions that guarantee the compactness of the manifold is reduced to the problem of finding the proper oscillation conditions of second-order linear differential equations. The proof of the theorems is based on the Riccati comparison theorem and some related oscillation conditions.

This paper investigates a class of uncertain fractional-order delayed cellular neural networks (UFODCNNs) with fuzzy operators and nonlinear activations. Both fuzzy AND and fuzzy OR are considered, which help to improve the robustness of the model when dealing with various uncertain problems. To achieve the finite-time (FT) synchronization and Mittag–Leffler synchronization of the concerned neural networks (NNs), a nonlinear adaptive controller consisting of three information feedback modules is devised, and each submodule performs its function based on current or delayed historical information. Based on the fractional-order comparison theorem, the Lyapunov function, and the adaptive control scheme, new FT synchronization and Mittag–Leffler synchronization criteria for the UFODCNNs are derived. Unlike previous feedback controllers, the control strategy proposed in this article can adaptively adjust the strength of the information feedback, and partial parameters only need to satisfy inequality constraints within a local time interval, which shows our control mechanism has a significant advantage in conservatism. The experimental results show that our mean synchronization time and variance are 11.397% and 12.5% lower than the second-ranked controllers, respectively.

ABSTRACTBased on the dynamic management of insulin, we consider the effects of insulin, glucose, and glucocorticoid concentration on the dynamics of the diabetes model with nonlinear impulsive control to simulate the physiological process of exogenous insulin therapy. Firstly, if , in type 1 diabetes mellitus (T1DM), the existence of positive periodic solution of the system is proved by using fixed point theory for difference equations, whereas the global asymptotic stability of positive periodic solution is proved by using the Floquet multiplier theory and comparison theorem. Next, if , in type 2 diabetes mellitus (T2DM), the permanence of the solution of the system is proved by using the comparison theorem. Furthermore, simulation results are performed to verify the correctness of the theoretical findings and to study the system's dynamic behavior under various conditions. By simulating the feedback regulation mechanism between insulin and glucose in a closed‐loop system, the model not only elucidates the dynamic equilibrium relationship between insulin and glucose but also offers a novel and feasible strategy for the treatment and control of diabetes, thereby demonstrating substantial theoretical and practical significance.

This paper deals with general mean-field reflected backward doubly stochastic differential equation (RBDSDE), whose coefficient depends not only on the solution but also on the law of the solution. First, we give the well-posedness of general mean-field RBDSDEs under uniformly Lipschitz conditions. Then, we present the existence and uniqueness of solution and a comparison theorem to general mean-field RBDSDEs with weak monotonicity coefficients.

ABSTRACTWe analyze nonlinear degenerate coupled partial differential equation (PDE)‐PDE and PDE‐ordinary differential equation (ODE) systems that arise, for example, in the modelling of biofilm growth. One of the equations, describing the evolution of a biomass density, exhibits degenerate and singular diffusion. The other equations are either of advection‐reaction‐diffusion type or ODEs. Under very general assumptions, the existence of weak solutions is proven by considering regularized systems, deriving uniform bounds, and using fixed point arguments. Assuming additional structural assumptions we also prove the uniqueness of solutions. Global‐in‐time well‐posedness is established for Dirichlet and mixed boundary conditions, whereas, only local well‐posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems, we formulate sufficient conditions for finite‐time blow‐up or uniform boundedness of solutions. Finally, we show that solutions of the degenerate parabolic equation inherit additional global spatial regularity if the diffusion coefficient has a power‐law growth.

In this paper, solutions of the inhomogeneous Schrodinger equation are studied Lu = Δu − c(x)u = h(x), (9) where c(x), h(x) — locally continuous Holder functions, with variations of its potential c(x) ≥ 0 on quasi-model Riemannian manifolds. It is clear that if h(x) ≡ ≡ 0, then the equation (9) is a stationary Schrodinger equation. Therefore, we assume that h(x) ̸= 0. In this paper, we obtain the exact conditions under which the solvability of boundary value problems of the inhomogeneous Schrodinger equation is preserved with some variations of the coefficient c(x) ≥ 0 on M. At present one of the directions of the development of this topic is the question of the solvability on a non-compact manifold of the Dirichlet problem on the restoring the solution of the stationary Schrodinger equation from boundary data on "infinity" and other boundary value problems. It can be noted that the formulation of the Dirichlet problem on non-compact manifolds may be problematic. One of the possible approaches to solving this problem is based on the introduction of classes of equivalent functions on a non-compact manifold (see [2; 9]). In some cases, geometric compactification of a manifold allows us to do this in the same way as when setting the classical Dirichlet problem in bounded domains Rn (see [3; 6; 7; 11; 15; 16]). The proof of the main results of the work is based on classical statements of the theory of partial differential equations: the maximum principle, the comparison and uniqueness theorems for solutions of linear elliptic differential equations. This work we study the existence of solutions and the solvability of boundary value problems of the equation (9) on non-compact Riemannian manifolds of the following form. Let M be a complete Riemannian manifold without the partial, represented as a union M = B ∪ D, where B — some compact, and the subset D is isometric to the product [r0, +∞) × S1 × S2... × Sk (where r0 > 0, Si — compact Riemannian manifolds without partial) and has the metric Theorem 1. [10] Let the Riemannian manifold M and the right-hand side of the equation (9) be such that Ik < ∞ (Kh< ∞, if c(x) ≡ 0), I1 = ... = Is = = ∞, Li < ∞ for everyone i = s + 1, ..., K, 0 ≤ s ≤ k. Then for any continuous on S1 × S2 × ... × Sk function Φ(0s+1, ..., 0k) there is a unique bounded solution u(x) on M to the equation (9) such that on D holds. Theorem 2. Let the Riemannian manifold M and the coefficients of the equation (10) be such that Ih < ∞, I1 = ... = Is = ∞, Ii < ∞ for everyone i = s+1, ..., k, 0 ≤ s ≤ k. Then for any continuous on S1×S2×...×Sk function Ф(0s+1, ..., 0k) there is a unique bounded solution u(x) on M to the equation (10) such that on D holds.

Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent.

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability properties, derive the Feynman–Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semi-linear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and the certainty equivalent based on g-expectation. In these applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion.

Considering the hibernation of pests and impulsive nonlinear release of natural enemies, we construct a pest management system with hibernation of pests and impulsive nonlinear release of natural enemies. Using Floquet theory and the comparison theorem of impulsive differential equations, we derive the conditions under which the pest-eradication periodic solution is globally asymptotically stable (GAS) and the system is permanent. Furthermore, we obtain a control threshold that is critical for managing pest population. The validity of these conditions is rigorously tested and confirmed through numerical simulations. Our work provides ways to pest management.

In this study, we investigate the dynamic mechanisms of tumor progression in response to fluctuations and uncertainties within the tumor-immune microenvironment. Utilizing temporal single-cell data, we develop a novel stochastic reaction-convection model that captures the spatiotemporal dynamics of macrophage responses to tumor cells subjected to both multiplicative and additive noise generated by non-homologous microenvironmental fluctuations. We prove the existence and uniqueness of a global positive solution for the proposed stochastic model. Then, by combining the stochastic Lyapunov analysis and the comparison theorem, we explore the moment boundaries for cell populations, as well as the asymptotic behavior at the boundary equilibrium points; sufficient conditions for driving sustained tumor growth and clearance are derived by employing the ergodicity theorem and are interestingly found to be only related to multiplicative noise. Furthermore, we employ an upwind finite difference scheme to simulate the effects of different noise types on a cell population distribution and the persistence of tumor growth. Results show that while additive noise influences the multimodal distribution of early tumor cell phenotypes, it has minimal impact on the mean density of tumor cells, indicating that additive noise acts primarily as a diffusion factor. In contrast, increasing multiplication noise effectively inhibits the development without altering the number of peaks in a phenotypic distribution. Interestingly, when additive and multiplicative noises are correlated, stronger additive noise can have dual effects on the steady-state distribution of tumor cells, with increased correlation positively influencing tumor cell elimination. These results provide novel insight into the tumor-immune microenvironment dynamics.

In this paper, a glioma-immune stochastic model with logistic growth is studied to explore the growth of glioma cells in a microenvironment. We can understand the dynamics of glioma-immune interactions by deterministic differential equations. However, stochastic interference plays a pivotal role in glioma-immune interactions. Using the law of large numbers, the comparison theorem, and the Itô’s formula, we prove that a stochastic system has a unique global solution with arbitrary initial values and study the stationary ergodic distribution of the positive solution of the system, where the equilibrium of the solution fluctuates up and down in the deterministic case, resulting in stochastic glioma cells persistence, provide threshold conditions to ensure tumor elimination and persistence. The above results are verified by the numerical simulation. The results show that white noise affects the dynamics of gliomas, white noise intensity is a key factor in the treatment of gliomas, and the addition of stochastic conditions has theoretical guiding significance for clinical medicine to control tumor growth.

Let L/\mathbb{Q}_{p} be a finite extension. We introduce L -typical prisms , a mild generalization of prisms. Following ideas of Bhatt, Scholze, and Wu, we show that certain vector bundles, called Laurent F -crystals, on the L -typical prismatic site of a formal scheme X over \operatorname{Spf}\mathcal{O}_{L} are equivalent to \mathcal{O}_{L} -linear local systems on the generic fiber X_{\eta} . We also give comparison theorems for computing the étale cohomology of a local system in terms of the cohomology of its corresponding Laurent F -crystal. In the case X=\operatorname{Spf}\mathcal{O}_{K} for K/L a p -adic field, we show that this recovers the Kisin–Ren equivalence between Lubin–Tate (\varphi_{q},\Gamma) -modules and \mathcal{O}_{L} -linear representations of G_{K} , as well as the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of (\varphi_{q},\Gamma) -modules. We can thus regard Laurent F -crystals on the L -typical prismatic site as providing a suitable notion of relative (\varphi_{q},\Gamma) -modules.

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results.

. In this note, by an elementary use of Girsanov’s transform, we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Peköz and R. Righter in 2024, while the Brownian motion case is the continuous analogue as discussed in the same paper. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.

Posets, their incidence algebras and relative operads, and the cohomology comparison theorem

In this article, we establish a predator–prey model with fear factor and impulsive nonlinear effects. The globally asymptotically stable conditions for the pest extinction periodic solution and the permanence condition of the formulated model are derived using Floquet theory and the comparison theorem of the impulsive differential equations. Simulations confirm the correctness of the theoretical results obtained in this paper and reveal the complex dynamics of the investigated model. Our results may assist pest control experts in determining the appropriate impulsive control period and release quantity of natural enemies for more effective pest management.