MDPI - Publisher of Open Access Journals

38 pages, 16379 KiB

Open AccessArticle

Hyperbolic Sine Function Control-Based Finite-Time Bipartite Synchronization of Fractional-Order Spatiotemporal Networks and Its Application in Image Encryption

by Lvming Liu, Haijun Jiang, Cheng Hu, Haizheng Yu, Siyu Chen, Yue Ren, Shenglong Chen and Tingting Shi

Fractal Fract. 2025, 9(1), 36; https://doi.org/10.3390/fractalfract9010036 - 13 Jan 2025

Viewed by 870

Abstract

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be [...] Read more.

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application, 2nd Edition)

► Show Figures

Figure 1

10 pages, 253 KiB

Open AccessArticle

The Dual Hamilton–Jacobi Equation and the Poincaré Inequality

by Rigao He, Wei Wang, Jianglin Fang and Yuanlin Li

Mathematics 2024, 12(24), 3927; https://doi.org/10.3390/math12243927 - 13 Dec 2024

Viewed by 783

Abstract

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity [...] Read more.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations. Full article

17 pages, 304 KiB

Open AccessArticle

Heat-Semigroup-Based Besov Capacity on Dirichlet Spaces and Its Applications

by Xiangyun Xie, Haihui Wang and Yu Liu

Mathematics 2024, 12(7), 931; https://doi.org/10.3390/math12070931 - 22 Mar 2024

Viewed by 986

Abstract

In this paper, we investigate the Besov space and the Besov capacity and obtain several important capacitary inequalities in a strictly local Dirichlet space, which satisfies the doubling condition and the weak Bakry–Émery condition. It is worth noting that the capacitary inequalities in [...] Read more.

In this paper, we investigate the Besov space and the Besov capacity and obtain several important capacitary inequalities in a strictly local Dirichlet space, which satisfies the doubling condition and the weak Bakry–Émery condition. It is worth noting that the capacitary inequalities in this paper are proved if the Dirichlet space supports the weak

(1, 2)

-Poincaré inequality, which is weaker than the weak

(1, 1)

-Poincaré inequality investigated in the previous references. Moreover, we first consider the strong subadditivity and its equality condition for the Besov capacity in metric space. Full article

11 pages, 256 KiB

Open AccessArticle

A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

by Fabio Silva Botelho

AppliedMath 2023, 3(2), 406-416; https://doi.org/10.3390/appliedmath3020021 - 2 May 2023

Viewed by 1765

Abstract

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. [...] Read more.

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. Full article

31 pages, 397 KiB

Open AccessArticle

Ricci Curvature on Birth-Death Processes

by Bobo Hua and Florentin Münch

Axioms 2023, 12(5), 428; https://doi.org/10.3390/axioms12050428 - 26 Apr 2023

Viewed by 1463

Abstract

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s

CD (K, n)

[...] Read more.

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s

CD (K, n)

condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line

Z

with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than

- R^{2}

are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte’s result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound. Full article

(This article belongs to the Special Issue Discrete Curvatures and Laplacians)

15 pages, 320 KiB

Open AccessArticle

Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

by Alexander Sukhinov, Yulia Belova, Natalia Panasenko and Valentina Sidoryakina

Mathematics 2023, 11(3), 575; https://doi.org/10.3390/math11030575 - 21 Jan 2023

Cited by 3 | Viewed by 1316

Abstract

The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this [...] Read more.

The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L₂ as the time step τ tends to zero at the rate O(τ). Full article

(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)

9 pages, 244 KiB

Open AccessArticle

Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation

by Tosiya Miyasita

Axioms 2022, 11(9), 429; https://doi.org/10.3390/axioms11090429 - 25 Aug 2022

Viewed by 1229

Abstract

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate on the radial solutions in [...] Read more.

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate on the radial solutions in an annulus. First, we construct a time-local solution with an abstract theory of differential equations. Next, we show that decreasing energy exists in this problem. Finally, we establish a global solution for the sufficiently small initial value and parameter by Sobolev embedding and Poincaré inequalities together with some technical estimates. Moreover, when we take the smaller parameter, we prove that the global solution tends to zero as time goes to infinity. Full article

(This article belongs to the Special Issue Recent Developments in Ordinary and Partial Differential Equations)

22 pages, 377 KiB

Open AccessArticle

Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

by Tianjun Shen and Bo Li

Mathematics 2022, 10(7), 1112; https://doi.org/10.3390/math10071112 - 30 Mar 2022

Cited by 1 | Viewed by 1859

Abstract

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

[...] Read more.

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

and V be a Muckenhoupt weight belonging to a reverse Hölder class

R H_{q} (X)

for some

q \geq (Q + 1) / 2

. In this paper, we consider the Dirichlet problem for the Schrödinger equation

- \partial_{t}^{2} u + 𝓛 u + V u = 0

on the upper half-space

X \times R_{+}

, which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space

R^{Q}

to the metric measure space X and improves the reverse Hölder index from

q \geq Q

to

q \geq (Q + 1) / 2

. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

16 pages, 327 KiB

Open AccessFeature PaperArticle

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

by Gani Stamov, Ivanka Stamova and Cvetelina Spirova

Entropy 2021, 23(12), 1631; https://doi.org/10.3390/e23121631 - 3 Dec 2021

Cited by 9 | Viewed by 2913

Abstract

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single [...] Read more.

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

9 pages, 270 KiB

Open AccessFeature PaperArticle

Skewed Jensen—Fisher Divergence and Its Bounds

by Takuya Yamano

Foundations 2021, 1(2), 256-264; https://doi.org/10.3390/foundations1020018 - 16 Nov 2021

Cited by 3 | Viewed by 3466

Abstract

A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of [...] Read more.

A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of the skewed Jensen–Shannon divergence and of the variational distance. The defined measure coincides with the definition from the skewed Jensen–Shannon divergence via the de Bruijn identity. Our results follow from applying the logarithmic Sobolev inequality and Poincaré inequality. Full article

(This article belongs to the Section Information Sciences)

18 pages, 561 KiB

Open AccessArticle

Trajectory Tracking Control for Reaction–Diffusion System with Time Delay Using P-Type Iterative Learning Method

by Yaqiang Liu, Jianzhong Li and Zengwang Jin

Actuators 2021, 10(8), 186; https://doi.org/10.3390/act10080186 - 5 Aug 2021

Cited by 2 | Viewed by 2633

Abstract

This paper has dealt with a tracking control problem for a class of unstable reaction–diffusion system with time delay. Iterative learning algorithms are introduced to make the infinite-dimensional repetitive motion system track the desired trajectory. A new Lyapunov–Krasovskii functional is constructed to deal [...] Read more.

This paper has dealt with a tracking control problem for a class of unstable reaction–diffusion system with time delay. Iterative learning algorithms are introduced to make the infinite-dimensional repetitive motion system track the desired trajectory. A new Lyapunov–Krasovskii functional is constructed to deal with the time-delay system. Picewise distribution functions are applied in this paper to perform piecewise control operations. By using Poincaré–Wirtinger inequality, Cauchy–Schwartz inequality for integrals and Young’s inequality, the convergence of the system with time delay using iterative learning schemes is proved. Numerical simulation results have verified the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Control Systems in the Presence of Time Delays)

► Show Figures

Figure 1

23 pages, 447 KiB

Open AccessArticle

Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element

by Xuqing Zhang, Yu Zhang and Yidu Yang

Mathematics 2020, 8(8), 1252; https://doi.org/10.3390/math8081252 - 31 Jul 2020

Cited by 3 | Viewed by 2137

Abstract

This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the [...] Read more.

This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the nonconforming Crouzeix–Raviart finite element to the planar linear elastic eigenvalue problem with the pure traction boundary condition, and obtain the guaranteed lower eigenvalue bounds. Finally, numerical experiments with MATLAB program are reported. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

18 pages, 299 KiB

Open AccessArticle

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

by Pierre Hodara and Ioannis Papageorgiou

Mathematics 2019, 7(6), 518; https://doi.org/10.3390/math7060518 - 6 Jun 2019

Cited by 4 | Viewed by 2582

Abstract

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their [...] Read more.

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework. Full article

(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)

15 pages, 325 KiB

Open AccessArticle

Poincaré and Log–Sobolev Inequalities for Mixtures

by André Schlichting

Entropy 2019, 21(1), 89; https://doi.org/10.3390/e21010089 - 18 Jan 2019

Cited by 16 | Viewed by 4792

Abstract

This work studies mixtures of probability measures on

R^{n}

and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the

χ^{2}

-distance. The estimation [...] Read more.

This work studies mixtures of probability measures on

R^{n}

and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the

χ^{2}

-distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method. Full article

(This article belongs to the Special Issue Entropy and Information Inequalities)

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI