Applications of Mathematics最新文献

We study the self-consistent chemotaxis-fluid system with nonlinear resource consumption

under no-flux boundary conditions in a bounded domain Ω ⊂ ℝ³ with smooth boundary. It is proved that this system possesses a global weak solution provided m > 1 and (alpha>{4 over 3}).

A fractional model is developed to study the transmission dynamics of tuberculosis disease. The use of a fractional model provides a memory effect and long-term dynamics often observed in chronic infectious diseases such as tuberculosis, which is characterized by a prolonged incubation period and risks of reactivation. The basic reproduction number is computed and we derive the qualitative stability analysis of equilibria. A sensitivity analysis is conducted to assess the impact of the model parameters. Three control strategies are applied, namely treatment, vaccination, and infection rate management, to minimize the number of infected individuals. Numerical simulations are carried out to illustrate the theoretical results obtained.

The tensor eigenvalue problem has been widely studied in recent years. In this paper, several new properties of eigenvalues and determinants of tensors are explored. We also proposed a formula to compute the determinant of a tensor as a mimic of the matrix determinant. The Perron-Frobenius theorem, one of the most important results in non-negative matrix theory, is proposed for the class of non-negative tensors in the Einstein product framework. Further, the power method, a widely used matrix iterative method for finding the largest eigenvalue, is framed for tensors using the Einstein product. The proposed higher-order power method is applied to calculate the largest eigenvalue of the Laplacian tensors associated with hyper-stars and hyper-trees. The numerical results show that the higher-order power method with the Einstein product is stable.

We propose a weighted HS (Hestenes-Stiefel)-FR (Fletcher-Reeves) hybrid conjugate gradient method for unconstrained multiobjective optimization problem, in which a new positive coefficient of the multiobjective steepest descent direction is adaptively updated to keep its positiveness. The method takes advantage of a weighted hybrid of our modified HS and FR parameters and under the Armijo-type backtracking line search, it has global convergence to a Pareto critical point (point satisfying the first-order necessary condition for Pareto optimality) without convexity assumption on the objectives. Numerical experiments show that the practical performance of the method is competitive with the existing methods such as conjugate gradient method, steepest descent method, Newton method, and quasi-Newton method for unconstrained multiobjective optimization.

We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions, the so-called one-troughed travelling waves. The main goal of this paper is to show the existence of at least two different one-troughed travelling waves for particular wave speeds and input parameters of the studied problem. We present the upper bounds for the maximal number of one-troughed solutions together with a visualisation of obtained results and corresponding solutions. Finally, we list several open questions regarding this topic.

We represent the point clouds of objects and audio signals as manifolds of Gaussian Mixture Models, and analyze the shape variation and compare the audio patterns using three divergence measures, namely the Kullback-Leibler Divergence, Jensen-Shannon Divergence, and Modified Symmetric Kullback-Leibler Divergence. Experiments are conducted on basic geometric shapes, 3D human body shapes, animal shapes, point clouds of the same object produced from the dense point clouds in the PU-GAN (Point Cloud Upsampling Adversarial Network) dataset. Then, we present a method to generate a point cloud of an audio signal using the Short-Time Fourier Transform. The audio-derived point clouds represent frequency, time, and magnitude relationships, enabling analysis of speech and audio patterns. The results across all datasets show that the Modified Symmetric Kullback-Leibler Divergence provides the most distinct and stable comparison between different point clouds, demonstrating its robustness for point cloud comparison.

One of the fundamental applications of artificial neural networks is solving Partial Differential Equations (PDEs) which has been considered in this paper. We have created an effective method by combining the spectral methods and multi-layer perceptron to solve Generalized Fitzhugh-Nagumo (GFHN) equation. In this method, we have used Chebyshev polynomials as activation functions of the multi-layer perceptron. In order to solve PDEs, independent variables, which are collocation points, have been used as input dataset. Furthermore, the loss function has been constructed from the residual of the equation and its boundary condition. Minimizing the loss function has adjusted the appropriate values for the parameters of the network. Hence, the network has shown an outstanding performance not only on the training dataset but also on the unseen data. Some numerical examples and a comparison between the results of our proposed method and other existing approaches have been provided to show the efficiency and accuracy of the proposed method. For this purpose different cases such as linear, nonlinear and multi dimensional equations are considered.

HIV is known for causing the destruction of the immune system by affecting different types of cells, while SARS-CoV-2 is an extremely contagious virus that leads to the development of COVID-19. Understanding how these two viruses interact in coinfected individuals is essential, especially in populations under antiretroviral treatment. In this study, we develop and analyze a novel mathematical model capturing the coinfection dynamics of HIV and SARS-CoV-2 under the influence of highly active antiretroviral therapy (HAART). In contrast to previous models, our formulation includes the effect of HAART on both infections and derives the basic reproduction numbers for each virus. We prove that transcritical bifurcations occur when the basic reproduction numbers cross the threshold value of 1, and we establish the conditions for stability of the disease-free equilibria. Numerical simulations show that HAART, although designed to control HIV, also reduces SARS-CoV-2 proliferation in coinfected hosts, which, as far as we know, has not been fully addressed in previous models in the literature. These findings reveal a potentially beneficial indirect effect of antiretroviral therapy on SARS-CoV-2 dynamics, offering new theoretical insights into the control of viral coinfections.

We investigate a class of Kirchhoff-type equations characterized by critical growth within Orlicz-Sobolev spaces. The main result establishes the existence of infinitely many solutions with negative energy. Using an adapted concentration-compactness principle and advanced variational methods, we overcome key challenges such as non-compactness and non-differentiability to the associated functionals. This work extends existing results to more general functional spaces, offering new insights into nonlocal nonlinear equations.

We consider a projection-based model reduction approach to computing the maximal impact, one agent or a group of agents has on the cooperative system. As a criterion for measuring the agent-team impact on multi-agent systems, we use the H_∞ norm, and output synchronization is taken as the underlying cooperative control scheme. We investigate a projection-based model reduction approach that allows efficient H_∞ norm calculation. The convergence of this approach depends on initial interpolation points, so we present approaches to their determination. Since the analysis of multi-agent systems is important from different perspectives, several comparisons are presented in the section on numerical experiments. A graph Laplacian matrix of an inter-agent interaction graph is a foundational element in modeling and analyzing multi-agent systems. We consider various graph topology matrices, system parameters, and excitations of different agents. Different strategies for selecting initial interpolation points are also compared with baseline approaches for calculating the H_∞ norm.