Applied Mathematics and Computation最新文献

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph $G\left[S\right]$ dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where $\psi ⩾\lceil 3n/4\rceil$. Finally, we identify all the connected n-vertex graphs with generalized 4-independence number $\psi \in \{3,\lceil 3n/4\rceil ,n-1,n-2\}$ having the minimum spectral radius.

In this article, we present an error estimation in the $L^2$ norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

Given $G=(g_1,\cdots ,g_t)$ with $g_i\in N$ for $1\le i\le t$, let $G{K}_s^{\le t}$ denote the non-uniform complete hypergraph on s vertices, whose edge set contains $g_i$ copies of every i-subset of vertex set for $1\le i\le t$. Let $K_s^{\le t}$ denote $G{K}_s^{\le t}$ for $g_i=1$ for $1\le i\le t$. Recently, He et al. determined all $s,t$ such that $K_s^{\le t}$ has a 1-factorization. In this manuscript, we consider the 1-factorization of $G{K}_s^{\le t}$ and obtain the following results. (1) If $2g_t\ge g_j$ for $1\le j\le t-1$ and $s\equiv 0\left(modt\right)$, then $G{K}_s^{\le t}$ has a 1-factorization for sufficiently large s. (2) If $G{K}_s^{\le t}$ has a 1-factorization for sufficiently large s, then $s\equiv 0,-1\left(modt\right)$.

This paper deals with the Turing bifurcation and pattern dynamics of a Lotka-Volterra model with the predator-taxis and the homogeneous no-flux boundary conditions. To investigate the pattern dynamics, we first give the occurrence conditions of the Turing bifurcation. It is found that there is no Turing bifurcation when predator-taxis disappears, while the Turing bifurcation occurs as predator-taxis is presented. Next, we establish the amplitude equation by virtue of weakly nonlinear analysis. Our theoretical result suggests the Lotka-Volterra model admits the supercritical or subcritical Turing bifurcation. In this manner, we can determine the stability of the bifurcating solution. Finally, some numerical simulation results verify the validity of the theoretical analysis. The stripe pattern, the mixed patterns, and wave patterns are performed. Interestingly, the stable stripe patterns will be broken and become wave patterns when the predator-taxis parameter is far from the Turing bifurcation critical point.

This paper considers the problem of trajectory tracking and collision avoidance for a class of high-order nonlinear strict feedback systems with unknown nonlinearities. The main issue is how to ensure collision avoidance and tracking performance simultaneously in the presence of unknown nonlinear functions. To address the issue, an integral-multiplicative barrier Lyapunov function (BLF) is integrated into the backstepping procedure to remove the dynamic mismatching issue of the existing SUM-type BLF. It has been proven that the proposed adaptive approach ensures both collision avoidance and tracking performance of high-order nonlinear systems in multi-obstacle environments, and all the signals in the closed-loop system are uniformly ultimately bounded (UUB). Simulation results confirm the effectiveness of the proposed method.

In this paper, a dynamic event-driven optimal control scheme is proposed for the zero-sum differential graphical games in nonlinear multiagent systems with full-state constraints. Initially, to address the dual demands of optimality and state constraints, a set of system transformation functions are introduced to satisfy the state constraints of the agents. Then, by applying the principle of differential game theory, the distributed optimal control problem affected by external disturbances is formulated as a zero-sum differential graphical game, and the performance index function related to neighbor informations and disturbances is designed for each follower. Afterwards, to enhance the utilization of communication resource, a novel dynamic event-triggered mechanism characterized by a dynamic threshold parameter and an auxiliary dynamic variable is developed, which not only exhibits greater flexibility but also diminishes the frequency of triggers. Furthermore, the approximate optimal control strategies are obtained by employing an event-driven adaptive dynamic programming algorithm. Ultimately, a simulation example is presented to verify the applicability of the proposed control approach.

This paper focuses on the problem of dynamic event-triggered predefined-time adaptive attitude control of quadrotor unmanned aerial vehicle (QUAV) suffering from unknown deception attacks. The command filter is utilized to avoid the “explosion of complexity” problem, while concurrently eliminating the effect of filtered error by constructing the fractional power error compensation signals. By using the Nussbaum gain technique, the unknown control coefficients generated by unknown deception attacks have been resisted. A dynamic event-triggered predefined-time attitude control scheme is proposed by introducing internal dynamic variables, which reduces the data transmissions and avoids the Zeno behavior. It is proved that the closed-loop system is practically predefined-time stable, and the attitude of QUAV is driven into a small region near the origin in a predefined time. Finally, a simulation example is provided to show the effectiveness and superiority of the developed predefined-time attitude control algorithm.

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h ($h\le H$). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

A signed graph Σ is a graph whose edges yield the signs ±1. Let $K_n$ be the complete graph with n vertices and $\Sigma =(K_n,F^{-})$ be a signed complete graph, where F is a subgraph induced by the negative edges of Σ. The least Laplacian eigenvalue of Σ is the least eigenvalue of its Laplacian matrix. A unicyclic graph is a connected graph containing exactly one cycle. In this paper, we focus on the least Laplacian eigenvalue of $\Sigma =(K_n,F^{-})$, where F is a unicyclic graph.

In this paper, a new computational topological framework for hypergraph analysis and recognition is developed. “Topology provides scale” is the principle at the core of this set of algebraic topological tools, whose fundamental notion is that of a scale-space topological model ($s^2$-model). The scale of this parameterized sequence of algebraic hypergraphs, all having the same Euler-Poincaré characteristic than the original hypergraph G, is provided by its relational topology in terms of evolution of incidence or adjacency connectivity maps. Its algebraic homological counterpart is again an $s^2$-model, allowing the computation of new topological characteristics of G, which far exceeds current homological analytical techniques. Both scale-space algebraic dynamical systems are hypergraph isomorphic invariants. The hypergraph isomorphism problem is attacked here to demonstrate the power of the proposed framework, by proving the ability of $s^2$-models to differentiate challenging cases that are difficult or even infeasible for state-of-the-art practical polynomial solvers. The processing, analysis, classification and learning power of the $s^2$-model, at both combinatorial and algebraic levels, augurs positive prospects with respect to its application to physical, biological and social network analysis.