arXiv subjects

Jiandong Zhu

Publications and source records attributed to Jiandong Zhu.

8 recordsLinked to original sources

arXiv2026

From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors

The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.

math.RA↗

arXiv2024

Synchronization of High-Dimensional Linear Networks over Finite Fields

This paper investigates the synchronization problems for general high-dimensional linear networks over finite fields. By using the technique of linear transformations and invariant subspaces for linear spaces over finite fields, several necessary and sufficient conditions for the synchronization of high-dimensional linear networks over finite fields are proposed. This paper not only generalizes the existing results from 1-dimensional to high-dimensional linear networks but also adopts a new approach. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.

math.OC↗

arXiv2022

A Necessary and Sufficient Condition for Complete phase synchronization of high-dimensional nonidentical Kuramoto oscillators

For original Kuramoto models with nonidentical oscillators, it is impossible to realize complete phase synchronization. However, this paper reveals that complete phase synchronization can be achieved for a large class of high-dimensional Kuramoto models with nonidentical oscillators. Under the topology of strongly connected digraphs, a necessary and sufficient condition for complete phase synchronization is obtained. Finally, some simulations are provided to validate the obtained theoretic results.

math.DS↗

arXiv2022

On Matrix Method of Symmetric Games

This paper provides a new version of matrix semi-tensor product method based on adjacent transpositions to test symmetric games. The advantage of using adjacent transpositions lies in the great simplification of analysis of symmetric games. By using the new method, new necessary and sufficient conditions for symmetric games are proposed, and a group of bases of a symmetric game space can be easily calculated. Moreover, the testing equations with the minimum number can be concretely determined. Finally, two examples are displayed to show the effectiveness of the proposed method.

cs.GT↗

arXiv2021

Exact exponential synchronization rate of high-dimensional Kuramoto models with identical oscillators and digraphs

For the high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although exponential synchronization was proved under some initial state constraints, the exact exponential synchronization rate has not been revealed until now. In this paper, the exponential synchronization rate is precisely determined as the smallest non-zero real part of Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.

math-ph↗

arXiv2018

Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. %due to the asymmetry of the adjacency matrices. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting spanning trees. Finally, some numerical simulations are given to illustrate the obtained theoretical results.

math.OC↗

arXiv2015

On Potential Equations of Finite Games

In this paper, some new criteria for detecting whether a finite game is potential are proposed by solving potential equations. The verification equations with the minimal number for checking a potential game are obtained for the first time. Some connections between the potential equations and the existing characterizations of potential games are established. It is revealed that a finite game is potential if and only if its every bi-matrix sub-game is potential.

cs.GT↗

arXiv2014

Decomposition with respect to outputs for Boolean control networks

This paper investigates the problem of decomposition with respect to outputs for Boolean control networks (BCNs). First, with the linear expression of BCNs and the matrix semi-tensor product, some algebraic equivalent conditions for the decomposition are obtained. Second, a necessary and sufficient graphical condition for the decomposition with respect to outputs is given. Third, an effective method is proposed to reduce the computational burden in the realization of the decomposition. Finally, some examples are addressed to validate the effectiveness of the proposed method.

math.OC↗