Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From t-Product to t-STP of Cubic Matrices With Application to Hyper-Networked Systems

Daizhan Cheng, Zhengping Ji

TL;DR

This work addresses the limitations of the t-product for cubic matrices in control applications by introducing a dimension-free, hybrid framework. It develops the DK-STP to preserve dimension while enabling linear and nonlinear dynamics, then overcomes subsystems’ decoupling by introducing the t-STP, which combines t-product with DK-STP to produce coupled, scalable dynamic models. The authors establish a comprehensive algebraic foundation, including monoid/ring/module structures, a universal homomorphism, and Lie-theoretic constructs ($gl_*$, $GL_*$), and demonstrate a hyper-networked evolutionary game as a concrete application. The approach offers a versatile toolkit for modeling large-scale, multi-way data in hyper-networks and related systems, with potential extensions to higher-order hyper-matrices and STP-CS integrations.

Abstract

From the perspective of dynamic (control) systems over cubic matrices, this paper proposes some new products on cubic matrices and uses them to construct linear and nonlinear control systems over cubic matrices. First, the t-product of cubic matrices is surveyed, and some algebraic properties are investigated. When t-product based dynamic (control) systems are considered, its two weaknesses are found as follows: (i) it has rigorous dimension restriction; (ii) it is unable to formulate nonlinear (control) systems. Then the dimension keeping semi-tensor product (DK-STP) of cubic matrices is proposed, which overcomes the aforementioned weaknesses. Unfortunately, all subsystems, corresponding to every frontal slides, of a DK-STP based dynamic (control) system are decoupled, which makes it unsuitable for formulating systems with coupled subsystems. Finally, the t-STP, a combination of t-product with DK-STP, is presented. The t-STP related algebraic structures over cubic matrices, such as group, ring, module, and Lie group/Lie algebra, etc. are investigated. Finally, t-STP based dynamic (control) systems over cubic matrices are obtained, and its application to hyper-networked evolutionary games is investigated.

From t-Product to t-STP of Cubic Matrices With Application to Hyper-Networked Systems

TL;DR

This work addresses the limitations of the t-product for cubic matrices in control applications by introducing a dimension-free, hybrid framework. It develops the DK-STP to preserve dimension while enabling linear and nonlinear dynamics, then overcomes subsystems’ decoupling by introducing the t-STP, which combines t-product with DK-STP to produce coupled, scalable dynamic models. The authors establish a comprehensive algebraic foundation, including monoid/ring/module structures, a universal homomorphism, and Lie-theoretic constructs (, ), and demonstrate a hyper-networked evolutionary game as a concrete application. The approach offers a versatile toolkit for modeling large-scale, multi-way data in hyper-networks and related systems, with potential extensions to higher-order hyper-matrices and STP-CS integrations.

Abstract

From the perspective of dynamic (control) systems over cubic matrices, this paper proposes some new products on cubic matrices and uses them to construct linear and nonlinear control systems over cubic matrices. First, the t-product of cubic matrices is surveyed, and some algebraic properties are investigated. When t-product based dynamic (control) systems are considered, its two weaknesses are found as follows: (i) it has rigorous dimension restriction; (ii) it is unable to formulate nonlinear (control) systems. Then the dimension keeping semi-tensor product (DK-STP) of cubic matrices is proposed, which overcomes the aforementioned weaknesses. Unfortunately, all subsystems, corresponding to every frontal slides, of a DK-STP based dynamic (control) system are decoupled, which makes it unsuitable for formulating systems with coupled subsystems. Finally, the t-STP, a combination of t-product with DK-STP, is presented. The t-STP related algebraic structures over cubic matrices, such as group, ring, module, and Lie group/Lie algebra, etc. are investigated. Finally, t-STP based dynamic (control) systems over cubic matrices are obtained, and its application to hyper-networked evolutionary games is investigated.

Paper Structure

This paper contains 22 sections, 31 theorems, 103 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. t-Product of Cubic Matrices
  4. DK-STP
  5. Hypergraph
  6. Algebraic Structure of Cubic Matrices
  7. Universal Homomorphism
  8. Dynamic System over Cubic Matrices
  9. From S-System to Dynamic System
  10. Analytic Functions of Cubic Matrices
  11. t-Product Based Dynamic (Control) Systems
  12. Dynamic Systems of Cubic Matrices via DK-STP Approach
  13. Kronecker Product and DK-STP of Cubic Matrices
  14. DK-STP Based Dynamic Control Systems
  15. t-STP and t-STP Based Systems
  16. ...and 7 more sections

Key Result

Proposition 2.6

Let $A\in {\mathcal{M}}_{m\times n}$, $B\in {\mathcal{M}}_{p\times q}$. Then where

Figures (5)

  • Figure 1: Cubic Matrices
  • Figure 2: A Hypergraph
  • Figure 3: Dual Hypergraph
  • Figure 4: A Supply Network $H$
  • Figure 5: Dual Hyper-graph $H^*$

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 54 more