n-Linear Algebra of type I and its applications

TL;DR

This paper introduces n-linear algebra of type I, a generalization of classical algebraic structures, and explores its applications in areas like Markov chains, fuzzy models, and structural analysis.

Contribution

It defines n-linear algebra of type I, develops its properties, and demonstrates its applications in various fields such as coding theory, structural analysis, and fuzzy models.

Findings

n-linear algebra of type I generalizes bilinear algebra and bivector spaces

Applications include error correction, structural analysis, and fuzzy models

Several properties of n-linear algebra are proved

Abstract

With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of n-vector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces, we can define several types of linear transformations. The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further, it is suggested that the concept of n-matrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of these algebraic structures are given in the third chapter. The fourth chapter suggests problems to further a reader's understanding of the subject.