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Determinant of Some Matrices of Field Elements

Yatsuka Nakamura — 2006

Formalized Mathematics

Here, we present determinants of some square matrices of field elements. First, the determinat of 2 * 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.

Connectedness and Continuous Sequences in Finite Topological Spaces

Yatsuka Nakamura — 2006

Formalized Mathematics

First, equivalence conditions for connectedness are examined for a finite topological space (originated in [9]). Secondly, definitions of subspace, and components of the subspace of a finite topological space are given. Lastly, concepts of continuous finite sequence and minimum path of finite topological space are proposed.

Definition and some Properties of Information Entropy

Bo ZhangYatsuka Nakamura — 2007

Formalized Mathematics

In this article we mainly define the information entropy [3], [11] and prove some its basic properties. First, we discuss some properties on four kinds of transformation functions between vector and matrix. The transformation functions are LineVec2Mx, ColVec2Mx, Vec2DiagMx and Mx2FinS. Mx2FinS is a horizontal concatenation operator for a given matrix, treating rows of the given matrix as finite sequences, yielding a new finite sequence by horizontally joining each row of the given matrix in order...

Determinant and Inverse of Matrices of Real Elements

Nobuyuki TamuraYatsuka Nakamura — 2007

Formalized Mathematics

In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, and the inverse matrix. The new concept discussed in the case of matrices of real numbers is the property of matrices as operators acting on finite sequences of real numbers from both sides. The relations...

Basic Properties and Concept of Selected Subsequence of Zero Based Finite Sequences

Yatsuka NakamuraHisashi Ito — 2008

Formalized Mathematics

Here, we develop the theory of zero based finite sequences, which are sometimes, more useful in applications than normal one based finite sequences. The fundamental function Sgm is introduced as well as in case of normal finite sequences and other notions are also introduced. However, many theorems are a modification of old theorems of normal finite sequences, they are basically important and are necessary for applications. A new concept of selected subsequence is introduced. This concept came from...

A Theory of Matrices of Real Elements

Yatsuka NakamuraNobuyuki TamuraWenpai Chang — 2006

Formalized Mathematics

Here, the concept of matrix of real elements is introduced. This is defined as a special case of the general concept of matrix of a field. For such a real matrix, the notions of addition, subtraction, scalar product are defined. For any real finite sequences, two transformations to matrices are introduced. One of the matrices is of width 1, and the other is of length 1. By such transformations, two products of a matrix and a finite sequence are defined. Also the linearity of such product is shown....

Invertibility of Matrices of Field Elements

Yatsuka NakamuraKunio OniumiWenpai Chang — 2008

Formalized Mathematics

In this paper the theory of invertibility of matrices of field elements (see e.g. [5], [6]) is developed. The main purpose of this article is to prove that the left invertibility and the right invertibility are equivalent for a matrix of field elements. To prove this, we introduced a special transformation of matrix to some canonical forms. Other concepts as zero vector and base vectors of field elements are also introduced as a preparation.MML identifier: MATRIX14, version: 7.9.01 4.101.1015

Cell Petri Net Concepts

Mitsuru JitsukawaPauline KawamotoYasunari ShidamaYatsuka Nakamura — 2009

Formalized Mathematics

Based on the Petri net definitions and theorems already formalized in [8], with this article, we developed the concept of "Cell Petri Nets". It is based on [9]. In a cell Petri net we introduce the notions of colors and colored states of a Petri net, connecting mappings for linking two Petri nets, firing rules for transitions, and the synthesis of two or more Petri nets.MML identifier: PETRI 2, version: 7.11.01 4.117.1046

The Real Vector Spaces of Finite Sequences are Finite Dimensional

Yatsuka NakamuraArtur KorniłowiczNagato OyaYasunari Shidama — 2009

Formalized Mathematics

In this paper we show the finite dimensionality of real linear spaces with their carriers equal Rn. We also give the standard basis of such spaces. For the set Rn we introduce the concepts of linear manifold subsets and orthogonal subsets. The cardinality of orthonormal basis of discussed spaces is proved to equal n.MML identifier: EUCLID 7, version: 7.11.01 4.117.1046

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