A Time-Reversible Formulation of the H-Theorem

Article信州大学工学部紀要 44: 1-10 (1978)departmental bulletin pape

On Markov Channels

Article信州大学工学部紀要 37: 1-10 (1974)departmental bulletin pape

Embedding Paths and Circuits into the Hypercube

Article信州大学工学部紀要 65: 1-16 (1989)departmental bulletin pape

An Algorithm for Solving Alive Situation Puzzles in GO Game

Article信州大学工学部紀要 65: 17-24 (1989)departmental bulletin pape

A Theory of Finite Topology and Image Processing

Article信州大学工学部紀要 69: 11-24 (1991)departmental bulletin pape

A Self-Supporting Quadruped Walking-Robot

Article信州大学工学部紀要 63: 19-28 (1988)departmental bulletin pape

Cell Petri Net Concepts

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.Mitsuru Jitsukawa - Chiba-ken Asahi-shi, Kotoda 2927-13 289-2502 JapanPauline Kawamoto - Shinshu University, Nagano, JapanYasunari Shidama - Shinshu University, Nagano, JapanYatsuka Nakamura - Shinshu University, Nagano, Japa

The Cauchy-Riemann Differential Equations of Complex Functions

In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.Yamazaki Hiroshi - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanNakamura Yatsuka - Shinshu University, Nagano, JapanPacharapokin Chanapat - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Chanapat Pacharapokin, Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Complex function differentiability. Formalized Mathematics, 17(2):67-72, 2009, doi:10.2478/v10037-009-0007-9.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Logical correctness of vector calculation programs

Summary. In C-program, vectors of n-dimension are sometimes represented by arrays, where the dimension n is saved in the 0-th element of each array. If we write the program in non-overwriting type, we can give Logical-Model to each program. Here, we give a program calculating inner product of 2 vectors, as an example of such a type, and its Logical-Model. If the Logical-Model is well defined, and theorems tying the model with previous definitions are given, we can say that the program is logically correct. In case the program is given as implicit function form (i.e., the result of calculation is given by a variable of one of arguments of a function), its Logical-Model is given by a definition of a new predicate form. Logical correctness of such a program is shown by theorems following the definition. As examples of such programs, we presented vector calculation of add, sub, minus and scalar product. and [7] provide the terminology and notation for this paper. In this paper m, n, i are natural numbers and D is a set. The following proposition is true (1) For all n, m holds n ∈ m iff n < m. Let D be a non empty set. One can check that there exists a finite 0-sequence of D which is non empty. The following proposition is true (2) For every non empty set D and for every non empty finite 0-sequence f of D holds len f > 0. Let D be a set and let q be a finite sequence of elements of D. The functor FS2XFS(q) yields a finite 0-sequence of D and is defined by: 37

The Real Vector Spaces of Finite Sequences are Finite Dimensional

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.Yatsuka Nakamura - Shinshu University Nagano, JapanNagato Oya - Shinshu University Nagano, JapanYasunari Shidama - Shinshu University Nagano, JapanArtur Korniłowicz - Institute of Computer Science, University of Białystok, Sosnowa 64, 15-887 Białystok, Polan