Volume 15 (2007): Issue 4 (January 2007)

In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact spaces and one point compactification are proved.

A formalization of the first proof from [6].

We have formalized the BCI-algebras closely following the book [7] pp. 16-19 and pp. 58-65. Firstly, the article focuses on the properties of the element and then the definition and properties of congruences and quotient algebras are given. Quotient algebras are the basic tools for exploring the structures of BCI-algebras.

In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.

In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.

I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m x n matrix A by the condition: A has the rank r if and only if, there is a r x r submatrix of A with a non-zero determinant, and for every k x k submatrix of A with a non-zero determinant we have k ≤ r.

At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.

This Mizar paper presents the definition of a "Preordered Coherent Space" (PCS). Furthermore, the paper defines a number of operations on PCS's and states and proves a number of elementary lemmas about these operations. PCS's have many useful properties which could qualify them for mathematical study in their own right. PCS's were invented, however, to construct Scott domains, to solve domain equations, and to construct models of various versions of lambda calculus.

For more on PCS's, see [11]. The present Mizar paper defines the operations on PCS's used in Chapter 8 of [3].

In this article, we extended properties of sequences of real numbers to sequences of extended real numbers. We also introduced basic properties of the inferior limit, superior limit and convergence of sequences of extended real numbers.

In this article the general theory of Commutative BCK-algebras and BCI-algebras and several classes of BCK-algebras are given according to [2].

In this article, we prove a series of differentiation identities [3] involving the secant and cosecant functions and specific combinations of special functions including trigonometric, exponential and logarithmic functions.