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Czes{\l}aw Byli{\'n}ski.
Alexandroff One Point Compactification,
Formalized Mathematics 15(4),
pages 167-170, 2007. MML Identifier: COMPACT1 Summary: 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.
Freek Wiedijk.
Arrow's Impossibility Theorem,
Formalized Mathematics 15(4),
pages 171-174, 2007. MML Identifier: ARROW Summary: A formalization of the first proof from \cite{Geanakoplos:1996}.
Yuzhong Ding, Zhiyong Pang.
Congruences and Quotient Algebras of {BCI}-algebras,
Formalized Mathematics 15(4),
pages 175-180, 2007. MML Identifier: BCIALG_2 Summary: We have formalized the BCI-algebras closely following the book \cite{BCIAlgebras2} 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.
Xiquan Liang, Li Yan, Junjie Zhao.
Linear Congruence Relation and Complete Residue Systems,
Formalized Mathematics 15(4),
pages 181-187, 2007. MML Identifier: INT_4 Summary: 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.
Cuiying Peng, Fuguo Ge, Xiquan Liang.
Several Integrability Formulas of Special Functions,
Formalized Mathematics 15(4),
pages 189-198, 2007. MML Identifier: INTEGRA8 Summary: In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.
Karol P\c{a}k.
Basic Properties of the Rank of Matrices over a Field,
Formalized Mathematics 15(4),
pages 199-211, 2007. MML Identifier: MATRIX13 Summary: In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.\par 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 $m\times n$ matrix $A$ by the condition: $A$ has the rank $r$ if and only if, there is a $r\times r$ submatrix of $A$ with a non-zero determinant, and for every $k\times k$ submatrix of $A$ with a non-zero determinant we have $k\leq r$.\par 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 $A$.
Klaus E. Grue, Artur Korni{\l}owicz.
Basic Operations on Preordered Coherent Spaces,
Formalized Mathematics 15(4),
pages 213-230, 2007. MML Identifier: PCS_0 Summary: 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 \cite{Krivine93}. The present Mizar paper defines the operations on PCS's used in Chapter 8 of \cite{berline97kappadenotational}.
Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, Hiroyuki Okazaki.
Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers,
Formalized Mathematics 15(4),
pages 231-235, 2007. MML Identifier: RINFSUP2 Summary: 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.
Tao Sun, Dahai Hu, Xiquan Liang.
Several Classes of {BCK}-algebras and their Properties,
Formalized Mathematics 15(4),
pages 237-242, 2007. MML Identifier: BCIALG_3 Summary: In this article the general theory of Commutative BCK-algebras and BCI-algebras and several classes of {BCK}-algebras are given according to \cite{BCIAlgebras}.
Bo Li, Pan Wang.
Several Differentiation Formulas of Special Functions. {P}art {VI},
Formalized Mathematics 15(4),
pages 243-250, 2007. MML Identifier: FDIFF_10 Summary: In this article, we prove a series of differentiation identities \cite{Chemnitius:1956} involving the secant and cosecant functions and specific combinations of special functions including trigonometric, exponential and logarithmic functions.