DSpace Kolekcja: http://hdl.handle.net/11320/8761 2026-08-02T12:54:07Z 2026-08-02T12:54:07Z Solutions of Linear Equations Pąk, Karol http://hdl.handle.net/11320/20685 2026-07-30T08:31:06Z 2008-01-01T00:00:00Z Tytuł: Solutions of Linear Equations Autorzy: Pąk, Karol Abstrakt: In this paper I present the Kronecker-Capelli theorem which states that a system of linear equations has a solution if and only if the rank of its coefficient matrix is equal to the rank of its augmented matrix. 2008-01-01T00:00:00Z Complete Spaces Pąk, Karol http://hdl.handle.net/11320/20671 2026-07-29T11:56:10Z 2008-01-01T00:00:00Z Tytuł: Complete Spaces Autorzy: Pąk, Karol Abstrakt: This paper is a continuation of [12]. First some definitions needed to formulate Cantor’s theorem on complete spaces and show several facts about them are introduced. Next section contains the proof of Cantor’s theorem and some properties of complete spaces resulting from this theorem. Moreover, countable compact spaces and proofs of auxiliary facts about them is defined. I also show the important condition that every metric space is compact if and only if it is countably compact. Then I prove that every metric space is compact if and only if it is a complete and totally bounded space. I also introduce the definition of the metric space with the well metric. This article is based on [13]. 2008-01-01T00:00:00Z Elementary Number Theory Problems. Part XIX Pąk, Karol http://hdl.handle.net/11320/19740 2026-02-02T13:53:08Z 2025-01-01T00:00:00Z Tytuł: Elementary Number Theory Problems. Part XIX Autorzy: Pąk, Karol Abstrakt: In this paper, we present formal solutions to twelve problems selected from Wacław Sierpiński’s book 250 Problems in Elementary Number Theory. The selected problems are: 108, 112–114, 118–119, 127, 129, 130, and 132–134 formalized in the Mizar system. 2025-01-01T00:00:00Z Elementary Number Theory Problems. Part XVIII Grabowski, Adam http://hdl.handle.net/11320/19737 2026-02-02T12:02:22Z 2025-01-01T00:00:00Z Tytuł: Elementary Number Theory Problems. Part XVIII Autorzy: Grabowski, Adam Abstrakt: In this paper another seven problems from Wacław Sierpiński’s book “250 Problems in Elementary Number Theory” are formalized, using the Mizar formalism, namely: 53, 61, 81, 90, 100, 156, and 167. 2025-01-01T00:00:00Z