http://hdl.handle.net/11320/3332 2026-08-14T11:11:46Z http://hdl.handle.net/11320/20706 Tytuł: Uniform Boundedness Principle Autorzy: Sakurai, Hideki; Kunimune, Hisayoshi; Shidama, Yasunari Abstrakt: Inthisarticle at first, we proved the lemma of the inferior limit and the superior limit. Next, we proved the Baire category theorem (Banach space version) [20], [9], [3], quoted it and proved the uniform boundedness principle. Moreover, the proof of the Banach-Steinhaus theorem is added. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/20702 Tytuł: Gauss Lemma and Law of Quadratic Reciprocity Autorzy: Yan, Li; Liang, Xiquan; Zhao, Junjie Abstrakt: In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems [14], [12]. Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/20695 Tytuł: Regular Expression Quantifiers– at least m Occurrences Autorzy: Trybulec, Michał Abstrakt: This is the second article on regular expression quantifiers. [4] introduced the quantifiers m to n occurrences and optional occurrence. In the sequel, the quantifiers: at least m occurrences and positive closure (at least 1 occurrence) are introduced. Notation and terminology were taken from [8], several properties of regular expressions from [7]. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/20690 Tytuł: The Vector Space of Subsets of a Set Based on Symmetric Difference Autorzy: Alama, Jesse Abstrakt: For each set X, the power set of X forms a vector space over the field Z2 (the two-element field {0,1} with addition and multiplication done modulo 2): vector addition is disjoint union, and scalar multiplication is definedby the two equations (1·x := x, 0·x := ∅ for subsets x of X). See [10], Exercise2.K, for more information. 2008-01-01T00:00:00Z