DSpace Kolekcja:

Embedded Lattice and Properties of Gram Matrix

2017-10-05T23:14:34Z

Tytuł: Embedded Lattice and Properties of Gram Matrix Autorzy: Futa, Yuichi; Shidama, Yasunari Abstrakt: In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [16] and cryptographic systems with lattice [17].

Group of Homography in Real Projective Plane

2017-10-05T23:14:32Z

Tytuł: Group of Homography in Real Projective Plane Autorzy: Coghetto, Roland Abstrakt: Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group. Then, we prove that, using the notations of Borsuk and Szmielew in [3] “Consider in space ℝℙ2 points P1, P2, P3, P4 of which three points are not collinear and points Q1,Q2,Q3,Q4 each three points of which are also not collinear. There exists one homography h of space ℝℙ2 such that h(Pi) = Qi for i = 1, 2, 3, 4. ”(Existence Statement 52 and Existence Statement 53) [3]. Or, using notations of Richter [11] “Let [a], [b], [c], [d] in ℝℙ2 be four points of which no three are collinear and let [a′],[b′],[c′],[d′] in ℝℙ2 be another four points of which no three are collinear, then there exists a 3 × 3 matrix M such that [Ma] = [a′], [Mb] = [b′], [Mc] = [c′], and [Md] = [d′] ”Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs [10], [9].

Ordered Rings and Fields

2017-10-05T23:14:33Z

Tytuł: Ordered Rings and Fields Autorzy: Schwarzweller, Christoph Abstrakt: We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].

Introduction to Liouville Numbers

2017-10-05T23:14:30Z

Tytuł: Introduction to Liouville Numbers Autorzy: Grabowski, Adam; Korniłowicz, Artur Abstrakt: The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and 0 < x − p q < 1 q n . It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum X∞ k=1 ak b k! for a finite sequence {ak}k∈N and b ∈ N. Based on this definition, we also introduced the so-called Liouville number as L = X∞ k=1 10−k! = 0.110001000000000000000001 . . . , substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https: //en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.