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1
Content available remote Differentiability of Polynomials over Reals
100%
Korniłowicz A.
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nr 1
31-37
EN
In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].
2
Content available remote Mazur-Ulam Theorem
100%
Korniłowicz A.
Formalized Mathematics
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2011
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tom 19
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nr 3
127-130
EN
The Mazur-Ulam theorem [15] has been formulated as two registrations: cluster bijective isometric -> midpoints-preserving Function of E, F; and cluster isometric midpoints-preserving -> Affine Function of E, F; A proof given by Jussi Väisälä [23] has been formalized.
3
Content available remote Cayley's Theorem
100%
Korniłowicz A.
Formalized Mathematics
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2011
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tom 19
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nr 4
223-225
EN
The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.
4
Content available remote Cayley-Dickson Construction
100%
Korniłowicz A.
Formalized Mathematics
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2012
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tom 20
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nr 4
281-290
EN
Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.
5
Content available remote Products in Categories without Uniqueness of cod and dom
100%
Korniłowicz A.
Formalized Mathematics
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2012
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tom 20
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nr 4
303-307
EN
The paper introduces Cartesian products in categories without uniqueness of cod and dom. It is proven that set-theoretical product is the product in the category Ens [7].
6
Content available remote Arithmetic Operations on Functions from Sets into Functional Sets
100%
Korniłowicz A.
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nr 1
43-60
EN
In this paper we introduce sets containing number-valued functions. Different arithmetic operations on maps between any set and such functional sets are later defined.MML identifier: VALUED 2, version: 7.11.01 4.117.1046
7
Content available remote Collective Operations on Number-Membered Sets
100%
Korniłowicz A.
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nr 2
99-115
EN
The article starts with definitions of sets of opposite and inverse numbers of a given number membered set. Next, collective addition, subtraction, multiplication and division of two sets are defined. Complex numbers cases and extended real numbers ones are introduced separately and unified for reals. Shortcuts for singletons cases are also defined.
8
Content available remote Miscellaneous Facts about Open Functions and Continuous Functions
100%
Korniłowicz A.
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tom 18
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nr 3
171-174
EN
In this article we give definitions of open functions and continuous functions formulated in terms of "balls" of given topological spaces.
9
Content available remote The Correspondence Betweenn-dimensional Euclidean Space and the Product ofnReal Lines
100%
Korniłowicz A.
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tom 18
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nr 1
81-85
EN
In the article we prove that a family of open n-hypercubes is a basis of n-dimensional Euclidean space. The equality of the space and the product of n real lines has been proven.
10
Content available remote On the Continuity of Some Functions
100%
Korniłowicz A.
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tom 18
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nr 3
175-183
EN
We prove that basic arithmetic operations preserve continuity of functions.
11
Content available remote The First Isomorphism Theorem and Other Properties of Rings
63%
Korniłowicz A. , Schwarzweller C.
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nr 4
291-301
EN
Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial
12
Content available remote Pseudo-Canonical Formulae are Classical
63%
Caminati M. B. , Korniłowicz A.
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nr 2
99-103
EN
An original result about Hilbert Positive Propositional Calculus introduced in [11] is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see [17]) are a subset of the classical tautologies.
13
Content available remote Vieta’s Formula about the Sum of Roots of Polynomials
63%
Korniłowicz A. , Pąk K.
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nr 2
87-92
EN
In the article we formalized in the Mizar system [2] the Vieta formula about the sum of roots of a polynomial anxn + an−1xn−1 + ··· + a1x + a0 defined over an algebraically closed field. The formula says that [...] x1+x2+⋯+xn−1+xn=−an−1an $x_1 + x_2 + \cdots + x_{n - 1} + x_n = - {{a_{n - 1} } \over {a_n }}$ , where x1, x2,…, xn are (not necessarily distinct) roots of the polynomial [12]. In the article the sum is denoted by SumRoots.
14
Content available remote Basel Problem
63%
Pąk K. , Korniłowicz A.
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nr 2
149-155
EN
A rigorous elementary proof of the Basel problem [6, 1] ∑n=1∞1n2=π26 $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.
15
Content available remote Basel Problem – Preliminaries
63%
Korniłowicz A. , Pąk K.
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nr 2
141-147
EN
In the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.
16
Content available remote Introduction to Liouville Numbers
63%
Grabowski A. , Korniłowicz A.
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nr 1
39-48
EN
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 [...] 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 [...] for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as [...] 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.
17
Content available remote Characteristic of Rings. Prime Fields
63%
Schwarzweller C. , Korniłowicz A.
Formalized Mathematics
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2015
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tom 23
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nr 4
333-349
EN
The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.
18
Content available remote Fundamental Group of n-sphere for n ≥ 2
63%
Riccardi M. , Korniłowicz A.
Formalized Mathematics
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2012
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tom 20
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nr 2
97-104
EN
Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17]
19
Content available remote The Borsuk-Ulam Theorem
63%
Korniłowicz A. , Riccardi M.
Formalized Mathematics
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2012
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tom 20
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nr 2
105-112
EN
The Borsuk-Ulam theorem about antipodals is proven, [18, pp. 32-33].
20
Content available remote Niven’s Theorem
63%
Korniłowicz A. , Naumowicz A.
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tom 24
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nr 4
301-308
EN
This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].
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