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1
Content available Automatic search of rational self-equivalences
Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2018
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Vol. 23
67--74
EN
Two Witt rings that are not strongly isomorphic (i.e., two Witt rings over two fields that are not Witt equivalent) have different groups of strong automorphisms. Therefore, the description of a group of strong automorphisms is different for almost every Witt ring, which requires the use various tools in proofs. It is natural idea to use computers to generate strong automorphisms of the Witt rings, which is especially effective in the case of the finitely generated Witt rings, where a complete list of strong automorphisms can be created. In this paper we present the algorithm that was used to generate strong automorphisms from the infinite group of strong automorphisms of the Witt ring of rational numbers W(Q).
2
Content available Applications of python programs in solving of equations based on selected numerical methods
Ziółkowski M., Stępień L., Stępień M. R., Gola A.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2017
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Vol. 22
31--45
EN
In this paper we present the mathematical background of the four most used numerical methods of solving equations and few examples of Python applications that find the approximations of the roots of the given equations. We also compare the exact and approximate solutions of polynomial equations of third degree. Exact solutions are obtained with usage of Cardano formulae by the help of Mathematica environment, the approximate ones – based on the selected numerical methods by the help of applications written in Python language.
3
Content available remote The usage of monotonic and injective function concepts in solving simple equations and inequalities
Ziółkowski M., Stępień L., Stępień M. R.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
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2016
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Nr 2(12)
37--46
EN
In this paper, we draw attention to the existing interesting connection between the concepts of injective or monotonic function and the process of solving equations and inequalities. In our opinion students of the fourth educational level in Poland or higher education often make mistakes because they do not understand the concepts of monotonic or injective functions or simply do not use the above concepts in the correct way in solving mathematical tasks. These mistakes appear especially in the process of solving simple power equations and inequalities. We present some examples of applications of these concepts in typical and more complicated tasks for students.
PL
W prezentowanym artykule zwracamy uwagę na istniejący związek pomiędzy pojęciami funkcji różnowartościowej oraz monotonicznej a procesem rozwiązywania prostych równań i nierówności. Uczniowie czwartego etapu edukacyjnego w Polsce oraz studenci popełniają błędy w procesie rozwiązywania równań i nierówności, co jest niewątpliwie związane z nieznajomością lub niezrozumieniem tych pojęć. Typowe błędy pojawiają się w szczególności w prostych równaniach i nierównościach potęgowych. Przedstawiamy szereg przykładów wykorzystania tych pojęć przy rozwiązywaniu prostych oraz bardziej skomplikowanych zadań dla uczniów i studentów oraz proponujemy alternatywną, być może trudniejszą, metodę rozwiązywania, która jednak pokazuje związek tych pojęć z procesem rozwiązywania równań i nierówności.
4
Content available From arithmetic expressions to propositional formulae
Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2016
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Vol. 21
135--143
EN
In papers [3], [4], [5] Authors presented a new method of solving some kinds of computational tasks in the area of linear algebra by applying SAT-solver as the highly optimized algorithms for solving the problem of propositional satisfiability. On input SAT-solver (cf. [1], [2]) takes a propositional formula in the clause form. In this paper we show in detail how any arithmetical expression can be translated into propositional formula in the CNF form skipping out its traditional form. For this, we define the notion of consistency of arithmetic and boolean valuations.
5
Content available Time calculations in school
Ziółkowski M., Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2015
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Vol. 20
23--28
EN
In this article we want to present some simple method of doing time calculations which is not often used by teachers and show that the discussed way does not have to be difficult for students and is based on strict mathematical rules.
6
Content available Automorphisms of Witt rings of local type
Stępień M. R.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 3
109-119
EN
We present a description of the group of strong automorphisms of Witt rings of local type (Witt rings of local fields) using quaternionic structures.
7
Content available On teaching of geometric transformations in school
Stępień L., Stępień M. R., Ziółkowski M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2014
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Vol. 19
153--164
EN
The current core curriculum in mathematics for lower secondary school (3-rd educational level in Poland) omits formal definitions of concepts related to geometric transformations in the plane and is based on their intuitive sense. Practice shows that the current approach makes teaching very difficult and the students solve the typical tasks, not understanding the meaning of geometrical concepts. The article contains basic concepts connected with geometric transformations and examples of geometric tasks that are solved in the third and also in the fourth educational level in an intuitive way, sometimes deviating or even incompatible with the mathematical definition. We show how they could be solved in easier way with introducing definitions of geometric transformations in a simple and understandable for students way sometimes using vector calculus. We take into account isometries: reflection and point symmetry, rotation and translation and similarities with particular consideration on homothetic transformation.
8
Content available On strong automorphisms of direct products of Witt rings. Part 2
Stępień M. R., Stępień L.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 4
119--125
EN
In the paper strong automorphisms of finitely generated Witt rings are considered. Every finitely generated Witt ring can be expressed in terms of Z / 2Z and basic indecomposable Witt rings using the operations of group ring formation and direct product. Groups of strong automorphisms of basic indecomposables and their direct products and the description of all strong automorphisms of any group Witt ring are known. In this paper the strong automorphisms of direct products of group Witt rings are considered. Presented are two wide classes of Witt rings where the group of strong automorphisms is isomorphic to the direct product of groups of strong automorphisms of Witt rings which are factors in the direct product.
9
Content available On strong automorphisms of direct products of Witt rings. Part 1
Stępień M. R., Stępień L.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 3
123--133
EN
The notion of Witt ring is fundamental in bilinear algebra. Automorphisms of Witt rings have been investigated until recent years. In this paper we consider Witt rings which are direct products of finite number of other Witt rings. We shall present a necessary condition in order to group of all strong automorphisms of direct product of Witt rings be a direct product of groups of strong automorphisms of Witt rings which are factors in the direct product. Subsequently, there are considered some examples of Witt rings, where described condition is fulfilled.
10
Content available remote Uncountability of the group of strong automorphisms of Witt ring of rational numbers
Stępień M. R.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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Vol. 11, nr 1
117-128
EN
We use the notion of rational self-equivalence which is a special case of Hilbert symbol equivalence of fields, where both fields are considered to be the field Q of rational numbers. We define a small self-equivalence of the field Q as a special case of small equivalence of fields - a tool for constructing Hilbert-symbol equivalence of fields. We shall show, that one can choose initial sets of prime numbers and then control the processes of extending of small self-equivalence such that uncountable many rational self-equivalences can be constructed. The final conclusion is the corollary deciding that the group of strong automorphisms of Witt ring W(Q) of rational numbers is uncountable.
11
Content available Automorphisms of Witt rings of finite fields
Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2011
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Vol. 16
67--70
EN
The problem of general description of the group of automorphisms of any Witt ring W seems to be very difficult to solve. However, there are many types of Witt rings, which automorphism are described precisely (e.g. [1], [2], [4], [5], [6],[7], [8]). In our paper we characterize automorphisms of abstract Witt rings (cf. [3]) isomorphic to powers of Witt rings of quadratic forms with coefficients in finite fields with characteristic different from 2.
12
Content available Automatic search of automorphisms of Witt rings
Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2011
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Vol. 16
141--146
EN
The investigation of strong automorphisms of Witt rings is a difficult task because of variety of their structures. Cordes Theorem, known in literature as Harrison-Cordes criterion (cf. [1, Proposition 2.2], [3, Harrison's Criterion]), makes the task of describing all the strong automorphisms of a given (abstract) Witt ring W = (G, R) easier. By this theorem, it suffices to find all such automorphisms ơ of the group G that map the distiguished element -1 of the group G into itself (i.e. ơ(-1) = -1) in which the value sets of 1-fold Pfister forms are preserved in the following sense: ơ(D(1, α)) = D(1, ơ(α)) for all α ∈ G. We use the above criterion and the well-known structure of the group G as a vector space over two-element field F2 for searching all automorphisms of this group. Then we check Harrison-Cordes criterion for found automorpisms and obtain all the automorpisms of a Witt ring W. The task is easy for small rings (with small groups G). For searching of all strong automorpisms of bigger Witt rings we use a computer which automatizes the procedure described above. We present the algorithm for finding strong automorphisms of a Witt rings with finite group G and show how this algorithm can be optimized.
13
Content available remote Automorphisms of Witt rings and quaternionic structures
Stępień M. R.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2011
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Vol. 10, nr 1
231-237
EN
M. Marshall introduced the notion of quaternionic structure and he showed that the categories of Witt rings and quaternionic structures are naturally equivalent. Quaternionic structures turn out to bea useful tool for the investigation of Witt rings, since it suffices to handle the structure of a group. In our paper we shall describe precisely the one-to-one correspondence between automorphisms of quaternionic structures and strong the automorphisms of Witt rings.
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