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O rozwiązywaniu zadań konkursu matematyczno-informatycznego Koala

Dębicka Donata

MINUT Matematyka i Informatyka na Uczelniach Technicznych

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2022

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Nr 4

37-46

PL

W artykule prezentowany jest drużynowy konkurs matematyczno-informatyczny Koala dla młodzieży z Wielkopolski, którego celem jest rozbudzanie i rozwijanie zainteresowania matematyką i informatyką, w szczególności takimi dziedzinami jak: kombinatoryka, algorytmika i logika (stąd nazwa konkursu: Koala). Ponadto celem konkursu jest doskonalenie umiejętności pracy w grupie, dyskusji oraz prezentowania własnych rozwiązań zadań i uzasadniania ich poprawności. W roku szkolnym 2021/2022 odbyła się już IX edycja konkursu. W artykule przedstawione zostaną metody (strategie) przydatne podczas rozwiązywania niektórych typów zadań konkursowych.

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Kiedy można wygrać i jak wtedy grać, żeby wygrać?

Słanina Piotr

MINUT Matematyka i Informatyka na Uczelniach Technicznych

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2022

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Nr 4

1-15

PL

ednym z głównych problemów w grze kombinatorycznej jest wskazanie gracza, który jest w stanie zapewnić sobie zwycięstwo (tak zwane rozwiązanie gry). Jeżeli już to się uda, to czasami nadal nie będziemy w stanie wskazać temu graczowi, jak ma grać, aby wygrać. W tym artykule czytelnik zapozna się z najpopularniejszymi matematycznymi narzędziami, stosowanymi przy poszukiwaniu najlepszych strategii w grach kombinatorycznych. Ich opisy będą uzupełnione o przykłady gier. Będą się one pojawiały według stopnia trudności ich rozwiązania od najłatwiejszych do nierozwązanych po dzień dzisiejszy.

3

On Rough Approximations of Languages under Infinite Index Indiscernibility Relations

Martín Torres Gabriela

Fundamenta Informaticae

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2021

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Vol. 179, nr 3

275--293

EN

In the paper [13] Păun, Polkowski and Skowron introduce several indiscernibility relations among strings that are infinite index equivalence or tolerance relations, and study lower and upper rough approximations of languages defined by them. In this paper we develop a further study of some of these indiscernibility relations among strings. We characterize the classes defined by them, and the rough approximations of general and context free languages under them. We also compare some of the rough approximations these relations produce to the ones given by the congruences defining testable, reverse testable, locally testable, piecewise testable and commutative languages. Those yield languages belonging to that families. Next, we modify some of the relations to obtain congruences, and study the families of languages the rough approximations under them give rise to. One of these modificated relations turns out to be the k-abelian congruence, that was defined by J. Karhumäki in [7], in the context of combinatorics on words. We show that it defines a pseudo-principal +-variety, a term defined in [9]. Our results in that work are then applied to determine when a given language has a best upper approximation in that family. Finally, we make some comments on the accuracy of the rough approximations obtained in each case.

4

Combinatorial Testing of Context Aware Android Applications

Piparia Shraddha, Adamo David, Bryce Renee, Do Hyunsook, Bryant Barrett

Annals of Computer Science and Information Systems

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2021

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Vol. 25

17--26

EN

Mobile devices such as smart phones and smart watches utilize apps that run in context aware environments and must respond to context changes such as changes in network connectivity, battery level, screen orientation, and more. The large number of GUI events and context events often complicate the testing process. This work expands the AutoDroid tool to automatically generate tests that are guided by PairwiseInterleaved coverage of GUI event and context event sequences. We systematically weave context and GUI events into testing using the pairwise interleaved algorithm. The results show that the pairwise interleaved algorithm achieves up to five times higher code coverage compared to a technique that generates test suites in a single predefined context (without interleaving context and GUI events), a technique that changes the context at the beginning of each test case (without interleaving context and GUI events), and Monkey-Context-GUI (which randomly chooses context and GUI events). Future work will expand this strategy to include more context variables and test emerging technologies such as IoT and autonomous vehicles.

5

Rotating machinery single plane balancing non-contact method

Kędzierski Łukasz, Sikorski Rafał

Przegląd Elektrotechniczny

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2020

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R. 96, nr 4

162--167

EN

The paper introduces a method for single plane balancing of rotating machinery using order analysis of data acquired by the proximity probes. It provides a detailed description of calculating the unbalance magnitude and location followed by the description of the algorithm responsible for distribution of discrete trim weights values in discrete locations in order to counter the unbalance. The method could be used when the bearings of the rotating machinery are not accessible for accelerometers installation, when great accuracy is needed or when there are only a few angular locations available for installation of few possible trim weights. Experiments conducted using the method helped to reduce the vibration caused by unbalance by the order of three to ten times (depending on the initial vibration) in every case which proves its effectiveness.

PL

Artykuł ten opisuje metodę wyważania jednopłaszczyznowego maszyn wirujących wykorzystującą analizę rzędów na podstawie danych z czujników zbliżenia. Artykuł przedstawia dokładny opis obliczania amplitudy i kąta niewyważenia oraz opis algorytmu odpowiedzialnego za rozmieszczenie dostępnych mas w dostępnych kątowych lokalizacjach w celu minimalizacji niewyważenia. Zaprezentowana metoda może zostać wykorzystana, gdy nie ma możliwości zainstalowania akcelerometrów na łożyskach maszyny wirującej, gdy potrzebna jest duża dokładność lub gdy konstrukcja maszyny wirującej pozwala jedynie na instalację określonej kombinacji mas w określonych miejscach. W przypadku wszystkich prób wyważania za pomocą przedstawionej metody udało się ograniczyć wibracje spowodowane niewyważeniem o trzy do dziesięciu razy (w zależności od początkowego poziomu wibracji), co dowodzi skuteczności metody.

6

Studying Word Equations by a Method of Weighted Frequencies

Saarela A.

Fundamenta Informaticae

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2018

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Vol. 162, nr 2/3

223--235

EN

We briefly survey some results and open problems on word equations, especially on those equations where the right-hand side is a power of a variable. We discuss a method that was recently used to prove one of the results, and we prove improved versions of some lemmas that are related to the method and can be used as tools when studying word equations. We use the method and the tools to give new, simple proofs for several old results.

7

How Hungarian teachers think about combinatorics and its teaching

Ambrus G., Rácz N. K., Vancsó Ö.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2017

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Vol. 22

11--22

EN

Teachers’ attitudes and beliefs have a fundamental influence on the way of teaching and the efficiency of learning. This topic was also brought up in the didactic research in recent years, in connection with mathematics teachers as well.– To the best of our knowledge, however, a study that investigates this specific area (combinatorics) was not included. As combinatorics has a special role in connecting the development of mathematical thinking and teaching different areas, our short-time Project (Didactics Grant of the Hungarian Academy: Complex Mathematics Education in the 21st Century) focused on the teaching of combinatorics in Hungary. Our aim was to examine some aspects of this field in connection with Tamás Varga’s method and also teachers’ beliefs towards combinatorics with the help of a questionnaire. The main results of this survey are discussed below.

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Elements transpositions and their impact on the cyclic structure of permutations

Grebennik I., Chorna O.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2015

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Vol. 4, No 3

33--38

EN

The objective of this paper is the investigation of the cyclic structure and permutation properties based on neighbor elements transposition properties and the properties of the permutation polyhedron. In this paper we consider special type of transpositions of elements in a permutation. A feature of these transpositions is that they corresponding to the adjacency criterion in a permutation polyhedron. We will investigate permutation properties with the help of the permutation polyhedron by using the immersing in the Euclidian space. Six permutation types are considered in correspondence with the location of arbitraży components. We consider the impact of the corresponding components on the cyclic structure of permutations depending on the type of a permutation. In this paper we formulate the assertion about the features of the impact of transpositions corresponding to the adjacency criterion on the permutations consisting of the one cycle. During the proof of statement all six types of permutations are considered and clearly demonstrated that only two types arrangement of the elements in the cycle contribute to the persistence a single cycle in the permutation after the impast of two transpositions. Research conducted in the Niven work, will be further employed in mathematical modeling and computational methods. Especially for solving combinatorial optimization problems and for the generation of combinatorial objects with a predetermined cyclic structure.

9

Epichristoffel Words and Minimization of Moore Automata

Castiglione G., Sciortino M.

Fundamenta Informaticae

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2014

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Vol. 134, nr 3/4

319--333

EN

This paper is focused on the connection between the combinatorics of words and minimization of automata. The three main ingredients are the epichristoffel words, Moore automata and a variant of Hopcroft’s algorithm for their minimization. Epichristoffel words defined in [14] generalize some properties of circular sturmian words. Here we prove a factorization property and the existence of the reduction tree, that uniquely identifies the structure of the word. Furthermore, in the paper we investigate the problem of the minimization of Moore automata by defining a variant of Hopcroft’s minimization algorithm. The use of this variant makes simpler the computation of the running time and consequently the study of families of automata that represent the extremal cases of the minimization process. Indeed, such a variant allows to use the above mentioned factorization property of the epichristoffel words and their reduction trees in order to find an infinite family of Moore automata such that the execution of the algorithm is uniquely determined and tight.

10

Araucaria Trees: Construction and Grafting Theorems

Schmitt D., Spehner J-C.

Fundamenta Informaticae

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2013

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Vol. 123, nr 4

417--445

EN

Araucarias have been introduced by Schott and Spehner as trees which appear in the minimal automaton of the shuffle of words. We give here a new definition of araucarias which is more constructive and we prove that our definition of araucarias is equivalent to the original one. From the new definition we derive an optimal algorithm for the construction of araucarias and a new method for calculating their size. Moreover we characterize araucarias by properties of their maximal paths, by associating a capacity to every edge. We then show that every araucaria can be obtained by grafting and merging smaller araucarias. We prove also that every directed tree can be embedded in an araucaria. Moreover we define a capacity for every vertex of an araucaria, which leads to different new enumeration formulas for araucarias.

11

O mostach, oficerach, uczennicach i barwach, czyli słynne zadania z kombinatoryki. Cz. 2

Biernacki W.

Wiadomości Naftowe i Gazownicze

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2008

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R. 11, Nr 9 (125)

9--13

12

O mostach, oficerach, uczennicach i barwach, czyli słynne zadania z kombinatoryki. Cz. 1

Biernacki W.

Wiadomości Naftowe i Gazownicze

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2008

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R. 11, Nr 8 (124)

10--14

13

O mostach, oficerach, uczennicach i barwach, czyli słynne zadania z kombinatoryki. Cz. 3

Biernacki W.

Wiadomości Naftowe i Gazownicze

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2008

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R. 11, Nr 10 (126)

8--14

14

Multiple perspectives and generalizations of the Desargues configuration

Prażmowska M.

Demonstratio Mathematica

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2006

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Vol. 39, nr 4

887-906

EN

We introduce a class of finite confgurations, which we call combinatorial Grassmannians, and which generalize the Desargues configuration. Fundamental geome- tric properties of them are established, in particular we determine their automorphisms, correlations, mutual embedability, and prove that no one of them contains a Pascal or Pappus figure.

15

On arcs in quadruple systems

Wilczyńska K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 3

275-278

EN

In this paper we deal with arcs in the quadruple systems. The arcs have been constructed.

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Procedura obliczania wartości liczbowych wielomianów symetrycznych wielu zmiennych

Piwowarczyk T.

Czasopismo Techniczne. Elektrotechnika

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2000

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R. 97, z. 4-E

80--98

EN

The theory of symmetric polynomials of multiple variables produces an unlimited set of algebra identities. These formulas contain numerous coefficients. For the most part their evaluation is difficult. This article presents a certain method of finding the coefficients which is based on some theorems of the theory. As a result, we obtain a linear system with only one solution. This method has been checked for the degree not greater that 6. Hypothetically, we can assume that the method is correct for any natural degree

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Podstawy mnogościowe przestrzeni wektorowej wielomianów symetrycznych wielu zmiennych w notacji spektralnej : zarys teorii

Piwowarczyk T.

Czasopismo Techniczne. Elektrotechnika

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2000

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R. 97, z. 4-E

61--79

EN

All symmetric polynomials of multiple variables, with the variables belonging to the fixed, final set, are the elements of the vector space of infinite dimension. The vector space contains numerous finite subspaces. These subspaces are subjected to a standard analysis. Set theory classifications and ordering relations indicate how to choose the basis. The investigation of transient matrices reveals sets of algebra identities. The unlimited set of these identities is particularly useful when it comes to engineering applications.