XII th Czech-Polish-Slovak Mathematical School

XIIth Czech-Polish-Slovak Mathematical School Solution of a quadratic equation. Martin Billich Abstract: This paper is focused on methods of solving quadratic equations. We present some of the geometric solutions to a quadratic equation such as using the power of a point with respect to a circle. We will take the properties of roots of a quadratic equation (Viete’s theorem) in our considerations. We want to show that this can be solved by ordinary geometry by using straight lines and circles on a plane. Literature: (1) Čižmár, J.: Euklidovské konštrukcie, Proceedings of Seminars on Computational Geometry SCG’99, Volume 8, pp. 31-47. (2) Šedivý, O. - Božek, M.- Duplák, J. – Kršňák, P. - Trenkler, M.: Geometria 2, Bratislava, SPN, 1987. Author: Martin Billich, RNDr. PF KU v Ružomberku, katedra matematiky a fyziky Námestie A. Hlinku 56/1, 034 01 Ružomberok [email protected] Special Block Factorization of Linear Systems With Indefinite Matrices Daniela Bittnerová Abstract: The paper presents a view of some useful properties needed to solving systems of linear equations with indefinite matrices. These systems occur in many technical applications. The main idea is to find a special LDLT-factorization that could be removed zeros from the main diagonal. This method can be also used to great dense matrices if the block decomposition is found. Author: RNDr. Daniela Bittnerová, CSc. Department of Mathematics and Didactics of Mathematics Technical Univerzity in Liberec Hálkova 6, 461 17 Liberec, Czech Republic. [email protected] Inductive proofs Jiří Cihlář Abstract: The paper contains specimens of inductive proofs of theorems belonging to various spheres of mathematics, and methodological recommendations for removing students´ difficulties in learning inductive proofs. Literature: (1) Balcar, B.: Teorie množin, Academia, Praha, 1986 (2) Steinhaus, G.: Matematický kaleidoskop, 1953 (3) Conway, J.: On numbers and games, Academic Press, London, 1976 Author: Jiří Cihlář, Prof. RNDr. CSc. Katedra matematiky PF UJEP, České mládeže 8, 400 96 Ústí nad Labem, Česká republika [email protected] 1

Investigating mathematical connections with technology Gunnar Gjone Abstract: Investigating number patterns, and geometrical properties can efficiently be performed with technology. We will look at how number patterns can be investigated using spreadsheets. Examples will be figurate (polygonal) numbers, Fibonacci-numbers and other number sequences. In geometry we will look at some of the possibilities with „Cabri-like“ software. We will especially demonstrate such investigations on the calculator ClassPad 300 by Casio, which has a spreadsheet and other programs built in. Literature: (1) Gjone, G. & T. Andersen (2003) Shapes and Numbers. Mathematical Activities on ClassPad 300. Bornbach, Germany: Casio Europe. Author: Gunnar Gjone, Professor dr. philos University of Oslo Department of Teacher Training and School development P.O. Box 1099 Blindern NO-0317 Oslo Norway [email protected] Derivative of a function at the point Ján Gunčaga, Štefan Tkačik Abstract: Paper presents the concept of differentiable functions and derivates. The notion of a differentiable function f at the point x is based on existence the function ϕ(u) which is continuous at the point 0. ϕ(u) is defined by formula f(x + y) – f(x) = ϕ(u) u for all real number u at the neighbourhood of the point 0. Literature (1) Fulier J.: Funkcie a funkčné myslenie vo vyučovaní matematickej analýzy. Nitra, UKF, 2001. (2) Gunčaga J.: Limitné procesy v školskej matematike. Dizertačná práca. Nitra, UKF, 2004. (3) Hischer H., Scheid H.: Grundbegriffe der Analysis: Genese und Beispiele aus didaktischer Sicht. Heidelberg; Berlin; Oxford, Spektrum Akademischer Verlag, 1995. (4) Kluvánek I.: Prípravka na diferenciálny a integrálny počet. Žilina, VŠDS, 1991. (5) Kluvánek I.: Čo nie je dobré vo vyučovaní matematickej analýzy?. In: Matematické obzory, 1991, č. 36, s. 23 - 49/č. 37, s. 47 - 66. (6) Tkačik Š.: Spojitosť a limity trochu inak. In: Zborník konferencie Setkání kateder matematiky České a Slovenské republiky připravující budoucí učitele, Ústí nad Labem, 2004, s. 85 – 89. Author: PaedDr. Ján Gunčaga, PhD., RNDr. Štefan Tkačik, PhD. Katedra matematiky a fyziky PF KU v Ružomberku Námestie Andreja Hlinku 56 034 01 Ružomberok, Slovensko [email protected] [email protected]

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Survival analysis and the unemployment data. Karel Hrach Abstract: Survival analysis is a set of methods allowing to analyse time to any event, even by the presence of so called censored observations. Censoring means, that before the event occurred, the following of some statistical units had been stopped. This contribution shortly introduces selected methods of survival analysis (estimators of the survival function, tests comparing survival functions for two or more groups, Cox model for the dependency of the survival function on selected regressors) and performs the application of these methods to the real data using SW STATISTICA. The data were obtained from the Labour Offices in frame of the grant GAČR 402/04/0263 “Statistical methods and unemployment”. Literature: (1) Fleming, T.R., Harrington, D.P.: Counting Processes and Survival Analysis. John Willey, USA, 1991. (2) Zvárová, J. a kol.: Statistické metody v epidemiologii. Karolinum, Praha, 2003. (3) Učebnice StatSoft. www.statsoft.cz/textbook/stathome.html. Author: Karel Hrach, RNDr., Ph.D. Vysoká škola ekonomie a managementu Národního odboje 17 40003 Ústí nad Labem, Czech Rep. [email protected]; [email protected] On Symmetry Reduction of Some New Classes of the First-Order Differential Equations in the Space M(1,4)× × R(u) Vasyl Fedorchuk, Volodymyr Fedorchuk Abstract: We consider classes of the first-order differential equations in the space M(1,4) × R(u), which are invariant under splitting subgroups of the generalized Poincare′ group P(1,4). To study these classes we used some continuous subgroups of the symmetry groups of considered classes. Using the functional bases of the invariants of some splitting subgroups of the symmetry groups of considered classes we have constructed the ansatzes, which reduce some new classes investigated to differential equations with fewer independent variables. The corresponding symmetry reduction is done. Literature: (1) Fedorchuk V.M., Splitting subalgebras of the Lie algebra of the generalized Poincare′ group P(1,4), Ukr. Mat. Zh., 1979, V.31, N6, 717--722. (2) Fedorchuk V., Fedorchuk V., Some new differential equations of the first-order in the spaces M(1,3) × R(u)and M(1,4) × R(u) with given symmetry groups, Functional Analysis and its Applications, North-Holland Mathematics Studies, 197, Elsevier, Amsterdam et all, 2004, 85--95. (3) Fedorchuk V.I., On Symmetry Reduction of Some Classes of the First-Order Differential Equations in the Space M(1,4) × R(u), Proceedings of Institute of Mathematics of NAS of Ukraine, 2004, 50, Part 1, 92--97. Authors: Vasyl Fedorchuk, Institute of Mathematics, Pedagogical Academy, Podchorazych 2, 30-084 Krakow, Poland;

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Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine, 3b Naukova Str., 79601 Lviv, Ukraine [email protected] Volodymyr Fedorchuk Franko Lviv National University, 1 Universytets'ka Str., 79000 Lviv, Ukraine [email protected] Remarks on metrics Roman Frič Abstract: The notions of metrics and metric space are important tools when dealing with multidimensional models and spaces of functions. Besides, it is our experience that some rather elementary constructions involving metrics and their analysis help the students to appreciate the beauty of abstraction and to understand situations leading to mathematical structures. We present two applications of the triangle inequality and discuss metrics in products. Literature: Adámek, J.: Mathematical Structures and Categories, SNTL, Praha, 1982. (In Czech.) Author: Roman Frič, Doc.,RNDr.,DrSc. Pedagogická fakulta Katolické university Nám. A. Hlinku 56/1, 034 01 Ružomberok Slovenská republika [email protected] Developing mathematical thinking through investigating problems Len Frobisher Abstract: Investigating problems (or Investigations) is an approach to teaching and learning school mathematics for all ages of student. It supplements and complements other approaches which aim to teach the concepts, knowledge and skills which form the school mathematics curriculum. Investigations balance the formal teaching approach common in most schools by involving students in mathematical processes such as collecting evidence or data, sorting, ordering, searching for and describing patterns, explaining, reasoning, conjecturing and generalizing. The most important aspect of Investigations is the way that it requires students to behave as mathematicians resulting in them developing their mathematical thinking. Literature: (1) Chambers, J., Frobisher, L., Kurta, J and Mumford, J. Maths Investigations, Books 3 to 6, Oxford, Heinemann Education, 2003 (2) Frobisher, L. Problems, Investigations and an Investigative Approach. In A. Orton and G. Wain (eds), Issues in Teaching Mathematics. London, Cassell, 1994 (3) Kirkby, D. and Patilla, P. Maths Investigations, London, Hutchinson Education, 1987 (4) Kmetic, S. and Frobisher, L. Izzivi za Mlade Matematike, Maribor, Zalozba Obzorja, 1996 (5) Orton, A. and Frobisher, L. Insights into Teaching Mathematics, London, Continuum, 2005 Author: Leonard John Frobisher Cert Maths, Dip Maths, BSc, BA, MA. 58 Buckstone Avenue, Leeds LS17 5HP England [email protected] 4

Sentence indicative and logical semantics Karel Kamiš Abstract: The principal aim is the analysis of methodologically important aspects of scientific language. This model is based on the scheme of the dialogue and on the discrimination of the syntactic semantic and pragmatic level in the analysis of language communication. The central organizing element of the semantic structure of sentences is to be seen in the predicate with its semantic valency. Literature: (1) Kamiš, K.: Lokálové konstrukce ve struktuře věty. Acta Universitatis Purkynianae, 90, Studia Linguistica, UJEP, 2003, ISBN 80-7044-498-3 Author: Karel Kamiš, Prof.,PhDr., CSc. Katedra bohemistiky, Pedagogická fakulta UJEP České mládeže 8, 400 96 Ústí nad Labem,Česká republika [email protected] Certain class of formulas of some normal form in modal system T Marcin Kaptur Abstract: The normal form is a formula in the special form, which is equivalent to all formulas in the considered system. Let us consider the modal system T axiomatically defined with the following primitive connectives: ~ (negation), L (necessity), ∨ (disjunction). This article deals with introducing certain rules to convert the same class formulas of the system T into the normal form. The conversion is made by the defined functions on a set of formulas from the system T. Of course, the normal form may be useful if it is easy to decide whether such a form is provable or not. This paper includes the algorithm converting the formula into the normal form. Literature: (1) G. E. Hughes & M. J. Cresswell, An introduction to modal logic, Methuen & Co Ltd, 1968. (2) J. Słupecki, G. Bryll, Pewien dowód pełności systemu S5 Lewisa, Zeszyty naukowe WyŜszej Szkoły Pedagogicznej w Opolu, Matematyka XIII, 1973. Author: Marcin Kaptur, mgr Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] Exhibits from the book: The Investigative Approach in School Mathematics Jan Kopka Abstract: Investigation is not only a method for a teacher to use in teaching mathematics, but also for a student to use in learning the subject. Thus, this book is for both teachers and students. The book has a practical character. We wish to demonstrate the method of investigation to teachers and to help them see school mathematics from this perspective. To this end, we present several investigated situations and many problems solved in detail. We feel that the situations

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and problems we present can help the student to learn and appreciate the true nature of mathematics and indeed, begin investigations of his own. Literature: (1) Cofman, J.: What to solve? New York, Oxford univerzity press, 1989. (2) Kopka, J: Hrozny problémů ve školské matematice, Ústí nad Labem, UJEP, 1999 (3) Kopka, J.: Výzkumný přístup při výuce matematiky, Ústí nad Labem, UJEP, 2004 (4) Orton, A., Frobisher, L.: Insightes into Teaching Mathematics, London, Cassell, 1996 (5) Schultz, J.: Mathematics for Elementary School Teachers, A Bell & Howell Company, 1982 Author: Jan Kopka, Prof., RNDr., CSc. Faculty of Education of University J. E. Purkyně České mládeže 8, 400 96 Ústí nad Labem, Czech Republic [email protected] On some integrals associated with the Hermite School GraŜyna Krech Abstract: We consider, among others, the integral G(f;x,y) = ∫R P(x, y, z) f (z) dz, where x>0, yєR, P(x, y, z) =(2πsinh2x)-½ exp(-½(tanh2x)-1(y2 + z2)+(sinh2x)-1yz) and we prove an approximation theorem for functions f such that ƒ є Lp (R) , where 1 ≤ p ≤ ∞ . Literature: (1) Goselin, J., Stempak, K. : Conjugate expansions for Hermite functions, III. J. of Math., 38(1994), 177-197. (2) Muckenhoupt, B. : Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139(1969), 231-242. Author: GraŜyna Krech, magister. Instytut Matematyki Akademii Pedagogicznej Ul. PodchorąŜych 2, 30-084 Kraków, Polska [email protected] Engel's reductions of stochastic graphs Ireneusz Krech Abstract: In this work we present a method of calculating the weight factor of all traces leading to a fixed boundary node in a stochastic graph. These weights are the probabilities of some events in countable probabilistic spaces. Literature: (1) A. Engel, Wahrscheinlichkeitsrechnung und Statistik, Band 1. Ernst Klett Verlag, Stuttgart 1980. (2) N. Deo, Teoria grafów i jej zastosowania w technice i informatyce, PWN, Warszawa, 1980. Author: Ireneusz Krech, magister Instytut Matematyki Akademii Pedagogicznej w Krakowie ul. PodchorąŜych 2, 30-084 Kraków [email protected]

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On Some Relations on Regular Expressions Mirosław Kurkowski, Anna Sowik Abstract: In optimalization and verification of algorithms and software systems automatic methods play an important role. Often for a given (discrete and distributed) system or algorithm an adequate abstract model is created and suitable automatically searched. Usually these systems/algorithms are modeled as automata. This is one of the most imporatnt reasons why in computer science studies it is necessery to have a course on Automata and Formal Language Theory. In this paper we investigate some problems which studying computer science students have when they need to create an automaton from a given regular expression. We will define and discuss some relations on regular expressions and their properties which simplify a process of creating an automaton from a given regular expression. Literature: (1) E. Clarke, O. Grumberg, D. Peled, Model Checking, MIT Press, 2000. (2) J. Hopcroft, J. Ullman, Wprowadzenie do teorii automatów, języków i obliczeń, Wydawnictwo Naukowe PWN 2003. 1st author

Mirosław Kurkowski, PhD Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] 2nd author Anna Sowik Student at Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] On the understanding of the notion of absolute value by secondary school graduates Joanna Major Abstract: The main aim of this article is to present some questions connected to applying the knowledge of absolute value in solving problems by secondary school graduates. Author: Joanna Major, Mgr. Instytut Matematyki, Akademia Pedagogiczna Podchorazych 2, 30-084 Krakow [email protected]

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On probability problems for the secondary school students Maciej Major Abstract: This paper presents an the analysis of some examination problems from the point of view of the government standards of teaching for the secondary school students. Author: Maciej Major, Dr. Instytut Matematyki, Akademia Pedagogiczna Podchorazych 2, 30-084 Krakow [email protected] The explicit form of no arbitrage condition for the multifactor term structure model Gennady Medvedev Abstract: The no arbitrage conditions are derived in the explicit form for the market where the zero coupons bonds of various maturities are accessible for the investors to draw up the portfolios. It is supposed that the investor at any moment of time has a possibility to make the self-financed portfolio of given cost. It is considered that the processes of the short interest rate and rates of inflation follow the stochastic differential equations. These random processes are described by some objective probability measure. The no arbitrage condition, usually used in such situation, is the assumption about existence of the equivalent martingale measure. This condition is quite general but difficult verifiable, as at present the problem on construction equivalent martingale measure waits for its solution. Therefore deriving of the no arbitrage conditions in the explicit form, without resorting to construction of equivalent martingale measure, is useful. The no arbitrage condition for multifactor models of a term structure of the interest rates is considered. The important special case when the zero coupons bond price depends on the factors only through the nominal interest rate is discussed. Author: Gennady Medvedev, prof. Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Czestochowa, Poland [email protected] Some Attributes of Reverse Numbers Recorded on Z-adic Numerical System Jan Melichar Abstract: Differences of two-digit and three-digit natural numbers recorded on random z-adic numerical system are divisible by the same number depending on natural base of this numerical system. Even palindromes have similar attributes. There are examples of activities suitable for education of methodology of mathematics for students studying mathematics for primary school. There are shown clusters of problems and verification of given attributes. Literature: (1) Kopka, J.: Výzkumný přístup při výuce matematiky, UJEP, Ústí nad Labem, 2004, ISBN 807044-604-8 (2) Kopka, J.: Hrozny problémů ve školské matematice, UJEP, Ústí nad Labem, 1999, ISBN 807044-247-6 (3) Kopka, J.: Konkretizace a zobecňování, In: Podíl matematiky na přípravě učitele elementární školy, str.86-91, UP, Olomouc, 2002, ISBN 80-244-0440-0 8

(4) Melichar, J.: Rozvoj matematického myšlení I, UJEP, Ústí nad Labem, 2003, ISBN 80-7044512-2 Author: Jan Melichar, Prof., RNDr., CSc. Pedagogická fakulta UJEP, Hoření 13, 400 96 Ústí nad Labem, Česká republika [email protected] The history of mathematics and its role Ondřej Moc Abstract: The method of genetic parallel (cited in (1)) claims that the history of mathematics allows us to gain a better insight into the understanding of the development of a pupils´s ideas about mathematics. It means that the history of mathematics provides an important help to a teacher. The history of mathematics also has a similar role for students. It allows them understand a subject matter in an easier manner, together with the connection and background of mathematical discoveries. I want to mention some significant historical articles and point to some ways of work with them. Literature: (1) Hejný, M. a kol.: Teória vyučovania matematiky 2. SPN, Bratislava 1990 (2) Edwards, Jr., C.H.: The historical development of the calculus. Springer, New York 1979. Author: Ondřej Moc, Mgr. Katedra matematiky a statistiky Fakulta sociálně ekonomická UJEP Moskevská 54, 400 96 Ústí nad Labem [email protected] A few remarks concerning the reactions of students of pedagogics to their futur pupils mistakes Barbara Nawolska Abstract: This paper is on attempt to analyse how students of early elementary school pedagogics are prepared to teach mathematics in classes with young pupils. They should be able to not only properly react to their pupils mistakes but have the ability to use these mistakes in order to teach them mathematics. This paper has many examples of students reactions to their pupils mistakes. Author: Barbara Nawolska, Dr. Instytut Pedagogiki Przedszkolnej i Szkolnej Akademia Pedagogiczna w Krakowie [email protected] Free radical rings Petr Němec Abstract: Commutative rings equal to their Jacobson radical are investigated. In particular, free objects in this class are constructed. Author: Petr Němec, Prof., RNDr., Ing., DrSc. Faculty of Education of University J. E. Purkyně 9

České mládeže 8, 400 96 Ústí nad Labem, Czech Republic [email protected] Properties of (n,k)-cube Sergiej Novikov The (n,k)-cube is the graph G(V,R) with |V| = kⁿ. The words p(vi) with length n in alphabet {0,1,2,...,k-1} are ascribed to vertexs vi ∈ V. Vertex vi and vertex vj are united by an edge if for p(vi) = α1 α2…αn

and

p(vj)= β1β2….β n. we have

α1 - β1+ α2 - β2+...+αn - βn=1 We have obtained properties for (n,k)-cube: 1. n

≤

deg(v) ≤ 2 n v∈ V(G) 2. D 2 n−i (G) = C in 2 i (k-2) n−i for 3. |R| = ½

0≤i≤n

n

∑

(2n − i ) C in 2 i (k-2) n−i

i =0

Theorem 1. The (n,k)-cube for n ≥ 2, k ≥ 3 contains the Hamiltonian circuit. Theorem 2. For n ≥ 2, k ≥ 4, where k is even-numbered, the (n,k)-cube is Hamilton’s graph.

Author: Sergiej Novikov, prof. Dr hab Institute of Computer Science University of Podlasie, Poland Sienkiewicza Str. 51, 08-110 Siedlce [email protected] On a random walk Martin Papčo Abstract: There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. This leads to fuzzy random variable, as a map of probability measures on the sample space into probability measures on the real line satisfying certain measurability condition. We present more complex example of a natural fuzzy random situation in which random input is, due to to some random fluctuations, distorted and hence from the given output only a fuzzy conclusion about input can be drawn. This case is described as a random walk with no returns in a finite set (straightforwardly generalized to a countable set). The fuzzy random variable resulting from our model of the random walk sends each output to the conditional probability that the walk proceeds trough the respective points of the input set given that we have finished our walkk at the given output; it can be calculated via the Bayes rule. Supported by VEGA Grant 1/9056/24. 10

Literature: [1] BUGAJSKI, S.: Statiscical maps I., Basic properties Math. Slovaca 51 (2001), 321-342 [2] BUGAJSKI, S.: Statiscical maps II. Operational random variables, Math. Slovaca 51 (2001), 343-361 [3] FRIČ, R.: Remarks on statiscical maps and fuzzy (operational) random variables, preprint, 2004 [4] PAPČO, M.: On measurable spaces and measurable maps, Tatra Mountains Math. Publ. 28 (2004), 125-140 [5] PAPČO, M.: On fuzzy random variables: examples and generalizations, Tatra Mountains Math. Publ. (To appear) Author: Martin Papčo, Mgr. Faculty of Education of Catholic University Námestie A. Hlinku 56, 034 01 Ružomberok, Slovakia [email protected] Approximation of Functions Using Interpolation Polynomials Štěpán Pelikán Abstract: This application is used to approximate functions using table data with appropriate interpolation polynomials - Lagrange's, Newton's and cubic spline functions. Individual points from the table data can be input directly or they can be randomly generated according to given conditions. After data input, the application display concrete step-by-step calculations and a graph of the interpolation function. Author: Štěpán Pelikán, PaedDr. Faculty of Education of University J. E. Purkyně České mládeže 8, 400 96 Ústí nad Labem, Czech Republic [email protected] Development of Geometrical Imagination of Pupils Jaroslav Perný Abstract: The contribution deals with geometrical, mainly space imagination, as an important man skill and with some possibilities of its development both at students of pedagogy and particularly basic school pupils. Model tasks and topics used for development of this skill are presented. Author: Jaroslav Perný, Doc.,PaedDr.,Ph.D. Department of Mathematics and Didactics of Mathematics Technical Univerzity in Liberec Hálkova 6, 461 17 Liberec, Czech Republic [email protected] Stochastic Methodology in Mathematics „for Everybody“ Adam Płocki Abstract: The amount of probabilities consists not only of groups of definitions and sentences but of special judgements. Those judgements concerns the process of decision making, validation of hypotheses, estimation of chances and risks. It arises from the extent of probability of some 11

events, from the average value of random variable etc. To form those judgements and proofs is a mathematical activity typical for the stochastic. Development of the ability to form judgements, which in practice arises from the fact that some event has low probability (in fact is imposible), is an important aim of education of stochastic at school. There are examples of some statistic judgements suitable for mathematical lessons. Literature: [1] Płocki, A: Pravděpodobnost kolem nás, Acta Universitatis Purkynianae 68, Studia Matematica IV., Univerzita J. E. Purkyně, Ústí nad Labem, 2001, ISBN 80-7044-355-3 Author: Adam Płocki, Prof. dr hab. Akademia Pedagogiczna, PodchorąŜych 2, Pl-30-084 Kraków, Polska [email protected] Two-dimensional balance equations for defects Jurij Povstenko Abstract: Interfacial region is considered as two-dimensional continuum with its own physical characteristics (1). It is well known that the surface layers of materials have their own defect structure which differs from that in the bulk (2). Two-dimensional balance equations for surface defects are obtained taking into account the interaction between the interface and threedimensional bulk phases. Literature: (1) Y.S. Podstrigach, Y.Z. Povstenko, Introduction to Mechanics of Surface Phenomena in Deformable Solids, Naukova Dumka, Kiev, 1985 (in Russian). (2) V.P. Alekhin, Physics of Strength and Plasticity of the Surface Layers of Materials, Nauka, Moscow, 1983 (in Russian). Author: Jurij Povstenko, prof. dr hab. Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] Space imagination in mathematics teaching GraŜyna Rygał Abstract: The space imagination plays an important part in solving geometrical problems at every level of teaching. Practical training allowing development of imagination from the youngest years is highly efficient. There is a set of tasks and practical training allowing us to develop the pupils’ imagination. Some of them are presented. Author: GraŜyna Rygał, dr Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] 12

An assessment of materials for distance learning by students Anna Stopenová, Jindřiška Eberová Abstract: Study materials for distance learning were developed for the purpose of self-directed learning. It is not common for textbooks in mathematics to be designed in this specific style. Distance learning materials for attendees of the distance educational program Primary School Teaching were created by Stopenová (Fundaments of Mathematics 1) and Eberová (Fundaments of Mathematics 2). A questionnaire was used to examine the students’ opinions on the textbooks. Literature: (1) EBEROVÁ, J. Základy matematiky 2. Olomouc: VUP, 2003. 60 s. ISBN 80-244-0759-0 (2) STOPENOVÁ, A. Základy matematiky 1. Olomouc: VUP, 2003. 60 s. ISBN 80-244-0758-2 (3) EBEROVÁ, J. STOPENOVÁ, A. Několik zkušeností s psaním distančních opor pro matematiku. In: Cesty (k) poznávání v matematice primární školy - sborník příspěvků. Olomouc: VUP Olomouc, 2004. S. 72-76. ISBN 80-244-0818-X (4) NOVÁK, B. Integrace prvků distančního vzdělávání do přípravy učitelů matematiky. In: XXI. Mezinárodní kolokvium o řízení osvojovacího procesu. CD ROM. Vyškov: VVŠPV, 2003. ISBN 80-7231-105-0. (5) EBEROVÁ, J. STOPENOVÁ, A. Jak psát distanční texty. In: Sborník příspěvků - Malino, v tisku. Autors: Jindřiška Eberová, RNDr. Katedra matematiky Pedagogické fakulty Univerzity Palackého Žižkovo nám. 5, 771 40 Olomouc, Česká republika [email protected] Anna Sropenová, PaedDr.,Ph.D. Katedra matematiky Pedagogické fakulty Univerzity Palackého Žižkovo nám. 5, 771 40 Olomouc, Česká republika [email protected] Qualocation Methods-Methods for Solving of Integral Equations Jana Šimsová Abstract: Boundary integral equations are usually solved with Galerkin method or Collocation method. The high order of convergence is reached by Galerkin method but collocation method is much more easier as to as implementation. There are introduced qualocation methods, which connect both of these aspects. Qualocation method is Galerkin method in which the outer integrals are performed numerically by special quadrature rules and it has the order of convergence O(h5) in a suitable negative norm. Literature: (1) I.H.Sloan: A Quadrature-Based Approach to Improving the Collocation Method. Numer.Math.54,41-56,1988 (2)G.A.Chandler, I.H.Sloan: Spline qualocation methods for boundary integral equations. Numer.Math.58, 537-567, 1990 (3)I.H.Sloan, W.L.Wendland: Qualocation methods for elliptic boundary integral equations. NumerMath. 79, 451-483,1998 Author Jana Šimsová, RNDr. FSE UJEP Ústí nad Labem, Moskevská 54 [email protected] 13

Cabri Geometry in the Beltrami-Klein’s Model of a Lobachevskian Geometry Petr Rys , Tomáš Zdráhal Abstract: Non – Euclidean geometries run counter to common sense. In order to understand them, students need to draw pictures. First the teacher must give them models, however. Once students have models, they can draw pictures. Why do not they use here the well known program Cabri Geometry? It was found just this program is both attractive and comfortable for them and enable them to solve many problems concerning e.g. a Lobachevskian geometry. Authors: Petr Rys, Mgr. Faculty of Education of University J. E. Purkyně České mládeže 8, 400 96 Ústí nad Labem, Czech Republic [email protected] Tomáš Zdráhal, Doc.,RNDr.,CSc. Faculty of Education of University J. E. Purkyně České mládeže 8, 400 96 Ústí nad Labem, Czech Republic [email protected] Queueing Systems with Non-Homogeneous Internal and External Demands Oleg Tikhonenko, Marcin Ziółkowski Abstract: We study a single-server queueing model without losses with non-homogeneous demands of two following types: 1) external demands serving by the system under consideration, 2) internal demands occurring during external demands service and interrupting this process. The external demands occur according to a stationary Poisson process. Demands of each abovementioned type are characterized by some random volume. Service time of the demand arbitrarily depends on its volume. Two schemes of demands service in the system are analyzed. The stationary total demands volume distribution is determined in terms of Laplace-Stieltjes transforms. The first moment of total demands volume in the system is calculated for each scheme. Literature (1) Tikhonenko O. M. Queueing Models in Computer Systems. Minsk: University Edition, 1990 (in Russian). (2) Tikhonenko O. Modele obsługi masowej w systemach informacyjnych. Warszawa: Akademicka Oficyna Wydawnicza EXIT, 2003. (3) Klimov G. P. Stochastic service systems. Moscow: Nauka, 1966 (in Russian). Authors Oleg Tikhonenko, prof. dr hab. Marcin Ziółkowski, mgr Institute of Mathematics and Computer Science Jana Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-201 Częstochowa, Poland [email protected]

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Some remarks about the random triangles Pavel Tlustý, Hana Štěpánková Abstract: The article deals with the attributes of triangles which arise from a straight line „random broken“ into three pieces. Literature: (1) Plocki, A., Tlustý, P.: Počet pravděpodobnosti pro začátečníky a mírně pokročilé, Prometheus, Praha 2005 (in czech) (2) Tlustý, P.: Some remarks about the random number generators, Acta Universitas Purkynianae 72, 2001, 58-61 Autors: Pavel Tlustý, doc. RNDr. CSc., Hana Štěpánková, Mgr. Katedra matematiky PF JU, Jeronýmova 10, 371 15 České Budějovice [email protected] [email protected] On magic graphs and magic stars Marián Trenkler Abstract: In 1962 Jiří Sedláček found out connestions between magic squares and one special class of edge-labeled graphs and called them magic graphs. Since this year many mathematiciens studied magic graphs defined by different ways. In our contribution we will applicate some results of magic graphs at brain-twisters called magic stars. Literature: (1) M.Trenkler: Magické hviezdy, Obzory matematiky, fyziky a informatiky 51(1998), 1-7 http://kosice.upjs.sk/~trenkler/98-HVIEZ.PDF (2) M.Trenkler: Magic stars, The PME Journal (USA), 11(Spring 2004), 549-554 Author: Marián Trenkler, m.prof Pedagogická fakulta KU, nám. A.Hlinku 56, 034 01 Ružomberok, Slovakia [email protected]

Some consequences of locally defined operators Małgorzata Wróbel Abstract: Let n, m ∈ N be fixed. For a closed interval D ⊂ R n by C m (D ) denote the set of m times continuously differentiable functions ϕ : D → R . An operator K : C m (D ) → C k (D ) is said to be locally defined if for every two functions ϕ ,ψ ∈ C m (D ) and for every open interval J ⊂ D the relation ϕ | J = ψ | J implies that K (ϕ ) | J = K (ψ ) | J . A characterization of the locally defined operators mapping C m (D ) into C 1 (D ) will be presented.

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Author: Małgorzata Wróbel, dr Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Al. Armii Krajowej 13/15 42-200 Częstochowa, Poland [email protected] Projects of Ministry of Education in SIPVZ Milan Zelenka Abstract: The Ministry of Education gives emphasis on the implementation of ICT into the education in the present days. One of the means of the implementation are projects resulting from the realization of the State Informative Policy in the education which are intended on this area. Basic school Pod Vodojemem in Ústí nad Labem joined in this activity with their projects “Use of the graphic calculator in Mathematics teaching at lower secondary school” and “Mathematics and its aplication in the connection with ICT at lower secondary school”. There will be detailed information about the realization of these projects in the report. Also students of Mathematics at the Faculty of Education of Jan Evangelista Purkyně University in Ústí nad Labem will be continuously acquainted with the results of these projects during their didactic seminars. Author Milan Zelenka, PaedDr. Katedra matematiky Pedagogické fakulty UJEP, České mládeže 8, 40096 Ústí nad Labem [email protected]

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Students’ ideas of the limit Petr Eisenmann Abstract: This contribution describes the students’ results in a test of the “infinitesimal feeling”. In their minds there are not enough models that would serve as the basis of the set of abstract notions and relations passed on during lessons. Literature: Blum, W., Törner, G.: Didaktik der Analysis, Vandenhoeck and Ruprecht, Göttingen 1983 Eisenmann, P.: Endlich oder unendlich viele? Über eine Diskussion in der gymnasialen Oberstufe, Mathematik in der Schule 36 (1998), 275 – 277 Hayen, J.: Zur Entwicklung des Begriffverständnisses vom Grenzwert, mathematica didactica 11 (1988), 72 – 95 Hejný, M. a kol.: Teória vyučovania matematiky 2, Slovenské pedagogické nakladatelstvo, Bratislava, 1989. Herget, W.: Konvergenz – Experimente mit dem Computer, Mathematik lehren, Heft 39 (1990), 49 - 56 Herget, W., Sperner, P.: Die harmonische Reihe konvergiert gegen 8,449?, Praxis der Mathematik 19 (1978), 281 – 285 Hischer, H., Scheid, H.: Grundbegriffe der Analysis, Spektrum Akademischer Verlag Heidelberg – Berlin – Oxford 1995 King, L.: The Epsilon – delta Connection, Two – Year College Mathematics Journal 14 (1983), 42 – 47 Renschler, Ch.: Wirklich oder nur scheinbar konvergent?, Mathematik in der Schule 35 (1997), 88 – 89 Schweiger, F.: Vom unauffällig Unendlichen zum auffällig Unendlichen, ÖMG Didaktik – Reihe, Heft 3 (1980), 255 – 280 Wigand, K.: Grenzprozesse in der Unterstufe, MU 5 (1959), 7 – 16 Zeitz, G.: Untersuchungen zur Vorbereitung und Behandlung zentraler Begriffe der Analysis, Dissertationsarbeit, Humboldt – Universität Berlin 1973 Zeitz, G.: Zur Entwicklung und Überprüfung des „infiniten Denkens“, Mathematik in der Schule 28 (1990), 778 – 783 Author: Petr Eisenmann, Doc., PaedDr., CSc. Katedra matematiky PF UJEP České mládeže 8, Ústí nad Labem [email protected]

On logarithmic functions Jitka Laitochová Abstract: The unusual function y = ln x a and its uncommonly encountered inverse y = x a are studied from a calculus viewpoint, using only the usual calculus background about logarithmic and exponential functions. Literature [1] Odvárko, O.: Matematika pro gymnázia. Funkce. Prometeus, Praha, 1993. [2] Hrubý, D. – Kubát, J.: Matematika pro gymnázia. Diferenciální a integrální počet. Prometeus, Praha, 1997. Author Jitka Laitochová, RNDr., CSc. Department of Mathematics Pedagogical Faculty, Palacký University Žižkovo nám. 5 Olomouc Czech Republic [email protected]

Linear recurrence relations and the binomial coefficients George Grossman Abstract: In this work we have established the solution to (real) linear, nonhomogeneous, recurrence relations,

−a +a + 2a = P ( n) , n = 0, 1, 2, 3... n n +1 n+2 when P(n) is an arbitrary binomial coefficient. This solution takes the form as ‘sum of sums’ and is recursive in nature. Literature: [1] Grossman, G. and Narayan, S.: On the characteristic polynomial of the jth order Fibonacci sequence, Applications of Fibonacci Numbers, Vol 8., ed. Fredric T. Howard, Kluwer Academic Publishers, The Netherlands, 1999, pp. 165-177. [2] Grossman, G. and Zeleke, A: On linear recurrence relations, , Journal of Concrete and Applicable Mathematics, Vol. 1, No. 3, 1994, pp. 229-245. Author: George Grossman, Prof. Dr. Department of Mathematics Central Michigan University Mt Pleasant, MI 48859 [email protected]

Expressing a map of two variables as a sum of maps of one variable. Eva Trenklerová Abstract. If a compact set K in the plane contains no triple if points forming a vertical and a horizontal segment, then for each continuous real valued map f on K there exist real valued continuous maps of one real variable g, h such that f ( x, y ) = g ( x ) + h( y ) for all points ( x, y ) in K. Using an example, we describe a new constructive proof of this result. Literature: (1) A. N. Kolmogorov, On the representations of continuous functions of many variables by superpositions of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR, 114(1957), 953-956 (2)

P. A. Ostrand, Dimension of metric spaces and Hilbert's problem 13, Bull. Amer. Math. Soc., 71(1965), 619-622.

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2 A. Skopenkov, A description of continua basically embeddable in R , Topology Appl., 65(1995), 29-48.

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Y. Sternfeld, Dimension, superposition of functions and separation of points, in compact metric spaces, Israel J. Math., 50(1985), 13-52.

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Y. Sternfeld, Y., Hilbert's 13th problem and dimension, Lect. Notes Math., 1376(1989), 1-49.

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N. Mramor-Kosta and E. Trenklerová, On basic embeddings of compacta into the plane. Bulletin of the Australian Mathematical Society, 68(2003), 471-480.

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N. Mramor-Kosta and E. Trenklerová, On constructing basic embeddings in the plane. Preprint series of University of Ljubljana, 2005.

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D. Repovš and M. Željko, On basic embeddings into the plane. Rocky Mountain Journal of Mathematics, to be published.

Author: Eva Trenklerová, RNDr. Fakulty of Science PJ Šafárik University 04154 Košice Slovakia [email protected]