September 2012, Volume 12, Number 3
Summary: 
BEREŽNÝ, ŠTEFAN Processing data for time series analysis and inputs of algorithms for airport simulations [full paper] This article focuses on the application of given data of flight delays at the airport Koˇsice and its adaptation for further processing.
These data were recorded from 2006 till 2009. The airport Koˇsice did not do the dataprocessing of delays yet. Since the results of
this process are useful in the planning or scheduling, we try to establish a methodology for analysing these data. The values of basic
statistical parameters for different airlines and different types of flights are shown. We publish their short analysis and commentary.
We try to show the problem of prediction of development of these delays at the airport using statistical test.

BUŠA, JÁN Solving quadratic programming problem with linear constraints containing absolute values [full paper] In this paper the quadratic programming problem with linear constraints containing absolute values of variables (QPPLCAV) is
considered. Hessian matrix is presumed to be positive definite. The problem is transformed to the larger problem with double number
of variables with the same number of linear constraints without absolute values and with additional nonnegativity conditions (one
inequality containing n absolute values could be ‘directly’ substituted by the system of 2n inequalities without absolute values). This
problem may have several solutions. The relations between the original and the transformed problems are studied. In order to obtain
stable approximations to the normal solution to the transformed problem corresponding to the unique solution of the original problem
a regularization technique is proposed. A numerical example is given.

DRAŽENSKÁ, EMÍLIA The crossing numbers of products with cycles [full paper] The crossing numbers of Cartesian products of all graphs of order at most four with cycles are known. The crossing numbers of
Cartesian products G2Cn for several graphs G on five and six vertices and the cycle Cn are also given. In this paper, we extend these
results by determining crossing numbers of Cartesian products G2Cn for some specific six vertex graphs G and for some fixed number
n = 3; 4; 5.

GYÖNGYÖSI WIERSUM , ERIKA Teaching and learning mathematics through games and activities [full paper] In this paper we first present a theoretical approach to study mathematics teacher knowledge and conditions for developing it. Then
some interesting activities and games are presented. As a result, this paper supplies teachers with information that may be useful in
better understanding the nature of games, activities and their role in teaching and learning mathematics.
At the age of 10 pupils can concentrate less than 20 minutes during a lesson. However, a lesson in primary and secondary schools
lasts for 45 minutes and 50 at universities. What a contradiction! To solve this problem we try to attract their attention with different
techniques. As children and adults enjoy playing games we can teach and learn mathematics through games and activities. Experience
reveals that games can be very productive learning activities. Are some games better than others? What educational benefits are there
to be gained from games? How to integrate games in mathematics lessons? How to distinguish between ‘activity’ and ‘game’? How to
teach students specialising in electrical engineering and informatics using such an approach?

KLEŠČ, MARIÁN — SCHRÖTTER , ŠTEFAN On the packing chromatic number of semiregular polyhedra [full paper] Packing colouring of a graph G is a partitioning of the vertex set of G with the property that vertices in ith class have pairwise distance greater than i. The packing chromatic number of G is the smallest integer k such that the vertex set of G can be partitioned as X1;X2; : : : ;Xk where Xi is an ipacking for each i. In the paper, the packing chromatic numbers for all Platonic solids as well as for all prisms are given.

KLEŠČ, MARIÁN — VALO, MATÚŠ Minimum crossings in join of graphs with paths and cycles [full paper] The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. Only few results
concerning crossing numbers of graphs obtained as join product of two graphs are known. There was collected the exact values of
crossing numbers for join of all graphs of at most four vertices and of several graphs of order five with paths and cycles. We extend
these results by giving the crossing numbers for join products of the special graph on six vertices with n isolated vertices as well as
with the path on n vertices and with the cycle on n vertices.

KÖRTESI, PÉTER Modeling hypercomplex numbers [full paper] The present paper offers a generalization of the modeling by matrices for the complex numbers and quaternions to hypercomplex
numbers of dimensions 2, 4 and 8. The given matrix model seems to be a suitable tool to study further properties of the hypercomplex
numbers too. The matrix model used here was well known for modeling 2 dimensional complex and hypercomplex numbers, even for
quaternions (see [4]), and we extend its use to the case of 4 and 8 dimensional hypercomplex numbers.

KRAVECOVÁ , DANIELA — PETRILLOVÁ, JANA The crossing number of P^{2}_{N}C^{4} [full paper] The exact crossing number is known only for few specific families of graphs. According to their special structure, Cartesian products
of two graphs are one of few graph classes for which the exact values of crossing numbers were obtained. Let Pn be a path with n+1
vertices and P^{k}_{N}
n be the kpower of the graph P_{n}. Very recently, some results concerning crossing numbers of P^{k}_{N}
were obtained. For the
Cartesian product of P^{2}_{N}
n with the cycle of length three, the value 3n3 for its crossing number is given. In this paper, we extend this
result by proving that the crossing numbers of the Cartesian product P^{2}_{
n} C4 is 4n4.

LUČIĆ, DANKA — VARGA , MARIO Simulation of the twobody problem in GeoGebra [full paper] Classical problem of the motion of two bodies under gravitational interaction will be analyzed and simulated in GeoGebra. The
twobody problem will be reduced to the singlebody problem in central force field. Solutions of the singlebody problem will be mapped
onto solutions of the twobody problem and their correspondence will be analyzed. Finally, simulation of the twobody problem in a
moving frame will be shown.

MYŠKOVÁ, HELENA Weak stability of interval orbits of circulant matrices in fuzzy algebra [full paper] Fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by and
, where ab =
maxfa;bg; a
b = minfa;bg. An orbit of A generated by x is called stable if per(A;x) = 1. An interval orbit of an interval matrix
A and an interval vector X and the weak stability of an interval orbit are defined. A necessary and sufficient condition for the weak
stability of interval orbits of circulant matrices is introduced and justified.

MYŠKOVÁ, HELENA Interval eigenvectors of circulant matrices in fuzzy algebra [full paper] Fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by and
, where ab =
maxfa;bg; a
b = minfa;bg. A vector x is an eigenvector of a matrix A if A
x = x.
An interval vector X and the possible and universal eigenvectors are defined. A necessary and sufficient condition for the possible
and universal eigenvectors of a circulant matrix are proved and several examples are given.

OSTERTAGOVÁ, EVA — OSTERTAG, OSKAR Forecasting using simple exponential smoothing method [full paper] In the paper a relatively simple yet powerful and versatile technique for forecasting time series data – simple exponential smoothing
is described. The simple exponential smoothing (SES) is a shortrange forecasting method that assumes a reasonably stable mean in
the data with no trend (consistent growth or decline). It is one of the most popular forecasting methods that uses weighted moving
average of past data as the basis for a forecast. The procedure gives heaviest weight to more recent observations and smaller weight to
observations in the more distant past. The accuracy of the SES method strongly depends on the optimal value of the smoothing constant
a. To determine the optimal a value in the paper was used a traditional optimalization method based on the lowest mean absolute
error (MAE), mean absolute percentage error (MAPE) and root mean square error (RMSE).

RONTÓ, ANDRÁS — RONTÓ, MIKLÓS — SHCHOBAK, NATALIYA On numericalanalytic techniques for boundary value problems [full paper] We discuss several facts related to numericalanalytic methods for boundary value problems for first order ordinary and functional
differential equations. A numericalanalytic scheme of investigation of a twopoint boundary value problem for functional differential
equations is stated.

RONTÓ, MIKLÓS Numericalanalytic investigation of solutions of nonlinear integral boundary value problems [full paper] We consider the integral boundaryvalue problem for a certain class of nonlinear system of ordinary differential equations.
We give a new approach for studying this problem, namely by using an appropriate parametrization technique the given problem is
reduced to the equivalent parametrized twopoint boundaryvalue problem with linear boundary conditions without integral term.
To study the transformed problem we use a method based upon a special type of successive approximations, which are constructed
analytically.

STOJANČEVIĆ , TIJANA — DŽALETA, NATAŠA Mathematical modeling of options using GeoGebra [full paper] The aim of this contribution is to analyze the application of informatics software package GeoGebra in the modeling of options
strategy. Several specific examples are presented. The object of this study is behavior of overall profit in different options strategies,
observed from graphic point of view. Based on the analyzed examples using GeoGebra slider feature we conclude that bull spread and
bear spread both provide limited profit and loss. In addition, for specific values of parameters profit is strictly positive.
