Theory and Practice in Mathematics Teaching (TVMd)
Mathematical preparation of future undergraduate students
Analysis of mathematical preparation of secondary school pupils for university level mathematics with the aim to align the expectation of university lecturers with the possibilities of mathematical education at secondary schools. Development of proposals for the improvement of the status quo.
prof. RNDr. Jozef Doboš, CSc.
Discrete Mathematics (DMd)
From proper to strong edge-coloring of graphs
To study the proper edge coloring of graphs that require stronger conditions for some of the colors. Strong coloring for each color requires that the end vertices of the edges with that color induce matching. It is known that Δ (G) +1 colors are sufficient for regular coloring of the graph G. For strong coloring, it is conjectured that 1.25.Δ(G)^2 colors are sufficient, but currently the best known upper bound is 1,772.Δ(G)^2. Try to find new bounds for some classes of graphs (regular graphs, bipartite graphs, planar graphs, etc.).
N. Gastineau and O. Togni. On S-packing edge-colorings of cubic graphs. Discrete Appl. Math., 259:63–75, 2019 H. Hocquard, D. Lajou, and B. Lužar. Between proper and strong edge-colorings of subcubic graphs. In L. Gasieniec, R. Klasing, and T. Radzik, editors, Combinatorial Algorithms, IWOCA 2020, volume 12126 of Lecture Notes in Comput. Sci., pages 355–367, Springer, 2020.
doc. RNDr. Roman Soták, PhD.
Theory and Practice in Mathematics Teaching (TVMd)
Support for the development of students' geometric imagination in mathematics teaching
Geometric imagination can be understood as a set of abilities that are important for orientation and movement in the surrounding world and when performing various professions in real practice. Important components of geometric imagination are the abilities to recognize and imagine geometric shapes, their properties and mutual position based on planar diagrams and pictures. Geometric imagination of children develops by performing various activities and gaining experience. In the mathematics teaching, it is possible to stimulate and guide the development of the geometric imagination of students at all school levels with illustrative teaching aids and suitable learning activities. This thesis will be focused on the analysis of the possibilities of developing students' geometric imagination in the mathematics teaching, the design of manipulative learning activities and interactive dynamic teaching materials and the evaluation of their influence on the development of students' geometric imagination.
Objectives: to analyze the possibilities of developing students' geometric imagination in mathematics teaching, to design learning activities and to develop interactive dynamic teaching materials for the development of geometric imagination and to evaluate their impact on the improvement of students' geometric imagination.
doc. RNDr. Stanislav Lukáč, PhD.
Discrete Mathematics (DMd)
Relations on unary algebras
The aim is to investigate relations on unary algebras, especially relations compatible with operations (that is, preserving all operations of a given algebra). To study properties of selected compatible relations, including congruences and quasiorder relations.
1. S.Burris.H.P.Sankappanavar: A course in universal algebra, The millennium edition, www.math.waterloo.ca/snburris/htdocs/UALG/uni-algebra.pdf 2. D.Jakubíková-Studenovská, J.Pócs: Monounary algebras, UPJŠ Košice, 2009
doc. RNDr. Miroslav Ploščica, CSc.
RNDr. Lucia Janičková, PhD.
Discrete Mathematics (DMd)
Structural properties of embedded graphs
Study the various local and global properties of embedded graphs. Known properties of planar graphs (eg the existence of vertices of degree at most 5 or edges with the sum of degrees of its end vertices at most 13, etc.) can be proved in a similar form for graphs embedded on different surfaces. Focus on the structural properties of embedded graphs with different conditions on simple graph invariants (minimum graph degree, graph girth, etc.). Point out the use of the obtained results in some coloring of graphs.
J.L. Gross, T.W. Tucker, Topological Graph Theory, Dover Publications (2001). S. Jendrol', H.-J. Voss, Light subgraphs of graphs embedded in the plane - A survey, Discrete Math. 313 (2013), 406-421.
doc. RNDr. Roman Soták, PhD.
Applied Mathematics (AMd)
Test statistics in special multivariate models
Investigate properties and practical applications of tests in multivariate statistical models with special variance structures, especially of those which can be represented as product of beta distributions.
current journal articles
prof. RNDr. Ivan Žežula, CSc.
Applied Mathematics (AMd)
Mean testing in elliptical distributions with special variance structures
Study the tests of the mean in elliptical distributions with special variance structures, their derivation and properties.
1. Gupta, A.K., Nagar, D.K. (1999): Matrix variate distributions, Chapman and Hall 2. Gupta, A.K., Varga, T., Bodnar, T. (2013): Elliptically Contoured Models in Statistics and Portfolio Theory, Springer 3. Papers from scientific journals.
doc. RNDr. Daniel Klein, PhD.