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Date Name
4/30/18 14:00Vladimír Baláž (STU Bratislava): On Generalized Notion of Convergence by Means of Ideal and Its Applications
Description:The starting point of this talk is the notion of I–convergence, in this way introduced by P. Kostyrko, T. Šalát and W. Wilczynski by means of ideal I [3], (see also [1] where I–convergence is defined by means of filter - the dual notion to ideal). I–convergence is the natural generalization of the notion of statistical convergence. The notion of statistical convergence was independently introduced by H. Fast (1951) [2] and I. J. Schoenberg (1959) [4] which generalized the notion of classical convergence and has been developed in several directions and used in various parts of mathematics, in particular in number theory, mathematical analysis and ergodic theory. This talk would like to point out the usefulness of I–convergence mainly in number theory.
www:https://web.cs.cas.cz/mku/hora/aktualni.pdf
4/27/18 10:00Frederik Garbe (Czech Academy of Sciences, MI): Contagious sets in degree-proportional bootstrap percolation
Description:We study the following bootstrap percolation process: given a connected graph G, we infect an initial set A of vertices, and in each step a vertex v becomes infected if at least a ρ-proportion of its neighbours are infected. Once infected, a vertex remains infected forever. A set A which infects the whole graph is called a contagious set. It is natural to ask for the minimal size of a contagious set. Our main result states that for every ρ between 0 and 1, every connected graph G on n vertices has a contagious set of size less than 2ρn or a contagious set of size 1 (note that allowing the latter possibility is necessary since every contagious set has size at least one). This result is best-possible and improves previous results of Chang, Chang and Lyuu, and Gentner and Rautenbach. We also provide a stronger bound in the case of graphs of girth at least five. Both proofs exploit the structure of a minimal counterexample in a randomised fashion. This is joint work with Andrew McDowell and Richard Mycroft.
www:http://uivty.cs.cas.cz/ExtrA/seminar.html
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