The Hitchhiker’s Guide to measuring investor reactions

In this thesis I present the idea to use the so-called log-periodic power law (LPPL) to identify the influence of larger exogenous shocks on investor behaviour during antibubbles. I apply this methodology to look at the influence of bank rescues after the 2007 financial crisis. The results I obtain are consistent for the same kinds of bank rescues and can …

Colouring of the plane, lines and circles

Tasks about colouring often occurs in mathematics, e.g. the four colour map theorem or the chromatic number of the plane. My problem begins with a case from Mathematical Olympiad: how many colours at most we can use to colour points of the plane in such a way that each line is at most two-coloured. The answer is three, but if …

Motivic Symbols and Classical Multiplicative Functions

The notion of a multiplicative function is one of the central building blocks of modern number theory, and were studied by Euler, Gauss and Ramanujan among others. Classical multiplicative functions work as machines giving key information about various properties of positive integers. For example, there is a function which outputs 1 if the input is a square number, and 0 …

The regularity of cycles in permutation groups

In general, ivestigated problem can be interpreted as the probability, that a randomly selected element from permutation group generates a transitive group. This is a fundamental research that goes to the statistical group theory. But it may attract mathematicians of different specializations, because the various techniques from number theory, combinatorics and graph theory were used and new interesting results were …