A Beta-splitting model for evolutionary trees

In this article, we construct a generalization of the Blum-Fran\c{c}ois Beta-splitting model for evolutionary trees, which was itself inspired by Aldous' Beta-splitting model on cladograms. The novelty of our approach allows for asymmetric shares of diversification rates (or diversification `potential') between two sister species in an evolutionarily interpretable manner, as well as the addition of extinction to the model in a natural way. We describe the incremental evolutionary construction of a tree with n leaves by splitting or freezing extant lineages through the Generating, Organizing and Deleting processes. We then give the probability of any (binary rooted) tree under this model with no extinction, at several resolutions: ranked planar trees giving asymmetric roles to the first and second offspring species of a given species and keeping track of the order of the speciation events occurring during the creation of the tree, unranked planar trees, ranked non-planar trees and finally (unranked non-planar) trees. We also describe a continuous-time equivalent of the Generating, Organizing and Deleting processes where tree topology and branch-lengths are jointly modeled and provide code in SageMath/python for these algorithms.Comment: 23 pages, 3 figures, 1 tabl

On the tunnel number and the Morse-Novikov number of knots

We prove that the Morse-Novikov number of a link L in a 3-sphere is less than or equal to twice the tunnel number of L.Comment: 8 pages, revised and extende

An Analysis of the Company Norgren CZ Using Time Series

Předmětem bakalářské práce je analýza vybraných ukazatelů společnosti Norgren CZ pomocí statistických metod, která zachycuje souvislosti mezi přesuny výroby z podniků v západní Evropě do České republiky a vývojem jednotlivých ukazatelů. Obsahuje doporučení, která po zavedení do praxe přispějí ke zlepšení firemních procesů a snížení stavu zásob i teoretická východiska nezbytná k pochopení všech prostředků, které byly v práci využity.The financial analysis of chosen indicators of Norgren CZ Company based on statistical methods, which captures links between production transfer from the Western European companies into the Czech Republic and development of particular indicators, is the object of presented thesis. The thesis contains recommendations, which after putting themselves into practice contribute to the betterment of business process and to the stock reduction, and theoretical background needed for full understanding of all measures used in the thesis as well.

Decision Diagrams for Qualitative Biological Models

The modelisation of large biological systems such that metabolic networks or gene interaction networks, implies interpretations of heterogeneous biological knownledge. A common feature of observations accumulated by biologist on these systems is their qualitative nature. More over it often happens that the knownledge is based on comparisons between different experimental conditions. In previous papers variations in observed variables between two conditions was interpreted in terms of equilibria shift. This leads to use qualitative equations in a sign algebra. The present report gives details on the implementation and main algorithms involved

Circular thin position for knots in the 3-sphere

A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as simple as possible. To make this precise the concept of circular width for E(K) is introduced. When E(K) is endowed with a handle decomposition which realizes the circular width we will say that the knot K is in circular thin position. We use this to show that many knots have more than one non-isotopic incompressible Seifert surface. We also analyze the behavior of the circular width under some knot operations.Comment: 25 pages, 10 figure