Browsing by Author "Pajala, Aapo"
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- The Eremenko and Gabrielov proof for the Shapiro conjecture in the n=2 case
Perustieteiden korkeakoulu | Master's thesis(2024-03-12) Pajala, AapoThe Shapiro conjecture, presented in the mid-1990s by Boris and Michael Shapiro, gives a sufficient condition for the existence of a real basis for a finite-dimensional vector subspace of the space of univariate complex polynomials in terms of the Wronski polynomial associated to the subspace. The result was first proved in full generality in 2009 by Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko, and is nowadays known as a result of theirs. The simplest non-trivial special case of the conjecture is a result about the critical points of rational functions. This case was first proved by Alexander Eremenko and Andrei Gabrielov in 2002. In 2005, they presented another, "elementary", proof that utilised simpler machinery than their 2002 proof. This thesis aims to present the 2005 proof in a manner that is approachable for a non-expert. This involves presenting the required geometric background with the aim of reformulating the conjecture as a result about the Wronski map, which is an analytic map between complex analytic manifolds. Additionally, this thesis presents proofs that were omitted in the original 2005 article and adds a significant amount of detail to the ones presented. - Euclidean Distance Geometry and Its Applications in 3D Genome Reconstruction
Perustieteiden korkeakoulu | Bachelor's thesis(2020-12-18) Pajala, Aapo