Deformations of Border Bases

Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to the degree form ideal works only under additional hypotheses, we introduce border basis schemes and universal border basis families. With their help the problem can be rephrased as the search for a certain rational curve on a border basis scheme. We construct the system of generators of the vanishing ideal of the border basis scheme in different ways and study the question of how to minimalize it. For homogeneous ideals, we also introduce a homogeneous border basis scheme and prove that it is an affine space in certain cases. In these cases it is then easy to write down the desired deformations explicitly.Comment: 21 page

The Geometry of Border Bases

The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^\mu(A^n) by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal component

Dual-to-kernel learning with ideals

In this paper, we propose a theory which unifies kernel learning and symbolic algebraic methods. We show that both worlds are inherently dual to each other, and we use this duality to combine the structure-awareness of algebraic methods with the efficiency and generality of kernels. The main idea lies in relating polynomial rings to feature space, and ideals to manifolds, then exploiting this generative-discriminative duality on kernel matrices. We illustrate this by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and feature learning, and test their accuracy on synthetic and real world data.Comment: 15 pages, 1 figur

Equity valuation of Gerresheimer AG

This master thesis performs an equity valuation of Gerresheimer AG, a global manufacturer for the pharma and healthcare industry, and determines its ordinary share price as of 30.11.2018. The state-of-the-art valuation approaches are presented and the industry- and macroeconomic environment of Gerresheimer is analyzed. Afterwards, the equity value of Gerresheimer is determined using the sum of the parts DCF approach, combined with a relative valuation consisting of trading multiples. The Advanced Technologies division of Gerresheimer is valued based on the fair market value of the purchase price. The author issued a buy recommendation with a target price of 78€ as of Nov 30, 2018, with an upside potential of 24% compared to a share price of 63€ as of Nov 30, 2019. The results are subject to a sensitivity analysis, consisting of different scenarios and variations of Gerresheimer´s expected operating performance, completed with a Monte Carlo analysis. Finally, the methodologies and results are compared to the equity report provided by Credit Suisse, a leading multinational investment bank.A problemática da presente Tese de Mestrado consiste na avaliação do capital próprio da empresa Gerresheimer AG que, sendo um produtor a nível global, atua nas indústrias Farmacêutica e de Saúde. O preço das ações ordinárias da referida empresa é infra determinado à data de 30.11.2018. Distintas metodologias de avaliação são apresentadas, bem como uma análise da indústria e do ambiente macroeconómico em que se insere a Gerresheimer. Seguidamente, o valor do capital próprio da empresa é calculado através da combinação de duas abordagens: DCF (soma das partes) e avaliação relativa recorrendo a múltiplos de transação. O valor da divisão de Tecnologias Avançadas da Gerresheimer é determinado com base no justo valor de mercado do preço de aquisição. O autor apresenta uma recomendação de compra a um preço-alvo de 78€ à data de 30.11.2018, com um potencial de retorno de 24% comparativamente com o preço por ação de 63€ a 30 de novembro de 2018. Os resultados obtidos foram sujeitos a uma análise de sensibilidade relativa a diferentes cenários expectáveis da performance operacional da Gerresheimer, análise essa que é complementada com uma análise Monte Carlo. Finalmente, as metodologias e resultados são comparados ao relatório de capital próprio da Gerresheimer AG emitido pela Credit Suisse, um banco de investimentos líder multinacional