Orientador: Paulo Roberto Brumatti / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T19:30:30Z (GMT). No. of bitstreams: 1 Veloso_MarceloOliveira_D.pdf: 662198 bytes, checksum: f119f8026ebe09649fca4a175b7cec47 (MD5) Previous issue date: 2009 / Resumo: Este trabalho é dedicado ao estudo das derivações localmente nilpotentes de certas K-álgebras finitamente geradas, onde K é um corpo de característica zero. Estes domínios são generalizações de anéis bem conhecidos na literatura sendo um deles o anel de Fermat. Mais precisamente, caracterizamos o conjunto das derivações localmente nilpotentes destes domínios ou de um subconjunto deste conjunto. Também calculamos o ML invariante destes domínios e como aplicação direta destas informações encontramos um conjunto de geradores para o grupo dos automorfismos de um destes domínos. No caso do anel de Fermat mostramos que nem sempre temos um domíno rígido. Além disso, verificamos que a Conjectura de Nakai é verdadeira para o anel de Fermat. / Abstract: This work is dedicated to the study of locally nilpotent derivations of certain finitely generated K-algebras, where K is a field of zero characteristic. These domains are generalizations of the well-known rings in the literature. One of this is the Fermat ring. More precisely, we characterize the set of locally nilpotent derivations of these domains or some subsets of this set. We also calculate the ML invariant of these domains and as a direct application of these results we find a set of generators for the group of automorphisms of some of these domains. We show that the Fermat ring is not always a rigid domain. Furthermore, we prove that Nakai's conjecture is true for the ring Fermat. / Doutorado / Algebra Comutativa / Doutor em Matemática

We discuss Lojasiewicz's beautiful proof of the division theorem for smooth functions. The standard proofs are based on the Weierstrass preparation theorem for analytic functions and use techniques from the theory of partial differential equations. Lojasiewicz's approach is more geometric and syn¬thetic. In the appendices appear new proofs of results which are required for the theorem. / Dissertation (MSc (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

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In this thesis, we consider hom-associative algebras, which is an algebra with multiplication that is not necessarily commutative nor associative, but obeys a twisted version of associativity by a linear homomorphism. We will give some conditions for associativity, which helps us determine commuting elements. Under other conditions, such as different types of unitality conditions, we can also state some results regarding commuting elements in the general, non-associative case.

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring. / <p>QC 20131114</p>

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<p>This thesis has three major topics. The first is on generalized multiplicities. The second is on height bounds for ideals of minors of matrices with a given rank. The last topic is on the ideal of minors of generic generalized diagonal matrices.</p> <p>In the first part of this thesis, we discuss various generalizations of Hilbert-Samuel multiplicity. These include the Buchsbaum-Rim multiplicity, mixed multiplicities, $j$-multiplicity, and $\varepsilon$-multiplicity. For $(R,m)$ a Noetherian local ring of dimension $d$ and $I$ a $m$-primary ideal in $R$, Lech showed the following bound for the Hilbert-Samuel multiplicity of $I$, $e(I) \leq d!\lambda(R/I)e(m)$. Huneke, Smirnov, and Validashti improved the bound to $e(mI) \leq d!\lambda(R/I)e(m)$. We generalize the improved bound to the Buchsbaum-Rim multiplicity and to mixed multiplicities. </p> <p>For the second part of the thesis we discuss bounds on heights of ideals of minors of matrices. A classical bound for these heights was shown by Eagon and Northcott. Bruns' bound is an improvement on the Eagon-Northcott bound taking into consideration the rank of the matrix. We prove an analogous bound to Bruns' bound for alternating matrices. We then discuss an open problem by Eisenbud, Huneke, and Ulrich that asks for height bounds for symmetric matrices given their rank. We show a few reduction steps and prove some small cases of this problem. </p> <p>Finally, for the last topic we explore properties of the ideal of minors of generic generalized diagonal matrices. Generalized diagonal matrices are matrices with two ladders of zeros in the bottom left and top right corners. We compute their initial ideals and give a description of the facets of their Stanley-Reisner complex. Using this description, we characterize when these ideals are Cohen-Macaulay. In the special case where the ladders of zeros are triangles, we compute the height and multiplicity</p>

We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed. / Ph. D.

In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015 / FAU Electronic Theses and Dissertations Collection

After presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order. / text