<div> This dissertation is based on joint work with Lindsey Hill. There are two main parts, which are linked by the common theme of the integral closure of the Rees algebra.</div><div> </div><div> In the first part of this dissertation, comprised of Chapter 3 and Chapter 4, we study the integral closure of the Rees algebra directly. In Chapter 3 we identify a bound for the multiplicity of the Rees algebra R[It] of a homogeneous ideal I generated in the same degree, and combine this result with theorems of Ulrich and Vasconcelos to obtain upper bounds on the number of generators of the integral closure of the Rees algebra as a module over R[It]. We also find various other upper bounds for this number, and compare them in the case of a monomial ideal generated in the same degree. In Chapter 4, inspired by the large depth assumption on the integral closure of R[It] in the results of Chapter 3, we obtain a lower bound for the depth of the associated graded ring and the Rees algebra of the integral closure filtration in terms of the dimension of the Cohen-Macaulay local ring R and the equimultiple ideal I. We finish the first part of this dissertation with a characterization of when the integral closure of R[It] is Cohen-Macaulay for height 2 ideals. </div><div> </div><div> In the second part of this dissertation, Chapter 5, we use the integral closure of the Rees algebra as a tool to discuss specialization of the integral closure of an ideal I. We prove that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo general elements of I. This result is analogous to a result of Itoh and an extension by Hong and Ulrich which show that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo generic elements of I. We then discuss specialization modulo a general element of the maximal ideal, rather than modulo a general element of the ideal I itself. In general it is not the case that the operations of integral closure and specialization modulo a general element of the maximal ideal are compatible, even under the assumptions of our main theorem. We prove that the two operations are compatible for local excellent algebras over fields of characteristic zero whenever R/I is reduced with depth at least 2, and conclude with a class of ideals for which the two operations appear to be compatible based on computations in Macaulay2.</div>

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Esta tese representa uma sequencia natural a trabalhos de vários autores, em que se busca obter resultados novos sobre aplicações birracionais usando técnicas de álgebra comutativa. Uma das lacunas conhecidas é o problema da característica do corpo de base. Habitualmente tratados separadamente, o caso de característica zero e de característica prima, deixam a desejar do ponto de vista da unificação dos resultados gerais. Outro aspecto relevado é o do enunciado de critérios de birracionalidade alternativos ao tradicional cálculo do grau de uma aplicação racional. O principal objetivo deste trabalho é discutir um invariante numérico de birracionalidade válido em característica arbitrária, denominado posto Jacobiano dual. Este invariante depende fortemente da estrutura graduada da álgebra de Rees do ideal de base da aplicação racional, a qual permite uma análise mais precisa do que o tratamento geométrico habitual do gráfico como variedade "b lowup". _________________________________________________________________________________________ ABSTRACT: This thesis stands as a natural sequence to the work of several authors, seeking to obtain new results on birational maps using techniques from commutative algebra. One of the classical problems in the theory of birational maps is the case where the characteristic of the base field is positive. The usual separate treatment of the case of characteristic zero and characteristic prime falls short of unifying general results. Another aspect scarcely dealt with is the statement of a birationality criterion which stands as an alternative to the traditional calculation of the degree of a rational map. The main objective of this work is a numerical invariant of birationality valid in arbitrary characteristic, called the Jacobian dual rank. This invariant depends strongly on the structure of the graded Rees algebra of the base ideal of a rational map, which allows a more precise analysis than the usual geometric treatment of the graph as a "blowup".

Made available in DSpace on 2014-06-12T18:27:56Z (GMT). No. of bitstreams: 2 arquivo3272_1.pdf: 331050 bytes, checksum: 3ee7ab47fc1bfe97f30f2ba3991996cb (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2011 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esta tese representa uma sequência natural a trabalhos de vários autores, em que se busca obter resultados novos sobre aplicações birracionais usando técnicas de álgebra comutativa. Uma das lacunas conhecidas é o problema da característica do corpo de base. Habitualmente tratados separadamente, o caso de característica zero e de característica prima, deixam a desejar do ponto de vista da unificação dos resultados gerais. Outro aspecto relevado é o do enunciado de critérios de birracionalidade alternativos ao tradicional cálculo do grau de uma aplicação racional. O principal objetivo deste trabalho é discutir um invariante numérico de birracionalidade válido em característica arbitrária, denominado posto Jacobiano dual. Este invariante depende fortemente da estrutura graduada da álgebra de Rees do ideal de base da aplicação racional, a qual permite uma análise mais precisa do que o tratamento geométrico habitual do gráfico como variedade \blowup

Made available in DSpace on 2015-05-15T11:46:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 955645 bytes, checksum: fb2e571981d8360440e11732c479b02b (MD5) Previous issue date: 2013-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present the notion of Rees algebra of an ideal and some of its basic properties. Such concept is related to the normality of rings and ideals, and to reductions of ideals as well. Finally, we shall exhibit the Rees algebra of a module, proving some generalizations of results in the case of ideals. / Neste trabalho, apresentaremos a noção de álgebra de Rees de um ideal e propriedades básicas. Tal conceito será relacionado com normalidade de anéis e ideais, e redução de ideais. Por fim, exibiremos a álgebra de Rees de um módulo, mostrando algumas generalizações de resultados do caso de ideais.

Thesis (Ph.D.) - James Cook University, 2005. / Typescript (photocopy) Biblography: leaves 270-275.

We study the Cohen-Macaulay property of Rees algebras of modules of Kähler differentials. When the module of differentials has projective dimension one, it is known that condition $F_1$ is sufficient for the Rees algebra to be Cohen-Macaulay. The converse was proved if the module of differentials is already $F_0$. We weaken the condition $F_0$ globally by assuming some homogeneity condition.<br> <br> We are also interested in the defining ideal of the Rees algebra of a Jacobian module. If the Jacobian module is an ideal, we prove a formula for computing the defining ideal. Using the formula, we give an explicit description of the defining ideal in the monomial case. From there, we characterize the Cohen-Macaulay property of the Rees algebra.<br> <br> In the last chapter, we study Gorenstein linkage mostly in the graded case. In particular, we give an explicit example of a class of monomial ideals that are in the homogeneous Gorenstein linkage class of a complete intersection. To do so, we prove a Gorenstein double linkage construction that is analogous to Gorenstein biliaison.

The demands for copper and rare earth elements (REEs) in the U.S. will keep rising due to their applications in green energy technologies. Meanwhile, copper production in the U.S. has been declining over the past five years due to the depletion of high-grade ore deposits. The situation for REEs is worse; there is no domestic supply chain of REEs in the U.S. since the demise of Molycorp, Inc. in 2016. Studies have shown that the rejected materials from copper and coal processing plants contain significant amounts of valuable metals. As such, this rejected material can be considered as potential secondary sources for extracting copper and REEs, which may help combat future supply risks for the supply of copper and REEs in the U.S. However, the valuable mineral particles in these resources are ultrafine in size, which poses considerable challenges to the most widely used fine particle beneficiation technique, i.e., froth flotation. A novel technology called the Hydrophobic-Hydrophilic Separation (HHS) process, developed at Virginia Tech, has been successfully applied to recover fine coal in previous research. The results of research into the HHS process showed that the process has no lower particle size limit, similar to solvent extraction. Therefore, the primary objective of this research is to explore the feasibility of using the new process to recover ultrafine particles of coal, copper minerals, and rare earth minerals (REMs) associated with coal byproducts. In the present work, a series of laboratory-scale oil agglomeration and HHS tests have been carried out on coal with the objectives of assisting the HHS tests in pilot-scale, and the scale-up of the process. The knowledge gained from this study was successfully applied to solving the problems encountered in the pilot-scale tests. Additionally, a new and more efficient equipment known as the Morganizer has been designed and constructed to break up the agglomerates in oil phase as a means to remove entrained gangue minerals and water. The effectiveness of the new Morganizers has been demonstrated in laboratory-scale HHS tests, which may potentially result in the reduction of capital costs in commercializing the HHS process. Furthermore, the prospect of using the HHS process for processing high-sulfur coals has been explored. The results of this study showed that the HHS process can be used to increase the production of cleaner coal from waste streams. Application of the HHS process was further extended to recover the micron-sized REMs from a thickener underflow sample from the LW coal preparation plant, Kentucky. The results showed that the HHS process was far superior to the forced-air flotation process. In one test conducted during the earlier stages of the present study, a concentrate assaying 17,590 ppm total REEs was obtained from a 300 ppm feed. In this test, the Morganizer was not used to upgrade the rougher concentrate due to the lack of proper understanding of the fundamental mechanisms involved in converting oil-in-water (o/w) Pickering emulsions to water-in-oil (w/o) Pickering emulsions. Many of the studies has, therefore, been focused on the studies of phase inversion mechanisms. The results showed that phase inversion requires that i) the oil contact angles (θo) of the particles be increased above 90o, ii) the phase volume of oil (ϕo) be increased, and iii) the o/w emulsion be subjected to a high-shear agitation. It has been found that the first criterion can be readily met by using a hydrophobicity-enhancing agent. These findings were applied to produce high-grade REM concentrates from an artificial mixture of micron-sized monazite and silica. Based on the improved understanding of phase inversion, a modified HHS process has been developed to recover ultrafine particles of copper minerals. After successfully demonstrating the efficacy and effectiveness of this process on a series of artificial copper ore samples, the modified HHS process was used to produce high-grade copper concentrates from a series of cleaner scavenger tails obtained from operating plants. / Doctor of Philosophy / Recovery and dewatering of ultrafine particles have been the major challenges in the minerals and coal industries. Based on the thermodynamic advantage that oil droplets form contact angles about twice as large as those obtainable with air bubbles, a novel separation technology called the hydrophobic-hydrophilic separation (HHS) process was developed at Virginia Tech to address this issues. The research into the HHS process previously was only conducted on the recovery of ultrafine coal particles; also, the fundamental aspects of the HHS process were not fully understood, particularly the mechanisms of phase inversion of oil-in-water emulsions to water-in-oil emulsions. As a follow-up to the previous studies, emulsification tests have been conducted using ultrafine silica and chalcopyrite particles as emulsifiers, and the results showed that phase inversion requires high contact angles, high phase volumes, and high-shear agitation. These findings were applied to improve the HHS process for the recovery of ultrafine particles of coal, copper minerals, and rare earth minerals (REMs). The results obtained in the present work show that the HHS process can be used to efficiently recover and dewater fine particles without no lower particle size limits.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-31T13:40:38Z No. of bitstreams: 1 arquivo total.pdf: 1581712 bytes, checksum: 55cfc2e330d11ed8545538014daa3873 (MD5) / Made available in DSpace on 2016-03-31T13:40:38Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 1581712 bytes, checksum: 55cfc2e330d11ed8545538014daa3873 (MD5) Previous issue date: 2015-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we initially present an overview about the symmetric and the Rees algebras in the wide context of modules, and we consider particularly the special situation in which the given module possesses a linear presentation. In the sequel, the main goal is the study of such blowup algebras in the case where the module is the celebrated module of K ahler di erentials, the focus being given on the investigation of an interesting version of the long-standing Berger's Conjecture for the symmetric algebra, as well as on the study of fundamental properties such as: integrality, Cohen- Macaulayness and normality; these properties are also investigated in a special way in the case of the Rees algebra (of the di erential module), highlighting the connection to the so-called Fitting conditions. / Nesta disserta c~ao, inicialmente apresentamos no c~oes gerais sobre a algebra sim etrica e a algebra de Rees no contexto amplo de m odulos, e consideramos particularmente a situa c~ao especial na qual o dado m odulo possui apresenta c~ao linear. Na sequ^encia, o principal objetivo e o estudo de tais algebras de blowup no caso em que o m odulo e o celebrado m odulo de diferenciais de K ahler, tendo como foco a investiga c~ao de uma interessante vers~ao da persistente Conjectura de Berger para a algebra sim etrica, bem como o estudo de propriedades fundamentais como: integridade, Cohen-Macaulicidade e normalidade; tais propriedades s~ao tamb em investigadas de forma especial no caso da algebra de Rees (do m odulo de diferenciais), evidenciando inclusive a conex~ao com as chamadas condi c~oes de Fitting.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we will present a projective parameterization class that resemble the classic maps of Jonquières. We will have the aim to show how the main properties of your ideal base, such as the structure of sizígias and the free presentation of this ideal. In this sense, we obtain the implicit equation that defines the your image and we explicit the formula the degree of the equation. Finally, we will determine Rees algebra equations associated with the base ideal of the parameterization, using computational methods. This parameterization, called parameterization of Jonquières, is constructed from a Cremona and defines a birational map P^n in a hypersurface W of P^n+1. / Nesta dissertação apresentaremos uma classe de parametrizações projetivas que se assemelham aos mapas clássicos de Jonquières e teremos como objetivos mostrar as principais propriedades do seu ideal base, tais como a estrutura das sizígias e a apresentação livre desse ideal, consequentemente obteremos a equação implícita que define a sua imagem, bem como explicitar a fórmula do grau dessa equação. Por fim, determinaremos as equações da álgebra de Rees associada ao ideal base da parametrização, com o uso de métodos computacionais. Tal parametrização, denominada parametrização de Jonquières, é construída a partir de uma Cremona e define um mapa birracional de P^n em uma hipersuperfície W de P^{n+1}.