We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

This diploma thesis is focused on the analysis of security tools, which are commonly used for automatic verification of security protocols mainly. These security protocols are used for creation of private communication channells in insecured networks. Security will never be perfect, so finding of weaknesses in security protocols is always necessary and we need to improve these protocols. In this diploma thesis I have focused on looking for various verification tools on Internet. Subsequently, I have described each tool in this thesis, in way that each reader can easily find out, which tool is for him useful and which is not. I have summarized main benefits and drawbacks of each tool at the end of his description.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, under the view of commutative algebra, we will study reductions of an ideal, the concept was introduced by Northcott and Rees. First of all, we will give preliminary no- tions about dimension theory, Hilbert’s polynomial, Hilbert-Samuel’s polynomial, regularity of modules and superficial elements. Next we will discuss the main theme of this dissertation, where we will talk about integral closure of ideal, reduction and the Rees algebra, moreover, we will establish connections between these concepts. Finally, we will discuss some applica- tions in Hilbert-Samuel's polynomial and multiplicity theory, in which some recent results will be presented. / Neste trabalho, sob a luz da álgebra comutativa, estudaremos reduções de um ideal, tal conceito foi introduzido por Northcott e Rees. Em um primeiro momento, daremos noções preliminares sobre teoria de dimensão, polinômio de Hilbert, polinômio de Hilbert-Samuel, regularidade de módulos e elementos superficiais. Na sequência discutiremos o tema principal da dissertação, no qual falaremos de fecho integral de um ideal, redução e a álgebra de Rees, além disso, estabeleceremos conexões entre esses conceitos. Por fim, discutiremos algumas aplicações na teoria de multiplicidade e polinômio de Hilbert-Samuel, no qual será apresentado alguns resultados recentes. / São Cristóvão, SE

The diploma thesis is based on the study of hydrophobic interactions of the native hyaluronan with selected solutes. On the basis of a literature search were chosen fluorescent probes and fluorescing biologically active substances, which are useful for investigation of colloids as 6-(p-toluidino)-2-naphthalenesulfonic acid (polarity probe), lipophilic vitamin (±)-alpha-tocopherol, pyrene (polarity probe) and finally hydrophilic vitamin riboflavin. In the experimental part of this thesis was studied the influence of solvents with different polarities, or more precisely dielectric constant, on the emission spectra, as well. There were investigated interactions of native hyaluronan with TNS and then interactions, which were influenced by the ionic strength. Such influenced interactions were not observed, that was probably due to the strong solvation´s wrapping of the hyaluronan. Interactions were observed after the process of lyophilisation followed-up by the rehydratation of the samples. For the next study of interactions the riboflavin was chosen and was investigated the REES effect in the native hyaluronan in different concentrations of its different molecular weights. In this case were not observed any shifts in the emission maximum with the excitation wavelenght shift and that is why the interactions of hyaluronan with riboflavin were not demonstrated in the field of chosen concentrations. By using another probe alpha-tocopherol was investigated the associative behaviour of hyaluronan and moreover was observed anisotropy of alpha-tocopherol in different concentrations of different molecular weights of native hyaluronan. The anisotropy reached high values in contrast to the reference solute that was the mixture of glycerol and ethanol. The anisotropy depended more on the molecular weight than on the concentration of hyaluronan. Interactions of hyaluronan were also studied by using the polarity probe pyrene in different concentrations of different molecular weights of the hyaluronan. The pyrene 1:3 ratio did not show the concentration dependence within the chosen concentrations except for the molecular weight 253.9 kg mol–1. Both probes alpha-tocopherol and pyrene were performed by the process of lyophilisation followed-up by the rehydratation, which improved interactions of these probes with hyaluronan.

In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

Submitted by Danielle Karla Martins Silva (danielle.martins@ufpe.br) on 2015-03-12T16:03:51Z No. of bitstreams: 2 TESE Maral Mostafazadehfard.pdf: 759835 bytes, checksum: 0db918f26f85cab03090a30fba1d2b36 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Made available in DSpace on 2015-03-12T16:03:51Z (GMT). No. of bitstreams: 2 TESE Maral Mostafazadehfard.pdf: 759835 bytes, checksum: 0db918f26f85cab03090a30fba1d2b36 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Previous issue date: 2014 / CNPq / Os resultados desta tese se enquadram na teoria dos polin^omios homaloidais, com ^enfase no caso de determinantes. O objetivo principal e o estudo das propriedades homol ogicas do determinante da matriz gen erica de Hankel e de uma de suas degenera c~oes, como um m etodo de abordar o seu comportamento de natureza homal oide. No caso da matriz de Hankel gen erica, em caracter stica zero, concluimos que o Hessiano do determinante e n~ao nulo (equivalentemente, o mapa polar associado e dominante), mas o determinante n~ao e homal oide. No caso degenerado, sabese que o determinante e homal oide (provado por Cilibert-Russo-Simis [3]); aqui, determinamos os invariantes num ericos e homol ogicos do respectivo ideal gradiente (polar), esses podendo ser usados para simpli car algumas passagens no argumento de [3]. Os principais resultados da tese s~ao baseados em ferramentas n~ao triviais da algebra comutativa e a natureza do uso dessas ferramentas e um dos recursos importantes desta tese.

<p>The defining equations of Rees algebras provide a natural pathway to study these rings. However, information regarding these equations is often elusive and enigmatic. In this dissertation we study Rees algebras of particular classes of ideals and modules over hypersurface rings. We extend known results regarding Rees algebras of ideals and modules to this setting and explore the properties of these rings.</p> <p><br></p> <p>The majority of this thesis is spent studying Rees algebras of ideals in hypersurface rings, beginning with perfect ideals of grade two. After introducing certain constructions, we arrive in a setting similar to the one encountered by Boswell and Mukundan in [3]. We establish a similarity between Rees algebras of ideals with linear presentation in hypersurface rings and Rees algebras of ideals with <em>almost</em> linear presentation in polynomial rings. Hence we adapt the methods developed by Boswell and Mukundan in [3] to our setting and follow a path parallel to theirs. We introduce a recursive algorithm of <em>modified Jacobian dual iterations</em> which produces a minimal generating set for the defining ideal of the Rees algebra.</p> <p><br></p> <p>Once success has been achieved for perfect ideals of grade two, we consider perfect Gorenstein ideals of grade three in hypersurface rings and their Rees algebras. We follow a path similar to the one taken for the previous class of ideals. A recursive algorithm of <em>gcd-iterations</em> is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. </p> <p><br></p> <p>Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using <em>generic Bourbaki ideals</em> we study Rees algebras of modules with projective dimension one over hypersurface rings. For such a module $E$, we show that there exists a generic Bourbaki ideal $I$, with respect to $E$, which is perfect of grade two in a hypersurface ring. We then adapt the techniques used by Costantini in [9] to our setting in order to relate the defining ideal of $\mathcal{R}(E)$ to the defining ideal of $\mathcal{R}(I)$, which is known from the earlier work mentioned above.</p> <p><br></p> <p>In all three situations above, once the defining equations have been determined, we investigate certain properties of the Rees algebra. The depth, Cohen-Macaulayness, relation type, and Castelnuovo-Mumford regularity of these rings are explored.</p>

First in this dissertation we make a brief overview about basic tools of commutative algebra required for understanding the rest of the text. Then, we present the definition of symbolic powers and we discuss their basic properties, mainly emphasizing questions such as primary decomposition and calculation of generators. We conclude this work by showing actual results that relate the symbolic powers with other notions in commutative algebra and algebraic geometry. / Nesta dissertação fazemos inicialmente um breve apanhado sobre ferramentas básicas de álgebra comutativa úteis para o entendimento do resto do texto. Em seguida, apresentamos a definição de potências simbólicas e discutimos suas propriedades mais elementares, destacando sobretudo questões como decomposição primária e cálculo de geradores. Finalizamos o trabalho mostrando resultados atuais que relacionam as potências simbólicas com outras noções da álgebra comutativa e geometria algébrica.

Rare earth elements (REEs) and their isotopes (such as Nd isotopes) can be potentially used to trace a wide range of oceanic processes in both modern and ancient oceans, but their successful application as tracers requires a comprehensive understanding of REE cycling in the modern ocean. Previous studies of REEs in seawater were largely constrained by analytical difficulties in generating accurate and precise REE data from seawater, which typically contain REE concentrations at a sub-ppt to ppt level. A new, and relatively simple, analytical method for precise and accurate determination of all dissolved REE concentrations in reasonably small (&Tilde;100 ml) seawater samples is presented in this thesis. With the application of the new method, this thesis reports the first full-depth, zonal ocean section of all dissolved REE concentrations, collected during the CoFeMUG cruise along &Tilde;12&deg;S in the South Atlantic. The section approach of this study places the distribution of dissolved REE concentrations in a well-constrained hydrographic context, allowing the first quantitative assessment (by an inverse model) of the relative importance of hydrographic controls resulting from advection/mixing of ocean circulation, together with non-conservative controls resulting from local particle scavenging and remineralization, in controlling the distribution of dissolved REEs in this region. A noteworthy decoupling of Ce and Mn with respect to their cycling in the water column was also observed in this study. The application of Nd isotopes as a tracer to reconstruct changes in ocean circulation in the NW European chalk shelf sea during rapid climatic events, including the mid-Cenomanian Event and oceanic anoxic event 2 (OAE 2), suggests a tight coupling between ocean circulation and transient climatic cooling during the general warm Late Cretaceous. An advected volcanic signal during OAE 2 was registered in the seawater Nd-isotope record from the English Chalk, probably suggesting a period of enhanced ocean ventilation/mixing at this time.