Energia é um recurso que historicamente apresenta tendência de crescimento de demanda. Projeções indicam que, para suprir as necessidades energéticas do futuro, será necessário um uso massivo do hidrogênio como combustível. O uso de sistemas de célula a combustível baseada no uso de membrana polimérica condutora de prótons (PEMFC) tem características que permitem sua aplicação para geração de energia elétrica em aplicações estacionárias, automotivas e portáteis. O uso de hidrogênio como combustível para PEMFC apresenta a vantagem de resultar em baixa emissão de poluentes quando comparado às dos combustíveis fósseis. Para que ocorram as reações em uma PEMFC é necessária a construção de conjuntos eletrodo-membrana-eletrodo (MEA), sendo o processo de produção de MEAs e os materiais utilizados, relevantes no custo final do kW instalado para geração de energia por sistemas de célula a combustível, o que é, atualmente, uma barreira tecnológica e financeira para a aplicação em grande escala desta tecnologia. Nesse trabalho foi desenvolvido um processo de produção de MEAs por impressão a tela que apresenta alta reprodutibilidade, rapidez e baixo custo. Foram desenvolvidos o processo de impressão a tela e a composição de uma tinta precursora da camada catalisadora (TPCC), que permitem o preparo de eletrodos para confecção de MEAs com a aplicação da massa exata de eletrocatalisador adequada para cátodos 0,6 miligramas de platina por centímetro quadrados (mgPt.cm-2) e ânodos 0,4 mgPt.cm-2 em apenas uma aplicação por eletrodo. A TPCC foi desenvolvida, produzida, aplicada e caracterizada, apresentando características semelhantes a de tintas de impressão a tela para outras aplicações. Os MEAs produzidos apresentaram desempenho de até 712 mA.cm-2 a 600 mV para MEAs de 25 cm2 e o custo para produção de MEAs de 247,86 cm2 capazes de gerar 1 kW de energia foi estimado em R$ 13.939,45, considerando custo de equipamentos, materiais e mão de obra. / Energy is a resource that presents historical trend of growth in demand. Projections indicate that future energy needs will require a massive use of hydrogen as fuel. The use of systems based on the use of proton exchange membrane fuel cell (PEMFC) has features that allow its application for stationary applications, automotive and portable power generation. The use of hydrogen as fuel for PEMFC has the advantage low pollutants emission, when compared to fossil fuels. For the reactions in a PEMFC is necessary to build membrane electrode assembly (MEA). And the production of MEAs and its materials are relevant to the final cost of kW of power generated by systems of fuel cell. This represent currently a technological and financial barriers to large-scale application of this technology. In this work a process of MEAs fabrication were developed that showed high reproducibility, rapidity and low cost by sieve printing. The process of sieve printing and the ink composition as a precursor to the catalyst layer were developed, which allow the preparation of electrodes for MEAs fabrication with the implementation of the exact catalyst loading, 0.6 milligrams of platinum per square centimeters (mgPt.cm-2) suitable for cathodes and 0.4 mgPt.cm-2 for anode in only one application step per electrode. The ink was developed, produced, characterized and used with similar characteristics to ink of sieve printing build for other applications. The MEAs produced had a performance of up to 712 mA.cm-2 by 600 mV to 25 cm2 MEA area. The MEA cost production for MEAs of 247.86 cm2, that can generate 1 kilowatt of energy was estimated to US$ 7,744.14 including cost of equipment, materials and labor.

A glicerina é um composto de grande utilidade em muitas áreas de aplicação. Atualmente, a maior fonte de glicerina é como subproduto da produção do biodiesel. Em 2010 o Brasil foi o segundo maior produtor de biodiesel no mundo com 2,4 milhões de m3. Estima-se que 10% da produção de biodiesel é glicerina bruta e destes 10%, aproximadamente entre 30% e 60% correspondem à glicerol. As impurezas são formadas por água, sais orgânicos e inorgânicos, ésteres e álcoois, e traços de glicerídeos. Como as aplicações mais nobres da glicerina requerem uma glicerina isenta de impurezas, novas rotas de purificação da glicerina bruta vem sendo estudadas. Neste sentido, o Laboratório de Separações Térmicas e Mecânicas da EPUSP propôs uma nova rota de purificação em 2008, na qual a ultima etapa é a desidratação de uma solução glicerol-água por meio de destilação trifásica usando tolueno como arrastador. O objetivo do presente trabalho foi estudar o comportamento deste tipo de destilação em uma coluna modulada com três pratos perfurados. Nesta coluna avaliou-se o layout de pratos e as melhores condições de operação. Os resultados, obtidos com este estudo, constituem uma contribuição importante para o projeto básico de coluna de destilação trifásica. As séries experimentais foram planejadas sequencialmente empregando-se planejamento estatístico de experimentos (DOE). Como variáveis de processo foram estudadas a vazão de vapor do tolueno (kg/h), vazão de alimentação de glicerina (kg/h) e concentração de glicerol na alimentação (% em massa). Como variáveis geométricas foram estudadas a área livre do prato () e a altura de vertedouro (Hw). O desempenho da coluna foi avaliado mediante o incremento da concentração de glicerol (em relação à concentração de glicerol na alimentação) e a concentração de glicerol no fundo da coluna (estas duas variáveis de resposta são dependentes entre si). A influência das variáveis de processo e geométricas nas respostas estudadas permitiram a construção de modelos estatísticos, e o melhor deles foi comparado com os modelos de não equilíbrio (NEQ) e de equilíbrio (EQ) obtidos por simulação no programa ASPEN PLUS. O modelo de não equilíbrio está baseado nas equações de Maxwell-Stefan, que utiliza a abordagem de Eckert e Vanek (2001) e a correlação de Chen-Chuang (1993), para estimar os coeficientes binários de transporte de massa. O modelo de equilíbrio utiliza as equações MESH (Material balance, Equilibrium, mole fraction Summation, and Heat balance). Conclusivamente, pode-se afirmar que as variáveis operacionais exercem maior influência do que as variáveis geométricas na desidratação da glicerina. As melhores condições foram: a vazão de vapor de tolueno de 23,5 kg/h, vazão de alimentação de glicerina de 2,2 kg/h e concentração de glicerol na alimentação de 50 % glicerol em massa, usando o layout L5 com área livre de 0,04 e altura de vertedouro de 70 mm. Os valores preditos pelo modelo estatístico (obtido com dados experimentais) e pelo modelo de NEQ representaram o comportamento da desidratação da glicerina por destilação trifásica à pressão atmosférica usando tolueno como arrastador, na coluna de pratos perfurados estudada. O modelo de EQ superestimou os valores reais. / Glycerine is a material of outstanding utility with many areas of application. Currently, the largest source of glycerine is as a by-product of biodiesel production. In 2010, Brazil was the second largest biodiesel productor in the world with 2.4 million m3. It is estimated that 10% of biodiesel is raw glycerine, and of this 10%, approximately between 30% and 60% corresponds to glycerol. The impurities are formed by water, organic and inorganic salts, esters and alcohols, and traces of glycerides. Although many attempts have been made to use raw glycerine, it is still necessary to purify it for of the most applications; hence new routes of glycerine purification have been studied. In this way, the Laboratory of Thermal and Mechanical Separations of EPUSP have proposed a new route of purification in 2008, where the last step is the dehydration of glycerol-water solution by three-phase distillation using toluene as entrainer. The aim of the present work was to study the behavior of the three-phase distillation using a modulated column with three sieve trays. In this column, the tray layout and the best operating conditions were evaluated. The results achieved in this study are an important contribution to the basic design of three-phase distillation column. Experimental series were designed sequentially employing a statistical design of experiments (DOE). The process variables studied were the vapor flow rate of toluene (kg/h), feed flow rate of glycerine (kg/h) and the feed glycerol concentration (% wt.). The geometric variables studied were the fractional hole area () and the weir height (Hw). The column performance was evaluated by the increment of glycerol concentration (in relation to the feed glycerol concentration) and the bottom glycerol concentration (these two variables are mutually dependent responses). The influence of process and geometric variables allowed the empirical models building, in which the best model was compared with the non-equilibrium (NEQ) and equilibrium (EQ) models obtained by the simulator ASPEN PLUS. The non-equilibrium model is based on Stefan-Maxwell equations, which uses the approach of Eckert and Vanek\'s (2001) and the Chen-Chuang correlation (1993), to estimate the binary coefficients of mass transport. The equilibrium model is based on the MESH equations (Material balance, Equilibrium, mole fraction Summation, and Heat balance). Finally, it can be stated that the operating variables have more influence than the geometric variables in the glycerine dehydration. The best conditions were: vapor flow rate of toluene of 23.5 kg/h, feed flow rate of glycerine of 2.2 kg/h and feed glycerol concentration of 50% wt., using the fractional hole area of L5 of 0.04 and weir height of 70 mm. The predicted values obtained by the statistical model and by the non-equilibrium model represented the behavior of the glycerine dehydration by three-phase distillation at atmospheric pressure using toluene as entrainer in the sieve tray column studied. The equilibrium model (EQ) overpredicted the real values.

La sécurité des systèmes cryptographiques à clef publique repose sur la difficulté de résoudre certains problèmes mathématiques, parmi lesquels se trouve le problème du logarithme discret sur les corps finis GF(p^n). Dans cette thèse, nous étudions les variantes de l’algorithme de crible algébrique, number field sieve (NFS) en anglais, qui résolvent le plus rapidement ce problème, dans le cas où la caractéristique du corps est dite moyenne. NFS peut être divisé en quatre étapes principales : la sélection polynomiale, la recherche de relations, l’algèbre linéaire et le calcul d’un logarithme individuel. Nous décrivons ces étapes, en insistant sur la recherche de relations, une des étapes les plus coûteuses. Une des manières efficaces de réaliser cette étape est d’utiliser des algorithmes de crible. Contrairement au cas classique où la recherche de relations est réalisée dans un espace à deux dimensions, les corps finis que nous ciblons requièrent une énumération d’éléments dans un espace de plus grande dimension pour atteindre la meilleure complexité théorique. Il existe des algorithmes de crible efficaces en deux dimensions, mais peu pour de plus grandes dimensions. Nous proposons et analysons deux nouveaux algorithmes de crible permettant de traiter n’importe quelle dimension, en insistant particulièrement sur la dimension trois. Nous avons réalisé une implémentation complète de la recherche de relations pour plusieurs variantes de NFS en dimensions trois. Cette implémentation, qui s'appuie sur nos nouveaux algorithmes de crible, est diffusée au sein du logiciel CADO-NFS. Nous avons pu valider ses performances en nous comparant à des exemples de la littérature. Nous avons également été en mesure d’établir deux nouveaux records de calcul de logarithmes discrets, l'un dans un corps GF(p^5) de taille 324 bits et l'autre dans un corps GF(p^6) de taille 422 bits / The security of public-key cryptography relies mainly on the difficulty to solve some mathematical problems, among which the discrete logarithm problem on finite fields GF(p^n). In this thesis, we study the variants of the number field sieve (NFS) algorithm, which solve the most efficiently this problem, in the case where the characteristic of the field is medium. The NFS algorithm can be divided into four main steps: the polynomial selection, the relation collection, the linear algebra and the computation of an individual logarithm. We describe these steps and focus on the relation collection, one of the most costly steps. A way to perform it efficiently is to make use of sieve algorithms. Contrary to the classical case for which the relation collection takes place in a two-dimensional space, the finite fields we target require the enumeration of elements in a higher-dimensional space to reach the best theoretical complexity. There exist efficient sieve algorithms in two dimensions, but only a few in higher dimensions. We propose and study two new sieve algorithms allowing us to treat any dimensions, with an emphasis on the three-dimensional case. We have provided a complete implementation of the relation collection for some variants of the NFS in three dimensions. This implementation relies on our new sieve algorithms and is distributed in the CADO-NFS software. We validated its performances by comparing with examples from the literature. We also establish two new discrete logarithm record computations, one in a 324-bit GF(p^5) and one in a 422-bit GF(p^6)

Energia é um recurso que historicamente apresenta tendência de crescimento de demanda. Projeções indicam que, para suprir as necessidades energéticas do futuro, será necessário um uso massivo do hidrogênio como combustível. O uso de sistemas de célula a combustível baseada no uso de membrana polimérica condutora de prótons (PEMFC) tem características que permitem sua aplicação para geração de energia elétrica em aplicações estacionárias, automotivas e portáteis. O uso de hidrogênio como combustível para PEMFC apresenta a vantagem de resultar em baixa emissão de poluentes quando comparado às dos combustíveis fósseis. Para que ocorram as reações em uma PEMFC é necessária a construção de conjuntos eletrodo-membrana-eletrodo (MEA), sendo o processo de produção de MEAs e os materiais utilizados, relevantes no custo final do kW instalado para geração de energia por sistemas de célula a combustível, o que é, atualmente, uma barreira tecnológica e financeira para a aplicação em grande escala desta tecnologia. Nesse trabalho foi desenvolvido um processo de produção de MEAs por impressão a tela que apresenta alta reprodutibilidade, rapidez e baixo custo. Foram desenvolvidos o processo de impressão a tela e a composição de uma tinta precursora da camada catalisadora (TPCC), que permitem o preparo de eletrodos para confecção de MEAs com a aplicação da massa exata de eletrocatalisador adequada para cátodos 0,6 miligramas de platina por centímetro quadrados (mgPt.cm-2) e ânodos 0,4 mgPt.cm-2 em apenas uma aplicação por eletrodo. A TPCC foi desenvolvida, produzida, aplicada e caracterizada, apresentando características semelhantes a de tintas de impressão a tela para outras aplicações. Os MEAs produzidos apresentaram desempenho de até 712 mA.cm-2 a 600 mV para MEAs de 25 cm2 e o custo para produção de MEAs de 247,86 cm2 capazes de gerar 1 kW de energia foi estimado em R$ 13.939,45, considerando custo de equipamentos, materiais e mão de obra. / Energy is a resource that presents historical trend of growth in demand. Projections indicate that future energy needs will require a massive use of hydrogen as fuel. The use of systems based on the use of proton exchange membrane fuel cell (PEMFC) has features that allow its application for stationary applications, automotive and portable power generation. The use of hydrogen as fuel for PEMFC has the advantage low pollutants emission, when compared to fossil fuels. For the reactions in a PEMFC is necessary to build membrane electrode assembly (MEA). And the production of MEAs and its materials are relevant to the final cost of kW of power generated by systems of fuel cell. This represent currently a technological and financial barriers to large-scale application of this technology. In this work a process of MEAs fabrication were developed that showed high reproducibility, rapidity and low cost by sieve printing. The process of sieve printing and the ink composition as a precursor to the catalyst layer were developed, which allow the preparation of electrodes for MEAs fabrication with the implementation of the exact catalyst loading, 0.6 milligrams of platinum per square centimeters (mgPt.cm-2) suitable for cathodes and 0.4 mgPt.cm-2 for anode in only one application step per electrode. The ink was developed, produced, characterized and used with similar characteristics to ink of sieve printing build for other applications. The MEAs produced had a performance of up to 712 mA.cm-2 by 600 mV to 25 cm2 MEA area. The MEA cost production for MEAs of 247.86 cm2, that can generate 1 kilowatt of energy was estimated to US$ 7,744.14 including cost of equipment, materials and labor.

The purpose of this study was to evaluate four different measuring methods as potentialquality control tests for Tungsten Carbide (WC) raw material used in the hard metalmanufacturing at Seco Tools Fagersta. This was warranted as the current quality testcalled the HcK test, is very time consuming and the results provided that are used formilling time calculations cannot be relied on. The four measuring methods chosen wasLaser diffraction, X-Ray diffraction, Brunauer Emmett Teller analysis and Fisher Sub-Sieve Sizer measurement. These measuring techniques were chosen by Seco ToolsThe project followed the general structure of the project cycle modified to fit the projectand involving the steps project planning, present state analysis, goals and requirementdetermination, evaluation elimination of options and final of choice of option. Thealternative methods were evaluated using an evaluation matrix containing therequirements determined to be the most relevant for the quality control process. Therequirements were measurement precision, operator influence, ease of use, time requiredand operator time required. The ability of the measurement method to distinguishbetween all of the different WC powders used a Seco Tools was put as a must requirementas the new method is meant to serve as a verification test as well as a quality test.The analysis methods were evaluated practically through experiments and therefore atesting strategy was established. Initial test on two powder types was performed so thatan early evaluation could be done. During this stage both the Laser diffraction and the XRaydiffraction were eliminated due to poor results. The BET analysis and the Fisher Sub-Sieve Sizer performed better and was therefore tested further. Randomized testing wasthe method chosen when testing with all powders to minimize the environmental impacton the testing as this could affect the results of the study.The testing showed that only the Fisher SSS was able to distinguish between all thedifferent powders making it the only viable option as a new quality test. This was onlypossible when measuring on agglomerated material and the values extracted couldtherefore not be used further in the production. The Fisher SSS also performed the bestwhen ranked against the rest of the requirements, showing a high precision whenmeasuring on deagglomerated material and a combination of the two tests was thereforeconsidered.When compared to the HcK test the Fisher SSS showed a significant reduction in lead timebut also a reduction in information gathered and eventual defects controlled. The FisherSSS provides only an average particle size measurement and since no correlation has yetbeen established between the measured particle size and the required milling times it wasrecommended that the HcK test remains.It was recommended that a study is started aimed at establishing a correlation betweenthe measured particle size and the required milling time is started to investigate if this ispossible.

The main theme of this dissertation is to engineer nanoporous materials and nanostructured surfaces for applications in gas separation membranes. Tunable methods have been developed to create inorganic hydroxide nanostructures on zeolite surfaces, and used to control the inorganic/polymer interfacial morphology in zeolite/polymer composite membranes. The study of the structure-property relationships in this material system showed that appropriate tuning of the surface modification methods leads to quite promising structural and permeation properties of the membranes made with the modified zeolites. First, a facile solvothermal deposition process was developed to prepare roughened inorganic nanostructures on zeolite pure silica MFI crystal surfaces. The functionalized zeolite crystals resulted in high-quality ̒mixed matrix̕ membranes, wherein the zeolite crystals were well-adhered to the polymeric matrix. Substantially enhanced gas separation characteristics were observed in mixed matrix membranes containing solvothermally modified MFI crystals. Gas permeation measurements on membranes containing nonporous uncalcined MFI revealed that the performance enhancements were due to significantly enhanced MFI-polymer adhesion and distribution of the MFI crystals. Solvothermal deposition of inorganic nanostructures was successfully applied to aluminosilicate LTA surfaces. Solvothermal treatment of LTA was tuned to deposit smaller/finer Mg(OH)₂ nanostructures, resulting in a more highly roughened zeolite surface. Characterization of particles and mixed matrix membranes revealed that the solvothermally surface-treated LTA particles were promising for application in mixed matrix membranes. Zeolite LTA materials with highly roughened surfaces were also successfully prepared by a new method: the ion-exchange-induced growth of Mg(OH)₂ nanostructures using the zeolite as the source of the Mg²⁺ ions. The size/shape of the inorganic nanostructures was tuned by adjusting several parameters such as the pH of the reagent solution and the amount of magnesium in the substrates and systematic modification of reaction conditions allowed generation of a good candidate for application in mixed matrix membranes. Zeolite/polymer adhesion properties in mixed matrix membranes were improved after the surface treatment compared to the untreated bare LTA. Surface modified zeolite 5A/6FDA-DAM mixed matrix membranes showed significant enhancement in CO₂ permeability with slight increases in CO₂/CH₄ selectivity as compared to the pure polymer membrane. The CO₂/CH₄ selectivity of the membrane containing surface treated zeolite 5A was much higher than that of membrane with untreated zeolite 5A. In addition, the use of metal organic framework (MOF) materials has been explored in mixed matrix membrane applications. ZIF-90 crystals with submicron and 2-μm sizes were successfully synthesized by a nonsolvent induced crystallization technique. Structural investigation revealed that the ZIF-90 particles synthesized by this method had high crystallinity, microporosity and thermal stability. The ZIF-90 particles showed good adhesion with polymers in mixed matrix membranes without any compatibilization. A significant increase in CO₂ permeability was observed without sacrificing CO₂/CH₄ selectivity when Ultem® and Matrimd® were used as the polymer matrix. In contrast, mixed matrix membranes with the highly permeable polymer 6FDA-DAM showed substantial enhancement in both permeability and selectivity, as the transport properties of the two phases were more closely matched.

Soit $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ la suite des nombres premiers, et soient $q \ge 3$ et $a$ des entiers premiers entre eux. R\'ecemment, Daniel Shiu a d\'emontr\'e une ancienne conjecture de Sarvadaman Chowla. Ce dernier a conjectur\'e qu'il existe une infinit\'e de couples $p_n,p_$ de premiers cons\'ecutifs tels que $p_n \equiv p_{n+1} \equiv a \bmod q$. Fixons $\epsilon > 0$. Une r\'ecente perc\'ee majeure, de Daniel Goldston, J\`anos Pintz et Cem Y{\i}ld{\i}r{\i}m, a \'et\'e de d\'emontrer qu'il existe une suite de nombres r\'eels $x$ tendant vers l'infini, tels que l'intervalle $(x,x+\epsilon\log x]$ contienne au moins deux nombres premiers $\equiv a \bmod q$. \'Etant donn\'e un couple de nombres premiers $\equiv a \bmod q$ dans un tel intervalle, il pourrait exister un nombre premier compris entre les deux qui n'est pas $\equiv a \bmod q$. On peut d\'eduire que soit il existe une suite de r\'eels $x$ tendant vers l'infini, telle que $(x,x+\epsilon\log x]$ contienne un triplet $p_n,p_{n+1},p_{n+2}$ de nombres premiers cons\'ecutifs, soit il existe une suite de r\'eels $x$, tendant vers l'infini telle que l'intervalle $(x,x+\epsilon\log x]$ contienne un couple $p_n,p_{n+1}$ de nombres premiers tel que $p_n \equiv p_{n+1} \equiv a \bmod q$. On pense que les deux \'enonc\'es sont vrais, toutefois on peut seulement d\'eduire que l'un d'entre eux est vrai, sans savoir lequel. Dans la premi\`ere partie de cette th\`ese, nous d\'emontrons que le deuxi\`eme \'enonc\'e est vrai, ce qui fournit une nouvelle d\'emonstration de la conjecture de Chowla. La preuve combine des id\'ees de Shiu et de Goldston-Pintz-Y{\i}ld{\i}r{\i}m, donc on peut consid\'erer que ce r\'esultat est une application de leurs m\'thodes. Ensuite, nous fournirons des bornes inf\'erieures pour le nombre de couples $p_n,p_{n+1}$ tels que $p_n \equiv p_{n+1} \equiv a \bmod q$, $p_{n+1} - p_n < \epsilon\log p_n$, avec $p_{n+1} \le Y$. Sous l'hypoth\`ese que $\theta$, le \og niveau de distribution \fg{} des nombres premiers, est plus grand que $1/2$, Goldston-Pintz-Y{\i}ld{\i}r{\i}m ont r\'eussi \`a d\'emontrer que $p_{n+1} - p_n \ll_{\theta} 1$ pour une infinit\'e de couples $p_n,p_$. Sous la meme hypoth\`ese, nous d\'emontrerons que $p_{n+1} - p_n \ll_{q,\theta} 1$ et $p_n \equiv p_{n+1} \equiv a \bmod q$ pour une infinit\'e de couples $p_n,p_$, et nous prouverons \'egalement un r\'esultat quantitatif. Dans la deuxi\`eme partie, nous allons utiliser les techniques de Goldston-Pintz-Yldrm pour d\'emontrer qu'il existe une infinit\'e de couples de nombres premiers $p,p'$ tels que $(p-1)(p'-1)$ est une carr\'e parfait. Ce resultat est une version approximative d'une ancienne conjecture qui stipule qu'il existe une infinit\'e de nombres premiers $p$ tels que $p-1$ est une carr\'e parfait. En effet, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $n = \ell_1\cdots \ell_r$, avec $\ell_1,\ldots,\ell_r$ des premiers distincts, et tels que $(\ell_1-1)\cdots (\ell_r-1)$ est une puissance $r$-i\`eme, avec $r \ge 2$ quelconque. \'Egalement, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n = \ell_1\cdots \ell_r \le Y$ tels que $(\ell_1+1)\cdots (\ell_r+1)$ est une puissance $r$-i\`eme. Finalement, \'etant donn\'e $A$ un ensemble fini d'entiers non-nuls, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $\prod_ (p+a)$ est une puissance $r$-i\`eme, simultan\'ement pour chaque $a \in A$. / Let $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ be the sequence of all primes, and let $q \ge 3$ and $a$ be coprime integers. Recently, and very remarkably, Daniel Shiu proved an old conjecture of Sarvadaman Chowla, which asserts that there are infinitely many pairs of consecutive primes $p_n,p_{n+1}$ for which $p_n \equiv p_{n+1} \equiv a \bmod q$. Now fix a number $\epsilon > 0$, arbitrarily small. In their recent groundbreaking work, Daniel Goldston, J\`anos Pintz and Cem Y{\i}ld{\i}r{\i}m proved that there are arbitrarily large $x$ for which the short interval $(x, x + \epsilon\log x]$ contains at least two primes congruent to $a \bmod q$. Given a pair of primes $\equiv a \bmod q$ in such an interval, there might be a prime in-between them that is not $\equiv a \bmod q$. One can deduce that \emph{either} there are arbitrarily large $x$ for which $(x, x + \epsilon\log x]$ contains a prime pair $p_n \equiv p_{n+1} \equiv a \bmod q$, \emph{or} that there are arbitrarily large $x$ for which the $(x, x + \epsilon\log x]$ contains a triple of consecutive primes $p_n,p_{n+1},p_{n+2}$. Both statements are believed to be true, but one can only deduce that one of them is true, and one does not know which one, from the result of Goldston-Pintz-Y{\i}ld{\i}r{\i}m. In Part I of this thesis, we prove that the first of these alternatives is true, thus obtaining a new proof of Chowla's conjecture. The proof combines some of Shiu's ideas with those of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, and so this result may be regarded as an application of their method. We then establish lower bounds for the number of prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n < \epsilon\log p_n$ and $p_{n+1} \le Y$. Assuming a certain unproven hypothesis concerning what is referred to as the `level of distribution', $\theta$, of the primes, Goldston-Pintz-Y{\i}ld{\i}r{\i}m were able to prove that $p_{n+1} - p_n \ll_{\theta} 1$ for infinitely many $n$. On the same hypothesis, we prove that there are infinitely many prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n \ll_{q,\theta} 1$. This conditional result is also proved in a quantitative form. In Part II we apply the techniques of Goldston-Pintz-Y{\i}ld{\i}r{\i}m to prove another result, namely that there are infinitely many pairs of distinct primes $p,p'$ such that $(p-1)(p'-1)$ is a perfect square. This is, in a sense, an `approximation' to the old conjecture that there are infinitely many primes $p$ such that $p-1$ is a perfect square. In fact we obtain a lower bound for the number of integers $n$, up to $Y$, such that $n = \ell_1\cdots \ell_r$, the $\ell_i$ distinct primes, and $(\ell_1 - 1)\cdots (\ell_r - 1)$ is a perfect $r$th power, for any given $r \ge 2$. We likewise obtain a lower bound for the number of such $n \le Y$ for which $(\ell_1 + 1)\cdots (\ell_r + 1)$ is a perfect $r$th power. Finally, given a finite set $A$ of nonzero integers, we obtain a lower bound for the number of $n \le Y$ for which $\prod_{p \mid n}(p+a)$ is a perfect $r$th power, simultaneously for every $a \in A$.

Artins Vermutung über Primitivwurzeln besagt, dass es zu jeder ganzen Zahl a, die weder 0, ±1 noch eine Quadratzahl ist, unendlich viele Primzahlen p gibt, sodass a eine Primitivwurzel modulo p ist, d.h. a erzeugt eine multiplikative Untergruppe von Q*, dessen Reduktion modulo p Index 1 in (Z/pZ)* hat. Dies wirft die Frage nach Verteilung von Index und Ordnung dieser Reduktion in (Z/pZ)* auf, wenn man p variiert. Diese Arbeit widmet sich verallgemeinerten Fragestellungen in Zahlkörpern: Ist K ein Zahlkörper und Gamma eine endlich erzeugte unendliche Untergruppe von K*, so werden Momente von Index und Ordnung der Reduktion von Gamma sowohl modulo bestimmter Familien von Primidealen von K als auch modulo aller Ideale von K untersucht. Ist Gamma die Gruppe der Einheiten von K, so steht diese Fragestellung in engem Zusammenhang mit der Ramanujan Vermutung in Zahlkörpern. Des Weiteren werden analoge Probleme für rationale elliptische Kurven E betrachtet: Bezeichnet Gamma die von einem rationalen Punkt von E erzeugte Gruppe, so wird untersucht, wie sich Index und Ordnung der Reduktion von Gamma modulo Primzahlen verhalten. Teilweise unter Voraussetzung gängiger zahlentheoretischer Vermutungen werden jeweils asymptotische Formeln in manchen Fällen bewiesen und generelle Schwierigkeiten geschildert, die solche in anderen Fällen verhindern. Darüber hinaus wird eine weitere verwandte Fragestellung betrachtet und bewiesen, dass zu jeder hinreichend großen Primzahl p stets eine Primitivwurzel modulo p existiert, die sich als Summe von zwei Quadraten darstellen lässt und nach oben im Wesentlichen durch die Quadratwurzel von p beschränkt ist.

The current work focused on defining the material science options to fabricate novel, high performing ethylene/ethane (C₂H₄/C₂H₆) separation carbon molecular sieve (CMS) dense film membranes. Three polymer precursors: Matrimid®, 6FDA-DAM and 6FDA:BPDA-DAM were used as precursors to the CMS membranes. CMS performances were tailored by way of tuning pyrolysis conditions such as the pyrolysis temperature, heating rate, pyrolysis atmosphere etc. The CMS dense film membranes showed attractive C₂H₄/C₂H₆ separation performance far exceeding the polymeric membrane performances. Semi-quantitative diffusion size pore distributions were constructed by studying the transport performance of a range of different penetrant gases as molecular sized probes of the CMS pore structure. This, in conjunction with separation performance data, provided critical insights into the structure-performance relationships of the CMS materials. The effects of testing conditions, i.e. the testing temperature, pressure and feed composition on C₂H₄/C₂H₆ separation performance of CMS dense films were also analyzed. These studies were useful not just in predicting the membrane behavior from a practical stand-point, but also in a fundamental understanding of the nature of CMS membrane separation. The study helped clarify why CMS membranes outperform polymeric membrane performance, as well as allowed comparison between CMS derived from different precursors and processing conditions. The effects on C₂H₄/C₂H₆ separation in the presence of binary gas mixture were also assessed to get a more realistic measure of the CMS performance resulting from competition and bulk flow effects. The current work thus establishes a framework for guiding research ultimately aimed at providing a convenient, potentially scalable hollow fiber membrane formation technology for C₂H₄/C₂H₆ separation

Olefin/paraffin separation is a large potential market for membrane applications. Carbon molecular sieve membranes (CMS) are promising for this application due to the intrinsically high separation performance and the viability for practical scale-up. Intrinsically high separation performance of CMS membranes for olefin/paraffin separations was demonstrated. The translation of intrinsic CMS transport properties into the hollow fiber configuration is considered in detail. Substructure collapse of asymmetric hollow fibers was found during Matrimidﾮ CMS hollow fiber formation. To overcome the permeance loss due to the increased separation layer thickness, 6FDA-DAM and 6FDA/BPDA-DAM polyimides with higher rigidity were employed as alternative precursors, and significant improvement has been achieved. Besides the macroscopic morphology control of asymmetric hollow fibers, the micro-structure was tuned by optimizing pyrolysis temperature protocol and pyrolysis atmosphere. In addition, unexpected physical aging was observed in CMS membranes, which is analogous to the aging phenomenon in glassy polymers. For performance evaluation, multiple "proof-of-concept" tests validated the viability of CMS membranes under realistic conditions. The scope of this work was expanded from binary ethylene/ethane and propylene/propane separations for the debottlenecking purpose to mixed carbon number hydrocarbon processing. CMS membranes were found to be olefins-selective over corresponding paraffins; moreover, CMS membranes are able to effectively fractionate the complex cracked gas stream in a preferable way. Reconfiguration of the hydrocarbon processing in ethylene plants is possible based on the unique CMS membranes.