Efficient propylene and propane separation is a major challenge in the modern chemical industry. With current separation methods being highly energy-intensive, there is a pressing need to find alternative green technology. Membrane separation emerged as a promising candidate for propylene and propane separation. Their small footprint, low cost, reliability, and environmental friendliness give membrane separation systems a competitive edge in the race towards sustainable development. The continuous advancements in material science created avenues for new membrane materials such as carbon molecular sieve (CMS) membranes which exhibit exceptional gas separation performances for challenging applications due to their strong size-sieving capabilities. In this work, a carbon molecular sieve (CMS) membrane derived from a polymer of intrinsic microporosity (TPE-PIM) has been investigated for propylene/propane separation made by pyrolysis at 400, 450, 500, 550, 600, 650, and 700 ºC. TPE-PIM-derived CMS films showed excellent pure and mixed-gas permeability and selectivity, exceeding the upper bound limits for propylene and propane. Observed in this work was the presence of an optimal pyrolysis temperature at 600 ºC, where the film showed the best performance with a permeability of 41.6 Barrer and a selectivity of 197 based on pure-gas measurements but dropping to 34 Barrer and selectivity of 33 under equimolar mixed-gas conditions. Such performance could be attributed to the unique internal structural changes that occurred during the pyrolysis. In addition, propane permeability though the CMS films was slow and required long times to reach steady-state values. Such slow kinetics illustrates the molecular sieving capabilities of CMS membranes for bigger and more condensable gases. Several characterization techniques have been performed on the films to confirm CMS formation and showcase deeper molecular structure insights. X-ray diffraction of all TPE-PIM films showed a broad spectrum at each peak due to the material’s amorphous nature. Diffraction patterns also revealed a gradual peak shift for the (002) plane towards smaller values closer to that of pure graphite. Raman spectra showed the characteristic D and G peaks for carbon films prepared at 500 ºC and above. FTIR analysis was also performed to investigate the potential formation of triazine crosslinks in the thermally treated samples, but no conclusive results were obtained.

No description available.

With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a "large" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a "reasonable" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined. / Master of Science

<p>This thesis aims at implementing methods for factorisation of large numbers. Seeing that there is no deterministic algorithm for finding the prime factors of a given number, the task proves rather difficult. Luckily, there have been developed some effective probabilistic methods since the invention of the computer so that it is now possible to factor numbers having about 200 decimal digits. This however consumes a large amount of resources and therefore, virtually all new factorisations are achieved using the combined power of many computers in a distributed system. </p><p>The nature of the distributed system can vary. The original goal of the thesis was to develop a client/server system that allows clients to carry out a portion of the overall computations and submit the result to the server. </p><p>Methods for factorisation discussed for implementation in the thesis are: the quadratic sieve, the number field sieve and the elliptic curve method. Actually implemented was only a variant of the quadratic sieve: the multiple polynomial quadratic sieve (MPQS).</p>

Η διείσδυση των υπολογιστών, τόσο στα σπίτια μας, όσο και κυρίως στις επιχειρήσεις, κατά τα τελευταία χρόνια, καθώς επίσης και ο συνεχώς αυξανόμενος ρυθμός χρήσης του διαδικτύου, έχουν καταστήσει την ανάγκη για ασφαλείς ηλεκτρονικές επικοινωνίες και συναλλαγές κάτι παραπάνω από επιτακτική. Ένα από τα κυρίαρχα, σήμερα, συστήματα ασφαλούς ανταλλαγής δεδομένων είναι ο αλγόριθμος RSA, η ασφάλεια του οποίου βασίζεται στο γεγονός ότι είναι πολύ δύσκολο να παραγοντοποιήσουμε έναν «μεγάλο» αριθμό στους πρώτους παράγοντές του. Ο RSA αλγόριθμος θεωρείται αρκετά ασφαλής, αν βέβαια χρησιμοποιούμε κατάλληλο, για τα σημερινά δεδομένα, μέγεθος κλειδιού. Παρόλα αυτά, σε περίπτωση που βρεθεί κάποιος αποδοτικός αλγόριθμος που να μπορεί σε «λογικό» χρόνο να παραγοντοποιήσει οποιονδήποτε μεγάλο ακέραιο, τότε αυτομάτως η ασφάλεια του αλγορίθμου αυτού έχει παραβιαστεί και θα πρέπει να στραφούμε σε εναλλακτικές μεθόδους προστασίας της πληροφορίας. Ο πιο αποδοτικός σήμερα αλγόριθμος παραγοντοποίησης μεγάλων ακεραίων είναι ο Number Field Sieve. Η έρευνα που έχει γίνει πάνω σε αυτόν τον αλγόριθμο, έχει οδηγήσει σε σημαντική πρόοδο και έχει καταστήσει, πλέον, εφικτή την παραγοντοποίηση ακεραίων που υπό άλλες προϋποθέσεις θα απαιτούσε χιλιάδες χρόνια από cpu time σε supercomputers. Αν και ακόμη και σήμερα υπάρχουν αρκετά σημεία που θα μπορούσαν να βελτιωθούν στον αλγόριθμο, κάνοντάς τον ακόμη πιο αποδοτικό, ωστόσο η πολυπλοκότητά του αποτρέπει αρκετούς να ασχοληθούν με την βελτίωσή του. Με την εργασία αυτή θα προσπαθήσουμε αρχικά να διασαφηνίσουμε όλες τις πληροφορίες που απαιτούνται για την σωστή κατανόηση της λειτουργίας του αλγορίθμου. Θα γίνει λεπτομερής περιγραφή των διαφόρων βημάτων του αλγορίθμου και θα δοθεί αναλυτικό παράδειγμα παραγοντοποίησης. Τέλος, θα παρουσιαστεί η παράλληλη υλοποίησή του αλγορίθμου, η οποία μπορεί να εκτελεστεί τόσο σε supercomputer, όσο και σε cluster υπολογιστών που επικοινωνούν μεταξύ τους με χρήση του MPI. / The recent advances in computer science, in combination with the proliferation of computers in home and businesses and the explosive growth rate of the internet transactions, have increased the needs for secure electronic communications. One of the dominant systems of secure data transactions is the RSA algorithm. RSA’ s security relies on the fact that it is computationally difficult to factor a “large” integer into its component prime integers. RSA is considered secure as long as we use proper key length. However, if an efficient algorithm is developed that can factor any arbitrarily large integer in a “reasonable” amount of time, then the whole security of the algorithm will be broken, and we will have to use alternative methods to secure our systems. Today, the fastest known method for factoring large integers is the General Number Field Sieve algorithm. Research and development of the algorithm has enabled the factorization of integers that were once thought to require thousands of years of CPU time to accomplish. While there are still many possible optimizations that could increase the algorithm’s efficiency, however the complexity of the algorithm prevents many researchers from attempting to improve it. In this master thesis we present the information needed to understand the principles upon which the algorithm is based. The discrete steps of the algorithm are described in full detail, as well as a detailed factorization example, in order to enlighten the way each step works. Finally a parallel implementation is presented, able to be executed on a supercomputer or a computer cluster, with the use of MPI.

Orientador: José Plínio de Oliveira Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T11:40:03Z (GMT). No. of bitstreams: 1 Nascimento_MarcusViniciusSilva_M.pdf: 918557 bytes, checksum: de0f1627892732c764e7f5046966336f (MD5) Previous issue date: 2015 / Resumo: Nosso objetivo, nesse trabalho, é o de fazer um estudo sobre o método do crivo. A motivação reside no desejo de aplicar essas ideias a uma situação particular. Dividimos nosso trabalho em três partes. Na primeira fornecemos apenas as definições e con- ceitos básicos. Na segunda apresentamos o principio da inclusão-exclusão que embora sendo algo bastante conhecido merece destaque especial dada a sua importância como ferramenta no nosso trabalho. Na terceira e última parte, fazemos uma contextualização histórica e uma descrição da evolução das ideias do crivo de Eratóstenes-Legendre. A escolha desse crivo, dentre tantos outros, foi feita tendo em vista dois pontos. O primeiro é que o crivo de Eratóstenes-Legendre é o mais simples dentre os crivos estudados na teoria dos crivos. O segundo ponto está relacionado com o fato deste crivo fornecer a ideia geral dos crivos combinatoriais, uma vez que os crivos mais sofisticados são extensões de suas ideias básicas / Abstract: Our aim in this work is to make a study about the sieve method. The motivation lies in the intent of applying this idea in a particular situation. We splitted the study into three parts. The first part deals with definitions and basic concepts. In the second we present the principle of inclusion-exclusion while being something well known deserves special mention given its importance as a tool in our work. In the third and final part, we make a historical contextualization and a description of the evolution of the sieve Eratosthenes- Legendre ideas. The choice of sieve, among many others, has been made taking into account two points. The first is that the Eratosthenes-Legendre sieve is the simplest among the sieves studied the theory of sieves. The second point is related to the fact that this sieve provide the general idea of combinatorial sieve, since the more sophisticated sieves are extensions of its basic idea / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

This thesis aims at implementing methods for factorisation of large numbers. Seeing that there is no deterministic algorithm for finding the prime factors of a given number, the task proves rather difficult. Luckily, there have been developed some effective probabilistic methods since the invention of the computer so that it is now possible to factor numbers having about 200 decimal digits. This however consumes a large amount of resources and therefore, virtually all new factorisations are achieved using the combined power of many computers in a distributed system. The nature of the distributed system can vary. The original goal of the thesis was to develop a client/server system that allows clients to carry out a portion of the overall computations and submit the result to the server. Methods for factorisation discussed for implementation in the thesis are: the quadratic sieve, the number field sieve and the elliptic curve method. Actually implemented was only a variant of the quadratic sieve: the multiple polynomial quadratic sieve (MPQS).

Thesis (MScEng (Process Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Mass transfer efficiency in distillation, absorption and stripping depends on both thermodynamic efficiency and hydrodynamic behaviour. Thermodynamic efficiency is dependent on the system kinetics while hydrodynamics is the study of fluid flow behaviour. The focus of this thesis is the hydrodynamic behaviour in tray columns, which affects entrainment. In order to isolate hydrodynamic behaviour from the thermodynamic behaviour that occurs inside sieve tray columns, investigations are conducted under conditions of zero mass transfer. When the gas velocity is sufficiently high to transport liquid droplets to the tray above, entrainment occurs. The onset of entrainment is one of the operating limits that determines the design of the column and thus impacts on the capital cost. By improving the understanding of the parameters that affect entrainment, the design of the tray and column can be improved which will ultimately increase the operability and capacity while reducing capital costs. Existing correlations predicting entrainment in sieve tray columns are based on data generated mainly from an air/water system. Previous publications recommend that more testing should be performed over larger ranges of gas and liquid physical properties. An experimental setup was therefore designed and constructed to test the influence of the following parameters on entrainment: 1. gas and liquid physical properties 2. gas and liquid flow rates 3. tray spacing The experimental setup can also measure weeping rates for a continuation of this project. The hydrodynamic performance of a sieve tray was tested with air and water over a wide range of gas and liquid flow rates and at different downcomer escape areas. It was found that the downcomer escape area should be sized so that the liquid escaping the downcomer always exceeds a velocity of approximately 0.23 m/s in order to create a sufficient liquid seal in the downcomer. For liquid velocities between 0.23 and 0.6 m/s the area of escape did not have an effect on the percentage of liquid entrained. It was also established that entrainment increases with increasing gas velocity. The rate at which entrainment increases as the gas velocity increase depends on the liquid flow rate. As soon as the liquid flow rate exceeded 74 m3/(h.m) a significant increase in entrainment was noted and the gas velocity had to be reduced to maintain a constant entrainment rate. This is because the increased liquid load requires a longer flow path length for the froth to fully develop. The undeveloped froth, caused by the short (455 mm) flow path, then creates a non-uniform froth that is pushed up against the column wall above the downcomer. Consequently, the froth layer is closer to the tray above resulting in most of the droplets ejected from the froth reaching the tray above and increasing entrainment. By reducing the gas velocity, the froth height and ejecting droplet velocity is reduced, resulting in a decrease in entrainment. The results from the experiments followed similar trends to most of the entrainment prediction correlations found in literature, except for the change noted in liquid flow rates above 74 m3/(h.m). There was, however, a significant difference between the experimental results and the correlations developed by Hunt et al. (1955) and Kister and Haas (1988). Although the gas velocities used during the air/water experiments were beyond the suggested range of application developed by Bennett et al. (1995) their air/water correlation followed the results very well. The entrainment prediction correlation developed by Bennett et al. (1995) for non-air/water systems was compared with the experimental air/water results to test for system uniformity. A significant difference was noted between their non-air/water prediction correlation and the air/water results, which motivates the need for a general entrainment prediction correlation over a wider range of gas and liquid physical properties. Based on the shortcomings found in the literature and the observations made during the experiments it is suggested that the influence of liquid flow path length should be investigated so that the effect on entrainment can be quantified. No single correlation was found in the literature, which accurately predicts entrainment for a large range of liquid loads (17 – 112 m3/(h.m)), high superficial gas velocities (3 – 4.6 m/s) and different gas and liquid physical properties. It is therefore recommended that more work be done, as an extension of this project, to investigate the influence of gas and liquid physical properties on entrainment (under zero mass transfer conditions) for a large range of liquid (5 – 74 m3/(h.m)) and gas (2 – 4.6 m/s) flow rates. In order to understand the effect of droplet drag on entrainment, tray spacing should be varied and increased to the extent where droplet ejection velocity is no longer the mechanism for entrainment and droplet drag is responsible for droplet transport to the tray above. Since it is difficult and in most cases impossible to measure exact gas and liquid loads in commercial columns, another method is required to measure or determine entrainment. Since liquid hold-up was found to be directly related to the entrainment rate (Hunt et al. (1955), Payne and Prince (1977) and Van Sinderen et al. (2003) to name but a few), it is suggested that a correlation should be developed between the dynamic pressure drop (liquid hold-up) and entrainment. This will contribute significantly to commercial column operation from a hydrodynamic point of view.

Thesis (MScEng (Process Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Partially oxygenated aromatic compounds, e.g. quinones, hydroquinones and cresols, play a vital role in the fine chemical industry and were initially prepared by stoichiometric oxidation processes that produce toxic products that are hazardous towards the environment. As a result, it was important to investigate environmentally friendly processes for the hydroxylation of aromatic compounds. This resulted in newer methods using Ti-substituted microporous zeolites as catalysts with hydrogen peroxide as oxidant in the presence of a solvent. However, the methods were found to be ineffective for large, bulky substrates due to the small pore structure. This led to using Ti-mesoporous materials as catalysts but suffered from two drawbacks; the hydrophilic nature and low hydrothermal stability of the catalyst structure. Ti-microporous and Ti-mesoporous materials acting as catalysts for the oxidation of bulky substrates achieved environmentally friendly processes but obtained low conversions and quinone yields. Therefore, the challenge has been to develop a process that is environmentally friendly, achieves high conversions, where the catalyst acts truly heterogeneous and obtains high quinone yields for the hydroxylation of bulky substrates. Recently, micropores/mesopores catalysts incorporating advantages of both micropores and mesopores materials were synthesised and seemed promising for the hydroxylation of bulky substrates. This study focuses on synthesising and evaluating the feasibility of various Ti-substituted catalysts for improving the hydroxylation of the bulky substrate, 2-methylnaphthalene (2MN) with hydrogen peroxide as oxidant in the presence of a solvent, acetonitrile. The oxidation of 2MN produces 2-methyl-1,4-naphthoquinone (2MNQ). 2MNQ is also known as menadione or Vitamin K3 and acts as a blood coagulating agent. The catalysts synthesised for this study were mesoporous catalysts, Ti- MCM-41 and Ti-MMM-2 and microporous/mesoporous catalysts, Ti-MMM-2(P123) and a highly ordered mesoporous material. The main objective of this study was to design an efficient process that is environmentally friendly and achieves high 2MN conversions and 2MNQ yields. This was achieved by evaluating the various catalysts synthesised, reaction conditions, testing if the catalyst was truly heterogeneous and identifying the products formed from the process. The designed process was proved to be environmentally friendly because the system did not produce products that were harmful towards the environment. The products identified in this study were 2MNQ, 2-methyl-1-naphthol, 2-naphthaldehyde, 3-ethoxy-4-methoxybenzaldehyde and menadione epoxide. An investigation was conducted to determine which catalyst synthesised favoured this process by quantifying the effect reaction conditions have on the various catalysts. The reaction conditions were defined in terms of the hydrogen peroxide volume, catalyst amount, solvent volume, substrate amount, reaction time and reaction temperature. The desired catalyst for this study obtained the highest 2MN conversions in comparison with the other catalysts and favoured the formation of 2MNQ. The catalyst achieving the highest conversions and favouring 2MNQ in most cases for this investigation was the highly ordered mesoporous material. Improving operating conditions to obtain high 2MNQ yields for the oxidation of 2MN to 2MNQ over the highly ordered mesoporous material was determined by varying the reaction conditions with the one factor at a time approach and a factorial design. The one factor at a time approach showed that best 2MNQ yields were obtained at 1 g substrate when investigating a change in substrate amount between 0.5 g and 2 g. Best 2MNQ yields were obtained at 10 ml solvent when investigating a change of solvent volume between 5 ml and 20 ml. The 2MNQ yield increased with increasing the catalyst amount (50 mg to 200 mg), hydrogen peroxide volume (1 ml to 6 ml) and increasing the reaction times (2 hour to 6 hours) at reaction temperatures, 120°C and 150°C. The yield decreased with increasing the reaction time (2 hours to 6 hours) at reaction temperature, 180°C. A preliminary 2 level factorial design was prepared to observe if there were any important interactions affecting the 2MNQ yield. The results from the factorial design indicated that the hydrogen peroxide volume had the most influence on the 2MNQ yield followed by the reaction time-reaction temperature interaction and reaction temperature. From the factorial design, the yield increased by increasing the hydrogen peroxide volume and reaction temperature whilst decreasing the reaction temperature-reaction time interaction. The highest 2MNQ yields and 2MN conversions obtained for the hydroxylation of 2MN to 2MNQ over the highly ordered mesoporous material in this study were in the ranges 48-50 % and 97-99 %, respectively. This study indicates that the process system, reaction conditions and catalyst type have an impact on the products formed, 2MN conversion, 2MNQ selectivity and 2MNQ yield. The highly ordered mesoporous material was found to be truly heterogeneous because no leaching occurred and the catalyst could be recycled without losing its catalytic activity and selectivity for at least two catalyst cycles. It can be concluded that the highly ordered mesoporous material is therefore a promising catalyst for the selective oxidation of bulky substrates with aqueous H2O2 because it produces an environmentally friendly process, achieves high conversions, obtains high quinone yields and the catalyst truly acts heterogeneous. / AFRIKAANSE OPSOMMING: Gedeeltelik geoksideerde aromatiese verbindings (bv. kinone, hidrokinone en kresole) speel ‘n belangrike rol in die fynchemiebedryf. Hierdie verbindings is aanvanklik voorberei deur stoïchiometriese oksidasie prosesse wat gifstowwe nadelig vir die omgewing veroorsaak. Daarom is dit belangrik om omgewingsvriendelike prosesse vir die hidroksilering van aromatiese verbindings te ondersoek. Hierdie ondersoeke het gelei tot nuwe metodes wat Ti-vervangde mikroporeuse seoliete as katalisator met waterstofperoksied as oksideermiddel in die teenwoordigheid van ʼn oplosmiddel benut. Dit is egter gevind dat hierdie metodes oneffektief is vir groot, lywige substrate weens die fyn poriestruktuur van die katalisator. Dit lei tot die gebruik van Ti-mesoporeuse materiale as katalisators, maar toon twee tekortkominge, naamlik die hidrofiliese aard en lae hidrotermiese stabiliteit van die katalisatorstruktuur. Ti-mikroporeuse en Ti-mesoporeuse materiale benut as katalisators vir die oksidasie van lywige substrate lewer omgewingsvriendelike prosesse, maar vermag lae omsetting en kinoonopbrengs. ʼn Uitdaging is dus om ʼn omgewingsvriendelike proses te ontwikkel met hoë omsetting, waar die katalisator werklik heterogeen optree en hoë kinoonopbrengs lewer vir die hidroksilering van lywige substrate. Katalisators vir die hidroksilering van lywige substrate wat die voordele van beide mikroporieë/mesoporieë ten toon stel is onlangs gesintetiseer, met belowende resultate. Hierdie studie is ingestel op die sintetisering en evaluering van uitvoerbaarheid van verskeie Tivervangde katalisators vir die optimering van die hidroksilering van die lywige substraat, 2- metielnaftaleen (2MN), met waterstofperoksied as oksideermiddel met asetonitriel as oplosmiddel. Die oksidering van 2MN produseer 2-metiel-1,4-naftokinoon (2MNK), ook bekend as vitamien K3, ʼn bloedstollingsmiddel. Die katalisators vervaardig vir hierdie studie was die mesoporeuse katalisators, Ti-MCM-41 en Ti-MMM-2, en die mikroporeuse/mesoporeuse katalisor Ti-MMM-2(P123), sowel as ʼn hoogs geordende mesoporeuse materiaal. Die hoofdoel van hierdie studie was om ʼn doeltreffende, omgewingsvriendelike proses met hoë 2MN omsetting en 2MNK opbrengs te ontwerp. Voorgenoemde is vermag deur verskeie gesintetiseerde katalisators en reaksiekondisies te evalueer, om te toets of katalisators werklik heterogeen is, en om die prosesprodukte te identifiseer. Die ontwerpte proses kan beskou word as omgewingsvriendelik, aangesien die stelsel geen produkte lewer wat skade aan die natuur kan veroorsaak nie. 2MNK, 2-metiel-1-naftol, 2-naftaldehied, 3- etoksi-4-metoksibensaldehied en menadioonepoksied is in hierdie studie geïdentifiseer as prosesprodukte. Om te bepaal watter gesintetiseerde katalisators hierdie proses begunstig, is ʼn ondersoek geloods om die effek van reaksiekondisies op die verskeie katalisators te kwantifiseer. Die reaksiekondisies is omskryf in terme van waterstofperoksiedkonsentrasie, katalisatorhoeveelheid, oplosmiddelvolume, substraathoeveelheid, reaksietyd en reaksietemperatuur. Die gewenste katalistor vir hierdie proses was die katalisator wat die hoogste 2MN omsetting lewer en die vorming van 2MNK bevorder. Die hoogs geordende mesoporeuse materiaal was in hierdie ondersoek die katalisator met die hoogste omsetting wat ook 2MNK-vorming bevorder het in die meeste gevalle. Om die beste bedryfstoestande vir hoë 2MNK opbrengs vanaf die oksidering van 2MN oor hoogs geordende mesoporeuse materiaal te bepaal, is die reaksiekondisies verander deur met een faktor op ʼn slag te verander, sowel as faktorverandering volgens ʼn faktoriaalontwerp. Die een-faktor-op-‘nslag benadering het getoon dat die 2MNK opbrengs ʼn maksimum bereik waar die substraathoeveelheid tussen 0.5 g en 2 g wissel, met die oplosmiddelvolume tussen 5 ml en 20 ml. Die opbrengs het ietwat verbeter met ʼn groter hoeveelheid katalisatorhoeveelheid (van 50 mg na 200 mg), terwyl die opbrengs drasties verbeter het waar die waterstofperoksiedvolume van 3 ml tot 6 ml verhoog is. Die opbrengs het ook verbeter met ʼn styging in reaksietemperatuur (van 120°C tot 180°C) met reaksietydintervalle van 1 tot 6 ure. Die opbrengs het egter gedaal by 180°C waar reaksietye langer as 2 ure. Volgens die resultate van die een-faktor-op-‘n-slag benadering blyk dit dat reaksietemperatuur, waterstofperoksiedvolume, katalisatorhoeveelheid en reaksietyd faktore is wat verhoogde 2MNK opbrengs bevorder. Hierdie reaksiekondisies is geselekteer vir die faktoriaalontwerp. ʼn Voorlopige 2- vlak faktoriaalontwerp is voorberei om te bepaal of daar enige belangrike interaksies is wat die 2MNK opbrengs beïnvloed. Die resultate van die faktoriaalontwerp het aangetoon dat waterstofperoksiedvolume die grootste invloed op 2MNK opbrengs het, gevolg deur die interaksie van reaksietyd en reaksietemperatuur, en dan reaksietemperatuur. Die faktoriaalontwerp resultate toon verder dat opbrengs verhoog met toenemende waterstofperoksiedvolume en reaksietemperatuur, terwyl die opbrengs verlaag soos wat die reaksietyd-reaksietemperatuur interaksie toeneem. Hierdie studie het hoogste 2MNK opbrengs van 48-50% en 2MN omsetting van 97-99% vir die hidroksilering van 2MN na 2MNK oor hoogs geordende mesoporeuse materiale behaal. Hierdie studie bevestig bevindinge van die literatuur dat die prosesstelsel, reaksiekondisies en katalisatortipe ʼn groot impak het op prosesprodukte, 2MN omsetting, 2MNK selektiwiteit en 2MNK opbrengs. In hierdie navorsingstudie is bevind dat hoë 2MN omsetting en 2MNK opbrengs behaal word by hoë reaksietemperature met kort reaksietye en hoë waterstofperoksiedvolumes. Dit is gevind dat die hoogs geordende mesoporeuse materiaal werklik heterogeen is, aangesien geen loging plaasgevind het nie, en aangesien die katalisator hergebruik kon word sonder verlies aan katalisatoraktiwiteit en –selektiwiteit, vir ten minste twee katalisatorsiklusse. ʼn Gevolgtrekking kan gemaak word dat die hoogs geordende mesoporeuse materiaal ʼn belowende katalisator vir die selektiewe oksidering van lywige substrate met waterige H2O2 is, aangesien dit ʼn omgewingsvriendelike proses lewer met hoë omsetting, hoë kinoonopbrengs en katalisatorgedrag wat waarlik heterogeen is.

Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors. We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. We then give an exposition of the Siegel--Walfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions. Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the Bombieri--Vinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.