Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is the Ramanujan $\tau$-function. This is equivalent to the assertion that $\tau(n) \neq 0$ for any $n$. A related problem is to find the distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ } (\text{mod } p)$. These are open problems. However, the variant of estimating the number of integers $n$ for which $n$ and $\tau(n)$ do not have a non-trivial common factor is more amenable to study. More generally, let $f$ be a normalized eigenform for the Hecke operators of weight $k \geq 2$ and having rational integer Fourier coefficients $\{a(n)\}$. It is interesting to study the quantity $(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients $a(n)$ \begin{equation} \{ n \leq x \text { } | \text{ } (n,a(n))=1\} = \displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}} \end{equation} for some constant $U_f$. We extend this result to higher weight forms. \\ We also show that \begin{equation} \{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll \displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log \log x}}. \end{equation}

Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is the Ramanujan $\tau$-function. This is equivalent to the assertion that $\tau(n) \neq 0$ for any $n$. A related problem is to find the distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ } (\text{mod } p)$. These are open problems. However, the variant of estimating the number of integers $n$ for which $n$ and $\tau(n)$ do not have a non-trivial common factor is more amenable to study. More generally, let $f$ be a normalized eigenform for the Hecke operators of weight $k \geq 2$ and having rational integer Fourier coefficients $\{a(n)\}$. It is interesting to study the quantity $(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients $a(n)$ \begin{equation} \{ n \leq x \text { } | \text{ } (n,a(n))=1\} = \displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}} \end{equation} for some constant $U_f$. We extend this result to higher weight forms. \\ We also show that \begin{equation} \{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll \displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log \log x}}. \end{equation}

Orientador: Rodolpho Vilhena de Moraes / Banca: Maria Cecília França de Paula Santos Zanardi / Banca: Valdemir Carrara / Resumo: Devido à sua dependência a um grande número de parâmetros de difícil determinação, o cálculo de coeficiente de arrasto para satélites artificiais torna-se extremamente complexo. Teorias, como as desenvolvidas por Schamberg e Sehnal levam em consideração o coeficiente de acomodação térmico e os ângulos de incidência e reflexão das moléculas na superfície do satélite. Outras teorias como a de Stalder e Zurick utilizam nas suas formulações os seguintes parâmetros: a razão entre as velocidades do satélite e das moléculas da atmosfera, o coeficiente de acomodação térmico, o ângulo de ataque e a razão entre as temperaturas da superfície do satélite e a temperatura das moléculas incidentes. Neste trabalho algumas teorias para o cálculo do coefienciente de arrasto são analisadas comparativamente. Exemplos são exibidos, utilizando como modelo os satélites SCD1, SCD2 e CBERS1. / Abstract: Drag coefficient computation for artificial satellite is extremely complex due to its dependence on several parameters that are difficult to determine. Some theories for drag coefficient computation, such as those developed by Schamberg and Sehnal, took into account the thermal accommodation coefficient and the incident and reflection angles of the molecules in the satellite surface. Others theories, such as the Stalder and Zurick theories, use in their formulations the following parameters: the ratio between the satellite velocity and the velocity of the molecules in the atmosphere, the thermal accommodation coefficient, the attack angle and the ratio between the temperatures of the satellite surface and of the incident molecules. In this work some theories for drag coefficients computation are comparatively analyzed. Examples are exhibited using the satellites SCD1, SCD2 and CBERS1 as models. / Mestre

Reductions in the cost and size of integrated circuits are allowing more and more complex functions to be included in previously simple tools such as lawn-mowers, ovens, and thermostats. Because of this, the process of synthesizing such functions from their initial representation to an optimal VLSI implementation is rarely hand-performed; instead, automated synthesis and optimization tools are a necessity. The factors such tools must take into account are numerous, including area (size), power consumption, and timing factors, to name just a few. Existing tools have traditionally focused upon optimization of two-level representations. However, new technologies such as Field Programmable Gate Arrays (FPGAs) have generated additional interest in three-level representations and structures such as Kronecker Decision Diagrams (KDDs). The reason for this is that when implementing a circuit on an FPGA, the cost of implementing exclusive-or logic is no more than that of traditional AND or OR gates. This dissertation investigates the use of the autocorrelation coefficients in logic synthesis for these types of structures; specifically, whether it is possible to pre-process a function to produce a subset of its autocorrelation coefficients and make use of this information in the choice of a three-level decomposition or of decomposition types within a KDD. This research began as a general investigation into the properties of autocorrelation coefficients of switching functions. Much work has centered around the use of a function's spectral coefficients in logic synthesis; however, very little work has used a function's autocorrelation coefficients. Their use has been investigated in the areas of testing, optimization for Programmable Logic Arrays (PLAs), identification of types of complexity measures, and in various DD-related applications, but in a limited manner. This has likely been due to the complexity in their computation. In order to investigate the uses of these coefficients, a fast computation technique was required, as well as knowledge of their basic properties. Both areas are detailed as part of this work, which demonstrates that it is feasible to quickly compute the autocorrelation coefficients. With these investigations as a foundation we further apply the autocorrelation coefficients to the development of a classification technique. The autocorrelation classes are similar to the spectral classes, but provide significantly different information. The dissertation demonstrates that some of this information highlighted by the autocorrelation classes may allow for the identification of exclusive-or logic within the function or classes of functions. In relation to this, a major contribution of this work involves the design and implementation of algorithms based on these results. The first of these algorithms is used to identify three-level decompositions for functions, and the second to determine decomposition type lists for KDD-representations. Each of these implementations compares well with existing tools, requiring on average less than one second to complete, and performing as well as the existing tools about 70% of the time. / Graduate

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2001Bitstream added on 2014-06-13T19:06:47Z : No. of bitstreams: 1 silva_tr_me_guara.pdf: 448943 bytes, checksum: 69cd9d252dfbcf9b51952987c7904c0b (MD5) / Devido à sua dependência a um grande número de parâmetros de difícil determinação, o cálculo de coeficiente de arrasto para satélites artificiais torna-se extremamente complexo. Teorias, como as desenvolvidas por Schamberg e Sehnal levam em consideração o coeficiente de acomodação térmico e os ângulos de incidência e reflexão das moléculas na superfície do satélite. Outras teorias como a de Stalder e Zurick utilizam nas suas formulações os seguintes parâmetros: a razão entre as velocidades do satélite e das moléculas da atmosfera, o coeficiente de acomodação térmico, o ângulo de ataque e a razão entre as temperaturas da superfície do satélite e a temperatura das moléculas incidentes. Neste trabalho algumas teorias para o cálculo do coefienciente de arrasto são analisadas comparativamente. Exemplos são exibidos, utilizando como modelo os satélites SCD1, SCD2 e CBERS1. / Drag coefficient computation for artificial satellite is extremely complex due to its dependence on several parameters that are difficult to determine. Some theories for drag coefficient computation, such as those developed by Schamberg and Sehnal, took into account the thermal accommodation coefficient and the incident and reflection angles of the molecules in the satellite surface. Others theories, such as the Stalder and Zurick theories, use in their formulations the following parameters: the ratio between the satellite velocity and the velocity of the molecules in the atmosphere, the thermal accommodation coefficient, the attack angle and the ratio between the temperatures of the satellite surface and of the incident molecules. In this work some theories for drag coefficients computation are comparatively analyzed. Examples are exhibited using the satellites SCD1, SCD2 and CBERS1 as models.

We discuss the behavior of air lenses in lens design. The structural aberration coefficients of a thin air lens are derived and compared with their glass thin lens counterpart. Examples are provided for a telephoto lens and the Monochromatic Quartet where air lenses or aspheric surfaces are used.

Submitted in fulfillment of the requirements for the degree of Doctor of Technology: Chemistry, Durban University of Technology, Durban, South Africa, 2017. / The thermodynamic properties of mixtures involving ionic liquids (ILs) with organic acid (acetic acid or propanoic acid) or acetonitrile at different temperatures were determined. The ILs used were imidazolium-based: 1-ethyl-3-methylimidazolium ethyl sulphate [EMIM]+[EtSO4]-, 1-butyl- 3-methylimidazolium thiocyanate [BMIM]+[SCN]- and 1-butyl-3-methylimidazolium bis (trifluoromethylsulfonyl) imide ([BMIM]+[Tf2N]-. The ternary excess molar volume (V E ), isentropic compressibility (ks) and deviation in isentropic compressibility ( ks123 ) were determined for four ternary liquid mixtures of {[EMIM]+[EtSO4]- or [BMIM]+[SCN]− + acetic or propionic acid + acetonitrile} at different temperatures (293.15, 298.15, 303.15, 308.15 and 313.15) K and at a pressure of 0.1 MPa with aid of the experimental density (ρ), speed of sound (u) data. The calculated data were correlated by using the Cibulka equation with the help of Redlich–Kister parameters obtained from fitting the Redlich–Kister equation for the corresponding binary systems. Furthermore, the density and speed of sound were also measured for eight corresponding binary systems at the same experimental conditions. The binary excess molar volume, isentropic compressibility and deviation in isentropic compressibility were also calculated for measured systems and fitted to the Redlich–Kister equation to obtain the Redlich–Kister parameters as well as to check the accuracy of measured data which were used to correlated experimental data using Cibulka equation. These results were discussed, in terms of how the sign and magnitude of thermodynamic functions were influenced by the addition of a third component to liquid systems. Also, the possible molecular and pair-wise interactions between component molecules and the effect of temperature on the thermophysical and thermodynamic properties were predicted. In addition, the work focussed on application of ([BMIM]+[Tf2N]-) ionic liquid for the separations of (alkane/aromatic), (alkane/alk-1-ene), (cycloalkane/aromatic) and (water/alkan-1-ol) using gas- liquid chromatography (GLC) technique. The activity coefficients at infinite dilution, , for 31 organic solutes (alkanes, cycloalkanes, alkenes, alkynes, aromatics, alkanol and ketones) and water in ionic liquid were measured at temperatures of (323.15, 333.15, 343.15, 353.15 and 363.15) K. Stationary phase loadings of (42.83 and 68.66) % by mass were used to ensure repeatability of E , measurements. Partial molar excess enthalpies at infinite dilution, H1 , were also determined. The selectivities, S , and capacities, k , were determined for the above separations. The separating ij j ability of the investigated ionic liquid was compared with previously investigated ionic liquids and industrial solvents such as sulfolane, n-methyl-2-pyrrolidine (NMP) and n-formylmorpholine (NFM). The results obtained suggested that in general, the [BMIM]+[Tf2N]− had outperformed the conventional solvents such as sulfolane, NMP and NFM in terms of selectivity, while the [BMIM][Tf2N] had in general, performed better overall when the performance index was used for comparison. / D

Many methods exist in the literature for the prediction of pharmacokinetic parameters which describe drug distribution in rat and human, such as tissue-to-plasma partition coefficients (Kps) and volume of distribution (Vss). However, none of these methods make use of the in vivo information obtained at the early stages of the drug development process in the form of plasma concentration vs. time profiles. The overall aim of the presented study was to improve upon an existing Kp prediction method by making use of the distribution information contained within this experimental data. Chapter 2 shows that Kp values can be successfully obtained experimentally, but that this process is expensive and time-consuming. Chapter 3 compares six Kp prediction methods taken from the literature for their ability to predict the Kp values of 80 drugs. The Rodgers et al. model was found to be the most accurate, with over 77% of predictions within 3-fold of experimental values. This Chapter also discusses the Vss prediction ability of some of these methods, with the Poulin & Theil and Rodgers et al. models shown to be the most accurate predictors for rat Vss and human Vss respectively. Chapter 4 investigates the relationship between muscle Kp and the Kps of all other tissues, to show that experimental muscle Kp can be used as a surrogate from which all other non-adipose Kp values can be predicted. However, the predictions made using this method were shown to be less accurate than predictions made by the Rodgers et al. model for the same dataset of drugs. A relationship was identified between muscle Kp and tumour Kp in rat, suggesting a potential way to predict tumour Kp in the future. In Chapter 5, a novel method is developed whereby Kp predictions made by the Rodgers et al. model are updated using prior information obtained from the in vivo concentration-time profile. These updated values are then used within a physiologically-based pharmacokinetic (PBPK) model and are shown in Chapter 6 to generate improved predictions for other pharmacokinetic parameters such as Vss and clearance in both rat and human. 100% of human Vss predictions made by the most accurate of the novel methods presented here were within 3-fold of experimental values, compared to 68.8% of predictions made by the Rodgers et al. model. The work presented here has highlighted the need for a more accurate method for the prediction of Kp values, and has addressed this need by generating a model which improves upon the most accurate Kp prediction method currently found in the literature. This will lead to an increase in confidence in the use of predicted pharmacokinetic parameters within PBPK modelling.

The purpose of this thesis is to analyze the thermochemical properties of solutes in nonelectrolyte pure solvents and to develop mathematical expressions with the ability to describe and predict solution behavior using mobile order theory. Solubilities of pesticides (monuron, diuron, and hexachlorobenzene), polycyclic aromatic hydrocarbons (biphenyl, acenaphthene, and phenanthrene), and the organometallic ferrocene were studied in a wide array of solvents. Mobile order theory predictive equations were derived and percent average absolute deviations between experimental and calculated mole fraction solubilities for each solute were as follows: monuron in 21 non-alcoholic solvents (48.4%), diuron in 28 non-alcoholic solvents (60.1%), hexachlorobenzene (210%), biphenyl (13.0%), acenaphthene (37.8%), phenanthrene (41.3%), and ferrocene (107.8%). Solute descriptors using the Abraham solvation model were also calculated for monuron and diuron. Coefficients in the general solvation equation were known for all the solvents and solute descriptors calculated using multilinear regression techniques.

Philosophiae Doctor - PhD / One of the assumptions of conventional regression analysis is I that the parameters are constant over all observations. It has often been suggested that this may not be a valid assumption to make, particularly if the econometric model is to be used for economic forecasting0 Apart from this it is also found that econometric models, in particular, are used to investigate the underlying interrelationships of the system under consideration in order to understand and to explain relevant phenomena in structural analysis. The pre-requisite of such use of econometrics is that the regression parameters of the model is assumed to be constant over time or across different crosssectional units.