The aim of this thesis is primarily to investigate theadsorption of surfactants, in particular to the airwaterinterface. The effects of the head-group and the hydrocarbonchain are of special interest. Using thermodynamical models wewere able to predict that the transition from the Henry rangeto the liquid-expanded phase is controlled by the formation ofsmall clusters of surfactants oriented in the plane of thesurface. Hence, the transition from the Henry range can not beof first-order, and additionally the phase formed after theonset of the micelle formation does have a liquid-likecharacter. We refer to this phase as the dilute surfacemicellar range. Interactions between the micelles in the dilutesurface micellar range can be accounted for by excluded areainteractions using the harddisc model. The stability of themicelles can be attributed to the line tension acting on themicelle edge, which stops unlimited growth of the surfacemicelles. With increasing density of surface micelles a newphase is formed, referred to as the granular phase. Thegranular phase exhibits very low compressibilty and an almostconstant area/molecule in an extended surfactant bulkconcentration interval. For three sugar-based surfactantscarrying decyl hydrocarbon tails at room temperature thedefining area/molecule is in the close vicinity of 79 Å2. In the granular phase the lines of the surfacemicelles are intact, but distorted probably into a morehexagonal shape to cover the surface. For a sugar-basedsurfactant carrying dodecyl hydrocarbon tail the definingarea/molecule in the granular range was found to be close to 70Å2. For one of the decyl surfactants 70 Å2was also observed, indicating the existence of afew seemingly discrete adsorption modes for surfactants in thegranular range. The transition from the granular range to thetrue liquid-expanded phase, defined by a complete coverage ofthe entire surface of fluid hydrocarbon tails, is seemingly afirst-order transition. This is indicated in the surfacetension isotherm by a "knee" with an abrupt change from aconstant surface density (granular phase) to a lower molculararea and an increasing adsorption with increasing surfactantchemical potential (true liquid-expanded phase). In theliquid-expanded phase the role of the hydrocarbon chainrestriction in the surface was investigated by applying thehard-disc model on mixtures of sugar-based decyl surfactants toaccount for the head-group surface fractions obtained byapplying Gibbs surface tension equation to mixtures ofsugar-based surfactants. The behaviour of the head-groups atthe interface was accurately described by a hard-disc model,and hence this resulted in a prediction of the decylhydrocarbon chain configurational pressure as a function ofmolecular area. Applying the deduced configurational pressureon pure surfactants yielded accurate models on the surfacepressure and excess free energy contribution from thehead-groups. Futhermore, the limits of the predictioncapability of the model separation of head-group andhydrocarbon tail phase was tested by means of systemsexhibiting partial interpenetration of the head-group into thehydrocarbon tail phase. For ethylene-oxide-based surfactants aweak interpenetration effect was observed. However, for thethiomaltoside head-group a large effect due to interpenetrationwas encountered. This was deemed to be related to thereorientation of the head-group due to the preferential mixingof the sulphur with the hydrocarbon tail phase. Frommeasurements of surface tension after the cmc the averagemicelle size of n-dodecyl-ß-D-maltopyranoside wasdetermined to be 76 monomers. Surface forces measurements usingcrystalline hydrophobic surfaces in sugar-based surfactantsolution proved that surfactants desorb at low applied loads.Surface forces measurements on adsorbed layers of cationicpolymer containing ß-cyclodextrin (ß-CD-EPN) adsorbedonto mica in solutions of admantanegrafted polyethylene-oxidepolymers (Ad-PEO) were performed to investigate the interactionbetween ß-cyclodextrin and admantane. In the absence ofAd-PEO very low adhesion was observed between the ß-CD-EPNcoated mica surfaces, on addition of Ad-PEO a significantincrease was detected in both layer thickness and adhesion.This was attributed to the interaction between admantane andß-cyclodextrin.

A maximal operator over the bases $\mathcal{B}$ is defined as \[Mf(x) = \sup_{x \in B \in \mathcal{B}} \frac{1}{|B|}\int_B |f(y)|dy. \] The boundedness of this operator can be used in a number of applications including the Lebesgue differentiation theorem. If the bases are balls or rectangles parallel to the coordinate axes, the associated maximal operator is bounded from $L^p$ to $L^p$ for all $p>1$. On the other hand, Besicovitch showed that it is not bounded if the bases consists of arbitrary rectangles. In $\mathbb{R}^2$ we associate a subset $\Omega$ of the unit circle to the bases of rectangles in direction $\theta \in \Omega$. We examine the boundedness of the associated maximal operator $M_{\Omega}$ when $\Omega$ is lacunary, a finite sum of lacunary sets, or finite sets using the Fourier transform and geometric methods. The results are due to Nagel, Stein, Wainger, Alfonseca, Soria, Vargas, Karagulyan and Lacey.

In this thesis we calculated the coefficients of moment polynomials of the Riemann zeta function for k= 4, 5, 6...13 using cubic acceleration, which is an improved method from quadratic acceleration. We then numerically verified the moment conjectures. The results we obtained appear to support the conjectures. We also present a brief history of the moment polynomials by illustrating some of the important results of the field along with proofs for two of the classic results. The heuristics to find the integral moments of the Riemann zeta function is described in this thesis as well.

We present a largely self-contained exposition of Ehud Hrushovski's proof of the function field Mordell-Lang conjecture beginning from the Zilber Dichotomy for differentially closed fields and separably closed fields. Our account is based on notes from a series of lectures given by Rahim Moosa at a MODNET workshop at Humboldt Universitat in Berlin in September 2007. We treat the characteristic 0 and characteristic p cases uniformly as far as is possible, then specialize to characteristic p in the final stages of the proof. We also take this opportunity to work out the extension of Hrushovski's ``Socle Theorem'' from the finite Morley rank setting to the finite U-rank setting, as is in fact required for Hrushovski's proof of Mordell-Lang to go through in positive characteristic.

A maximal operator over the bases $\mathcal{B}$ is defined as \[Mf(x) = \sup_{x \in B \in \mathcal{B}} \frac{1}{|B|}\int_B |f(y)|dy. \] The boundedness of this operator can be used in a number of applications including the Lebesgue differentiation theorem. If the bases are balls or rectangles parallel to the coordinate axes, the associated maximal operator is bounded from $L^p$ to $L^p$ for all $p>1$. On the other hand, Besicovitch showed that it is not bounded if the bases consists of arbitrary rectangles. In $\mathbb{R}^2$ we associate a subset $\Omega$ of the unit circle to the bases of rectangles in direction $\theta \in \Omega$. We examine the boundedness of the associated maximal operator $M_{\Omega}$ when $\Omega$ is lacunary, a finite sum of lacunary sets, or finite sets using the Fourier transform and geometric methods. The results are due to Nagel, Stein, Wainger, Alfonseca, Soria, Vargas, Karagulyan and Lacey.

In this thesis we calculated the coefficients of moment polynomials of the Riemann zeta function for k= 4, 5, 6...13 using cubic acceleration, which is an improved method from quadratic acceleration. We then numerically verified the moment conjectures. The results we obtained appear to support the conjectures. We also present a brief history of the moment polynomials by illustrating some of the important results of the field along with proofs for two of the classic results. The heuristics to find the integral moments of the Riemann zeta function is described in this thesis as well.

We present a largely self-contained exposition of Ehud Hrushovski's proof of the function field Mordell-Lang conjecture beginning from the Zilber Dichotomy for differentially closed fields and separably closed fields. Our account is based on notes from a series of lectures given by Rahim Moosa at a MODNET workshop at Humboldt Universitat in Berlin in September 2007. We treat the characteristic 0 and characteristic p cases uniformly as far as is possible, then specialize to characteristic p in the final stages of the proof. We also take this opportunity to work out the extension of Hrushovski's ``Socle Theorem'' from the finite Morley rank setting to the finite U-rank setting, as is in fact required for Hrushovski's proof of Mordell-Lang to go through in positive characteristic.

<p>The aim of this thesis is primarily to investigate theadsorption of surfactants, in particular to the airwaterinterface. The effects of the head-group and the hydrocarbonchain are of special interest. Using thermodynamical models wewere able to predict that the transition from the Henry rangeto the liquid-expanded phase is controlled by the formation ofsmall clusters of surfactants oriented in the plane of thesurface. Hence, the transition from the Henry range can not beof first-order, and additionally the phase formed after theonset of the micelle formation does have a liquid-likecharacter. We refer to this phase as the dilute surfacemicellar range. Interactions between the micelles in the dilutesurface micellar range can be accounted for by excluded areainteractions using the harddisc model. The stability of themicelles can be attributed to the line tension acting on themicelle edge, which stops unlimited growth of the surfacemicelles. With increasing density of surface micelles a newphase is formed, referred to as the granular phase. Thegranular phase exhibits very low compressibilty and an almostconstant area/molecule in an extended surfactant bulkconcentration interval. For three sugar-based surfactantscarrying decyl hydrocarbon tails at room temperature thedefining area/molecule is in the close vicinity of 79 Ų. In the granular phase the lines of the surfacemicelles are intact, but distorted probably into a morehexagonal shape to cover the surface. For a sugar-basedsurfactant carrying dodecyl hydrocarbon tail the definingarea/molecule in the granular range was found to be close to 70Ų. For one of the decyl surfactants 70 Ųwas also observed, indicating the existence of afew seemingly discrete adsorption modes for surfactants in thegranular range. The transition from the granular range to thetrue liquid-expanded phase, defined by a complete coverage ofthe entire surface of fluid hydrocarbon tails, is seemingly afirst-order transition. This is indicated in the surfacetension isotherm by a "knee" with an abrupt change from aconstant surface density (granular phase) to a lower molculararea and an increasing adsorption with increasing surfactantchemical potential (true liquid-expanded phase). In theliquid-expanded phase the role of the hydrocarbon chainrestriction in the surface was investigated by applying thehard-disc model on mixtures of sugar-based decyl surfactants toaccount for the head-group surface fractions obtained byapplying Gibbs surface tension equation to mixtures ofsugar-based surfactants. The behaviour of the head-groups atthe interface was accurately described by a hard-disc model,and hence this resulted in a prediction of the decylhydrocarbon chain configurational pressure as a function ofmolecular area. Applying the deduced configurational pressureon pure surfactants yielded accurate models on the surfacepressure and excess free energy contribution from thehead-groups. Futhermore, the limits of the predictioncapability of the model separation of head-group andhydrocarbon tail phase was tested by means of systemsexhibiting partial interpenetration of the head-group into thehydrocarbon tail phase. For ethylene-oxide-based surfactants aweak interpenetration effect was observed. However, for thethiomaltoside head-group a large effect due to interpenetrationwas encountered. This was deemed to be related to thereorientation of the head-group due to the preferential mixingof the sulphur with the hydrocarbon tail phase. Frommeasurements of surface tension after the cmc the averagemicelle size of n-dodecyl-ß-D-maltopyranoside wasdetermined to be 76 monomers. Surface forces measurements usingcrystalline hydrophobic surfaces in sugar-based surfactantsolution proved that surfactants desorb at low applied loads.Surface forces measurements on adsorbed layers of cationicpolymer containing ß-cyclodextrin (ß-CD-EPN) adsorbedonto mica in solutions of admantanegrafted polyethylene-oxidepolymers (Ad-PEO) were performed to investigate the interactionbetween ß-cyclodextrin and admantane. In the absence ofAd-PEO very low adhesion was observed between the ß-CD-EPNcoated mica surfaces, on addition of Ad-PEO a significantincrease was detected in both layer thickness and adhesion.This was attributed to the interaction between admantane andß-cyclodextrin.</p>

We answer to a Wintner's question concerning the sequence of integers composed of primes from a given set. The results generalize and develop the answer to Wintner’s question due to Tijdeman.

We answer to a Wintner's question concerning the sequence of integers composed of primes from a given set. The results generalize and develop the answer to Wintner’s question due to Tijdeman.