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.

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.

This thesis focusses on the notion of analysis (&vàXuatç) in Stoic logic, that is to say on the procedure which the Stoic logicians followed in order to reduce all valid arguments to five basic patterns. By reconsidering the uses of its Aristotelian homonym and by examining the evidence on the classification of Stoic arguments, I distinguish two methods of Stoic analysis and I discuss their rules: (i) the analysis of non-simple indemonstrables, which constitutes a process of breaking up an argument by means of general logical principles (8 prlpcxta) ; and (ii) the analysis of <syllogistic> arguments, which replaces demonstration (&toösttç) and is effected by employing standard well-determined rules (Oata). The ancient sources provide us with concrete examples illustrating the first type of analysis; however, there is no single text that reports the exact procedure of analysing <syllogistic> arguments. Modern scholars have reconstructed in different ways this type of Stoic analysis; I deal with all of them separately and show that the proposed reconstructions are insightful but historically implausible. Based on the textual materiel concerning the notion of analysis not only in its Stoic context but also in some other of its uses, and especially in mathematical practice, I suggest an alternative reconstruction of the Stoic method of reducing valid arguments to the basic indemonstrables

This thesis consists of two parts. The first part deals with the orthogonal projections of pairs of smooth surfaces and of triples of smooth surfaces onto planes. We take as a model of pairwise smooth surfaces the variety X= {(x, 0, z): x> 0} U {(0, y, z): y> 0} and classify germs of maps R3,3,0 -º R2,2,0 up to origin preserving diffeomorphisms in the source which preserve the variety X and any origin preserving diffeomorphisms in the target. This yields an action of a subgroup xA of the Mather group A on C3 2, the set of map-germs R3,0 -º R2,0. We list the orbits of low codimensions of such an action, and give a detailed description of the geometry of each orbit. We extend these results to triples of surfaces. In the second part of the thesis we analyse the shape of smooth embedded closed curves in the plane. A way of picking out the local reflexional symmetry of a given curve -y is to consider the centres of bitangent circles to the curve. ° The closure of the locus of these centres is called the Symmetry Set of y. We present an equivalent way of tracing the local reflexional symmetry of -r by considering the lines with respect to which a point on y and its tangent line are reflected to another point on the curve and to its tangent line. The locus of all these lines form the dual curve of the symmetry set of -y. We study the singularities occurring on duals of symmetry sets and their generic transitions in 1-parameter families of curves 7. A first attempt to define an analogous theory to study the local rotational symmetry in the plane is given. The Rotational Symmetry Set of a curve y is defined to be the locus of centres of rotations taking a point -y(ti) together with its tangent line and its centre of curvature, to y(t2) together with its tangent line and its centre of curvature. We study the properties of the rotational symmetry set and list the generic transitions of its singularities in 1-parameter families of curves ry. In the final chapter we investigate the local structure of the midpoint locus of generic smooth surfaces