"Of considerable interest among mathematicians is the problem of the determination of primality of positive integers. For a small integer, N, we may say that N is prime or composite merely by trying to divide N by all primes less than or equal to the square root of N since if N is composite, one of its factors must be [less than or equal to] the square root of N. However, if N is large this test loses its practicality and we must resort to a more feasible method. It is the purpose of this paper to trace and show the development of such methods"--Introduction. / "June, 1959." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaf 28).

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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esse trabalho à baseado no artigo Analyticity Of Almost Everywhere Differentiable Functions, nele desenvolveremos um lema de partiÃÃo para funÃÃes superaditivas que permitirà uma demonstraÃÃo alternativa e simples dos teoremas de Besicovitch. / This work is based on the article Analyticity Of Almost Everywhere Differentiable Functions, it will develop a partitioning lemma for superadditive set functions which will lead to a simple alternative proof of Besicovitchâs theorems .

Tait showed in 1878 that the Four Color Theorem is equivalent to being able to three-color the edges of any planar, three-regular, two-edge connected graph. Not surprisingly, this equivalent problem proved to be equally difficult. We consider the problem of fractional colorings, which resemble ordinary colorings but allow for some degree of cheating. Happily, it is known that every planar three-regular, two-edge connected graph is fractionally three-edge colorable. Is there an analogue to Tait’s Theorem which would allow us to derive the Fractional Four Color Theorem from this edge-coloring result?

From fundamental forms to curvatures and geodesics, differential geometry has many special theorems and applications worth examining. Among these, the Gauss-Bonnet Theorem is one of the well-known theorems in classical differential geometry. It links geometrical and topological properties of a surface. The thesis introduced some basic concepts in differential geometry, explained them with examples, analyzed the Gauss-Bonnet Theorem and presented the proof of the theorem in greater detail. The thesis also considered applications of the Gauss-Bonnet theorem to some special surfaces.

This thesis is concerned with two different problems in harmonic analysis: the multilinear Kakeya theorem, and Wolff-type inequalities for paraboloids. Chapter 1 gives an overview of both of these problems. In Chapter 2 we investigate an important special case of the multilinear Kakeya theorem, the so-called “bush example”. While the endpoint case of the multilinear Kakeya theorem was recently proved by Guth, the proof is highly abstract; our aim is to provide a more elementary proof in this special case. This is achieved for a significant part of the three-dimensional case in the main result of the chapter. Chapter 3 is a study of the endpoint case of a mixed-norm Wolff-type inequality for the paraboloid. The main result adapts an example of Bourgain to show that the endpoint inequality cannot hold with an absolute constant; there must be a dependence on the thickening of the paraboloid. The remainder of the chapter is a series of case studies, through which we establish positive endpoint results for certain classes of function, as well as indicating specific examples which need to be better understood in order to obtain the full endpoint result.