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Thesis (M.S.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / Considerable study h8s been devoted to the multiplicity function of a real variable. For any real valued function of a real variable, f(x), define its multiplicity function, N(Y), as the cardinal number of roots, either finite or infinite of y = f(x), i.e., when the cardinal number of roots is transfinite, assign the value infinite where it is understood that the range of the multiplicity function is the extended real number system. Banach^1 was the first to relate this function to a continuous function of bounded variation. He realized that the integral of this function over the Entire real line was precisely the total variation. He demonstrated analogous theorems for curves and surfaces. His approach is to express the curve on the surface parametrically. In the case of a simple arc in the plane, the original theorem can be applied to the parametric equations. This results in an expression for rectifiable arcs. To determine a surface of finite area requires a more careful study. Here, too, the method of parametric equations simplifies the problem. The proof consists chiefly in defining and in organizing the expressions of the surface in a manner that will allow the multiplicity theorem to be applied [TRUNCATED]. / 2999-01-01

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.