The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

A well-known long standing problem in combinatorics and statistical mechanics is to find the generating function for self-avoiding walks (SAW) on a two-dimensional lattice, enumerated by perimeter. A SAW is a sequence of moves on a square lattice which does not visit the same point more than once. It has been considered by more than one hundred researchers in the pass one hundred years, including George Polya, Tony Guttmann, Laszlo Lovasz, Donald Knuth, Richard Stanley, Doron Zeilberger, Mireille Bousquet-Mlou, Thomas Prellberg, Neal Madras, Gordon Slade, Agnes Dit- tel, E.J. Janse van Rensburg, Harry Kesten, Stuart G. Whittington, Lincoln Chayes, Iwan Jensen, Arthur T. Benjamin, and many others. More than three hundred papers and a few volumes of books were published in this area. A SAW is interesting for simulations because its properties cannot be calculated analytically. Calculating the number of self-avoiding walks is a common computational problem. A recently proposed model called prudent self-avoiding walks (PSAW) was first introduced to the mathematics community in an unpublished manuscript of Pra, who called them exterior walks. A prudent walk is a connected path on square lattice such that, at each step, the extension of that step along its current trajectory will never intersect any previously occupied vertex. A lattice path composed of connected horizontal and vertical line segments, each passing between adjacent lattice points. We will discuss some enumerative problems in self-avoiding walks, lattice paths and walks with several step vectors. Many open problems are posted. / by Shanzhen Gao. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

In 1943 R. C. Buck showed that a sequence x is convergent if some regular matrix sums every subsequence of x. Thus, for example, if every subsequence of x is Cesaro summable, then x is actually convergent. Buck's result was quite surprising, since research in summability theory up to that time gave no hint of such a remarkable theorem. The appearance of Buck's result in the Bulletin of the American Mathematical Society (3) created immediate interest and has prompted considerable research which has taken the following directions: (i) to study regular matrix transformations in order to shed light on Buck's theorem, (ii) to extend Buck's theorem, (iii) to obtain analogs of Buck's theorem for sequence spaces other than the space of convergent sequences, and (iv) to obtain analogs of Buck's theorem involving processes other than subsequencing, such as stretching. The purpose of the present paper is to contribute to all facets of the problem, particularly to (i), (iii), and (iv).

Tsoi, Hung Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 88-90). / Abstracts in English and Chinese. / Preface --- p.7 / Chapter 1 --- Kahler Manifold --- p.10 / Chapter 1.1 --- Hermitian Manifold --- p.12 / Chapter 1.2 --- Kahler Manifold --- p.13 / Chapter 1.2.1 --- "Positive (l,l)-form" --- p.15 / Chapter 2 --- Vector Bundle --- p.16 / Chapter 2.1 --- Holomorphic Vector Bundle and Connection --- p.17 / Chapter 2.2 --- Hermitian Connection and Chern Connection --- p.18 / Chapter 2.2.1 --- Existence of Chern connection on a holomorphic vector bundle --- p.19 / Chapter 2.3 --- Curvature --- p.21 / Chapter 2.4 --- Positivity of Vector Bundles --- p.23 / Chapter 2.5 --- Chern Classes and Holomorphic Line Bundle --- p.24 / Chapter 2.5.1 --- Chern class in axiomatic approach --- p.25 / Chapter 2.5.2 --- Chern class in algebraic topology --- p.26 / Chapter 2.5.3 --- Chern class in terms of curvature --- p.27 / Chapter 2.5.4 --- In the case of hermitian line bundle --- p.28 / Chapter 3 --- Analytic Technique on Kahler Manifold --- p.30 / Chapter 3.1 --- Dolbeault Cohomology --- p.30 / Chapter 3.2 --- Commutator Relations on Kahler Manifold --- p.31 / Chapter 3.2.1 --- Commutator relation on a line bundle --- p.32 / Chapter 3.3 --- Hodge Theory --- p.33 / Chapter 3.4 --- Bochner Technique --- p.35 / Chapter 3.4.1 --- Bochner-Kodaira-Nakano identity --- p.36 / Chapter 4 --- Kodaira Vanishing Theorem and L2 estimate of d --- p.38 / Chapter 4.1 --- Kodaira Vanishing Theorem --- p.39 / Chapter 4.2 --- Extension of Kodaira Vanishing Theorem by L2 Method --- p.44 / Chapter 4.2.1 --- Plurisubharmonic functions and weakly pseudoconvex Kahler manifold --- p.47 / Chapter 5 --- Multiplier Ideal Sheaf --- p.55 / Chapter 5.1 --- Algebraic Properties of Multiplier Ideal Sheaf --- p.56 / Chapter 5.2 --- Some Calculations of Multiplier Ideal Sheaf --- p.59 / Chapter 6 --- Nadel Vanishing Theorem --- p.62 / Chapter 6.1 --- Nadel Vanishing Theorem by L2 Estimate of d --- p.62 / Chapter 6.2 --- The Original Setting of Nadel --- p.64 / Chapter 6.2.1 --- S-bounded and S-null sequence --- p.65 / Chapter 6.2.2 --- Multiplier ideal sheaf by Nadel --- p.67 / Chapter 6.3 --- Nadel Vanishing Theorem by Computation of Cech Cohomology --- p.69 / Chapter 6.3.1 --- L2 estimate of d --- p.69 / Chapter 6.3.2 --- Koszul cochain --- p.70 / Chapter 6.3.3 --- The cohomology vanishing theorem --- p.73 / Chapter 7 --- Kawamata-Viehweg Vanishing Theorem --- p.77 / Chapter 7.1 --- Numerically Effective Line Bundle --- p.77 / Chapter 7.2 --- Kawamata-Viehweg Vanishing Theorem --- p.85 / Bibliography --- p.88

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division. To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.

Ce travail met en jeu plusieurs systèmes dynamiques sur des tores en dimension finie, pour lesquels on sait établir des théorèmes limites, qui permettent de préciser leur comportement stochastique. On généralise d'abord le théorème limite local usuel sur un sous-shift de type fini, en ajoutant un terme de perturbation, en reprenant la preuve classique, par des techniques d'opérateurs. On en déduit un théorème limite local pour les sommes de « Riesz-Raïkov unitaires étendues », et des observables höldériennes. Pour cela, on reprend une méthode employée par Bernard Petit, en utilisant des codages symboliques, et le théorème limite local avec perturbation. Puis, on présente plusieurs situations de composées d'automorphismes hyperboliques du tore en dimension deux pour lesquelles on sait établir un théorème limite central quelque soit le choix de la composée. En particulier, on aborde le cas des matrices à coefficients entiers positifs.

M.V. Cerrillo, E.F. Bolinder. / "August 15, 1952." / Bibliography: p. 168. / Army Signal Corps Contract DA36-039 sc-100 Project 8-102B-0 Dept. of the Army Project 3-99-10-022

Manuel V. Cerrillo, Ernest A. Guillemin. / "June 4, 1952." / Bibliography: p. 46. / Army Signal Corps Contract DA36-039 sc-100 Project 8-102B-0 Dept. of the Army Project 3-99-10-022

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