Discrete Local Central Limit Theorems And Boolean Function Complexity Measures
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Discrete Local Central Limit Theorems and Boolean Function Complexity Measures
This thesis consists of 6 chapters (the first being an introduction). Two chapters relate to local central limit theorems, and three chapters relate to various boolean function complexity measures. Although the problems studied in this work originate from different areas of mathematics, the methods used to attack these problems are unified in their probabilistic and combinatorial nature. In Chapter 2 we prove a local central limit theorem for the number of triangles in the Erdos-Renyi random graph G(n, p) for constant edge probability p. In Chapter 6 we apply an existing local limit theorem for sums of independent random variables to estimate the density of a certain set of integers called happy numbers. In Chapters 3, 4, and 5 we will investigate the general question of how large one complexity measure of boolean functions can be relative to another. In one case we present a probabilistic construction of family of boolean functions which show tight (in the sense that there is a matching upper bound) separation between two measures, namely block sensitivity and certificate complexity. We also give partial results for upper bounding one measure in terms of another. This includes a new approach to the well known sensitivity conjecture which asserts that the degree of any boolean function is bounded above by some fixed power of its sensitivity.
The Probabilistic Method
One of the most powerful and popular tools used in combinatorics is the probabilistic method. Describes current algorithmic techniques, applying both the classical method and the modern tools it uses. Along with a detailed description of the techniques used in probabilistic arguments, it includes basic methods which utilize expectation and variance plus recent applications of martingales and correlation inequalities. Examines discrepancy and random graphs and covers such topics as theoretical computer science, computational geometry, derandomization of randomized algorithms and more. A study of various topics using successful probabilistic techniques is included along with an Open Problems Appendix by Paul Erdös, the founder of the probabilistic method.
Discrete Algorithmic Mathematics, Third Edition
Thoroughly revised for a one-semester course, this well-known and highly regarded book is an outstanding text for undergraduate discrete mathematics. It has been updated with new or extended discussions of order notation, generating functions, chaos, aspects of statistics, and computational biology. Written in a lively, clear style that talks to the reader, the book is unique for its emphasis on algorithmics and the inductive and recursive paradigms as central mathematical themes. It includes a broad variety of applications, not just to mathematics and computer science, but to natural and social science as well. A manual of selected solutions is available for sale to students; see sidebar. A complete solution manual is available free to instructors who have adopted the book as a required text.