Physical Sciences and Mathematics Commons^™

Recursive Methods In Number Theory, Combinatorial Graph Theory, And Probability, Jonathan Burns

USF Tampa Graduate Theses and Dissertations

Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms.

In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the …