TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementa...

Given a dominant rational self-map on projective variety over number field, we can define the arithmetic degree at point. It is known that any point less than or equal to first dynamical degree. In this paper, show there are densely many Q‾-rational points with maximal (i.e., whose degree) for self-morphisms varieties. For unirational varieties and Abelian varieties, sufficiently large field. W...

We show that for any computably enumerable (c.e.) degree a and any low n–c.e. degree l (n ≥ 1), if l < a, then there are n–c.e. degrees a0,a1 such that l < a0,a1 < a and a0 ∨ a1 = a. In particular, there is no low maximal d.c.e. degree.

In 1963, Moon and Moser gave a bipartite analogue to Ore’s famed theorem on hamiltonian graphs. While the sharpness examples of Ore’s Theorem have been independently characterized in at least four different papers, no similar characterization exists for the Moon-Moser Theorem. In this note, we give such a characterization, consisting of one infinite family and two exceptional graphs of order ei...

The projective degrees of strict partitions of n were computed for all n ≤ 100 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n “lie close to each other” in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n ≤ 220. The results ...