In this thesis we investigate the extension of certain theorems
of additive number theory to three algebraic systems. A generalization
of a theorem by Cauchy and Davenport on the cardinality of the
sum of two sets of residue classes is given. We obtain and compare
estimates for the order of...
Allen Freedman defined a density space to be the ordered pair
(S,𝓕) where S is a certain kind of semigroup called an s-set and 𝓕
is a special type of family of finite subsets of S called a fundamental
family on S. Several properties for density spaces are
obtained, and...
This thesis treats the problem of enumerating equivalence
classes of Euler paths of full graphs. A full graph
is a complete, unordered, graph with no loops or repeated
edges. Two Euler paths are equivalent if and only if one
can be transformed into the other by a finite sequence of...
An algorithm is described for determining the vertices
and supporting planes (or lines) of the convex hull
of a given set of N distinct points in 3-space. The
method involves determining a finite sequence of convex
hulls, each successive convex hull being a better approximation
to the convex hull of...
This thesis contains a collection of properties of the greatest integer part function which were obtained by an extensive literature search. A few original properties are stated and proved and some of the properties which were found unproved in the literature are proved.
Let A and B be two subsets of the set of all non-negative
integers with 0 ε A and O ε B. The sum of the sets A and B is
the set C = A + B = {a + b: a ε A, b ε B). For n...
This thesis brings together under one cover a survey of the
history of the real number pi along with a study of the present state
of its theory, application, and calculation.
In 1932 A. Ya. Khinchin gave the first partial solution of the celebrated 1931 αβ Conjecture of L.G. Schnirelmann and E. Landau of the density of sums of sets on integers, which was completely proved in 1942 by H.B. Mann.
Khinchin's theorem is proved along with theorems of P. Scherk...
This thesis contains a collection of summation formulas for
the greatest integer part function. Proofs are supplied for original
results and for those formulas which are stated without proof in the
literature. References are given for formulas and proofs which
appear in the literature.
In this thesis, we study conditions not involving density which
guarantee that a given positive integer is contained in a sum of sets
of nonnegative integers. We survey the literature, give more detailed
proofs of some known theorems, develop some new theorems,
and make some conjectures.