Mathematical analysis

In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.

It has been widely hypothesized that while doing arithmetic, individuals use two
distinct routes for phonological output. A direct route is used for exact arithmetic which
is language dependent, while an indirect route is used during arithmetic approximation
and thought to be language independent. The arithmetic double route has been
incorporated on the triple- code model that consists of visual arabic code for identifying
strings of digits, magnitude code for knowledge in numeral quantities, and verbal code
for rote arithmetic fact. Our goal is to investigate whether language experience has an
effect on the processing of exact/approximation math using bilingual participants who
have access to two languages, using a theoretical arithmetic processing model, which has
been validated across many studies. We have measured the two groups
(monolinguals/bilinguals) processing speed for completing the two tasks
(Exact/Approximation) in two codes (Arabic digit/Verbal). We hypothesized a faster
reaction time in exact arithmetic task in compared to approximation in accordance with the triple-code model. We alsoexpected a main effect for the task (Exact vs.Approximation) independent of the input code when the stimulus was presented in either Arabic digit and/or verbal codes. Our results show exact arithmetic is faster than
approximation of arithmetic facts in all codes supporting earlier theories. Also, there was
no significant difference in processing speed between monolinguals and bilinguals when
performing the arithmetic task in either Arabic and/or verbal codes. In addition, our
investigation suggests a modification to the triple-code model when interpreting
arithmetic facts in verbal code due to interference of two languages with bilingual
participants. Additions to the model can be suggested when the stimulus is expressed in
verbal code for visual identification, which may cause interference in bilinguals leading
to a first language advantage due to language experience.

We present several results involving three concepts: Prufer domains, the strong 2-generator property, and integer-valued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2-generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2-generator property . We note that, if D has the strong 2-generator property, then D is a Prufer domain. If Q is the field of fractions of D, and E is a finite nonempty subset of D; we define Int(E, D ) = {f(X) ∈ Q[ X] ∣ f(a) ∈ D for every a ∈ E} to be the ring of integer-valued polynomials on D with respect to the subset E. We show that D is a Prufer domain if and only if Int(E, D) is a Prufer domain. Our main theorem is that Int(E, D) has the strong 2-generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).

An integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every finite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed.

Definitions of convergence range from very strong (such as the concept of uniform convergence) to weaker and weaker versions like convergence in measure. We shall give a background in integration, measure theory, and convergence; then we move on to define convergence in measure with speed. We define three versions of convergence in measure with speed with an inner speed factor, an outer speed factor, and one with both speed factors. We then give examples of functions that fall under each of these categories and prove important results relating convergence in measure with speed to existing definitions.