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On The Group Of Transvections Of Ade-Diagrams, Marvin Jones

Theses and Dissertations

In this thesis we examine symplectic spaces with forms generated by the ADEdiagrams. Specifically, we determine the generators of the group of transvections for each space under the standard basis, S, of Kⁿ (where K is a field with characteristic 0) and the hyperbolic basis, H, we get from the classification theorem of symplectic spaces. Further, we examine how the generators of these groups are related via g : G_f,S ! SL(Z)_n where g(X) = P⁻¹XP where P is the change of basis matrix for S to H.

Selected Research In Covering Systems Of The Integers And The Factorization Of Polynomials, Joshua Harrington

Theses and Dissertations

In 1960, Sierpi\'{n}ski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. Such integers are known as Sierpi\'{n}ski numbers. Letting $f(x)=ax^r+bx+c\in\mathbb{Z}[x]$, Chapter 2 of this document explores the existence of integers $k$ such that $f(k)2^n+d$ is composite for all positive integers $n$. Chapter 3 then looks into a polynomial variation of a similar question. In particular, Chapter~\ref{CH:FH} addresses the question, for what integers $d$ does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ with $f(1)\neq -d$ such that $f(x)x^n+d$ is reducible for all positive integers $n$. The last two chapters of …

Sharp Bounds Associated With An Irreducibility Theorem For Polynomials Having Non-Negative Coefficients, Morgan Cole

Theses and Dissertations

Consider a polynomial f(x) having non-negative integer coefficients with f(b) prime for some integer b greater than or equal to 2. We will investigate the size of the coefficients of the polynomial and establish a largest such bound on the coefficients that would imply that f(x) is irreducible. A result of Filaseta and Gross has established sharp bounds on the coefficients of such a polynomial in the case that b = 10. We will expand these results for b in {8, 9, ..., 20}.

Coloring Pythagorean Triples And A Problem Concerning Cyclotomic Polynomials, Daniel White

Theses and Dissertations

One may easily show that there exist $O( \log n)$-colorings of $\{1,2, \ldots, n\}$ such that no Pythagorean triple with elements $\le n$ is monochromatic. In Chapter~\ref{CH:triples}, we investigate two analogous ideas. First, we find an asymptotic bound for the number of colors required to color $\{1,2,\ldots ,n\}$ so that every Pythagorean triple with elements $\le n$ is $3$-colored. Afterwards, we examine the case where we allow a vanishing proportion of Pythagorean triples with elements $\le n$ to fail to have this property.

Unrelated, in 1908, Schur raised the question of the irreducibility over $\Q$ of polynomials of the form …

3-D Computational Investigation Of Viscoelastic Biofilms Using Gpus, Paisa Seeluangsawat

Theses and Dissertations

A biofilm is a slimy colony of bacteria and the materials they secrete, collectively called “extracellular polymeric substances (EPS)”. The EPS consists mostly of bio-polymers, which cross link into a network that behave viscoelastically under deformation. We propose a single-fluid multi-component phase field model of biofilms that captures this behavior, then use numerical simulations on GPUs to investigate the biofilm’s growth and its hydrodynamics properties.