Other Mathematics Commons^™

Development Of Novel Compound Controllers To Reduce Chattering Of Sliding Mode Control, Mehran Rahmani

Theses and Dissertations

The robotics and dynamic systems constantly encountered with disturbances such as micro electro mechanical systems (MEMS) gyroscope under disturbances result in mechanical coupling terms between two axes, friction forces in exoskeleton robot joints, and unmodelled dynamics of robot manipulator. Sliding mode control (SMC) is a robust controller. The main drawback of the sliding mode controller is that it produces high-frequency control signals, which leads to chattering. The research objective is to reduce chattering, improve robustness, and increase trajectory tracking of SMC. In this research, we developed controllers for three different dynamic systems: (i) MEMS, (ii) an Exoskeleton type robot, and …

The Fundamental System Of Units For Cubic Number Fields, Janik Huth

Theses and Dissertations

Let $K$ be a number field of degree $n$. An element $\alpha \in K$ is called integral, if the minimal polynomial of $\alpha$ has integer coefficients. The set of all integral elements of $K$ is denoted by $\mathcal{O}_K$. We will prove several properties of this set, e.g. that $\mathcal{O}_K$ is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group $\mathcal{O}_K^\times$, defined as the set of all invertible elements of $\mathcal{O}_K$. We will prove Dirichlet's Unit Theorem and look at unit groups for …

Compactifications Of Manifolds With Boundary, Shijie Gu

Theses and Dissertations

This dissertation is concerned with compactifications of high-dimensional manifolds.

Siebenmann's iconic 1965 dissertation \cite{Sie65} provided necessary and

sufficient conditions for an open manifold $M^{m}$ ($m\geq6$) to be

compactifiable by addition of a manifold boundary. His theorem extends easily

to cases where $M^{m}$ is noncompact with compact boundary; however when

$\partial M^{m}$ is noncompact, the situation is more complicated. The goal

becomes a \textquotedblleft completion\textquotedblright\ of $M^{m}$, ie, a

compact manifold $\widehat{M}^{m}$ containing a compactum $A\subseteq\partial

M^{m}$ such that $\widehat{M}^{m}\backslash A\approx M^{m}$. Siebenmann did

some initial work on this topic, and O'Brien \cite{O'B83} extended that work

to an important special case. …

Z-Structures And Semidirect Products With An Infinite Cyclic Group, Brian Walter Pietsch

Theses and Dissertations

Z-structures were originally formulated by Bestvina in order to axiomatize the properties that an ideal group boundary should have. In this dissertation, we prove that if a given group admits a Z-structure, then any semidirect product of that group with an infinite cyclic group will also admit a Z-structure. We then show how this can be applied to 3-manifold groups and strongly polycyclic groups.

Splittings Of Relatively Hyperbolic Groups And Classifications Of 1-Dimensional Boundaries, Matthew Haulmark

Theses and Dissertations

In the first part of this dissertation, we show that the existence of non-parabolic local cut point in the relative (or Bowditch) boundary, $\relbndry$, of a relatively hyperbolic group $(\Gamma,\bbp)$ implies that $\Gamma$ splits over a $2$-ended subgroup. As a consequence we classify the homeomorphism type of the Bowditch boundary for the special case when the Bowditch boundary $\relbndry$ is one-dimensional and has no global cut points.

In the second part of this dissertation, We study local cut points in the boundary of CAT(0) groups with isolated flats. In particular the relationship between local cut points in $\bndry X$ and …

On Some One-Complex Dimensional Slices Of The Boundedness Locus Of A Multi-Parameter Rational Family, Matthew Hoeppner

Theses and Dissertations

Complex dynamics involves the study of the behavior of complex-valued functions when they are composed with themselves repeatedly. We observe the orbits of a function by passing starting values through the function iteratively. Of particular interest are the orbits of any critical points of the function, called critical orbits. The behavior of a family of functions can be determined by examining the change in the critical orbit(s) of the functions as the values of the associated parameters vary. These behaviors are often separated into two categories: parameter values where one or more critical orbits remain bounded, and parameter values where …

Cocompact Cubulations Of Mixed 3-Manifolds, Joseph Dixon Tidmore

Theses and Dissertations

In this dissertation, we complete the classification of which compact 3-manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical 3-manifold M is mixed if its JSJ decomposition has at least one JSJ torus and at least one hyperbolic block. We show the fundamental group of M is virtually compact special iff M is chargeless, i.e. each interior Seifert fibered block has a trivial Euler number relative to the fibers of adjacent blocks.

Asymptotic Expansion Of The L^2-Norm Of A Solution Of The Strongly Damped Wave Equation, Joseph Silvio Barrera

Theses and Dissertations

The Fourier transform, F, on R^N (N≥1) transforms the Cauchy problem for the strongly damped wave equation u_tt(t,x) - Δu_t(t,x) - Δu(t,x) = 0 to an ordinary differential equation in time t. We let u(t,x) be the solution of the problem given by the Fourier transform, and v(t,ƺ) be the asymptotic profile of F(u)(t,ƺ) = û(t,ƺ) found by Ikehata in [4].

In this thesis we study the asymptotic expansions of the squared L^2-norms of u(t,x), û(t,ƺ) - v(t,ƺ), and v(t,ƺ) as t → ∞. With suitable initial data u(0,x) and u_t(0,x), we establish the rate of growth or decay of …

The Root Finite Condition On Groups And Its Application To Group Rings, James Gollin

Theses and Dissertations

A group $G$ is said to satisfy the root-finite condition if for every $g \in G$, there are only finitely many $x \in G$ such that there exists a positive integer $n$ such that $x^n = g$. It is shown that groups satisfy the root-finite condition iff they satisfy three subconditions, which are shown to be independent. Free groups are root-finite. Ordered groups are shown to satisfy one of the subconditions for the root-finite condition. Finitely generated abelian groups satisfy the root-finite condition. If, in a torsion-free abelian group $G$, there exists a positive integer $r$ such that the subgroup …

Category O Representations Of The Lie Superalgebra Osp(3,2), America Masaros

Theses and Dissertations

In his seminal 1977 paper [Kac77], V. G. Kac classified the finite dimensional simple Lie superalgebras over algebraically closed fields of characteristic zero. However, over thirty years later, the representation theory of these algebras is still not completely understood, nor is the structure of their enveloping algebras.

In this thesis, we consider a low-dimensional example, osp(3,2). We compute the composition factors and Jantzen filtrations of Verma modules over osp(3,2) in a variety of cases.