Physical Sciences and Mathematics Commons^™

Spectral Properties Of A Non-Compact Operator In Ecology, Matthew Reichenbach

Department of Mathematics: Dissertations, Theses, and Student Research

Ecologists have used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. A priori, it is unclear whether these IPMs have an asymptotic growth rate λ, or a stable-stage distribution ψ. In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in …

Trisections Of Flat Surface Bundles Over Surfaces, Marla Williams

Department of Mathematics: Dissertations, Theses, and Student Research

A trisection of a smooth 4-manifold is a decomposition into three simple pieces with nice intersection properties. Work by Gay and Kirby shows that every smooth, connected, orientable 4-manifold can be trisected. Natural problems in trisection theory are to exhibit trisections of certain classes of 4-manifolds and to determine the minimal trisection genus of a particular 4-manifold.

Let $\Sigma_g$ denote the closed, connected, orientable surface of genus $g$. In this thesis, we show that the direct product $\Sigma_g\times\Sigma_h$ has a $((2g+1)(2h+1)+1;2g+2h)$-trisection, and that these parameters are minimal. We provide a description of the trisection, and an algorithm to generate a …

Optimal Allocation Of Two Resources In Annual Plants, David Mcmorris

Department of Mathematics: Dissertations, Theses, and Student Research

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season. Under the assumption that annual plants grow to maximize fitness, we can use techniques from optimal control theory to understand this process. We introduce two models for resource allocation in annual plants which extend classical work by Iwasa and Roughgarden to a case where both carbohydrates and mineral nutrients are allocated to shoots, roots, and fruits in annual plants. In each case, we use optimal control theory to determine the optimal resource allocation strategy for the plant …

Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ …

Exploring Pedagogical Empathy Of Mathematics Graduate Student Instructors, Karina Uhing

Department of Mathematics: Dissertations, Theses, and Student Research

Interpersonal relationships are central to the teaching and learning of mathematics. One way that teachers relate to their students is by empathizing with them. In this study, I examined the phenomenon of pedagogical empathy, which is defined as empathy that influences teaching practices. Specifically, I studied how mathematics graduate student instructors conceptualize pedagogical empathy and analyzed how pedagogical empathy might influence their teaching decisions. To address my research questions, I designed a qualitative phenomenological study in which I conducted observations and interviews with 11 mathematics graduate student instructors who were teaching precalculus courses at the University of Nebraska—Lincoln.

In the …