DRUM Collection: Mathematics Theses and Dissertations
http://hdl.handle.net/1903/2793
2014-10-01T14:16:04ZLift of the trivial representation to a nonlinear cover
http://hdl.handle.net/1903/15375
Title: Lift of the trivial representation to a nonlinear cover
Authors: Tsai, Wan-Yu
Abstract: Let G be the real points of a simply laced, simply connected complex Lie
group, and let G^~ be the nonlinear two-fold cover of G. We discuss a set of small genuine representations of G^~, denoted by Lift(C), which can be obtained from the trivial representation of G by a lifting operator. The representations in Lift(C) can be characterized by the following properties: (a) the inﬁnitesimal character is &rho/2; (b) they have maximal &tau-invariant; (c) they have a particular associated variety O.
When G is split and of type A or D , we have a full description for Lift(C). In
this case, these representations are parametrized by pairs (central character, real form of O), and exhaust all small representations with inﬁnitesimal character &rho/2 and maximal &tau-invariant.2014-01-01T00:00:00ZMinimal Models of Human-Nature Interaction
http://hdl.handle.net/1903/15363
Title: Minimal Models of Human-Nature Interaction
Authors: Motesharrei, Safa
Abstract: Over the last two centuries, the Human System went from having a small impact on the Earth System to becoming dominant, because both population and per capita consumption have grown extremely fast, especially since about 1950. We therefore argue that Human System Models must be included into Earth System Models through bidirectional couplings with feedbacks. In particular, population should be modeled endogenously, rather than exogenously as done currently in most Integrated Assessment Models. The growth of the Human System threatens to overwhelm the Carrying Capacity of the Earth System, and may be leading to collapse. Earth Sciences should be involved in the exploration of potential mitigation strategies including education, regulatory policies, and technological advances.
We describe a human population dynamics model developed by adding accumulated wealth and economic inequality to a predator-prey model of humans and nature. The model structure, and simulated scenarios that offer significant implications, are discussed. Four equations describe the evolution of Elites, Commoners, Nature, and Wealth. The model shows Economic Stratification or Ecological Strain can independently lead to collapse, in agreement with the historical record.
The measure ``Carrying Capacity'' is developed and its estimation is shown to be a practical means for early detection of a collapse. Mechanisms leading to two types of collapses are discussed. The new dynamics of this model can also reproduce the irreversible collapses found in history. Collapse can be avoided, and population can reach a steady state at maximum carrying capacity, if the rate of depletion of nature is reduced to a sustainable level, and if resources are distributed equitably.
Finally we present a Coupled Human-Climate-Water Model (COWA). Policies are introduced as drivers of the model so that the long-term effect of each policy on the system can be seen as we change its level. We have done a case study for the Phoenix AMA Watershed. We show that it is possible to guarantee the freshwater supply and sustain the freshwater sources through a proper set of policy choices.2014-01-01T00:00:00ZSpherical two-distance sets and related topics in harmonic analysis
http://hdl.handle.net/1903/15337
Title: Spherical two-distance sets and related topics in harmonic analysis
Authors: Yu, Wei-Hsuan
Abstract: This dissertation is devoted to the study of applications of
harmonic analysis. The maximum size of spherical few-distance sets
had been studied by Delsarte at al. in the 1970s. In particular,
the maximum size of spherical two-distance sets in $\mathbb{R}^n$
had been known for $n \leq 39$ except $n=23$ by linear programming
methods in 2008. Our contribution is to extend the known results
of the maximum size of spherical two-distance sets in
$\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46,
78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had
been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and
$20$ since 1973. We use the semidefinite programming method to
find the maximum size for equiangular line sets in $\mathbb{R}^n$
when $24 \leq n \leq 41$ and $n=43$.
We suggest a method of constructing spherical two-distance sets
that also form tight frames. We derive new structural properties
of the Gram matrix of a two-distance set that also forms a tight
frame for $\mathbb{R}^n$. One of the main results in this part is
a new correspondence between two-distance tight frames and certain
strongly regular graphs. This allows us to use spectral properties
of strongly regular graphs to construct two-distance tight
frames. Several new examples are obtained using this
characterization.
Bannai, Okuda, and Tagami proved that a tight spherical designs of
harmonic index 4 exists if and only if there exists an equiangular
line set with the angle $\arccos (1/(2k-1))$ in the Euclidean
space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We
show nonexistence of tight spherical designs of harmonic index $4$
on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite
programming method. We also derive new relative bounds for
equiangular line sets. These new relative bounds are usually
tighter than previous relative bounds by Lemmens and Seidel.2014-01-01T00:00:00ZA PDE approach to numerical fractional diffusion
http://hdl.handle.net/1903/15317
Title: A PDE approach to numerical fractional diffusion
Authors: Otarola, Enrique
Abstract: This dissertation presents a decisive advance in the numerical solution and analysis of fractional diffusion, a relatively new but rapidly growing area of research. We exploit the cylindrical extension proposed and investigated by X. Cabre and J. Tan, in turn inspired by L. Caffarelli and L. Silvestre, to replace the intricate integral formulation of the fractional Laplacian, in a bounded domain, by a local elliptic PDE in one higher dimension with variable coefficient. Inspired in the aforementioned localization results, we propose a simple strategy to study discretization and solution techniques for problems involving fractional powers of elliptic operators. We develop a complete and rigorous a priori and interpolation error analyses. We also design and study an efficient solver, and develop a suitable a posteriori error analysis. We conclude showing the flexibility of our approach by analyzing a fractional space-time parabolic equation.2014-01-01T00:00:00Z