Goldman and Turaev constructed a Lie bialgebra structure on the free Z-module generated by free homotopy classes of loops on an oriented surface. Turaev conjectured that the cobracket of A is zero if and only if A is a power of a simple class. Chas...
Si/SiGe quantum dots (QDs) are promising candidates for spin-based quantum bits (qubits) as a result of the reduced spin-orbit coupling as well as the Si isotopes with zero nuclear spin. Meanwhile qubit readout is a challenge related to...
Bayesian nonparametric methods have become increasingly popular in machine learning for their ability to allow the data to determine model complexity. In particular, Bayesian nonparametric versions of common latent variable models can learn as...
Breast cancer is the most common cancer in women. The discovery of breast tumor subtypes and the subsequent development of treatments aimed at them has allowed a reduction in the mortality of breast cancer. However, tumors with similar...
We study the bijective combinatorics of reduced words. These are fundamental objects in the study of Coxeter groups. We restrict our focus to reduced words of permutations and signed permutations. Our results can all be situated within the context...
In the past decade, the use of ordinal patterns in the analysis of time series and dynamical systems has become an important tool. Ordinal patterns (otherwise known as a permutation patterns) are found in time series by taking n data points at...
The ionosphere is the primary source for heavy ions which are ubiquitous in the terrestrial magnetosphere. Low-altitude energization in the auroral ionosphere results in bulk heating and transverse acceleration of ions, which begin to upwell and/or...
Siegel domains. Modular groups. Hecke algebras. Forms
In the 1960s Satake proved the existence of an isomorphism between the local Hecke algebra and the ring of polynomials invariant under the action of the signed permutation group W n (the Weyl group associated to Sp n over a local field)....
Riemannian manifolds. Singularities (Mathematics). Laplacian operator. Spectral theory (Mathematics). Riemann surfaces. Curves on surfaces. Geometry
Historically, inverse spectral theory has been concerned with the relationship between the geometry and the spectrum of compact Riemannian manifolds, where spectrum means the eigenvalue spectrum of the Laplace operator as it acts on smooth...
The demand for novel molecularly targeted drugs will continue to rise as we make progress toward personalizing cancer treatments to the molecular signatures of individual tumors. While the collection and analysis of genomic data has become routine,...
Radio relics are extended regions of synchrotron radio emission that have been found in the outskirts of a few dozen galaxy clusters. Relics are often associated with clusters undergoing merger activity. They are not associated optically with a...
Thermodynamics, and the classic balance between entropy and enthalpy, provides a proverbial zoo of exotic phase behaviour that chemists can harness to create new materials out of simple liquids and polymers. The diversity of self-assembling...
We have a limited understanding of how an opinion is originated, how an opinion and information supporting and explaining it gets conveyed, and how the communicated opinion is perceived and processed by others. One direction of current research...
After decades of searching we have yet to find the progenitor systems for type Ia supernovae. In fact most of what we know about this homogeneous class of supernovae is from spectral features associated with the incinerated remains of the C+O white...
Epilepsy is associated with cognitive impairments which often manifest as a higher prevalence of memory impairments. Memory impairments in patients with epilepsy may persist even with sufficient control of seizures, suggesting other factors may...
Although genome-wide association studies (GWAS) and other high-throughput initiatives have led to an information explosion in human genetics and genetic epidemiology, the mapping from genotype to phenotype remains challenging as most of the...
This thesis constitutes the first steps in the author's program to investigate the question of when a homotopy of 2-cocycles Ï‰ = {Ï‰[subscript t]}[subscript tâˆŠ[0,1]] on a locally compact Hausdorff groupoid Æ“ induces an isomorphism of the...
The focus of this thesis is the study of nuclearity and exactness for groupoid crossed product C*-algebras. In particular, we present generalizations of two well-known facts from group dynamical systems and crossed products to the groupoid setting....
Gene set testing has become a critical tool for interpreting the results of high-throughput genomic experiments. Despite the development of robust statistical methods and extensive gene set collections, however, the results from gene set testing...
In the past few years there has been a tremendous growth in the usage of digital images. Users can now access millions of photos, a fact that poses the need of having methods that can efficiently and effectively search the visual information of...
In this thesis, we consider several problems relating to cyclic subgroups of the group [mathematical equation]. Each element of [mathematical equation] has a unique representative in one of the two intervals [mathematical equation] and...
In this thesis, we characterize and enumerate the permutations which are realized by the orbits of signed shifts, a family of discrete dynamical systems on words. The permutations, which are called patterns of the signed shifts, are given by the...
In Chapter 2 we look at the distribution of permutation statistics in the context of pattern-avoiding permutations. The first part of this chapter deals with a recursively defined bijection of Robertson [37] between 123- and 132-avoiding...
This thesis comprises two separate but related studies, dealing with two strongly interacting nanoscale systems on the border between the quantum and classical domains. In Part 1, we use a Born-Markov approximated master equation approach to study...
Robustness to genetic perturbations is a fundamental property of all living things. The genetic code is degenerate, RNA secondary structure is robust to sequence changes, and protein structure is robust to amino acid substitutions. Complex systems...
The Euler '-function and Carmichael -function are extremely important in modern number theory, and much work has been devoted to studying the distribution and arithmetic properties of the values of each function. One interesting unresolved question...
Medical imaging methods have become increasingly important in diagnosing diseases and assisting therapeutic treatment. In particular, early detection of breast cancer is considered as a critical factor in reducing the mortality rate of women....
Earlier work has suggested that various neurological or neuropsychiatric disorders may result in characteristic spatial patterns in brain activation, potentially allowing their detection from maps of brain activity under different conditions....
Coherent states. Quantum field theory. Nonlinear theories.
In this thesis we study the properties of time-dependent, nontopological configurations and their effect on the macroscopic properties of a system described by a nonlinear field theory. These structures seem to be ubiquitous in relativistic field...
Magnetic nanoparticles are promising candidates for use in biomedical applications as remote sensors of biophysical properties, thermal therapy agents, and detectors of specific biomolecules. Many approaches have been used to model magnetic...
Boundary integral methods have long been used to solve boundary value problems for elliptic partial differential equations with piecewise constant coefficients, since they have several numerical advantages over conventional volume discretization....
This thesis centers around a generalization of the classical discrete Fourier transform. We first present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite...
In this thesis we study 1=k-geodesics, those closed geodesics that minimize on any subinterval of length L=k, where L is the length of the geodesic. These curves arise as critical points of the uniform energy, a function introduced in Morse theory...
The discovery of the Van Allen radiation belts in the 1958 was the first major discovery of the Space Age. There are two belts of energetic particles. The inner belt is very stable, but the outer belt is extremely variable, especially during...
The accelerated expansion of space in the Universe has been known for two decades. However, finding a plausible theory that describes this expansion is more difficult: Taking general relativity for granted requires us to stipulate the existence of...
For ordinary knots in R 3 , there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual...
Identifying the determinants of reproductive success in small-scale societies is critical for understanding how natural selection has shaped human evolution and behavior. The available evidence suggests that status-accruing behaviors such as...
We provide a theoretical model for a design involving a dc voltage biased Josephson junction (JJ) that strongly drives a high quality factor microwave cavity via the ac Josephson effect. We explore the rich classical dynamics of the resultant...
Motivated by quantum mechanics and geometric optics, it is a long-standing problem whether the length spectrum of a compact Riemannian manifold can be recovered from its Laplace spectrum. One route to proving that the length spectrum depends on the...
The performance of machine learning algorithms largely depends on data representation. As a critical step in machine learning, representation learning (feature learning) learns a transformation of training data to give a new representation that can...
'Orthogonal modular forms are algebraic modular forms arising from lattices in quadratic spaces. In this thesis, we define orthogonal modular forms, establish their basic properties, and then apply them to a case of ternary quadratic spaces to...
'The control of waves using periodic structures is crucial for modern optical, electromagnetic and acoustic devices such as diffraction gratings, filters, photonic crystals, solar cells, sensors, and absorbers. We present a high-order accurate...