Thesis (M.S.)--Boise State University, 2010. / Title from t.p. of PDF file (viewed July 30, 2010). Includes abstract. Includes bibliographical references (leaf 55).

Thesis (Ph. D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 2, 2007) Vita. Includes bibliographical references.

Representing and handling high-dimensional data has been increasingly ubiquitous in many real world-applications, such as computer vision, machine learning, and data mining. High-dimensional data usually have intrinsic low-dimensional structures, which are suitable for subsequent data processing. As a consequent, it has been a common demand to find low-dimensional data representations in many machine learning and data mining problems. Factorization methods have been impressive in recovering intrinsic low-dimensional structures of the data. When seeking low-dimensional representation of the data, traditional methods mainly face two challenges: 1) how to discover the most variational features/information from the data; 2) how to measure accurate nonlinear relationships of the data. As a solution to these challenges, traditional methods usually make use of a two-step approach by performing feature selection and manifold construction followed by further data processing, which omits the dependence between these learning tasks and produce inaccurate data representation. To resolve these problems, we propose to integrate feature learning and graph learning with factorization model, which allows the goals of learning features, constructing manifold, and seeking new data representation to mutually enhance and lead to powerful data representation capability. Moreover, it has been increasingly common that 2-dimensional (2D) data often have high dimensions of features, where each example of 2D data is a matrix with its elements being features. For such data, traditional data usually convert them to 1-dimensional vectorial data before data processing, which severely damages inherent structures of such data. We propose to directly use 2D data for seeking new representation, which enables the model to preserve inherent 2D structures of the data. We propose to seek projection directions to find the subspaces, in which spatial information is maximumly preserved. Also, manifold and new data representation are learned in these subspaces, such that the manifold are clean and the new representation is discriminative. Consequently, seeking projections, learning manifold and constructing new representation mutually enhance and lead to powerful data representation technique.

No description available.

Thesis advisor: Julia Elisenda Grigsby / In 1999, Khovanov constructed a combinatorial categorification of the Jones polynomial. Since then there has been a question of to what extent the topology of a link is reflected in his homology theory and how Khovanov homology can be used for topological applications. This dissertation compiles some of the authors contributions to these avenues of mathematical inquiry. In the first chapter, we prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen $s$-invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $\psi$ invariant is entirely determined by the $s$-invariant and the self-linking number for 3-braid closures. In the second chapter, we show if $L$ is any link in $S^3$ whose Khovanov homology is isomorphic to the Khovanov homology of $T(2,6)$ then $L$ is isotopic to $T(2,6)$. We show this for unreduced Khovanov homology with $\mathbb{Z}$ coefficients. Finally in the third chapter, we exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis consists of six chapters. The first chapter gives an introduction to the field of low-dimensional magnetic and electronic systems and relevant numerical techniques. The recent developments in molecular magnets are highlighted. The numerical techniques are reviewed along with their advantages and disadvantages from the present perspective. Study of entanglement of a system can give a great insight into the system. At the last part of this chapter a general overview is given regarding entanglement, its measures and its significance in studying many-body systems. Chapter 2 deals with the technique that has been developed by us for the full symmetry adaptation of non-relativistic Hamiltonians. It is advantageous both computationally and physically/chemically to exploit both spin and spatial symmetries of a system. It has been a long-standing problem to target a state which has definite total spin and also belongs to a definite irreducible representation of a point group, particularly for non-Abelian point groups. A very general technique is discussed in this chapter which is a hybrid method based on valence-bond basis and the basis of the z-component of the total spin. This technique is not only applicable to a system with arbitrary site spins and belonging to any point group symmetry, it is also quite easy to implement computationally. To demonstrate the power of the method, it is applied to the molecular magnetic system, Cu6Fe8, with cubic symmetry. In chapter 3, the extension of the previous hybrid technique to electronic systems is discussed. The power of the method is illustrated by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and is in the largest non-Abelian point group. All the eigenstates of the model are obtained using our technique. Chapter 4 deals with the thermodynamic properties of an important class of single-chain magnets (SCMs). This class of SCMs has alternate isotropic spin-1/2 units and anisotropic high spin units with the anisotropy axes being non-collinear. Here anisotropy is assumed to be large and negative, as a result, anisotropic units behave like canted spins at low temperatures; but even then simple Ising-type model does not capture the essential physics of the system due to quantum mechanical nature of the isotropic units. A transfer matrix (TM) method is developed to study statistical behavior of this class of SCMs. For the first time, it is also discussed in detail that how weak inter-chain interactions can be treated by a TM method. The finite size effect is also discussed which becomes important for low temperature dynamics. This technique is applied to a real helical chain magnet, which has been studied experimentally. In the fifth chapter a bipartite entanglement entropy of finite systems is studied using exact diagonalization techniques to examine how the entanglement changes in the presence of long-range interactions. The PariserParrPople model with long-range interactions is used for this purpose and corresponding results are com-pared with those for the Hubbard and Heisenberg models with short-range interactions. This study helps understand why the density matrix renormalization group (DMRG) technique is so successful even in the presence of long-range interactions in the PPP model. It is also investigated if the symmetry properties of a state vector have any significance in relation to its entanglement. Finally, an interesting observation is made on the entanglement profiles of different states, across the full energy spectrum, in comparison with the corresponding profile of the density of states. The entanglement can be localized between two noncomplementary parts of a many-body system by performing local measurements on the rest of the system. This localized entanglement (LE) depends on the chosen basis set of measurement (BSM). In this chapter six, an optimality condition for the LE is derived, which would be helpful in finding optimal values of the LE, besides, can also be of use in studying mixed states of a general bipartite system. A canonical way of localizing entanglement is further discussed, where the BSM is not chosen arbitrarily, rather, is fully determined by the properties of a system. The LE obtained in this way, called the localized entanglement by canonical measurement (LECM), is not only easy to calculate practically, it provides a nice way to define the entanglement length. For spin-1/2 systems, the LECM is shown to be optimal in some important cases. At the end of this chapter, some numerical results are presented for j1 −j2 spin model to demonstrate how the LECM behaves.

A major area of interest in condensed matter physics over the past decades has been the emergence of new states of matter from strongly correlated electron systems. A few limited examples would be the emergence of unconventional superconductivity in the high-T$_c$ superconductors and heavy-fermion systems, the appearance of the skyrmion magnetic vortex state in MnSi and magnetically mediated superconductivity in UGe$_2$. While detailed studies of many of the emergent phases have been made, there are still many gaps in understanding of the underlying states and mechanisms that allow them to form. This work aims to add to knowledge of the basic physics behind such states, and the changes within them as they are tuned to approach new phases. The cubic perovskite material SrTiO$_3$ has been studied for many decades and is well-documented to be an incipient ferroelectric, theorised to exist in the absence of any tuning in the proximity of a ferroelectric quantum critical point. This work presents the first high-precision dielectric measurements under hydrostatic pressure carried out on a quantum critical ferroelectric, leading to a full pressure-temperature phase diagram for SrTiO$_3$. The influence of quantum critical fluctuations is seen to diminish as the system is tuned away from the quantum critical point and a novel low temperature phase is shown to be emergent from it. The Néel Temperature of the two-dimensional antiferromagnet FePS$_3$ was found to increase linearly with applied hydrostatic pressure. Evidence of an insulator-metal transition is also presented, and an unexplained upturn in the resistivity at low temperatures in the metallic phase.

Embedded contact homology is an invariant of closed oriented contact 3-manifolds first defined by Hutchings, and is isomorphic to both Heegard Floer homology (by the work of Colin, Ghiggini and Honda) and Seiberg-Witten Floer cohomology (by the work of Taubes). The embedded contact chain complex is defined by counting closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the manifold. As part of their proof that ECH=HF, Colin, Ghiggini and Honda showed that if the contact form is suitably adapted to an open book decomposition of the manifold, then embedded contact homology can be computed by considering only orbits and differentials in the complement of the binding of the open book; this fact was then in turn used to define a knot version of embedded contact homology, denoted ECK, where the (null-homologous) knot in question is given by the binding. In this thesis we start by generalizing these results to the case of rational open book decompositions, allowing us to define ECK for rationally null-homologous knots. In its most general form this is a bi-filtered chain complex whose homology yields ECH of the closed manifold. There is also a hat version of ECK in this situation which is equipped with an Alexander grading equivalent to that in the Heegaard Floer setting, categorifies the Alexander polynomial, and is conjecturally isomorphic to the hat version of knot Floer homology. The main result of this thesis is a large negative $n$-surgery formula for ECK. Namely, we start with an (integral) open book decomposition of a manifold with binding $K$ and compute, for all $n$ greater than or equal to twice the genus of $K$, ECK of the knot $K(-n)$ obtained by performing ($-n$)-surgery on $K$. This formula agrees with Hedden's large $n$-surgery formula for HFK, providing supporting evidence towards the conjectured equivalence between the two theories. Along we the way, we also prove that ECK is, in many cases, independent of the choices made to define it, namely the almost complex structure on the symplectization and the homotopy type of the contact form. We also prove that, in the case of integral open book decompositions, the hat version of ECK is supported in Alexander gradings less than or equal to twice the genus of the knot.

The 5d transition metal oxides have drawn substantial interest for predictions of being suitable candidates for hosting exotic electronic and magnetic states, including unconventional superconductors, magnetic skyrmions, topological insulators, and Weyl semimetals. In addition to the electron-electron correlation notable in high-temperature 3d transition metal superconductors, the 5d oxides contain a large spin-orbit interaction term in their ground state, which is largely responsible for the intricate phase diagram of these materials. Iridates, or compounds containing 5d iridium bonded with oxygen, are of particular interest for their spin-orbit split Jeff = 1/2 state, which is partially filled without the presence of any additional electron correlation. However, the comparable energetics between a small, finite electron correlation energy and the spin-orbit interaction make the band structure of iridates amenable to small perturbations of the crystalline lattice and ideal for exploring the interplay between these two interactions. While altering the spin-orbit interaction strength of iridium is tenably not feasible, the electron correlation energy can be tuned using a variety of experimental techniques. In this dissertation, the electronic and magnetic properties of iridates at various electron correlation energies are studied by altering the epitaxial lattice strain, dimensionality, and the radius size of the A-site cation. These parameters tune the effective electronic bandwidth of the system, which is inversely proportional to the effective electron correlation energy. The lattice strain and the cationic radius size achieve this by altering the Ir-O-Ir bond angle between nearest neighbor Ir ions. In the case of dimensionality tuning, the effective bandwidth is controlled via the coordination number of each Ir ion. In the first study, a metal-to-insulator transition is observed in thin films of the semi-metallic SrIrO3 as in-plane compressive lattice strain is increased. This observation is consistent with the expectation of compressive lattice strain increasing the effective correlation energy; however, optical spectroscopy spectra reveal the increase is not sufficient for opening an insulating Mott gap. In the second part, the effective correlation energy is adjusted using a dimensional confinement of the layered iridate Sr2IrO4. Here, the coordination number of each Ir ion is reduced using an a-axis oriented superlattice of one-dimensional IrO2 quantum stripes, where several emergent features are revealed in its insulating Jeff = 1/2 state. In the final study, the effective correlation is tuned in a series of mixed-phase pyrochlore iridate thin films, where the Ir atoms take a corner-shared tetrahedral configuration. Here, a transition between conducting to insulating magnetic domain walls is revealed as the correlation energy is increased via A-site chemical doping. Each of these studies sheds light on the pronounced role the effective correlation energy plays in determining the local subset of phases predicted for iridates and related systems featuring strong spin-orbit interactions.

Nanostructures were formed on diced specimens of several silicon carbide polytypes and silicon using electron beam lithography. A general introduction to nanostructure synthesis and electron beam lithography,are presented. A scanning electron microscope was retrofitted with a commercially available electron beam lithography package and an electrostatic beam blanker to permit nanoscale lithography to be performed. A process was first developed and optimized on silicon substrates to expose, poly-methyl-methacrylate (PMMA) resist with an electron beam to make nanoscale nickel masks for reactive ion etching. The masks consist of an array of nickel dots that range in size from 20 to 100 nm in diameter. Several nanoscale structures were then fabricated in silicon carbide using electron beam lithography. The structures produced are characterized by field emission Scanning Electron Microscopy.