No description available.

A quandle is a set with a binary operation that satisfies three axioms that corresponds to the three Reidemeister moves on knot diagrams. Homology and cohomology theories of quandles were introduced in 1999 by Carter, Jelsovsky,Kamada, Langford, and Saito as a modification of the rack (co)homology theory defined by Fenn, Rourke, and Sanderson. Cocycles of the quandle (co)homology, along with quandle colorings of knot diagrams, were used to define a new invariant called the quandle cocycle invariants, defined in a state-sum form. This invariant is constructed using a finite quandle and a cocyle, and it has the advantage that it can distinguish some knots from their mirror images, and orientations of knotted surfaces. To compute the quandle cocycle invariant for a specific knot, we need to find a quandle that colors the given knot non-trivially, and find a cocycle of the quandle. It is not easy to find cocycles,since the cocycle conditions form a large, over-determined system of linear equations. At first the computations relied on cocycles found by computer calculations. We have seen significant progress in computations after Mochizuki discovered a family of 2- and 3-cocycles for dihedral and other linear Alexander quandles written by polynomial expressions. In this dissertation, following the method of the construction by Mochizuki, a variety of n-cocycles for n >̲ 2 are constructed for some Alexander quandles, given by polynomial expressions. As an application, these cocycles are used to compute the invariants for (2,n)-torus knots, twist knots and their r-twist spins. The calculations in the case of (2,n)-torus knots resulted in formulas that involved the derivative of the Alexander polynomial. Non-triviality of some quandle homology groups is also proved using these cocycles. Another application is given for tangle embeddings. The quandle cocycle invariants are used as obstructions to embedding tangles in links. The formulas for the cocycle invariants of tangles are obtained using polynomial cocycles, and by comparing the invariant values, information is obtained on which tangles do not embed in which knots. Tangles and knots in the tables are examined, and concrete examples are listed.

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

Io, the innermost Galilean satellite of Jupiter, exhibits a wide variety of complex phenomena such as interaction with Jupiter’s magnetosphere, volcanic activity, and a rarefied multi-species sublimating and condensing atmosphere with an ionosphere. Io’s orbital resonance with Jupiter and the other Galilean satellites produces intense tidal heating. This makes Io the most volcanically active body in the solar system with plumes that rise hundreds of kilometers above the surface. In the present work, the interaction of Io’s atmosphere with the Jovian plasma torus is simulated via the Direct Simulation Monte Carlo (DSMC) method and the aurora produced via electron-neutral excitation collisions is examined using electron transport Monte Carlo simulation. The electron-transport Monte Carlo simulation models the electron collisions with the neutral atmosphere and their transport along field lines as they sweep past Io, using a pre-computed steady atmosphere and magnetic field. As input to the Monte Carlo simulation, the neutral atmosphere was first modeled using prior 2D sunlit continuum simulations of Io’s atmosphere produced by others. In order to justify the use of a sunlit atmosphere for eclipse, 1D two-species (SO2 and a non-condensable) DSMC simulations of Io’s atmospheric dynamics during and immediately after eclipse were performed. It was found that the inclusion of a non-condensable species (SO or O2) leads to the formation of a diffusion layer which prevents rapid collapse. The degree to which the diffusion layer slowed the atmospheric collapse was found to be extremely sensitive to both the initial non-condensable mole fraction and the reaction (or sticking) probability on the surface of the “non-condensable”. Furthermore, upon egress, vertical stratification of the atmosphere occurred with the non-condensable species being lifted to higher altitudes by the rapid sublimation of SO2 as the surface warms. Simulated aurorae (specifically the [OI] 6300 Å and the S2, SO, and SO2 molecular band emission in the middle ultraviolet) show good agreement with observations of Io in eclipse and an attempt was made to use the simulations to constrain the upstream torus electron temperature and Io’s atmospheric composition, structure, and volcanic activity. It is found that the position of the bright [OI] 6300 Å wake spot relative to Io’s equator depends on the position of Io relative to the plasma torus’ equator and the asymmetric electron number flux that results. Using HST/STIS UV-Vis spectra, the upstream electron temperature is weakly constrained to be between 3 eV and 8 eV depending on the flux of a low energy (35 eV), non-thermal component of the plasma (more non-thermal flux requires lower thermal plasma temperatures to fit the spectrum). Furthermore, an upper limit of 5% of the thermal torus density (or 180 cm−3 based on the Galileo J0 plasma density at Io) is obtained for the low energy non-thermal component of the plasma. These limits are consistent with Galileo observations of the upstream torus temperature and estimates for the the non-thermal component. Finally, plume activity and S2 content during eclipse observations with HST/STIS were constrained by examining the emission intensity along the spatial axis of the aperture. During the August 1999 UV-Vis observations, the auroral simulations indicate that the large volcanoes Pele and Surt were inactive whereas Tvashtar was active and that Dazhbog and possibly Loki were also actively venting gas. The S2 content inferred for the large Pele-type plumes was between 5% (Tvashtar) and 30% (Loki, if active), consistent with prior observations (Spencer et al., 2000; Jessup et al., 2007). A 3D DSMC simulation of Io’s sublimation and sputtered atmosphere including photo- and plasma-chemistry was developed. In future work these atmospheric simulations will replace the continuum target atmosphere in the auroral model and thus enable a better match to the observed high altitude auroral emission. In the present work, the plasma interaction is modeled by a flux of ions and electrons which flow around and through Io’s atmosphere along pre-computed fields and interact with the neutral gas. A 3D DSMC simulation of Io’s atmosphere assuming a simple thermal model for the surface just prior to ingress into eclipse and uniform frost coverage has been performed in order to understand how Io’s general atmospheric dynamics are affected by the new plasma model with chemistry and sputtering. Sputtering was found to supply most of the nightside atmosphere (producing an SO2 column of ~5×1013 cm−2); however, the dense dayside sublimation atmosphere was found to block sputtering of the surface. The influence of the dynamic plasma pressure on the day-to-night circumplanetary flow was found to be quite substantial causing the day-to-night wind across the dawn terminator to flow slightly towards the equator. This results in a region of high density near the equator that extends far (~2000 km for the condensable species) onto the nightside across the dawn terminator. Thus, even without thermal lag due to rotation or variable surface frost, highly asymmetric equatorial column densities relative to the subsolar point are obtained. The non-condensable O2, which is a trace species on the dayside, is the dominant species on the nightside despite increased SO2 sputtering because the loss rate of O2 is slow. Finally, a very intriguing O2 flow feature was observed near the dusk terminator where the flow from the leading hemisphere (pushed by the plasma) meets the flow from the dayside trailing hemisphere. Since the O2 does not condense on the surface, it slowly convects towards the poles and then back onto the nightside, eventually to be dissociated or stripped away by the plasma. / text

One goal of music theory is to describe the resources of a pitch system. Traditionally, the study of pitch intervals was done using frequency ratios of the powers of small integers. Modern mathematical music theory offers an independent way of understanding the pitch system by considering intervals as transformations. This thesis takes advantage of the historical emergence of algebraic structures in musicology and, in the spirit of transformational theory, treats operations that form mathematical groups. Aspects of Neo-Riemannian theory are explored and developed, in particular the T/I and PLR groups as dual. Pitch class spaces, such as 12, can also be defined as torsors. In addition to surveying the group theoretical tools for music analysis, this thesis provides detailed proofs of many claims that are proposed but seldom supported.

No description available.

The goal of this thesis is to study the ground or metastable state structure of nematic liquid crystal systems confined inside handled shapes such as a torus or double torus. We begin our work by introducing a new method to generate a toroidal droplet from a Newtonian liquid inside another, immiscible, Newtonian liquid. In this situation, a toroidal droplet is unstable and follows one of two routes in transforming into a spherical droplet: (i) its tube breaks in a way reminiscent to the breakup of a cylindrical jet, or (ii) its tube grows until it finally coalesces onto itself. However, to be able to probe the nematic structure, we need to address the issue of instabilities. This is done by replacing the outer liquid with a yield stress material, which ultimately leads to the stabilization of the toroidal droplet. Through the experimental investigation, we are able to establish the stabilization conditions. Finally, we generate and stabilize toroidal droplets with a nematic liquid crystal as the inner liquid and a yield stress material as the outer medium. Here we observe that in the ground state, the nematic liquid crystal exhibits an intriguing twisted structure irrespective of the aspect ratio of the torus. While there are no defects observed in a toroidal droplet case, two defects with -1 topological charge each emerge each time we increase the number of handles.

This thesis consists of three articles treating topics in the theory of algebraic stacks. The first two papers deal with motivic invariants. In the first, we show that the class of the classifying stack BPGLn is the inverse of the class of PGLn in the Grothendieck ring of stacks for n ≤ 3. This shows that the multiplicativity relation holds for the universal torsors, although it is known not to hold for torsors ingeneral for the groups PGL2 and PGL3. In the second paper, we introduce an exponential function which can be viewed as a generalisation of Kapranov's motivic zeta function. We use this to derive a binomial theorem for a power operation defined on the Grothendieck ring of varieties. As an application, we give an explicit expression for the motivic class of a universal quasi-split torus, which generalises a result by Rökaeus. The last paper treats destackification. We give an algorithm for removing stackiness from smooth, tame stacks with abelian stabilisers by repeatedly applying stacky blow-ups. As applications, we indicate how the result can be used for destackifying general Deligne–Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.</p>

Universiẗat, Diss., 2001--Mannheim.

Universiẗat, Diss., 2003--Mainz.