Using scanning tunneling microscopy (STM) we have explored complex surface adsorbed molecules, nanocars, on Au(111) and the parameters related to the direct translation of these molecules by the STM tip. Specifically, the molecules focused on here were functionalized with C 60 or trans ruthenium complexes. With low tunneling currents the molecules could be imaged at room temperature. Increasing the tunneling current allowed us to bring the tip closer to individual molecules and reposition them on the surface. Below specific current and bias voltage conditions the molecules remained stationary, while in other cases the tip interaction was strong enough to drastically damage or eject the molecule from the field of view. High temperature scans revealed the effect of the wheel activation energy relative to the underlying surface as the different wheeled nanocars began diffusing at different temperatures confirming the manipulation measurements.

Examination of quantum critical points of itinerant electron systems will aid with understanding of d-electron magnetism that exhibits both local and itinerant characteristics in different families of compounds. Doping-induced quantum phase transition of the itinerant ferromagnet Sc 3.1 In that is composed of non-magnetic elements is the focus of our work. Polycrystalline samples of (Sc 1-x Lu x ) 3.1 In with 0≤ x≤ 0.08 were prepared by arcmelting and then annealing for an extended period of time. Susceptibility measurements were performed in an applied magnetic field H = 0.1 T for temperatures T = 1.85 K to 300 K. Linearity of Arrott plots in low-field region was significantly improved by implementing the non-mean-field Arrott-Noakes technique where plotting M 1/β vs. ( H/M ) 1/γ is used to determine both the Curie temperature and composition. Modified Arrott plot approach was used in order to determine the new critical exponents β, γ and δ that better describe this compound. The Curie temperature of the Sc 3.1 In compound was found to be T C = 4.4 K and the critical composition x c = 0.02. This work was supported by NSF DMR 0847681.

Quasi-one-dimensional Rydberg atoms exposed to alternating positive and negative electric field pulses (kicks) are an example of a chaotic atomic system. Chaotic ionization is predicted in this system via a phase space turnstile mechanism, and we have explored this experimentally. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study is the first to explicitly illuminate their relevance to atomic ionization. Two experiments are presented. In the first we show that the ionization of the electron depends not only on the initial electron energy, but also on the phase space position of the electron with respect to the turnstile--that part of the electron packet inside the turnstile ionizes quickly, after one period of the applied field, while that part outside the turnstile ionizes after multiple kicking periods. In the second experiment we show the signature of the turnstile manifests itself in the step-function-like behavior of the ionization fraction as a function of the kick strength. This behavior persists for different values of kicking periods and starting electron energies.

In this thesis we describe a new version of a magneto-friction model, which was developed for computing the magnetospheric equilibrium that includes an arbitrary Earth's dipole tilt and interplanetary magnetic field. We also describe the algorithms of this new friction code that trace magnetic field lines, locate the neutral sheet, and identify the magnetopause In addition, we present a generalized theory for calculating magnetic flux tube volume in the magnetotail, in an attempt to generalize the Wolf [2006] empirical formula, and describe a method for estimating flux tube volume from measurements at geosynchronous orbit. This new method has been tested against various equilibrated magnetospheres generated by the new friction code. Although still incomplete, the method exhibits promising features, and is to be completed in the future.

This thesis presents calculations of the magnetic helicity injection due to rotating sunspots and a determination of the characteristics of the rotating sunspots in the active regions with simple magnetic configurations. Four active regions are investigated to study the relationship between rotating sunspots and magnetic helicity. The observations indicate that significantly more helicity is injected during the period of rotation in polarities with strong magnetic field. This may be a result of the emergence of a magnetic flux rope from below the solar surface. Moreover, our preliminary study on a large sample of 90 active regions shows that the level of flaring activity increases with the rate of helicity injection. Finally, a statistical study is carried out to determine the relation between rotating sunspots and the emergence of magnetic flux tubes. Among 82 active regions which exhibit flux emergence, 93% are associated with rotating sunspots. Among 50 active regions without well-defined flux emergence, 60% of sunspots are observed to be rotating, though relatively slowly. In addition, we find that sense of the rotation (i.e., clockwise or counter-clockwise) of the sunspots shows a weak hemispherical tendency.

A compact semigroup is, roughly, a semigroup compactification of a locally compact group if it contains a dense homomorphic image of the group. The theory of semigroup compactifications has been developed in connection with subalgebras of continuous bounded functions on locally compact groups. The Eberlein algebra of a locally compact group is defined to be the uniform closure of its Fourier-Stieltjes algebra. In this thesis, we study the semigroup compactification associated with the Eberlein algebra. It is called the Eberlein compactification and it can be constructed as the spectrum of the Eberlein algebra. The algebra of weakly almost periodic functions is one of the most important function spaces in the theory of topological semigroups. Both the weakly almost periodic functions and the associated weakly almost periodic compactification have been extensively studied since the 1930s. The Fourier-Stieltjes algebra, and hence its uniform closure, are subalgebras of the weakly almost periodic functions for any locally compact group. As a consequence, the Eberlein compactification is always a semitopological semigroup and a quotient of the weakly almost periodic compactification. We aim to study the structure and complexity of the Eberlein compactifications. In particular, we prove that for certain Abelian groups, weak^{*}-closed subsemigroups of L^{\infty}[0, 1] may be realized as quotients of their Eberlein compactifications, thus showing that both the Eberlein and weakly almost periodic compactifications are large and complicated in these situations. Moreover, we establish various extension results for the Eberlein algebra and Eberlein compactification and observe that levels of complexity of these structures mimic those of the weakly almost periodic ones. Finally, we investigate the structure of the Eberlein compactification for a certain class of non-Abelian, Heisenberg type locally compact groups and show that aspects of the structure of the Eberlein compactification can be relatively simple.

The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is divided into the following two parts: the rst part is devoted to presenting moduli problems in algebraic geometry using a modern perspective, via the language of stacks and the second part is devoted to studying moduli problems from the perspective of symplectic geometry. The key motivation to the rst part is to present the theorem of Keel and Mori [20] which answers the classical question of under what circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is divided into the following two parts: the rst part is devoted to presenting moduli problems in algebraic geometry using a modern perspective, via the language of stacks and the second part is devoted to studying moduli problems from the perspective of symplectic geometry. The key motivation to the rst part is to present the theorem of Keel and Mori [20] which answers the classical question of under what circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure. It is shown that every differential algebraic group which is not of maximal differential type is definably isomorphic to the sharp points of a relative D-group. Pillay's generalized finite dimensional differential Galois theory is extended to the possibly infinite dimensional partial setting. Logarithmic differential equations on relative D-groups are discussed and the associated differential Galois theory is developed. The notion of generalized strongly normal extension is naturally extended to the partial setting, and a connection betwen these extensions and the Galois extensions associated to logarithmic differential equations is established. A geometric characterization, in the spirit of Pierce-Pillay, for the theory DCF_{0,m+1} (differentially closed fields of characteristic zero with m+1 commuting derivations) is given in terms of the differential algebraic geometry of DCF_{0,m} using relative prolongations. It is shown that this characterization can be rephrased in terms of characteristic sets of prime differential ideals, yielding a first-order geometric axiomatization of DCF_{0,m+1}. Using the machinery of characteristic sets of prime differential ideals it is shown that the theory of partial differential fields with an automorphism has a model companion. Some basic model theoretic properties of this theory are presented: description of its completions, supersimplicity and elimination of imaginaries. Differential-difference modules are introduced and they are used, together with jet spaces, to establish the canonical base property for finite dimensional types, and consequently the Zilber dichotomy for minimal finite dimensional types.

We consider the greatest prime factors of integers of certain form.