ACKNOWLEDGEMENTS I would like to thank my adviser Michael Zieve for teaching me to fish. I would like to thank Jeffrey Lagarias for helpful feedback in writing this thesis. I would like to thank my family (Peter, Susan, and Holly), my beautiful girlfriend Becky, and the community of Michigan graduate students for keeping me sane.

[1] The strong jet, with a speed between 500 and 600 m/s, is inferred in the equatorial region of Saturn by combining the nadir and limb observations of Composite Infrared Spectrometer (CIRS) aboard the Cassini spacecraft. A similar jet was discovered on Jupiter (F. M. Flasar et al., 2004a). These discoveries raise the possibility that intense jets are common in the equatorial stratospheres of ...

In this paper a class of quartic trigonometric Bézier curve, called QT Bézier curve, with two shape parameters is presented. These curves not only inherit most properties of the usual quartic Bézier curves in the polynomial space, but also enjoy some other advantageous properties for shape modelling. The shape of the curve can be adjusted by altering the values of shape parameters while the con...

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K0 be an arbitrary prime field and let r > 1 and s > 1 be integers that are coprime to one another. We show that one can find a finite extension K of K0, a degree-r extension L of K, a degr...

This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models.

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shim...

A simple and parsimonious model which originated from the Weibull frequency distribution was proposed to describe nonlinear survival curves of spores. This model was suitable for downward concavity curves (Bacillus cereus and Bacillus pumilus), as well as for upward concavity curves (Clostridium botulinum). It was shown that traditional F values calculated from this new model were no longer add...

By combining experiments with simple model calculations, we obtain new insight in spin transport through hybrid, CoFeB/Al2O3(1.5 nm)/tris(8-hydroxyquinoline)aluminium (Alq3)/Co spin valves. We have measured the characteristic changes in the I-V behavior as well as the intrinsic loss of magnetoresistance at the onset of multiple-step tunneling. In the regime of multiple-step tunneling, under the...

Sarabia et al. (1999) present a basic model to create ordered families of Lorenz curves, along with basic theorems that describe the conditions for the models to satisfy the definition of the Lorenz curve. Using these basic models, they suggest a family which includes several well-known Lorenz models as special cases. This paper first shows that their basic theorems can be generalized. The pape...