In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

The mean dynamic topography (MDT) of the ocean is closely related to the mean surface circulation. The objective of this thesis is to produce estimates of the MDT for the North Atlantic using newly available data from ocean and gravity observing systems, to evaluate these new estimates and so improve our understanding of the circulation. Oceanographic estimates of the MDT are based on the mean temperature and salinity (TS) fields of the ocean. These are typically averages from sparse observations collected over many decades. The ocean is a non-stationary system so it is more appropriate to define the mean for shorter, specific periods. Whilst the Argo observing system has increased the frequency and resolution of in situ oceanographic measurements, high frequency (eddy) variability remains. A new technique is described for removing this variability using satellite altimeter measurements of the sea surface height. A new TS mean is produced, relating to the period from 2000 to 2007, and this is used to map a new oceanographic estimate of the MDT using an ocean circulation model. New geodetic estimates of the MDT are produced using geoid models that incorporate gravity measurements from the ongoing GRACE and GOCE satellite missions. These are compared with the new oceanographic estimate and validated against independent observations such as drifter speeds. The geodetic method produces realistic estimates of the mean surface circulation, thereby realizing the long time dream of oceanographers to observe the ocean circulation from space. The new oceanographic estimates are not as accurate, but the new TS mean contributes to improvements in the performance of ocean models, a necessary step in understanding and predicting the oceans. Coastal tide gauges can provide an accurate estimate of the alongshore tilt of the coastal MDT and this has been used to evaluate the above estimates. Temporal variability of the tilt along the coast of the South Atlantic Bight is used, with statistical methods and an ocean circulation model, to identify the processes contributing to the tilt. A new opportunity to use tide gauges as part of an observing system for the ocean circulation is discussed.

In contrast to the studies of constant or time-varying correlations between stock and bond returns, in this thesis, I explore the regime-dependent correlations between stock and bond returns. Specifically, I start with a comprehensive asset pricing model, i.e., a regime-switching multifactor model, and then investigate the regime-dependent correlations between stock and bond returns. Based on the BIC, the number of regimes in the regime-switching model is optimally determined to be two. For the two regimes, the directions of the regime-dependent correlations appear to be significantly different. Also, the magnitudes of the regime-dependent correlations are substantially larger in these two regimes than the correlation in the single regime. With my findings in the regime-dependent correlations, I then examine the performance of portfolio strategies. Throughout the in-sample and out-of-sample tests, I find that the two portfolio strategies, regime inferred portfolio and probability implied portfolio, can outperform the benchmark, S&P 500.

No description available.

This paper provides methods to select pairs potentially profitable within the frame of statistical arbitrage. We employ a cointegration approach on pairwise combinations of five large energy companies listed on the New York Stock Exchange for the period 27th September 2012 to 22nd April 2014. We find one cointegrated pair, for which we further investigate both short and long run dynamics. A vector-error correction model is constructed, supporting a long run relationship between the two stocks, which is also supported by the mean-reverting characteristic of a stationary linear combination of the stocks. Impulse response functions and variance decomposition are also studied to further describe the interrelation of the stocks, supporting a unidirectional causality between the stocks.

I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in Kähler-Einstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^n. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities.

University, Diss., 2006--Augsburg. / Erscheinungsjahr an der Haupttitelstelle: 2005.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 121-123).

Techn. University, Diss., 2001--Berlin.