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It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution.(Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a

Free kicks play an important role in a modern soccer match. A skillful soccer player usually uses the opportunity of a free kick to score a goal. There are mainly two ways to give a free kick, falling ball and banana ball. The last thesis analyzes the path of falling soccer in two-dimensional plane. This thesis mainly focuses on using geometrical ways to generate a graph in three-dimensional coordinate system to analyze the path of banana ball and find out the most suitable distance to give a free kick.

Gene expression data from microarrays are being applied to predict preclinical and clinical endpoints, but the reliability of these predictions has not been established. In the MAQC-II project, 36 independent teams analyzed six microarray data sets to generate predictive models for classifying a sample with respect to one of 13 endpoints indicative of lung or liver toxicity in rodents, or of breast cancer, multiple myeloma or neuroblastoma in humans. In total,> 30,000 models were built using many combinations of analytical methods. The teams generated predictive models without knowing the biological meaning of some of the endpoints and, to mimic clinical reality, tested the models on data that had not been used for training. We found that model performance depended largely on the endpoint and team proficiency and that different approaches generated models of similar performance. The conclusions and

Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.