We survey standard topics in elementary differential geometry and complex analysis to build up the necessary theory for studying applications of univalent harmonic function theory to minimal surfaces. We then proceed to consider convex combination harmonic mappings of the form f=sf_1+(1-s) f_2 and give conditions on when f lifts to a one-parameter family of minimal surfaces via the Weierstrauss-Enneper representation formula. Finally, we demand two minimal surfaces M and M' be locally isometric, formulate a system of partial differential equations modeling this constraint, and calculate their symmetry group. The group elements generate transformations that when applied to a prescribed harmonic mapping, lift to locally isometric minimal surfaces with varying graphs embedded in mathbb{R}^3.

Thesis: M.S., Massachusetts Institute of Technology, Sloan School of Management, 1980 / Bibliography: leaf 53. / by Jai Yong Oum. / M.S. / M.S. Massachusetts Institute of Technology, Sloan School of Management

No description available.

In mathematical physics, Laplace's equation plays an especially significant role. It is fundamental to the solution of problems in electrostatics, thermodynamics, potential theory and other branches of mathematical physics. It is for this reason that this investigation concerns the development of some general properties of continuous solutions of this equation.

In treating the motion of a fluid mathematically, it is convenient to make some simplifying assumptions. The assumptions which are made will be justifiable if they save long and laborious computations in practical problems, and if the predicted results agree closely enough with experimental results for practical use. In dealing with the flow of air about an airfoil, at subsonic speeds, the fluid will be considered as a homogeneous, incompressible, inviscid fluid.

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science. Johannesburg, 2013 / Many of the problems in mathematics have very elegant solutions. As complex, real–world geometries come into play, however, this elegance is often lost. This is particularly the case with meshes of physical, real–world problems. Domain mapping helps to move problems from some geometrically complex domain to a regular, easy to use domain. Shape transformation, specifically, allows one to do this in 2D domains where mesh construction can be difficult. Numerical methods usually work over some mesh on the target domain. The structure and detail of these meshes affect the overall computation and accuracy immensely. Unfortunately, building a good mesh is not always a straight forward task. Finite Element Analysis, for example, typically requires 4–10 times the number of tetrahedral elements to achieve the same accuracy as the corresponding hexahedral mesh. Constructing this hexahedral mesh, however, is a difficult task; so in practice many people use tetrahedral meshes instead. By mapping the geometrically complex domain to a regular domain, one can easily construct elegant meshes that bear useful properties. Once a domain has been mapped to a regular domain, the mesh can be constructed and calculations can be performed in the new domain. Later, results from these calculations can be transferred back to the original domain. Using harmonic functions, source domains can be parametrised to spaces with many different desired properties. This allows one to perform calculations that would be otherwise expensive or inaccurate. This research implements and extends the methods developed in Voruganti et al. [2006 2008] for domain mapping using harmonic functions. The method was extended to handle cases where there are voids in the source domain, allowing the user to map domains that are not topologically equivalent to the equivalent dimension hypersphere. This is accomplished through the use of various boundary conditions as the void is mapped to the target domains which allow the user to reshape and shrink the void in the target domain. The voids can now be reduced to arcs, radial lines and even shrunk to single points. The algorithms were implemented in two and three dimensions and ultimately parallelised to run on the Centre for High Performance Computing clusters. The parallel code also allows for arbitrary dimension genus-0 source domains. Finally, applications, such as remeshing and robot path planning were investigated and illustrated.

by Lei Ka Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 70-71). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Dirichlet Problem at infinity --- p.9 / Chapter 1.1 --- The Geometric Boundary --- p.9 / Chapter 1.2 --- Dirichlet Problem --- p.15 / Chapter 2 --- The Martin Boundary --- p.29 / Chapter 2.1 --- The Martin Metric --- p.30 / Chapter 2.2 --- The Representation Formula --- p.31 / Chapter 2.3 --- Uniqueness of Representation --- p.36 / Chapter 3 --- The Geometric boundary and the Martin boundary --- p.42 / Chapter 3.1 --- Estimates for harmonic functions in cones --- p.42 / Chapter 3.2 --- A Harnack Inequality at Infinity --- p.49 / Chapter 3.3 --- The kernel function --- p.54 / Chapter 3.4 --- The Main Theorem --- p.55 / Chapter 4 --- Positive Harmonic Functions on Product of Manifolds --- p.61 / Chapter 4.1 --- Splitting Theorem --- p.61 / Chapter 4.2 --- Riemannian Halfspace and the parabolic Martin boundary --- p.62 / Chapter 4.3 --- Splitting of parabolic Martin kernels --- p.63 / Chapter 4.4 --- Proof of theorem 4.1 --- p.66 / Bibliography

No description available.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 65-68). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 65-68). Also available on the Internet.