Computational Equipment for Approximation Theory, Control Theory, Graph Theory

NSF提案DMS-0215442 PI：哈丁，Ellingham，霍恩，Saff和Schumaker摘要近似理论，控制理论和图论的计算设备范德比尔特大学数学系将购买计算机设备，专门用于支持数学科学的研究。这些设备将用于以下几个研究项目。 Ellingham教授将研究涉及曲面嵌入、可遍历性和独立集的图论。哈丁教授将构建和开发局部正交基的应用。霍恩教授将使用数值方法研究连接弹性结构的可控性。哈丁教授和萨夫教授将研究球面上的离散最小能量问题。萨夫教授和舒梅克教授将研究样条函数的理论和应用，特别是在球面上。埃林厄姆教授的研究涉及图，这是网络的数学模型。通信、交通和其他网络经常使用图论技术建模。 哈丁教授的研究涉及小波，小波是压缩图像、声音和其他真实世界信息的重要工具。它们可以应用于任何科学领域的数据点拟合方程或处理图像的问题，例如医学扫描设备。霍恩教授的研究涉及控制理论，它允许计算力应该如何施加到一个复杂的系统，以达到预期的结果。 它可以应用于许多现实世界的结构，如桁架、机器人手臂、太阳能电池板和吊桥。哈丁教授和萨夫教授的工作涉及到在球面上放置点的最佳方法，以使球面尽可能均匀地覆盖。这个问题在许多领域都有应用，包括病毒结构、晶体学、分子结构和静电学。 萨夫教授和舒梅克教授的研究试图在球体表面上提供精确但易于计算的数量表示。例如，这些技术可以用来建立一个精确的地球引力场模型。

NSF Proposal DMS-0215442PIs: Hardin, Ellingham, Horn, Saff, and Schumaker ABSTRACTComputational Equipment for Approximation Theory, ControlTheory, and Graph TheoryThe Department of Mathematics at Vanderbilt University willpurchase computer equipment dedicated to supporting researchin the mathematical sciences. The equipment will be used forseveral research projects, as follows. Professor Ellinghamwill investigate conjectures in graph theory involvingsurface embeddings, traversability, and independent sets.Professor Hardin will construct and develop applicationsof local orthogonal bases. Professor Horn will investigatethe controllability of linked elastic structurres usingnumerical methods. Professors Hardin and Saff willinvestigate discrete minimum energy problems on the sphere.Professors Saff and Schumaker will research the theory andapplications of splines, especially on the sphere.Professor Ellingham's research involves graphs, which aremathematical models of networks. Communication,transportation and other networks are often modeled usingtechniques from graph theory. Professor Hardin's researchinvolves wavelets, which are an important tool forcompressing images, sounds, and other real-world information.They may be applied to problems in fitting equations to datapoints in any area of science, or processing images, forexample from medical scanning equipment. Professor Horn'sresearch involves control theory, which allows computationof how forces should be applied to a complicated system toachieve a desired outcome. There are applications to manyreal-world structures such as trusses, robot arms, solarpanels, and suspension bridges. The work by ProfessorsHardin and Saff involves the best way to place points ona sphere so that the sphere is covered as uniformly aspossible. This problem has applications in many fields,including the structure of viruses, crystallography,structure of molecules, and electrostatics. The researchby Professors Saff and Schumaker involves trying to provideaccurate but easily computable representations of quantitieson the surface of a sphere. For example, these technique maybe used to construct an accurate model of the Earth'sgravitational field.