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The Frobenius Theorem, Hiroshi Nagao
The Frobenius Theorem, Hiroshi Nagao
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
Many theorems in differential geometry which deal with the existence of certain geometrical structures or properties depend upon various existence and uniqueness theorems for differential equations. Because of its wide range of applications one of the most important of these theorems is the Frobenius Theorem for systems of total differential equations. There are four different forms of the Frobenius Theorem. In applications of the theorem one form is often preferable to the others. In this report we -shall prove the Frobenius Theorem, establish the equivalence of these various forms, and discuss a few applications.
Classifications Of Plane Continua, Steven Ray Matthews
Classifications Of Plane Continua, Steven Ray Matthews
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
In the course of studying continua in the plane it has been asked if a given continuum has uncountably many disjoint duplications in the plane, and if so, what are the consequences of the existence of such a collection. The object of this paper is to study these problems and to develop some machinery useful in their resolution. In Section I, we review the definition of convergence and homeomorphic convergence of point sets in a metric space S. We then consider the space, π of all continuous functions from a compact metric space P to a separable metric space Q …