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The Poincaré Conjecture, Joseph D. Peck
The Poincaré Conjecture, Joseph D. Peck
All Graduate Theses and Dissertations, Spring 1920 to Summer 2023
The central theme for this paper is provided by the following three statements:
(1) Every compact connected 1-manifold is S1.
(2) Every compact connected simply connected 2-manifold is S2.
(3) Every compact connected simply connected 3-manifold is S3.
We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing …
Integral Representation Theorems, Leiko Hatta
Integral Representation Theorems, Leiko Hatta
All Graduate Theses and Dissertations, Spring 1920 to Summer 2023
Since F. Riesz showed in 1909 that the dual of C[0,1] is BV[0,1] (the functions of bounded variation on [0, 1] with || g ||BV = V(g)) via the Stieltjes integral, obtaining representations for linear operators in various settings has been a problem of interest. This paper shows the historical manner of representations, the road map type theorems and representations obtained via the v-integral.