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On Lane's IntegralHill, William James 08 1900 (has links)
The problem and purpose of this paper is to develop Lane's Integral in two-space, and then to expand these concepts into three-space and n-space. Lane's Integral can be used by both mathematicians and statisticians as one of the tools in the calculation of certain probabilities and expectations. The method of presentation is straightforward with the basic concepts of integration theory and Stieltjes Integral assumed.
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Euclidean N-spaceHorner, Donald R. 08 1900 (has links)
This study of the Euclidean N-space looks at some definitions and their characteristics, some comparisons, boundedness and compactness, and transformations and mappings.
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Simplicial HomologyChang, Chih-Chen 08 1900 (has links)
The purpose of this thesis is to construct the homology groups of a complex over an R-module. The thesis begins with hyperplanes in Euclidean n-space. Simplexes and complexes are defined, and orientations are given to each simplex of a complex. The chains of a complex are defined, and each chain is assigned a boundary. The function which assigns to each chain a boundary defines the set of r-dimensional cycles and the set of r—dimensional bounding cycles. The quotient of those two submodules is the r-dimensional homology group.
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