This thesis concerns cleavability. A space X is said to be cleavable over a space Y along a set A subset of X if there exists a continuous function f from X to Y such that f(A) cap f(X setminus A) = emptyset. A space X is cleavable over a space Y if it is cleavable over Y along all subsets A of X. In this thesis we prove three results regarding cleavability. First we discover the conditions under which cleavability of an infinite compactum X over a first-countable scattered linearly ordered topological space (LOTS) Y implies embeddability of X into Y. In particular, we provide a class of counter-examples in which cleavability does not imply embeddability, and show that if X is an infinite compactum cleavable over ω₁, the first uncountable ordinal, then X is embeddable into ω₁. We secondly show that if X is an infinite compactum cleavable over any ordinal, then X must be homeomorphic to an ordinal. X must also therefore be a LOTS. This answers two fundamental questions in the area of cleavability. We also leave it as an open question whether cleavability of an infinite compactum X over an uncountable ordinal λ implies X is embeddable into λ. Lastly, we show that if X is an infinite compactum cleavable over a separable LOTS Y such that for some continuous function f from X to Y, the set of points on which f is not injective is scattered, then X is a LOTS. In addition to providing these three results, we introduce a new area of research developed from questions within cleavability. This area of research is called almost-injectivity. Given a compact T₂ space X and a LOTS Y, we say a continuous function f from X to Y is almost-injective if the set of points on which f is not injective has countable cardinality. In this thesis, we state some questions concerning almost-injectivity, and show that if lambda is an ordinal, X is a T₂ compactum, and f is an almost-injective function from X to lambda, then X must be a LOTS.

The modern engineering design process includes computer software packages that require approximations to be made when representing geometries. These approximations lead to inherent discrepancies between the design geometry of a part or assembly and the corresponding manufactured geometry. Further approximations are made during the analysis portion of the design process. Manufacturing defects can also occur, which increase the discrepancies between the design and manufactured geometry. These approximations combined with manufacturing defects lead to discrepancies which, for high precision parts, such as jet engine compressor blades, can affect the modal analysis results. In order to account for the manufacturing defects during analysis, mesh morphing is used to morph a structural finite element analysis mesh to match the geometry of compressor blades with simulated manufacturing defects. The mesh morphing process is improved by providing a novel method to convert heteromorphic shape matching within Sculptor to homeomorphic shape matching. This novel method is automated using Java and the NX API. The heteromorphic to homeomorphic conversion method is determined to be valid due to its post-mesh morphing maximum deviations being on the same order as the post-mesh morphing maximum deviations of the ideal homeomorphic case. The usefulness of the automated heteromorphic to homeomorphic conversion method is demonstrated by simulating manufacturing defects on the pressure surface of a compressor blade model, morphing a structural finite element analysis mesh to match the geometry of compressor blades with simulated manufacturing defects, performing a modal analysis, and making observations on the effect of the simulated manufacturing defects on the modal characteristics of the compressor blade.

We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space.