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Banach Spaces On Topological Ramsey Structures, Cheng-Chih Ko
Banach Spaces On Topological Ramsey Structures, Cheng-Chih Ko
Electronic Theses and Dissertations
A Banach space T1(d, θ) with a Tsirelson-type norm is constructed on the top of the topological Ramsey space T1 defined by Dobrinen and Todorcevic [6]. Finite approximations of the isomorphic subtrees are utilised in constructing the norm. The subspace on each “branch” of the tree is shown to resemble the structure of an ℓ∞n+1 -space where the dimension corresponds to the number of terminal nodes on that branch. The Banach space T1(d, θ) is isomorphic to (∑n∊ℕ⊕ℓ∞n+1)p , where d ∈ ℕ with d ≥ 2, …
Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly
Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly
Electronic Theses and Dissertations
Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …