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Full-Text Articles in Mathematics
Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius
Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius
Graduate College Dissertations and Theses
In this work, the current state of the field of edge-colored graphs is surveyed. Anew concept of unshrinkable edge colorings is introduced which is useful for rainbow subgraph problems and interesting in its own right. This concept is analyzed in some depth. Building upon the linear edge ordering described in a recent work from Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer, edge-ordering graphs with the cyclic group is introduced and some results are given on this and a related counting problem.
Computing The Canonical Ring Of Certain Stacks, Jesse Franklin
Computing The Canonical Ring Of Certain Stacks, Jesse Franklin
Graduate College Dissertations and Theses
We compute the canonical ring of some stacks. We first give a detailed account of what this problem means including several proofs of a famous historical example. The main body of work of this thesis expands on our article \cite{Franklin-geometry-Drinfeld-modular-forms} in describing the geometry of Drinfeld modular forms as sections of a specified line bundle on a certain stacky modular curve. As a consequence of that geometry, we provide a program: one can compute the (log) canonical ring of a stacky curve to determine generators and relations for an algebra of Drinfeld modular forms, answering a problem posed by Gekeler …
Ring Learning With Errors, Sarah Days-Merrill
Ring Learning With Errors, Sarah Days-Merrill
Graduate College Dissertations and Theses
Over the last twenty years, lattice-based cryptosystems have gained interest due to their levelof security against attacks from quantum computers. The main cryptosystems are based on the hardness of Ring Learning with Errors (RLWE). The Learning with Errors (LWE) problems were first introduced in 2005 by Regev [Reg09] and in 2010, [LPR10] developed the Ring Learning with Errors (RLWE) problems as candidates for safe encryption against quantum computers. Let K be a number field with ring of integers OK. For a prime q, the RLWE problems rely on samples of the form (a, b) ∈ OK/qOK × OK/qOK where a …