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Title
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An Oblate Spheroid Model of Cortical Folding.
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Creator
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Grazzini, Courtney, Department of Mathematics
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Abstract/Description
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In previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reaction-diffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is...
Show moreIn previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reaction-diffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is activated. In turn, the IP cell duplicates into one or two neuroblasts. These cells form the cortical layer. The amplifications due to activated radial glial cells create gyral walls, and the inhibited cells create sulcal valleys. In this research, we modify Striegel and Hurdal's model to use an oblate spheroid domain. We develop mathematical equations using this new domain and investigate the role of various parameters through numerical stimulations. We suggest how these results can be applied to diseases, such as ventriculomegaly and holoprosencephaly, which alter the shape and size of the brain.
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Date Issued
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2015
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Identifier
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FSU_migr_uhm-0487
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Format
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Thesis
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Title
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Dirichlet's Theorem and Analytic Number Theory.
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Creator
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Frey, Thomas W., Department of Mathematics
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Abstract/Description
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In 1837 Dirichlet proved the infinitude of primes in all arithmetic co-prime sequences. This was done by look at Dirichlet L-functions, Dirichlet series, Dirichlet characters (modulo k), and Euler Products. In this thesis, the necessary facts, theorems, and properties are shown in order to prove Dirichlet's Theorem, concluding with a proof of Dirichlet's Theorem.
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Date Issued
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2015
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Identifier
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FSU_migr_uhm-0560
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Format
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Thesis
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Title
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An Overview of Homotopy Type Theory and the Univalent Foundations of Mathematics.
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Creator
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Dunn, Lawrence, Department of Mathematics
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Abstract/Description
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Homotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces'' -- not unstructured sets --- are the most primitive objects. This perspective offers...
Show moreHomotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces'' -- not unstructured sets --- are the most primitive objects. This perspective offers both a computational foundational for mathematics and a direct method for reasoning about homotopy theory. We present here a broad contextual overview of homotopy type theory, including a sufficiently thorough examination of the classical foundations which it replaces as to make clear the extent of its innovation. We will explain that homotopy type theory is, loosely speaking and among other things, a programming language for mathematics, especially one with native support for homotopy theory.
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Date Issued
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2014
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Identifier
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FSU_migr_uhm-0304
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Format
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Thesis
