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Physical Sciences and Mathematics Commons™
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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz
Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz
Department of Mathematics: Dissertations, Theses, and Student Research
Let R be a commutative ring, (f) an ideal of R, and E = K(f; R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T, the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.
Advisers: Luchezar L. Avramov and Srikanth B. Iyengar
Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, Caitlyn Parmelee
Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, Caitlyn Parmelee
Department of Mathematics: Dissertations, Theses, and Student Research
Mathematical modeling has broad applications in neuroscience whether we are modeling the dynamics of a single synapse or the dynamics of an entire network of neurons. In Part I, we model vesicle replenishment and release at the photoreceptor synapse to better understand how visual information is processed. In Part II, we explore a simple model of neural networks with the goal of discovering how network structure shapes the behavior of the network.
Vision plays an important role in how we interact with our environments. To fully understand how visual information is processed requires an understanding of the way signals are …
Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad
Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad
Department of Mathematics: Dissertations, Theses, and Student Research
We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.
Advisors: Mark Brittenham and Susan Hermiller
A Caputo Boundary Value Problem In Nabla Fractional Calculus, Julia St. Goar
A Caputo Boundary Value Problem In Nabla Fractional Calculus, Julia St. Goar
Department of Mathematics: Dissertations, Theses, and Student Research
Boundary value problems have long been of interest in the continuous differential equations context. However, with the advent of new areas like Nabla Fractional Calculus, we may consider such problems in new contexts. In this work, we will consider several right focal boundary value problems, involving a Caputo fractional difference operator, in the Nabla Fractional Calculus context. Properties of the Green's functions for each of these boundary value problems will be investigated and, in the case of a particular boundary value problem, used to establish the existence of positive solutions to a nonlinear version of the boundary value problem.
Adviser: …
Stable Local Cohomology And Cosupport, Peder Thompson
Stable Local Cohomology And Cosupport, Peder Thompson
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation consists of two parts, both under the overarching theme of resolutions over a commutative Noetherian ring R. In particular, we use complete resolutions to study stable local cohomology and cotorsion-flat resolutions to investigate cosupport.
In Part I, we use complete (injective) resolutions to define a stable version of local cohomology. For a module having a complete injective resolution, we associate a stable local cohomology module; this gives a functor to the stable category of Gorenstein injective modules. We show that this functor behaves much like the usual local cohomology functor. When there is only one non-zero local cohomology …
Cohen-Macaulay Dimension For Coherent Rings, Rebecca Egg
Cohen-Macaulay Dimension For Coherent Rings, Rebecca Egg
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation presents a homological dimension notion of Cohen-Macaulay for non-Noetherian rings which reduces to the standard definition in the case that the ring is Noetherian, and is inspired by the homological notion of Cohen-Macaulay for local rings developed by Gerko. Under this notion, both coherent regular rings (as defined by Bertin) and coherent Gorenstein rings (as defined by Hummel and Marley) are Cohen-Macaulay.
This work is motivated by Glaz's question regarding whether a notion of Cohen-Macaulay exists for coherent rings which satisfies certain properties and agrees with the usual notion when the ring is Noetherian. Hamilton and Marley gave …
Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey
Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey
Department of Mathematics: Dissertations, Theses, and Student Research
Let R be a commutative, Noetherian ring of characteristic p >0. Denote by f the Frobenius endomorphism, and let R^(e) denote the ring R viewed as an R-module via f^e. Following on classical results of Peskine, Szpiro, and Herzog, Marley and Webb use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor_i^R(R^(e),M) = 0 for all i >0 and infinitely many e >0. Using methods involving the derived category, we show that one only needs vanishing for dim R +1 consecutive values of …