Numerical Analysis and Computation Commons^™

Analysis And Construction Of Artificial Neural Networks For The Heat Equations, And Their Associated Parameters, Depths, And Accuracies., Shakil Ahmed Rafi

Graduate Theses and Dissertations

This dissertation seeks to explore a certain calculus for artificial neural networks. Specifically we will be looking at versions of the heat equation, and exploring strategies on how to approximate them.

Our strategy towards the beginning will be to take a technique called Multi-Level Picard (MLP), and present a simplified version of it showing that it converges to a solution of the equation (^∂/_∂t u_d ) (t, x) = (∇² _x u_d)(t, x).

We will then take a small detour exploring the viscosity super-solution properties of solutions to such …

Efficient Hierarchical Space-Time Models For Large Areal Datasets With Application To Forest Inventory Mapping Using Remote Sensing Imagery, Md Kamrul Hasan Khan

Graduate Theses and Dissertations

The focus of this dissertation is development of a novel hierarchical framework, that can be used for predictive modeling of Forest Inventory and Analysis (FIA) data over large regions. This dissertation has two significant contributions. Based on a study region in north-central Wisconsin, we analyze satellite imagery, along with a sample of national forest inventory field plots, to monitor and predict changes in forest conditions over time. The auxiliary data from the satellite imagery of this region are relatively dense in space and time, and can be used to learn how forest conditions changed over that decade. However, these records …

Lyubeznik Numbers Of Unmixed Edge Ideals, Sara Rae Jones

Graduate Theses and Dissertations

Lyubeznik numbers, defined in terms of local cohomology, are invariants of local rings that are able to detect many algebraic and geometric properties. Notably they recognize topological behaviors of various structures associated to their rings. We will discuss computations of these numbers for unmixed edge ideals by giving a completely combinatorial construction which realizes the connectedness information captured by these numbers.

Cohomology Of The Symmetric Group With Twisted Coefficients And Quotients Of The Braid Group, Trevor Nakamura

Graduate Theses and Dissertations

In 2014 Brendle and Margalit proved the level $4$ congruence subgroup of the braid group, $B_{n}[4]$, is the subgroup of the pure braid group generated by squares of all elements, $PB_{n}^{2}$. We define the mod $4$ braid group, $\Z_{n}$, to be the quotient of the braid group by the level 4 congruence subgroup, $B_{n}/B_{n}[4]$. In this dissertation we construct a group presentation for $\Z_{n}$ and determine a normal generating set for $B_{n}[4]$ as a subgroup of the braid group. Further work by Kordek and Margalit in 2019 proved $\Z_{n}$ is an extension of the symmetric group, $S_{n}$, by $\mathbb{Z}_{2}^{\binom{n}{2}}$. A …

Grid Homology Invariants For Singular Legendrian Links, Richard Michael Shumate

Graduate Theses and Dissertations

If $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ are two oriented singular Legendrian links that are Legendrian isotopic, we first construct front diagram representations of $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ that have a natural allowable singular gird diagram associated to them. These allowable singular grid diagrams will always correspond to singular Legendrian links. The grid Legendrian invariants, $\lambda^{\pm}$, in the nonsingular grid homology theory have a natural extension to the singular grid theory, and are natural under the newly defined singular grid moves. This gives an invariant of singular Legendrian links, and in fact, a broader class of singular links.

Development Of An Effect Size To Classify The Magnitude Of Dif In Dichotomous And Polytomous Items, James D. Weese

Graduate Theses and Dissertations

A standardized effect size for the SIBTEST/POLYSIBTEST procedure is proposed, allowing for Differential Item Functioning (DIF) to be classified with a single set of DIF heuristics regardless of whether data are dichotomous or polytomous. This proposed standardized effect size accounts for both variability in responses and whether participants are included in the SIBTEST/POLYSIBTEST calculations. First, a new set of unstandardized effect size heuristics are established for dichotomous data that are more aligned with Educational Testing Service (ETS) standards using two and three parameter logistic (2PL and 3PL) models. Second, a standardized effect size is proposed and compared to other DIF …

Closed Range Composition Operators On Bmoa, Kevser Erdem

Graduate Theses and Dissertations

Let φ be an analytic self-map of the unit disk D. The composition operator with symbol φ is denoted by Cφ. Reverse Carleson type conditions, counting functions and sampling sets are important tools to give a complete characterization of closed range composition operators on BMOA and on Qp for all p ∈ (0,∞).

Let B denote the Bloch space, let H2 denote the Hardy space. We show that if Cφ is closed range on B or on H2 then it is also closed range on BMOA. Closed range composition operators Cφ : B → BMOA are also characterized. Laitila found …

General Sampling Schemes For The Bergman Spaces, Newton Foster

Graduate Theses and Dissertations

A characterization of sampling sequences for the Bergman spaces was originally provided by Seip and later expanded upon by Schuster. We consider a generalized notion of sampling using the infimum norm of the quotient space. Adapting some old techniques, we provide a characterization of general sampling sequences in terms of the lower uniform density.

Hardy Space Properties Of The Cauchy Kernel Function For A Strictly Convex Planar Domain, Belen Espinosa Lucio

Graduate Theses and Dissertations

This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\mathbb{C}^N$, the Henkin-Ram\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\in(0,N)$.

The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\subset\mathbb{C}$ of class $C^2$ if we fix $\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function

\mathcal{K}(\zeta,z)=\frac{1}{2\pi i}\frac{1}{\zeta-z}

as a …

Pointwise Schauder Estimates Of Parabolic Equations In Carnot Groups, Heather Arielle Griffin

Graduate Theses and Dissertations

Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown …