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Full-Text Articles in Physical Sciences and Mathematics

Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin Nov 2023

Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin

Doctoral Dissertations

In this thesis we study the intersection cohomology of arrangement Schubert varieties with coefficients in a rank one local system on a hyperplane arrangement complement. We prove that the intersection cohomology can be computed recursively in terms of certain polynomials, if a local system has only $\pm 1$ monodromies. In the case where the hyperplane arrangement is generic central or equivalently the associated matroid is uniform and the local system has only $\pm 1$ monodromies, we prove that the intersection cohomology is a combinatorial invariant. In particular when the hyperplane arrangement is associated to the uniform matroid of rank $n-1$ …


When To Hold And When To Fold: Studies On The Topology Of Origami And Linkages, Mary Elizabeth Lee Nov 2023

When To Hold And When To Fold: Studies On The Topology Of Origami And Linkages, Mary Elizabeth Lee

Doctoral Dissertations

Linkages and mechanisms are pervasive in physics and engineering as models for a
variety of structures and systems, from jamming to biomechanics. With the increase
in physical realizations of discrete shape-changing materials, such as metamaterials,
programmable materials, and self-actuating structures, an increased understanding
of mechanisms and how they can be designed is crucial. At a basic level, linkages
or mechanisms can be understood to be rigid bars connected at pivots around which
they can rotate freely. We will have a particular focus on origami-like materials, an
extension to linkages with the added constraint of faces. Self-actuated versions typ-
ically start …


Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu Nov 2023

Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu

Doctoral Dissertations

In this thesis, we study the mathematical model of one-dimensional microscopic heat conduction of gas particles, applying both both analytical and numerical approaches. The macroscopic law of heat conduction is the renowned Fourier’s law J = −k∇T, where J is the local heat flux density, T(x, t) is the temperature gradient, and k is the thermal conductivity coefficient that characterizes the material’s ability to conduct heat. Though Fouriers’s law has been discovered since 1822, the thorough understanding of its microscopic mechanisms remains challenging [3] (2000). We assume that the microscopic model of heat conduction is a hard ball system. The …


Positive Factorizations Via Planar Mapping Classes And Braids, Richard E. Buckman Nov 2023

Positive Factorizations Via Planar Mapping Classes And Braids, Richard E. Buckman

Doctoral Dissertations

In this thesis we seek to better understand the planar mapping class group in
order to find factorizations of boundary multitwists, primarily to generate and study
symplectic Lefschetz pencils by lifting these factorizations. Traditionally this method
is applied to a disk or sphere with marked points, utilizing factorizations in the stan-
dard and spherical braid groups, whereas in our work we allow for multiple boundary components. Dehn twists along these boundaries give rise to exceptional sections of Lefschetz fibrations over the 2–sphere, equivalently, to Lefschetz pencils with base points. These methods are able to derive an array of known examples …


Semi-Infinite Flags And Zastava Spaces, Andreas Hayash Nov 2023

Semi-Infinite Flags And Zastava Spaces, Andreas Hayash

Doctoral Dissertations

ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct …


Facets Of The Union-Closed Polytope, Daniel Gallagher Nov 2023

Facets Of The Union-Closed Polytope, Daniel Gallagher

Doctoral Dissertations

In the haze of the 1970s, a conjecture was born to unknown parentage...the union-closed sets conjecture. Given a family of sets $\FF$, we say that $\FF$ is union-closed if for every two sets $S, T \in \FF$, we have $S \cup T \in \FF$. The union-closed sets conjecture states that there is an element in at least half of the sets of any (non-empty) union-closed family. In 2016, Pulaj, Raymond, and Theis reinterpreted the conjecture as an optimization problem that could be formulated as an integer program. This thesis is concerned with the study of the polytope formed by taking …


An Exploration Of Absolute Minimal Degree Lifts Of Hyperelliptic Curves, Justin A. Groves Aug 2023

An Exploration Of Absolute Minimal Degree Lifts Of Hyperelliptic Curves, Justin A. Groves

Doctoral Dissertations

For any ordinary elliptic curve E over a field with non-zero characteristic p, there exists an elliptic curve E over the ring of Witt vectors W(E) for which we can lift the Frobenius morphism, called the canonical lift. Voloch and Walker used this theory of canonical liftings of elliptic curves over Witt vectors of length 2 to construct non-linear error-correcting codes for characteristic two. Finotti later proved that for longer lengths of Witt vectors there are better lifts than the canonical. He then proved that, more generally, for hyperelliptic curves one can construct a lifting over …


The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs Aug 2023

The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs

Doctoral Dissertations

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, …


Adaptive And Topological Deep Learning With Applications To Neuroscience, Edward Mitchell May 2023

Adaptive And Topological Deep Learning With Applications To Neuroscience, Edward Mitchell

Doctoral Dissertations

Deep Learning and neuroscience have developed a two way relationship with each informing the other. Neural networks, the main tools at the heart of Deep Learning, were originally inspired by connectivity in the brain and have now proven to be critical to state-of-the-art computational neuroscience methods. This dissertation explores this relationship, first, by developing an adaptive sampling method for a neural network-based partial different equation solver and then by developing a topological deep learning framework for neural spike decoding. We demonstrate that our adaptive scheme is convergent and more accurate than DGM -- as long as the residual mirrors the …


Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein May 2023

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

Doctoral Dissertations

Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …


Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson May 2023

Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson

Doctoral Dissertations

We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.


Large Deviations For Self Intersection Local Times Of Ornstein-Uhlenbeck Processes, Apostolos Gournaris May 2023

Large Deviations For Self Intersection Local Times Of Ornstein-Uhlenbeck Processes, Apostolos Gournaris

Doctoral Dissertations

In the area of large deviations, people concern about the asymptotic computation of small probabilities on an exponential scale. The general form of large deviations can be roughly described as: P{Yn ∈ A} ≈ exp{−bnI(A)} (n → ∞), for a random sequence {Yn}, a positive sequence bn with bn → ∞, and a coefficient I(A) ≥ 0. In applications, we often concern about the probability that the random variables take large values, that is we concern about the P{Yn ≥ λ}, where λ > 0. Here, we consider the Ornstein-Uhlenbeck process, study the properties of the local times and self intersection …


Advances In Differentially Methylated Region Detection And Cure Survival Models, Daniel Ahmed Alhassan Jan 2023

Advances In Differentially Methylated Region Detection And Cure Survival Models, Daniel Ahmed Alhassan

Doctoral Dissertations

"This dissertation focuses on two areas of statistics: DNA methylation and survival analysis. The first part of the dissertation pertains to the detection of differentially methylated regions in the human genome. The varying distribution of gaps between succeeding genomic locations, which are represented on the microarray used to quantify methylation, makes it challenging to identify regions that have differential methylation. This emphasizes the need to properly account for the correlation in methylation shared by nearby locations within a specific genomic distance. In this work, a normalized kernel-weighted statistic is proposed to obtain an optimal amount of "information" from neighboring locations …


Recurrent Event Data Analysis With Mismeasured Covariates, Ravinath Alahakoon Mudiyanselage Jan 2023

Recurrent Event Data Analysis With Mismeasured Covariates, Ravinath Alahakoon Mudiyanselage

Doctoral Dissertations

"Consider a study with n units wherein every unit is monitored for the occurrence of an event that can recur with random end of monitoring. At each recurrence, p concomitant variables associated to the event recurrence are recorded with q (q ≤ p) collected with errors. Of interest in this dissertation is the estimation of the regression parameters of event time regression models accounting for the covariates. To circumvent the problem of bias and consistency associated with model's parameter estimation in the presence of measurement errors, we propose inference for corrected estimating functions with well-behaved roots under additive measurement errors …


We Are Still Playing: A Meta-Analysis Of Game-Based Learning In Mathematics Education, Thomas Conmy Jan 2023

We Are Still Playing: A Meta-Analysis Of Game-Based Learning In Mathematics Education, Thomas Conmy

Doctoral Dissertations

The purpose of this meta-analysis was to investigate the effectiveness of the use of games as part of mathematics instruction on academic achievement in grades Kindergarten to 12 in the United States. There were 17 studies selected for investigation published from 2010 to 2023 that focused on game-based learning and mathematics. This meta-analysis fills the gap in the knowledge by examining classes that are using game based learning across three platforms of instruction: nondigital games, digital on computers, and mobile devices. The findings from this meta-analysis suggest that the usage of game-based learning in a classroom has a positive effect …


Efficient High Order Ensemble For Fluid Flow, John Carter Jan 2023

Efficient High Order Ensemble For Fluid Flow, John Carter

Doctoral Dissertations

"This thesis proposes efficient ensemble-based algorithms for solving the full and reduced Magnetohydrodynamics (MHD) equations. The proposed ensemble methods require solving only one linear system with multiple right-hand sides for different realizations, reducing computational cost and simulation time. Four algorithms utilize a Generalized Positive Auxiliary Variable (GPAV) approach and are demonstrated to be second-order accurate and unconditionally stable with respect to the system energy through comprehensive stability analyses and error tests. Two algorithms make use of Artificial Compressibility (AC) to update pressure and a solenoidal constraint for the magnetic field. Numerical simulations are provided to illustrate theoretical results and demonstrate …


Essays On Conditional Heteroscedastic Time Series Models With Asymmetry, Long Memory, And Structural Changes, K C M R Anjana Bandara Yatawara Jan 2023

Essays On Conditional Heteroscedastic Time Series Models With Asymmetry, Long Memory, And Structural Changes, K C M R Anjana Bandara Yatawara

Doctoral Dissertations

"The volatility of asset returns is usually time-varying, necessitating the introduction of models with a conditional heteroskedastic variance structure. In this dissertation, several existing formulations, motivated by the Generalized Autoregressive Conditional Heteroskedastic (GARCH) type models, are further generalized to accommodate more dynamic features of asset returns such as asymmetry, long memory, and structural breaks. First, we introduce a hybrid structure that combines short-memory asymmetric Glosten, Jagannathan, and Runkle (GJR) formulation and the long-memory fractionally integrated GARCH (FIGARCH) process for modeling financial volatility. This formulation not only can model volatility clusters and capture asymmetry but also considers the characteristic of long …