Mathematics Commons^™

May Graduation, Samuel Coskey

Mathematics Faculty Publications and Presentations

Here I narrate the story of the last few days of my graduate program in mathematics. After the completion of the thesis and the delivery of the defense, several twists and turns await in the hours and even minutes before the last deadline.

New Jump Operators On Equivalence Relations, John D. Clemens, Samuel Coskey

Mathematics Faculty Publications and Presentations

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ-jump. We study the elementary properties of the Γ-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ, the Γ-jump is proper in the sense that for any Borel equivalence relation E the Γ-jump of E is strictly higher than E in the Borel reducibility hierarchy. On the other hand, there are examples of groups Γ for which the Γ-jump is not proper. To establish properness, …

Implicit Surface Reconstruction With A Curl-Free Radial Basis Function Partition Of Unity Method, Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright

Mathematics Faculty Publications and Presentations

Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set) approximation for an oriented point cloud using only information about (approximate) normals to the surface. The technique exploits the fundamental result from vector calculus that the normals to an implicit surface are curl-free. By using curl-free radial basis function (RBF) interpolation of the normals, we can extract a potential for the vector field whose zero-level surface approximates the point cloud. We use curl-free …

Tcr-L: An Analysis Tool For Evaluating The Association Between The T-Cell Receptor Repertoire And Clinical Phenotypes, Meiling Liu, Juna Goo, Yang Liu, Wei Sun, Michael C. Wu, Li Hsu, Qianchuan He

Mathematics Faculty Publications and Presentations

Background: T cell receptors (TCRs) play critical roles in adaptive immune responses, and recent advances in genome technology have made it possible to examine the T cell receptor (TCR) repertoire at the individual sequence level. The analysis of the TCR repertoire with respect to clinical phenotypes can yield novel insights into the etiology and progression of immune-mediated diseases. However, methods for association analysis of the TCR repertoire have not been well developed.

Methods: We introduce an analysis tool, TCR-L, for evaluating the association between the TCR repertoire and disease outcomes. Our approach is developed under a mixed effect modeling, where …

Relative Primeness And Borel Partition Properties For Equivalence Relations, John D. Clemens

Mathematics Faculty Publications and Presentations

We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations ��₂ and ��₁. We also discuss dichotomies related to primeness, and see that many natural questions related to Borel reducibility of equivalence …

Joint Full-Waveform Ground-Penetrating Radar And Electrical Resistivity Inversion Applied To Field Data Acquired On The Surface, Diego Domenzain, John Bradford, Jodi Mead

Mathematics Faculty Publications and Presentations

We exploit the different but complementary data sensitivities of ground-penetrating radar (GPR) and electrical resistivity (ER) by applying a multiphysics, multiparameter, simultaneous 2.5D joint inversion without invoking petrophysical relationships. Our method joins full-waveform inversion (FWI) GPR with adjoint derived ER sensitivities on the same computational domain. We incorporate a stable source estimation routine into the FWI-GPR. We apply our method in a controlled alluvial aquifer using only surface-acquired data. The site exhibits a shallow groundwater boundary and unconsolidated heterogeneous alluvial deposits. We compare our recovered parameters to individual FWI-GPR and ER results, and we compare them to log measurements of …

An Efficient High-Order Meshless Method For Advection-Diffusion Equations On Time-Varying Irregular Domains, Varun Shankar, Grady B. Wright, Aaron L. Fogelson

Mathematics Faculty Publications and Presentations

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains …

Homflypt Skein Theory, String Topology And 2-Categories, Uwe Kaiser

Mathematics Faculty Publications and Presentations

We show that relations in Homflypt type skein theory of an oriented 3-manifold M are induced from a 2-groupoid defined from the fundamental 2-groupoid of a space of singular links M. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental 2-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

Lag Time Between State-Level Policy Interventions And Change Points In Covid-19 Outcomes In The United States, Jaechoul Lee

Mathematics Faculty Publications and Presentations

State-level policy interventions have been critical in managing the spread of the new coronavirus. Here, we study the lag time between policy interventions and change in COVID-19 outcome trajectory in the United States. We develop a stepwise drifts random walk model to account for non-stationarity and strong temporal correlation and subsequently apply a change-point detection algorithm to estimate the number and times of change points in the COVID-19 outcome data. Furthermore, we harmonize data on the estimated change points with non-pharmaceutical interventions adopted by each state of the United States, which provides us insights regarding the lag time between the …

Efficient Inversion Of 2.5d Electrical Resistivity Data Using The Discrete Adjoint Method, Diego Domenzain, John Bradford, Jodi Mead

Mathematics Faculty Publications and Presentations

We have developed a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We have used the adjoint method directly in the discretized Maxwell’s steady state equation to compute …

A Partition Of Unity Method For Divergence-Free Or Curl-Free Radial Basis Function Approximation, Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright

Mathematics Faculty Publications and Presentations

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive …

A Fast And Accurate Algorithm For Spherical Harmonic Analysis On Healpix Grids With Applications To The Cosmic Microwave Background Radiation, Kathryn P. Drake, Grady B. Wright

Mathematics Faculty Publications and Presentations

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, …

Meager Sets, Games And Singular Cardinals, Liljana Babinkostova, Marion Scheepers

Mathematics Faculty Publications and Presentations

We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.

The Classification Of Countable Models Of Set Theory, John Clemens, Samuel Coskey, Samuel Dworetzky

Mathematics Faculty Publications and Presentations

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of ZFC.

A Bound For The Waring Rank Of The Determinant Via Syzygies, Mats Boij, Zach Teitler

Mathematics Faculty Publications and Presentations

We show that the Waring rank of the 3 × 3 determinant, previously known to be between 14 and 18, is at least 15. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the symmetric cactus rank of the 3 × 3 permanent is at least 14.

Quantifying Cds Sortability Of Permutations By Strategic Pile Size, Marisa Gaetz, Bethany Flanagan, Marion Scheepers, Meghan Shanks

Mathematics Faculty Publications and Presentations

The special purpose sorting operation, context directed swap (CDS), is an example of the block interchange sorting operation studied in prior work on permutation sorting. CDS has been postulated to model certain molecular sorting events that occur in the genome maintenance program of some species of ciliates. We investigate the mathematical structure of permutations not sortable by the CDS sorting operation. In particular, we present substantial progress towards quantifying permutations with a given strategic pile size, which can be understood as a measure of CDS non-sortability. Our main results include formulas for the number of permutations in S_n with …

A Robust Hyperviscosity Formulation For Stable Rbf-Fd Discretizations Of Advection-Diffusion-Reaction Equations On Manifolds, Varun Shankar, Grady B. Wright, Akil Narayan

Mathematics Faculty Publications and Presentations

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds �� ⊂ ℝ³ of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and …

On The Classification Of Vertex-Transitive Structures, John Clemens, Samuel Coskey, Stephanie Potter

Mathematics Faculty Publications and Presentations

We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above E₀ in complexity.

Using Literature To Engage Students Mathematically, Ann Wheeler, Joe Champion, Holly Dybvig

Mathematics Faculty Publications and Presentations

In this article, the authors share two lessons that incorporate children’s literature with the Pythagorean theorem and area to engage preservice teachers (PSTs) mathematically. Sample responses, example texts, and future work are discussed.

On A Hypothesis For ℵ0-Bounded Groups, Marion Scheepers

Mathematics Faculty Publications and Presentations

We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [6] holds while the classical Borel Conjecture fails.

Conjugacy For Homogeneous Ordered Graphs, Samuel Coskey, Paul Ellis

Mathematics Faculty Publications and Presentations

We show that for any countable homogeneous ordered graph G, the conjugacy problem for automorphisms of G is Borel complete. In fact we establish that each such G satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of G is Borel reducible to the conjugacy relation on automorphisms of G.

Bottle Filling Task Reasoning: A Comparison Of Matching Versus Constructed Student Responses, Dominique Banner, Laurie Overman Cavey

Mathematics Faculty Publications and Presentations

In this paper, we compare the levels of reasoning elicited during the completion of three versions of a bottle filling task: high school level matching; middle school level matching and constructed response. The goal of the tasks was to make visible secondary students covariational reasoning methods. Video of students completing the task while explaining their reasoning during one-onone interviews were analyzed. Analysis demonstrated a wide range of reasoning when provided a matching version with a greater incidence of accuracy with students who exhibited lower levels of reasoning. Conversely, the constructed response task demonstrated higher levels of reasoning more consistently with …

Efficient Blind Image Deblurring Using Nonparametric Regression And Local Pixel Clustering, Yicheng Kang, Partha Sarathi Mukherjee, Peihua Qiu

Mathematics Faculty Publications and Presentations

Blind image deblurring is a challenging ill-posed problem. It would have an infinite number of solutions even in cases when an observed image contains no noise. In reality, however, observed images almost always contain noise. The presence of noise would make the image deblurring problem even more challenging because the noise can cause numerical instability in many existing image deblurring procedures. In this paper, a novel blind image deblurring approach is proposed, which can remove both pointwise noise and spatial blur efficiently without imposing restrictive assumptions on either the point spread function (psf) or the true image. It even allows …

Alzheimer’S Disease And Alpha-Synuclein Neuropathology In The Olfactory Bulbs Of Children And Young Adults ≤40years Exposed To High Levels Of Fine Particulate Matter Air Pollution In Metropolitan Mexico City: Apoe4 Carriers At Higher Risk Of Suicide Accelerate Their Olfactory Bulb Damage, Partha S. Mukherjee

Mathematics Faculty Publications and Presentations

There is growing evidence that air pollution is a risk factor for a number of neurodegenerative diseases, most notably Alzheimer’s (AD) and Parkinson’s (PD). It is generally assumed that the pathology of these diseases arises only later in life and commonly begins within olfactory eloquent pathways prior to the onset of the classical clinical symptoms. The present study demonstrates that chronic exposure to high levels of air pollution results in AD- and PD-related pathology within the olfactory bulbs of children and relatively young adults ranging in age from 11 months to 40 years. The olfactory bulbs (OBs) of 179 residents …

Mesh-Free Semi-Lagrangian Methods For Transport On A Sphere Using Radial Basis Functions, Varun Shankar, Grady B. Wright

Mathematics Faculty Publications and Presentations

We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new …

Secondary Mathematics Teachers’ Planned Approaches For Teaching Standard Deviation, Maryann E. Huey, Joe Champion, Stephanie Casey, Nicholas H. Wasserman

Mathematics Faculty Publications and Presentations

Research-based guidelines for learning variation exist (e.g., Franklin et al., 2007; Garfield, delMas, & Chance, 2007), but little is known about how teachers plan to teach standard deviation, or how these plans align with recent recommendations. In this article, we survey lesson plans designed by inservice and preservice secondary mathematical teachers. We report on the accuracy, technology usage, and visual representations in the lesson plans. We consider how many elements are used, the level of conceptual development, and the mathematical nature. Findings support differences between preservice and master’s level students in education, as well as a tendency by in-service teachers …

Statistics As Unbiased Estimators: Exploring The Teaching Of Standard Deviation, Nicholas H. Wasserman, Stephanie Casey, Joe Champion, Maryann Huey

Mathematics Faculty Publications and Presentations

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role …

Combustion-Derived Nanoparticles, The Neuroenteric System, Cervical Vagus, Hyperphosphorylated Alpha Synuclein And Tau In Young Mexico City Residents, Lilian Calderón-Garcidueñas, Rafael Reynoso-Robles, Beatriz Pérez-Guillé, Partha S. Mukherjee, Angélica Gónzalez-Maciel

Mathematics Faculty Publications and Presentations

Mexico City (MC) young residents are exposed to high levels of fine particulate matter (PM_2.5), have high frontal concentrations of combustion-derived nanoparticles (CDNPs), accumulation of hyperphosphorylated aggregated α-synuclein (α-Syn) and early Parkinson's disease (PD). Swallowed CDNPs have easy access to epithelium and submucosa, damaging gastrointestinal (GI) barrier integrity and accessing the enteric nervous system (ENS). This study is focused on the ENS, vagus nerves and GI barrier in young MC v clean air controls. Electron microscopy of epithelial, endothelial and neural cells and immunoreactivity of stomach and vagus to phosphorylated ɑ-synuclein Ser129 and Hyperphosphorylated-Tau (Htau) …

Cardinal Characteristics And Countable Borel Equivalence Relations, Samuel Coskey, Scott Schneider

Mathematics Faculty Publications and Presentations

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number ��. We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding to the splitting number �� coincides with smoothness. We then settle many of the implication relationships between the properties; …

A Multi-Resolution And Adaptive 3-D Image Denoising Framework With Applications In Medical Imaging, Partha Sarathi Mukherjee

Mathematics Faculty Publications and Presentations

Due to recent increase in the usage of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images (fMRI), research on 3-D image processing becomes important. Observed 3- D images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Most image denoising methods in the literature are for 2-D images. However, their direct generalizations to 3- D images can not preserve complicated edge structures well. Because, the edge structures in a 3-D edge surface …