Mathematics Commons^™

Conditional Quantization For Uniform Distributions On Line Segments And Regular Polygons, Tsianna Danielle Dominguez

Theses and Dissertations

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this thesis, we have investigated the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where m ≥ 3, inscribed in a unit circle.

A Study Of Quantitative Reasoning Instructors’ Choices And Motivations When Teaching Quantitative Reasoning For The First Time, Trish Ann Harding

Theses and Dissertations

This qualitative study delves into the instructional decision-making processes of post-secondary instructors teaching quantitative reasoning (QR) courses for the first time. The study aims to address the gap in understanding how first-time QR instructors navigate the complexities of curriculum design and pedagogical strategies, and how these experiences contribute to their professional development. The research questions center on identifying the instructional decisions made by these instructors, exploring the factors influencing their decision-making, and understanding the impact of exercising agency has on their professional development. Through in-depth exploration, this study seeks to shed light on the challenges and opportunities faced by first-time …

Weighted Ehrhart Theory: Extending Stanley's Nonnegativity Theorem, Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

School of Mathematical & Statistical Sciences Faculty Publications

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of …

Statistics For Iwasawa Invariants Of Elliptic Curves, Ii, Debanjana Kundu, Anwesh Ray

School of Mathematical & Statistical Sciences Faculty Publications

We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.

Study Of Hybrid Nanofluid Flow In A Stationary Cone-Disk System With Temperature-Dependent Fluid Properties, A. S. John, Mahanthesh Basavarajappa, G. Lorenzini

School of Mathematical & Statistical Sciences Faculty Publications

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation …

Normalized Ground State Of A Mixed Dispersion Nonlinear Schrodinger Equation With Combined Power-Type Nonlinearities, Zhouji Ma, Xiaojun Chang, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.

The Inextricability Of Students’ Mathematical And Physical Reasoning In Quantum Mechanics Problems, Kaitlyn Stephens Serbin, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

Reasoning with mathematics plays an important role in university students’ learning throughout their courses in the scientific disciplines, such as physics. In addition to understanding mathematical concepts and procedures, physics students often must mathematize physical constructs in terms of their associated mathematical structures and interpret mathematical entities in terms of the physical context. In this study, we investigate physics students’ reasoning about mathematics in relation to physics content addressed in two quantum mechanics problems. Through qualitative analysis of interview data from twelve students, results show that 1) students use intricate, nonuniform problem-solving methods with reasoning that moves fluidly between structural …

The Eliahou-Kervaire Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Alexis Hardesty

School of Mathematical & Statistical Sciences Faculty Publications

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

The Invariantring Package For Macaulay2, Luigi Ferraro, Federico Galetto, Francesca Gandini, Hang Huang, Matthew Mastroeni, Xianglong Ni

School of Mathematical & Statistical Sciences Faculty Publications

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups, as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

Regularization By Deep Learning In Signal Processing, Carlos Ramirez Villamarin, Erwin Suazo, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we explore a new idea of using deep learning representations as a principle for regularization in inverse problems for digital signal processing. Specifically, we consider the standard variational formulation, where a composite function encodes a fidelity term that quantifies the proximity of the candidate solution to the observations (under a physical process), and a second regularization term that constrains the space of solutions according to some prior knowledge. In this work, we investigate deep learning representations as a means of fulfilling the role of this second (regularization) term. Several numerical examples are presented for signal restoration under …

Extensions Of Polynomial Plank Covering Theorems, Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

School of Mathematical & Statistical Sciences Faculty Publications

We prove the complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.

Nontrivial Invariant Subspaces Of Linear Operator Pencils, Jaewoong Kim, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair T=(T1,T2) and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of T and the linear pencil of the original pair T of bounded operators with dense ranges.

Finite Element Solution Of Crack-Tip Fields For An Elastic Porous Solid With Density-Dependent Material Moduli And Preferential Stiffness, Hyun C. Yoon, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the …

Functional Data Learning Using Convolutional Neural Networks, Jose Galarza, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we show how convolutional neural networks (CNNs) can be used in regression and classification learning problems for noisy and non-noisy functional data (FD). The main idea is to transform the FD into a 28 by 28 image. We use a specific but typical architecture of a CNN to perform all the regression exercises of parameter estimation and functional form classification. First, we use some functional case studies of FD with and without random noise to showcase the strength of the new method. In particular, we use it to estimate exponential growth and decay rates, the bandwidths of …

Using A Two-Way Engagement Community- And Family-Centered Pedagogy To Prepare Pre-Service Mathematics Teachers In A Hispanic-Serving Institution, Olga Ramirez, Mayra Ortiz Galarza, Luis M. Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

Research on effective methods to prepare pre-service teachers (PSTs) in teaching mathematics to K-12 Latin* students has been gaining significant momentum. These efforts have focused, in part, on promoting pedagogical practices that recognize and incorporate the culture and language that K-12 Latin* students and their communities share. As teacher educators, we argue that if we are to further prepare PSTs to serve the needs of such increasingly diversifying K-12 student population, the same pedagogical focus on the learner’s cultural wealth should also be applied to the preparation of PSTs themselves, especially among Latin* PSTs in Hispanic Serving Institutions (HSI) like …

Gauss Circle Problem Over Smooth Integers, Ankush Goswami

School of Mathematical & Statistical Sciences Faculty Publications

For a positive integer 𝑛, let 𝑟2(𝑛) be the number of representations of 𝑛 as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that 𝑅(𝑥) ∶= ∑ 𝑛≤𝑥 𝑟2(𝑛) ∼ 𝜋𝑥. Let 𝑃(𝑛) denote the largest prime factor of 𝑛 and let 𝑆(𝑥, 𝑦) ∶= {𝑛 ≤ 𝑥 ∶ 𝑃(𝑛) ≤ 𝑦}. In this paper, we study the asymptotic behavior of 𝑅(𝑥, 𝑦) ∶= ∑ 𝑛∈𝑆(𝑥,𝑦) 𝑟2(𝑛) for various ranges of 2 ≤ 𝑦 ≤ 𝑥. For 𝑦 in a …

A Finite Element Model For Hydro-Thermal Convective Flow In A Porous Medium: Effects Of Hydraulic Resistivity And Thermal Diffusivity, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy’s law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element discretization for the hydro-thermal convective flow model. The well-posedness of the proposed discrete Galerkin finite element formulation is guaranteed due to the decoupling property and the linearity of the numerical method. Computational experiments confirm the optimal convergence rates for a manufactured solution. Several numerical results …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of n-points and the nth conditional quantization errors …

Machine Learning For Wireless Network Throughput Prediction, Gustavo A. Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

This paper analyzes a dataset containing radio frequency (RF) measurements and Key Performance Indicators (KPIs) captured at 1876.6MHz with a bandwidth of 10MHz from an operational 4G LTE network in Nigeria. The dataset includes metrics such as RSRP (Reference Signal Received Power), which measures the power level of reference signals; RSRQ (Reference Signal Received Quality), an indicator of signal quality that provides insight into the number of users sharing the same resources; RSSI (Received Signal Strength Indicator), which gauges the total received power in a bandwidth; SINR (Signal to Interference plus Noise Ratio), a measure of signal quality considering both …

On Modeling Arterial Blood Flow With Or Without Solute Transport And In Presence Of Atherosclerosis, Daniel N. Riahi, Saulo Orizaga

School of Mathematical & Statistical Sciences Faculty Publications

In this article, we review previous studies of modeling problems for blood flow with or without transport of a solute in a section of arterial blood flow and in the presence of atherosclerosis. Moreover, we review problems of bio-fluid dynamics within the field of biophysics. In most modeling cases, the presence of red blood cells in the plasma is taken into account either by using a two-phase flow approach, where blood plasma is considered as one phase and red blood cells are counted as another phase, or by using a variable viscosity formula that accounts for the amount of hematocrit …

Use Of Hydroxychloroquine In Multidrug Protocols For Sars-Cov-2, Eleftherios Gkioulekas, Peter A. Mccullough

School of Mathematical & Statistical Sciences Faculty Publications

We review the available evidence supporting the use of hydroxychloroquine-based multidrug protocols in the treatment of COVID-19, in response to a recently published editorial in the Tasman Medical Journal.

The Coulomb Gauge In Non-Associative Gauge Theory, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop L , its tangent algebra l , a finite-dimensional Lie group Ψ , that is the pseudoautomorphism group of L , a smooth manifold M with a principal Ψ -bundle P , and associated bundles Q and A with fibers L and l , respectively. A configuration in this theory is …

Conditional Constrained And Unconstrained Quantization For Probability Distributions, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we present the idea of conditional quantization for a Borel probability measure P on a normed space Rk. We introduce the concept of conditional quantization in both constrained and unconstrained scenarios, along with defining the conditional quantization errors, dimensions, and coefficients in each case. We then calculate these values for specific probability distributions. Additionally, we demonstrate that for a Borel probability measure, the lower and upper quantization dimensions and coefficients do not depend on the conditional set of the conditional quantization in both constrained and unconstrained quantization.

Modeling The Effect Of Observational Social Learning On Parental Decision-Making For Childhood Vaccination And Diseases Spread Over Household Networks, Tamer Oraby, Andras Balogh

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we introduce a novel model for parental decision-making about vaccinations against a childhood disease that spreads through a contact network. This model considers a bilayer network comprising two overlapping networks, which are either Erdős–Rényi (random) networks or Barabási–Albert networks. The model also employs a Bayesian aggregation rule for observational social learning on a social network. This new model encompasses other decision models, such as voting and DeGroot models, as special cases. Using our model, we demonstrate how certain levels of social learning about vaccination preferences can converge opinions, influencing vaccine uptake and ultimately disease spread. In addition, …

Flow And Stability Of Three-Dimensional Rotating Viscoelastic Jet, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

We consider rotationally driven nonlinear viscoelastic jet. In contrast to previous unrealistic planar studies with no gravity effect, we investigate here realistic three-dimensional jet with gravity effect. Using theoretical and computational methods, we develop models for such jet and its stability and determine the solutions. We find three-dimensional jet with gravity effect leads to faster and thinner jet. For jet arc length higher than its exit diameter, our calculation with typical parameter values shows that jet radius is smaller by a factor of more than 1.4 and jet speed is higher by factor of more than 2 as are compared …

Exploring International Educators' Learning About Local And Global Social Justice In A Virtual Community Of Practice, Bima Sapkota, Xuwei Luo, Muna Sapkota, Murat Akarsu, Emmanuel Deogratias, Daphne Fauber, Rose Mbewe, Fidelis Mumba, Ram Krishna Panthi, Jill Newton

School of Mathematical & Statistical Sciences Faculty Publications

In this chapter, the authors report themes that emerged when a cross-cultural team of researchers involved in a virtual international community of practice (Global Social Justice in Education-GSJE) investigated reflections on activities focused on social justice in local and global contexts. The findings suggested that the activities elicited GSJE community members' understandings of the complexities of social justice associated with naming practices, privilege, and the arts within their own and across contexts. The authors discuss implications of the activities to advance diverse educators' understanding of social justice in global and local contexts. They also unpack the opportunities and challenges that …

Assessing Concepts, Procedures, And Cognitive Demand Of Chatgpt-Generated Mathematical Tasks, Bima Sapkota, Liza Bondurant

School of Mathematical & Statistical Sciences Faculty Publications

In November 2022, ChatGPT, an Artificial Intelligence (AI) large language model (LLM) capable of generating human-like responses, was launched. ChatGPT has a variety of promising applications in education, such as using it as thought-partner in generating curricular resources. However, scholars also recognize that the use of ChatGPT raises concerns, such as outputs that are inaccurate, nonsensical, or vague. We, two mathematics teacher educators, engaged in a collaborative self-study using qualitative descriptive approaches to investigate the procedures, concepts, and cognitive demand of ChatGPT-generated mathematical tasks focused on fraction multiplication using the area model approach. We found that the ChatGPT-generated tasks were …

Ivermectin, Colleen Aldous, Eleftherios Gkioulekas, Philip Oldfield

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Brillouin Zones Of Integer Lattices And Their Perturbations, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, Mathijs Wintraecken

School of Mathematical & Statistical Sciences Faculty Publications

For a locally finite set, 𝐴⊆ℝ𝑑 , the 𝑘 th Brillouin zone of 𝑎∈𝐴 is the region of points 𝑥∈ℝ𝑑 for which ‖𝑥−𝑎‖ is the 𝑘 th smallest among the Euclidean distances between 𝑥 and the points in 𝐴 . If 𝐴 is a lattice, the 𝑘 th Brillouin zones of the points in 𝐴 are translates of each other, and together they tile space. Depending on the value of 𝑘 , they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our …

Heuristics For Anti-Cyclotomic ℤp-Extensions, Debanjana Kundu, Lawrence C. Washington

School of Mathematical & Statistical Sciences Faculty Publications

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model “how often” the p-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants 𝜆 and 𝜇 usually vanish for imaginary quadratic fields where p is non-split.