Mathematics Commons^™

Hilbert’S 10th Problem Via Mordell Curves, Somnath Jha, Debanjana Kundu, Dipramit Majumdar

School of Mathematical & Statistical Sciences Faculty Publications

We show that for 5 / 6 -th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of Q ( ζ 3 , 3 √ p ) . We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of Q ( ζ 3 , √ D , 3 √ p ) for every D ∈ S and for every prime p ≡ 2 , 5 ( mod 9 ) . We use the CM elliptic curves y 2 = x …

Semilinear Klein-Gordon Equation In Space-Time Of Black Hole Which Is Gaining Mass In The Universe With Accelerating Expansion, Karen Yagdjian, Anahit Galstyan

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we consider the propagation of waves in the space-time of a single black hole with a static Schwarzschild radius in the expanding universe, namely, the solutions of the linear and semilinear Klein-Gordon equations.

Modelling The Behavior Of Human Crowds As Coupled Active-Passive Dynamics Of Interacting Particle Systems, Thoa Thieu, Roderick Melnik

School of Mathematical & Statistical Sciences Faculty Publications

The modelling of human crowd behaviors offers many challenging questions to science in general. Specifically, the social human behavior consists of many physiological and psychological processes which are still largely unknown. To model reliably such human crowd systems with complex social interactions, stochastic tools play an important role for the setting of mathematical formulations of the problems. In this work, using the description based on an exclusion principle, we study a statistical-mechanics-based lattice gas model for active-passive population dynamics with an application to human crowd behaviors. We provide representative numerical examples for the evacuation dynamics of human crowds, where the …

Voronoi Conjecture For Five-Dimensional Parallelohedra, Alexey Garber, Alexander Magazinov

School of Mathematical & Statistical Sciences Faculty Publications

We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope P tiles ℝ5 with translations, then P is an affine image of the Dirichlet-Voronoi polytope for a five-dimensional lattice.
Our proof is based on an exhaustive combinatorial analysis of possible dual 3-cells and incident dual 4-cells encoding local structures around two-dimensional faces of five-dimensional parallelohedron P and their edges aiming to prove existence of a free direction for P paired with new properties established for parallelohedra (in any dimension) that have a free direction that guarantee the Voronoi conjecture for P.

Group Approach To Geometric Quantization Of Two Physical Groups, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Physical systems as a rule are associated with a symmetry group. One way of quantizing the system is to use this symmetry group to construct a geometric form of quantization procedure. This group quantization procedure is discussed and illustrated here on two different systems. Group cohomology and extensions of groups play a key role. Many important groups in physics can be extended by the local U(1) group. The process is introduced here and applied to two nontrivial systems.

Stochastic Games Of Parental Vaccination Decision Making And Bounded Rationality, Andras Balogh, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Vaccination is an effective strategy to prevent the spread of diseases. However, hesitancy and rejection of vaccines, particularly in childhood immunizations, pose challenges to vaccination efforts. In that case, according to rational decision-making and classical utility theory, parents weigh the costs of vaccination against the costs of not vaccinating their children. Social norms influence these parental decision-making outcomes, deviating their decisions from rationality. Additionally, variability in values of utilities stemming from stochasticity in parents' perceptions over time can lead to further deviations from rationality. In this paper, we employ independent white noises to represent stochastic fluctuations in parental perceptions of …

Order-2 Delaunay Triangulations Optimize Angles, Herbert Edelsbrunner, Alexey Garber, Morteza Saghafian

School of Mathematical & Statistical Sciences Faculty Publications

The local angle property of the (order-1) Delaunay triangulations of a generic set in R2" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">R2R2 asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">R2R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations …

Ramanujan–Fine Integrals For Level 10, Shaun Cooper, Timothy Huber, Jeffery Opoku

School of Mathematical & Statistical Sciences Faculty Publications

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as

∫e−2π/10√0q∏j=1∞(1−qj)3(1−q10j)8(1−q5j)7dq=14(10−45–√−−−−−−−−√−1). We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.

Image Of Mathematics: A Case Study Of Two Women's Early Mathematics Experiences, Lili Zhou, Elizabeth Suazo-Flores, Bima Sapkota, Rose Mbewe, Jill Newton

School of Mathematical & Statistical Sciences Faculty Publications

People often view mathematics as abstract, cold, and irrelevant to real life, and their school experiences likely influence such views. In this case study, we investigated the mathematics experiences of two women who participated in an afterschool girls-only STEM club 30 years ago when they were in fifth and sixth grades. We explored how the participants' early in- and out-of-school experiences informed their images of mathematics. We collected data through a survey, focus group interviews, and individual interviews. We employed oral history methods in interviews to grant participants a significant degree of freedom to retrospectively recount and reconstruct their experiences. …

The Taylor Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Desiree Martin, W. Frank Moore

School of Mathematical & Statistical Sciences Faculty Publications

Let k be a field and let I be a monomial ideal in the polynomial ring Q = k[x1, . . . , xn]. In her thesis, Taylor introduced a complex which provides a finite free resolution for Q/I as a Q-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring R. Under the hypothesis that the skew commuting parameters defining R are roots of unity, we prove as an application that …

Polar Decompositions And Spectral Properties Of Linear Operator Pencils, Slaviša Djordjević, 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) . Using this decomposition, we define the generalized spherical Aluthge transforms for the linear pencil of an ordered pair T=(T1,T2) and give spectral properties between the linear pencil and the generalized spherical Aluthge transform of it. In detail, we study whether the spectral properties of the linear pencil of an ordered pair T=(T1,T2) are invariant under the generalized spherical Aluthge transform.

Editorial Insights: Reflections On The Volume And Charge To The Field, Heather Howell, Meghan Shaughnessy, Barbara Stengal, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Gregory Benoit, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

In this, the final chapter of Section 3, the authors document an interview that took place in June of 2024, as the editorial team wrapped up reviews of the chapters submitted to this volume. The interview was led by two of the advisory board members, Meghan Shaughnessy and Barbara Stengal, and included the editorial group, Heather Howell, Carrie Wilkerson Lee, Liza Bondurant, and Bima Sapkota, and other members of the advisory board, Greg Benoit and Yvonne Lai. The motivation to create this interview-based chapter was to capture some of the authors' own learning and reflection as they brought the volume …

Generalizations Of Some Q-Product Identities Of Ramanujan And Others, Timothy J. Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

By considering certain limiting cases of a WP-Bailey chain discovered by Andrews, and also limiting cases of certain classical summation formulae for basic hypergeometric series, we derive new expressions for certain Lambert series in terms of basic hypergeometric series. In some cases, the resulting series involve an arbitrary Bailey pair. This allows for the derivation of new basic hypergeometric expansions for some q-products and series that Ramanujan expressed in terms of Lambert series. Some of Ramanujan’s identities are extended to more general relations containing one or more free parameters.

Understanding Women, Infant And Children Feeding Decisions In Emergencies: The Case Of Covid-19 And The Formula Shortage, Davida Mensah, Edmund F. Agyemang, Constance Gewa

School of Mathematical & Statistical Sciences Faculty Publications

Background: The advantages of breastfeeding and its role in emergencies are undeniable. Studies show that many women in developed countries choose formula over breastmilk, even in emergencies. Nevertheless, no one knows if postpartum women under the Women, Infants and Children (WIC) program turned out to breastfeed or re-lactate their infants during these crises. This study aims to better understand WIC participants' perceptions about breastfeeding their babies in emergencies.

Methods: A chi-square test of independence was used to assess the association between breastfeeding choice and demographic characteristics and potential covariates of respondents. A post hoc analysis using Dunn's multiple comparisons test …

Effects Of Missing Data Imputation Methods On Univariate Blood Pressure Time Series Data Analysis And Forecasting With Arima And Lstm, Nicholas Niako, Jesus D. Melgarejo, Gladys E. Maestre, Kristina Vatcheva

School of Mathematical & Statistical Sciences Faculty Publications

Background

Missing observations within the univariate time series are common in real-life and cause analytical problems in the flow of the analysis. Imputation of missing values is an inevitable step in every incomplete univariate time series. Most of the existing studies focus on comparing the distributions of imputed data. There is a gap of knowledge on how different imputation methods for univariate time series affect the forecasting performance of time series models. We evaluated the prediction performance of autoregressive integrated moving average (ARIMA) and long short-term memory (LSTM) network models on imputed time series data using ten different imputation techniques. …

On The Power Series Method For Solitons And Peakons Of Nonlinear Pde, Erika Olivares, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we discuss the power series method to obtain traveling wave solutions of nonlinear PDE in one space variable. The method, including a recursive procedure for the coefficients and error estimate for the partial sums, is presented for the Camassa-Holm equation. Numerical examples of approximate single soliton and single peakon solutions are included.

A Supervised Machine Learning Statistical Design Of Experiment Approach To Modeling The Barriers To Effective Snakebite Treatment In Ghana, Eric Nyarko, Edmund F. Agyemang, Ebenezer Kwesi Ameho, Louis Agyekum, José María Gutiérrez, Eduardo Alberto Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

Background

Snakebite envenoming is a serious condition that affects 2.5 million people and causes 81,000–138,000 deaths every year, particularly in tropical and subtropical regions. The World Health Organization has set a goal to halve the deaths and disabilities related to snakebite envenoming by 2030. However, significant challenges in achieving this goal include a lack of robust research evidence related to snakebite incidence and treatment, particularly in sub-Saharan Africa. This study aimed to combine established methodologies with the latest tools in Artificial Intelligence to assess the barriers to effective snakebite treatment in Ghana.

Method

We used a MaxDiff statistical experiment design …

On The Lagrange And Hermite Quadrature Formula, Shuxia Li, Yonghong Chen

School of Mathematical & Statistical Sciences Faculty Publications

This paper builds on the error analysis method for Newton-Cotes quadrature formulas developed by D. R. Hayes and L. Rubin in 1970, which utilizes Lagrange interpolation polynomials. By adopting and extending their approach, this work derives the error estimate for Hermite interpolation quadrature. Specifically, we construct a polynomial P(x) analogous to the scaling function A(x) used by Hayes and Rubin, and prove its non-negativity over the interval. This allows us to establish a precise error formula for Hermite interpolation quadrature. The results provide a novel application of Hayes and Rubin's methodology, offering new insights …

On Representations Of The Super-Yangian Of The Queer Lie Superalgebra, Elena Poletaeva

School of Mathematical & Statistical Sciences Faculty Publications

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two 1-dimensional modules over the super-Yangian of Q(n) can be extended nontrivially. We describe the dual modules of the simple finite-dimensional modules over YQ(1) . We use these results to describe blocks in the subcategory of finite-dimensional YQ(1) -modules admitting the zero generalized central character.

Modelling Tinnitus Functional Index Reduction Using Supervised Machine Learning Algorithms, Edmund F. Agyemang

School of Mathematical & Statistical Sciences Faculty Publications

This study aims to model the reduction in the Tinnitus Functional Index (TFI) utilizing supervised machine learning algorithms, focusing primarily on Ordinary Least Squares (OLS), K-Nearest Neighbor (KNN), Ridge, and Lasso regressions. Our analysis highlighted Group, ISI, and SWLS as significant predictors of TFI reduction, identified through the best subset selection and confirmed by both forward and backward selection criteria in the OLS regression. Notably, the shrinkage methods, Ridge and Lasso regressions, demonstrated superior performance compared to OLS and KNN, with the Ridge regression presenting the smallest test mean square error (MSE) of 318.30. This finding establishes the Ridge regression …

Bioconvection Dynamics In Rotating And Stationary Cone-Disk Systems, Puneet Rana, Mahanthesh Basavarajappa

School of Mathematical & Statistical Sciences Faculty Publications

This work focuses on the study of bioconvection in a conical region of rotating and stationary cone-disk systems utilizing nanofluids involving gyrotactic micro-organisms. The flow geometry encompasses two different configurations, namely, rotating cone-disk system (RCDS) and stationary cone-disk system (SCDS). For RCDS, four unique configurations are considered: rotating cone static disk (Model-I), static cone rotating disk (Model-II), co-rotating cone-disk (Model-III), and counter-rotating cone-disk (Model-IV), while SCDS includes both swirling and non-swirling flow scenarios. A total of six different physical configurations that differ in boundary conditions are investigated. The mathematical model comprises Navier–Stokes, energy, nanoparticle volume fraction (NVF), and micro-organism density …

On The Total Perimeter Of Pairwise Disjoint Convex Bodies, Arseniy Akopyan, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

In this note we introduce a pseudometric on closed convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shorter proof of the theorem by Pinchasi that the sum of perimeters of 𝑘 convex planar bodies with disjoint interiors contained in a convex body of perimeter 𝑝 and diameter 𝑑 is not greater than 𝑝 + 2(𝑘 − 1)𝑑.

Quantization Dimensions For The Bi-Lipschitz Recurrent Iterated Function Systems, Amit Priyadarshi, Mrinal Kanti Roychowdhury, Manuj Verma

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the quantization dimensions of the Borel probability measures supported on the limit sets of the bi-Lipschitz recurrent iterated function systems under the strong open set condition in terms of the spectral radius have been estimated.

Non-Vanishing Modulo P Of Hecke L-Values Over Imaginary Quadratic Fields, Debanjana Kundu, Antonio Lei

School of Mathematical & Statistical Sciences Faculty Publications

Let p and q be two distinct odd primes. Let K be an imaginary quadratic field over which p and q are both split. Let Ψ be a Hecke character over K of infinity type (k, j) with 0 ≤ − j < k. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters κ over K which factor through the Z2q -extension of K, the p-adic valuation of the algebraic part of the L-value L(κΨ¯¯¯¯¯¯¯,k+j) is a constant independent of κ. In addition, when j = 0 and certain technical hypothesis holds, this constant is zero.

Algebraic Structures On Parallelizable Manifolds, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L , there exists a global trivialization of the tangent bundle, which defines a map ρ p : l ⟶ T p L for each point p ∈ L , where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l . Furthermore, flows of these vector fields give rise to a product between elements of l and L , which in turn induces …

Topology And The Quantum Hall Effects, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

The quantum Hall effects are an excellent example of physical systems where topology plays a major role in accounting for the physical observations. It is shown that the conductivity that appears in the quantum Hall effect is a topological invariant. It is illustrated how a fiber bundle over a torus can be constructed producing a geometry in which the system can be referred. The fractional effect can be studied by introducing homotopy and associated braid groups. Filling fractions can be obtained as a consequence of commensurability relations.

Substitution Tilings With Transcendental Inflation Factor, Dirk Frettlöh, Alexey Garber, Neil Mañibo

School of Mathematical & Statistical Sciences Faculty Publications

For any λ>2, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor λ. In particular, we obtain the first class of examples of substitutive systems with transcendental inflation factors. We also show that both the associated subshift and tiling dynamical systems are strictly ergodic, which is related to the quasicompactness of the underlying substitution operator.

Pedagogical Moves Related To Analogy That Support A Unified Understanding Of Eigentheory Concepts In A Quantum Mechanics Class, Kaitlyn Stephens Serbin, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

It is beneficial for quantum mechanics students to have a unified understanding of eigentheory concepts, so they can recognize the shared structure of mathematized phenomena from the different quantum mechanical systems of spin, energy, or position and recognize those as instantiations of the same overarching concept. Quantum mechanics instructors should, therefore, provide opportunities for their class community to develop a shared unified understanding of eigentheory concepts. One such opportunity can arise by engaging students in analogizing eigentheory concepts in one context with those from another context. We investigate the pedagogical moves related to analogies that can be used by a …

Integrable Semi-Discretization For A Modified Camassa-Holm Equation With Cubic Nonlinearity, Bao-Feng Feng, Heng-Chun Hu, Han-Han Sheng, Wei Yin, Guo-Fu Yu

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by …

Stability Of Optimal Spherical Codes, Károly J. Böröczky, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

For many extremal configurations of points on a sphere, the linear programming approach can be used to show their optimality. In this paper we establish the general framework for showing stability of such configurations and use this framework to prove the stability of the two spherical codes formed by minimal vectors of the lattice E8 and of the Leech lattice.