Physical Sciences and Mathematics Commons^™

A Note On Umbilic Points At Infinity, Brendan Guilfoyle

Publications

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric …

Facilitating Mathematics And Computer Science Connections: A Cross-Curricular Approach, Kimberly E. Beck, Jessica F. Shumway, Umar Shehzad, Jody Clarke-Midura, Mimi Recker

Publications

In the United States, school curricula are often created and taught with distinct boundaries between disciplines. This division between curricular areas may serve as a hindrance to students' long-term learning and their ability to generalize. In contrast, cross-curricular pedagogy provides a way for students to think beyond the classroom walls and make important connections across disciplines. The purpose of this paper is a theoretical reflection on our use of Expansive Framing in our design of lessons across learning environments within the school. We provide a narrative account of our early work in using this theoretical framework to co-plan and enact …

Liouville Soliton Surfaces Obtained Using Darboux Transformations, S. C. Mancas, K. R. Acharya

Publications

We construct parametric Liouville surfaces corresponding to parametric soliton solutions of the Liouville equation and Darboux-transformed counterparts. We also use a modified variation of parameters method together with the elliptic functions method to obtain the traveling wave solutions to Liouville equation and express the centroaffine invariant in terms of the soliton Hamiltonian.

Geometry And Coding: Introducing An Interactive And Integrated Mathematics-Computer Science Unit, Kimberly Beck, Jessica F. Shumway

Publications

As part of a collaborative project between Utah State University, the Cache County School District, and Stanford, instructional units were designed for fifth-grade students. These units integrated math concepts of geometrical shapes and computer science concepts of sequences, conditionals, and loops. One component of the unit was implemented in math classrooms by math teachers, and the other component was implemented in computer labs. This presentation will focus on the math unit as presented at the National Council of Teachers of Mathematics (NCTM-V).

An Exploration Of Computational Text Analysis Of Co-Design Discourse In A Research-Practice Partnership, Mei Tan, Victor R. Lee

Publications

In combination with contextualized human interpretation, computational text analysis offers a quantitative approach to interrogating the nature of participation and social positioning in discourse. Using meeting transcript data from the development of a co-design research-practice partnership, we examine the roles and forms of participation that contribute to an effective collaboration between a multileveled school system and researcher partners. We apply computational methods to explore the language of co-design and multi-stakeholder perspectives in support of educational improvement science efforts and our theoretical understanding of partnership roles. Results indicate participation patterns align with documented roles in co- design partnerships and highlight the …

Co-Designing Elementary-Level Computer Science And Mathematics Lessons: An Expansive Framing Approach, Umar Shehzad, Jody Clarke-Midura, Kimberly Beck, Jessica Shumway, Mimi Recker

Publications

This study examines how a rural-serving school district aimed to provide elementary-level computer science (CS) by offering instruction during students’ computer lab time. As part of a research-practice partnership, cross-context mathematics and CS lessons were co-designed to expansively frame and highlight connections across – as opposed to integration within – the two subjects. Findings indicated that most students who engaged with the lessons across the lab and classroom contexts reported finding the lessons interesting, seeing connections to their mathematics classes, and understanding the programming. In contrast, a three-level logistic regression model showed that students who only learned about mathematics connections …

One-Parameter Darboux-Deformed Fibonacci Numbers, Stefani C. Mancas, H. C. Rosu

Publications

One-parameter Darboux deformations are effected for the simple ODE satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (non integer) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov-Lewis invariants for these sequences are also discussed.

Applying Expansive Framing To An Integrated Mathematics-Computer Science Unit, Kimberly Evagelatos Beck, Jessica F. Shumway

Publications

In this research report for the National Council of Teachers of Mathematics 2022 Research Conference, we discuss the theory of Expansive Framing and its application to an interdisciplinary mathematics-computer science curricular unit.

A New Geometric Structure On Tangent Bundles, Nikos Georgiou, Brendan Guilfoyle

Publications

For a Riemannian manifold (N, g), we construct a scalar flat neutral metric G on the tangent bundle TN. The metric is locally conformally flat if and only if either N is a 2- dimensional manifold or (N, g) is a real space form. It is also shown that G is locally symmetric if and only if g is locally symmetric. We then study submanifolds in TN and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph in the tangent bundle TN to have parallel mean curvature …

A New Non-Inheriting Homogeneous Solution Of The Einstein-Maxwell Equations With Cosmological Term, Ian M. Anderson, Charles G. Torre

Publications

We find a new homogeneous solution to the Einstein-Maxwell equations with a cos- mological term. The spacetime manifold is R × S3. The spacetime metric admits a simply transitive isometry group G = R × SU(2) and is Petrov type I. The spacetime is geodesically complete and globally hyperbolic. The electromagnetic field is non- null and non-inheriting: it is only invariant with respect to the SU(2) subgroup and is time-dependent in a stationary reference frame.

Exo-Sir: An Epidemiological Model To Analyze The Impact Of Exogenous Spread Of Infection, Nirmal Kumar Sivaraman, Manas Gaur, Shivansh Baijal, Sakthi Balan Muthiah, Amit Sheth

Publications

Epidemics like Covid-19 and Ebola have impacted people's lives significantly. The impact of mobility of people across the countries or states in the spread of epidemics has been significant. The spread of disease due to factors local to the population under consideration is termed the endogenous spread. The spread due to external factors like migration, mobility, etc. is called the exogenous spread. In this paper, we introduce the Exo-SIR model, an extension of the popular SIR model and a few variants of the model. The novelty in our model is that it captures both the exogenous and endogenous spread of …

Reduced-Order Dynamic Modeling And Robust Nonlinear Control Of Fluid Flow Velocity Fields, Anu Kossery Jayaprakash, William Mackunis, Vladimir Golubev, Oksana Stalnov

Publications

A robust nonlinear control method is developed for fluid flow velocity tracking, which formally addresses the inherent challenges in practical implementation of closed-loop active flow control systems. A key challenge being addressed here is flow control design to compensate for model parameter variations that can arise from actuator perturbations. The control design is based on a detailed reduced-order model of the actuated flow dynamics, which is rigorously derived to incorporate the inherent time-varying uncertainty in the both the model parameters and the actuator dynamics. To the best of the authors’ knowledge, this is the first robust nonlinear closed-loop active flow …

An Extension Of Asgeirsson's Mean Value Theorem For Solutions Of The Ultra-Hyperbolic Equation In Dimension Four, Guillem Cobos, Brendan Guilfoyle

Publications

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae. The broader context of this result is the geometrization of Fritz John’s 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the compatibility for functions on line space to …

Evolving To Non-Round Weingarten Spheres: Integer Linear Hopf Flows, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth convex rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to …

Transitioning To An Active Learning Environment For Calculus At The University Of Florida, Darryl Chamberlain, Amy Grady, Scott Keeran, Kevin Knudson, Ian Manly, Melissa Shabazz, Corey Stone

Publications

In this note, we describe a large-scale transition to an active learning format in first-semester calculus at the University of Florida. Student performance and attitudes are compared across traditional lecture and flipped sections.

Finite-Time State Estimation For An Inverted Pendulum Under Input-Multiplicative Uncertainty, William Mackunis, Sergey V. Drakunov, Anu Kossery Jayaprakash, Krishna Bhavithavya Kidambi, Mahmut Reyhanoglu

Publications

A sliding mode observer is presented, which is rigorously proven to achieve finite-time state estimation of a dual-parallel underactuated (i.e., single-input multi-output) cart inverted pendulum system in the presence of parametric uncertainty. A salient feature of the proposed sliding mode observer design is that a rigorous analysis is provided, which proves finite-time estimation of the complete system state in the presence of input-multiplicative parametric uncertainty. The performance of the proposed observer design is demonstrated through numerical case studies using both sliding mode control (SMC)- and linear quadratic regulator (LQR)-based closed-loop control systems. The main contribution presented here is the rigorous …

Discontinuity-Driven Mesh Alignment For Evolving Discontinuities In Elastic Solids, Mihhail Berezovski, Arkadi Berezovski

Publications

A special mesh adaptation technique and a precise discontinuity tracking are presented for an accurate, efficient, and robust adaptive-mesh computational procedure for one-dimensional hyperbolic systems of conservation laws, with particular reference to problems with evolving discontinuities in solids. The main advantage of the adaptive technique is its ability to preserve the modified mesh as close to the original fixed mesh as possible. The constructed method is applied to the martensitic phase-transition front propagation in solids.

Infusing Humanities In Stem Education: Student Opinions Of Disciplinary Connections In An Introductory Chemistry Course, Emily K. Faulconer, Beverly Wood, John C. Griffith

Publications

The Next Generation Science Standards and other educational reforms support the formation of deep connections across the STEM disciplines. Integrated STEM is considered as a best practice by the educational communities of the disparate disciplines. However, the integration of non-STEM disciplines is understudied and generally limited to the integration of art (STEAM). Humanistic STEM blends the study of STEM with interest in and concern for human affairs, welfare, values, or culture. This study looks at an infusion of the humanities into an online chemistry course to see if there is an influence on student connection between course content and cross-disciplinary …

Titchmarsh–Weyl Theory For Vector-Valued Discrete Schrödinger Operators, Keshav R. Acharya

Publications

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.

An Explicit Finite Volume Numerical Scheme For 2d Elastic Wave Propagation, Mihhail Berezovski, Arkadi Berezovski

Publications

The construction of the two-dimensional finite volume numerical scheme based on the representation of computational cells as thermodynamic systems is presented explicitly. The main advantage of the scheme is an accurate implementation of conditions at interfaces and boundaries. It is demonstrated that boundary conditions influence the wave motion even in the simple case of a homogeneous waveguide.

Action Of Complex Symplectic Matrices On The Siegel Upper Half Space, Keshav R. Acharya, Matt Mcbride

Publications

The Siegel upper half space, Sn, the space of complex symmetric matrices, Z with positive definite imaginary part, is the generalization of the complex upper half plane in higher dimensions. In this paper, we study a generalization of linear fractional transformations, ΦS, where S is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which ΦS(Z) is well defined. We also consider Sn and Sn as metric spaces and discuss distance properties of the map ΦS from Sn to Sn and Sn respectively.

A Design Of A Material Assembly In Space-Time Generating And Storing Energy, Mihhail Berezovski, Stan Elektrov, Konstantin Lurie

Publications

The paper introduces a theoretical background of the mechanism of electromagnetic energy and power accumulation and its focusing in narrow pulses travelling along a transmission line with material parameters variable in 1D-space and time. This mechanism may be implemented due to a special material geometry- a distribution of two different dielectrics in a spatio-temporal checkerboard. We concentrate on the practically reasonable means to bring this mechanism into action in a device that may work both as energy generator and energy storage. The basic ideas discussed below appear to be fairly general; we have chosen their electromagnetic implementation as an excellent …

A Global Version Of A Classical Result Of Joachimstha, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove the following global analogue of this result. Suppose that two closed convex surfaces intersect with constant angle along a curve that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both nonorientable. We prove this by characterizing the constant angle …

Weighted Composition Operators On Analytic Function Spaces: Some Recent Progress, Dip Acharyya

Publications

Theory of Composition Operators is a steady point of interest for almost 100 years. While studying these operators, our general goal is to describe their operator theoretic properties in terms of the associated function symbols. In this talk, I will discuss some recent results concerning linear combinations (sums, differences, etc.) of weighted composition operators in certain spaces of Analytic functions.

Weighted Composition Operators On Spaces Of Analytic Functions: A Survey, Soumyadip Acharyya

Publications

“Pure mathematics is, in its way, the poetry of logical ideas.” - Albert Einstein. Pure mathematicians study abstract entities and structures that underlie mathematics. Although their general perspective is “math for math’s sake”, sometimes even the most abstract mathematics can have unexpected applications! Come learn some of these astonishing discoveries in the history of science and mathematics! They might make you thrilled but keep in mind real-world usage is rarely the goal behind developing a new mathematical theory.

Welcome to the world of pure mathematics! In this talk, we will focus on the theory of composition operators which is a …

A Mean-Risk Mixed Integer Nonlinear Program For Transportation Network Protection, Jie Lu, Akshay Gupte, Yongxi Huang

Publications

This paper focuses on transportation network protection to hedge against extreme events such as earthquakes. Traditional two-stage stochastic programming has been widely adopted to obtain solutions under a risk-neutral preference through the use of expectations in the recourse function. In reality, decision makers hold different risk preferences. We develop a mean-risk two-stage stochastic programming model that allows for greater flexibility in handling risk preferences when allocating limited resources. In particular, the first stage minimizes the retrofitting cost by making strategic retrofit decisions whereas the second stage minimizes the travel cost. The conditional value-at-risk (CVaR) is included as the risk measure …

Almost-Bps Solutions In Multi-Center Taub-Nut, C. Rugina, A. Ludu

Publications

Microstates of multiple collinear black holes embedded in a non-collinear two-center Taub-NUT spacetime are sought in 4 dimensions. A set of coupled partial differential equations are obtained and solved for almost-BPS states, where some supersymmetry is preserved in the context of N = 2 supergravity in 4 dimensions. The regularity of solutions is carefully considered, and we ensure that no CTC (closed time-like curves) are present. The larger framework is that of 11-dimensional N = 2 supergravity, and the current theory is obtained by compactifying it down to 4 dimensions. This work is a generalization (to three non-collinear centers) of …

Elliptic Solutions And Solitary Waves Of A Higher Order Kdv-Bbm Long Wave Equation, S.C. Mancas, Ronald Adams

Publications

We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass ℘ functions of both third and fifth-order KdV–BBM (Korteweg-de Vries–Benjamin, Bona & Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV–BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic ℘ functions. For the fifth-order KdV–BBM equation we show that a parameter γ = 1/12 …

Hopf Hypersurfaces In Spaces Of Oriented Geodesics., Nikos Georgiou, Brendan Guilfoyle

Publications

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the space of oriented geodesics admits a canonical Kaehler–Einstein and para-Kaehler–Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in …

Generalized Thomas-Fermi Equations As The Lampariello Class Of Emden-Fowler Equations, Haret C. Rosu, S.C. Mancas

Publications

A one-parameter family of Emden-Fowler equations defined by Lampariello’s parameter p which, upon using Thomas-Fermi boundary conditions, turns into a set of generalized Thomas-Fermi equations comprising the standard Thomas-Fermi equation for p = 1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel equations whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of equations for the standard Thomas-Fermi equation and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.