Projects

Multiplicative LLN and CLT and their Applications

July 25, 2017

Group Members

Lowen Peng, Anthony Sisti, Rajeshwari Majumdar

Supervisors

Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo

Overview

We study the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturb the random matrices with a parameter and we look to find the dependence of the the Lyapunov exponent on this parameter. We also study the variance related to the multiplicative CLT. We prove and conjecture asymptotics of various parameter dependent plots.

Publication: “Lyapunov exponent and variance in the CLT for products of random matrices related to random
Fibonacci sequences” — arXiv:1809.02294, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), pp 21

Presentations:

Raji Majumdar and Anthony Sisti, will present posters Applications of Multiplicative LLN and CLT for Random Matrices and Black Scholes using the Central Limit Theorem on Friday, January 12 at the MAA Student Poster Session, and give talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.

 

Investigating the Interplay of Argumentation and Mathematics in Classroom Tasks

June 2, 2016

Group Members

Megan Brown, Grace WrightMath Ed

Supervisors

Steven Lemay

Overview

This project was motivated by the current emphasis on argumentation in the Common Core State Standards for Mathematics. In particular, we focused on tasks that address the third standard for Mathematical Practice that highlights argumentation as an expertise all students should develop. Our data consists of 157 tasks from 40 elementary and high school teachers participating in a professional development program focused on mathematical argumentation. We analyzed how argumentation affects the cognitive demand levels of the tasks, as well as how argumentation detracts or contributes to the mathematical content in the tasks. Our findings show a change in cognitive demand levels when tasks were analyzed with and without the argumentation component. In addition, five different themes emerged from our analysis with regards to the interplay of argumentation and mathematical concepts, some of which were common across elementary and high school tasks.

Presentation

Poster

Mathematics and Intercultural Competence in the Middle School

Group Members

Chris Bennett, Megan BrunnerDiscussion

Supervisors

Kyle Evans

Overview

As today’s world becomes increasingly globalized, there exists a greater need to develop intercultural competence (ICC) in children through education. Three interdisciplinary units were designed by graduate students at UConn in the past year to develop ICC in sixth grade students, but lack assessment tools that track this development over the course of a school year. With a focus on Michael Byram’s model of ICC, we created two assessment tools – a survey and a rubric. The survey contains Likert scale items measuring four dimensions – attitudes towards cultures, knowledge of cultures, mathematics learning, and interdisciplinary learning – along with open-ended questions to add a qualitative component for each dimension. The rubric, designed for teachers, provides a resource for evaluating ICC in students’ interactions and written reflections. The tools have the potential to form a new baseline for assessment of ICC in children and can be refined and adapted for use across school districts and grade levels. In addition, we created three examples of lesson plans – one for third grade, one for sixth grade, and one for ninth grade – that demonstrate how intercultural competence and global issues can be incorporated across different levels of student development while also adhering to the Common Core State Standards for Mathematics.

Presentation

ICC Presentation

Poster

Spectrum of the Magnetic Laplacian on the Diamond Fractal

May 22, 2016

Group MembersIMG_5606

Stephen Loew, Madeline Hansalik, Aubrey Coffey

Supervisors

Luke Rogers, Antoni Brzoska

Overview

The diamond fractal is a fractal that is obtained in the following manner.  Start with a graph with two vertices and an edge and replace the edge with two new vertices connected to our original vertices to obtain a diamond shaped graph.   The diamond fractal is defined to be the limiting object after continuing with the edge replacement indefinitely.  In the project, the spectrum of magnetic Laplacian operators on graph approximations to the diamond fractal was computed.

Given a level n approximation to the fractal with known magnetic field strengths through cells and holes, it is possible to determine the net magnetic field through the cells and holes of the preceding graph approximations.  The spectral similarity relation between the operators on successive graph approximations was worked out, with the corresponding spectral decimation polynomial depending on the magnetic field strengths.  A poster and talk on this work was presented at the REU Mini-Symposium at UConn.

Publication: Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 32

arXiv:1704.01609

Presentation

Magnetic Spectral Decimation

Poster

Minimal Length Maximal Green Sequences and Triangulations of Polygons

May 21, 2016

Group Members

Emily Cormier, Peter Dillery, Jill Resh, John WhelanAlgebraic Geometry

Supervisors

Khrystyna Serhiyenko

Overview

Maximal green sequences (MGS’s) are combinatorial objects that involve local transformations of directed graphs, also called quivers.  We studied minimal length MGS’s for quivers of type A.   It is know that such quivers are in bijection with triangulations of polygons.  Moreover, these local transformations of quivers behave well under this correspondence, and there is a related notion of mutation of triangulations.  This enabled us to study MGS’s both in terms of quivers and in terms of triangulations. We showed that any minimal length MGS has length n+t, where n is the number of vertices in the quiver and t is the number of 3-cycles.  We also developed an algorithm that yields such sequences of minimal length.

 

Journal of Algebraic Combinatorics volume 44, pages 905–930 (2016)

arxiv 1508.02954

Presentation

Minimal Length Maximal Green Sequences

Poster

Power Dissipation in Fractal AC Circuits

Group Members

Loren Anderson, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen LoewIMG_5534

Supervisors

Alexander Teplyaev, Luke Rogers, Antoni Brzoska, Ulysses Andrews

Overview

In this project, Feynman’s analysis of an infinite ladder circuit is extended to fractal circuits, in particular, a Sierpinski Ladder circuit and two variants of a Hanoi circuit.   These circuits are constructed by taking the limit of graph approximations with inductors and capacitors placed along edges in a well-defined manner.  Inductors, capacitors and resistors all contribute to the impedance within a circuit; but whereas a resistor imparts a real impedance, inductors and capacitors impart a purely imaginary impedance.

For each circuit, the following was accomplished.  First, the net impedance between certain boundary points of the circuit was computed.  Second, the filter conditions for each circuit were found.  A circuit becomes a filter when the net impedance between boundary vertices has a positive real part!  Third, it was proved that these impedances can be obtained by placing a small positive resistance epsilon on each edge of the graph approximations, finding the limiting impedance between the boundary vertices, and then taking epsilon to zero.  Finally, the construction of harmonic functions on these circuits was outlined.

Publication

arXiv:1605.03890

Presentation

Fractal AC Circuits

Poster

Stochastic Stability of Planar Flows

May 12, 2016

Publication: arXiv:1510.09221

Journal reference: Stochastics and Dynamics, Vol. 17, No. 6 (2017) 1750046
DOI: 10.1142/S0219493717500460

Group Members

Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’ConnellREU2015-Stochastics

Supervisors

Joe P Chen, FanNy Shum

Overview

We investigated systems of complex-valued ordinary differential equations (ODEs) that blows up in finite time, which we refer to as explosive systems. The goal is to understand for what initial conditions does the system explode and will the addition of noise stabilize it; that is, if we were to perturb the system with an additive Brownian motion, will the system of stochastic differential equation (SDE) still be explosive? In fact, we were able to prove a toy model of the stochastic Burgers’ equation to be ergodic; that is, the SDE is nonexplosive and it has a unique limiting distribution.

Presentation

Poster

Wave Propagation through a Fractal Medium

May 2, 2016

Group Members

Edith Aromando, Lee FisherPoster - Wave Propagation through a Fractal Medium

Supervisors

Alexander Teplyaev, Luke Rogers, Ulysses Andrews

Overview

We consider the wave equation on the unit interval with fractal measure, and use two numerical models to study wave speed and propagation distance.  The first approach uses a Fourier series of eigenfunctions of the fractal Laplacian, while the second uses a Markov chain to model the transmission and reflection of classical waves on an approximation of the fractal. These models have complementary advantages and limitations, and we conjecture that they approximate the same fractal wave.

 

Presentation

Poster

Spectrum of the Magnetic Laplacian on the Sierpinski Gasket

April 21, 2016

Jessica Hyde, Jesse Moeller

Group Members

Jessica Hyde, Jesse Moeller, and Luis Seda

Supervisors

Luke Rogers, Dan Kelleher

Overview

One project investigated magnetic gauge fields on the Sierpinski Gasket. After numerical experimentation, using Mathematica and MatLab, this team determined that specific portions of the spectrum of the Laplacian are unaffected by a perturbating magnetic field given by a harmonic 1-form of finite topological type and continued on to prove that this. In a specific case they also gave a description of the spectrum via a covering space and symmetry argument. They presented a poster and a talk on this work at the REU mini-conference at the University of Massachusetts, Amherst and gave a talk at the Young Mathematician’s Conference at Ohio State University.

Presentation

Spectrum of the Magnetic Laplacian

Publication: 2017 CPAA 16 6 2299- 2319 doi: 10.3934/cpaa.2017113

arXiv.org

Measurable Riemannian Structure on Higher Dimensional Sierpinski Gaskets

Josh Frisch, Sara Chari

Group Members

Sara Chari and Joshua Frisch

Supervisors

Luke Rogers and Dan Kelleher

Overview

This team worked on the existence of measurable Riemannian structures in the sense of Kigami on higher dimensional Sierpinski-type gaskets. Some time ago, Kusuoka proved existence of a measure, metric (in the Riemannian sense) and gradient operator on a class of fractals that includes these gaskets, such that these objects bear the same relation to the Dirichlet form as do the Riemannian volume, metric and gradient on Euclidean space. Kigami later completed this picture in the case of the usual 3-vertex Sierpinski gasket by constructing a geodesic length that is the analogue of that occuring in the Riemannian case and proving Gaussian heat kernel estimates, and Kajino has subsequently proved very re ned estimates for the heat kernel in this setting. It was believed that although Kigami’s approach relied on certain 2-dimensional techniques the results would also be valid on Sierpinski-type gaskets with more vertices, and that is what our research team have proved. They presented a poster on this work at the REU mini-conference at the University of Massachusetts, Amherst, and are writing the results up for publication.

Publication

arXiv:1703.03380

Presentation

Geodesics and a Riemannian Metric on Harmonic Sierpinski Gaskets