Digital Commons Network^™

Multinomial Trees And Multinomial Percolation, Michele Tienni

Yale Graduate School of Arts and Sciences Dissertations

We study multinomial percolation, which is bond percolation on a family of graphs ("blow-up graphs") defined in terms of a fixed graph G. We compute connection thresholds in terms of distances on G. Moreover, if the percolation probability is inversely proportional to the blow-up multiplicity, there is a phase transition above which a giant component emerges. We show that many properties of the giant component can be computed using an analytic function with variables indexed by the vertices of G. Moreover, we find that the giant component gives rise to a natural field on G. As the blow-up multiplicity grows, …

Framed Bundles, Abelianization, And The Ooguri-Vafa Space, Danny Nackan

Yale Graduate School of Arts and Sciences Dissertations

In this thesis we study the hyperkähler structure of a particular moduli space of Higgs bundles near the discriminant locus of the Hitchin fibration, which in this case consists of quadratic differentials with a double zero.

Gaiotto, Moore, and Neitzke predicted that the hyperkähler Ooguri-Vafa space Mov should provide a local model for Hitchin moduli spaces near the discriminant locus. To this end, Tulli identified Mov with a certain space of framed Higgs bundles with an irregular singularity.

We extend this result by identifying the Ooguri-Vafa holomorphic symplectic form with a regularized version of the Atiyah-Bott form on …

On The Analysis Of Average Hyperbolicity And Its Applications, Joon-Hyeok Yim

Yale Graduate School of Arts and Sciences Dissertations

This thesis bridges the ideas of hyperbolic geometry and geometric group theory with practical applications in graph theory and machine learning. A metric space (X, d) is said to be Gromov δ-hyperbolic if, for any points x, y, z, w ∈ X, the inequality d(x, y) + d(z, w) ≤ max(d(x, z) + d(y, w), d(x, w) + d(y, z)) + 2δ holds. This definition generalizes key properties of …

Rigidity Of Kleinian Groups Via Dynamics On Higher Rank Spaces, Dongryul Kim

Yale Graduate School of Arts and Sciences Dissertations

his thesis is a summary of results obtained on papers; the first three are joint with Hee Oh.

Let Γ < PSL2(C) = Isom+(H3 ) be a Zariski dense Kleinian group whose ordinary set Ω = S 2 − Λ has at least two components. Let ρ : Γ → PSL2(C) be a Zariski dense discrete faithful representation with boundary map f : Λ → S 2 on the limit set.

In, Oh and I obtained a new rigidity theorem: if f is conformal on Λ, in the sense that f maps every circular slice of Λ into a circle, then f extends to a Möbius transformation g on S 2 and ρ is the conjugation by g. Moreover, unless ρ is a conjugation, the set Λf of conformal points of f in Λ, defined as the union of every circular slice C ∩ Λ with C a round circle such …

Majority Dynamics On Random Graphs With Almost Balanced Initial Settings: The Power Of Few, Hoang Bao Linh Tran

Yale Graduate School of Arts and Sciences Dissertations

A community of n people participate in a parliamentary election with two major parties. Every person initially supports one side, and with each day, changes their opinion to match the majority among their friends. This process, commonly known as Majority Dynamics in literature, and similar processes, have been studied in many different contexts, including Statistical Physics, Computer Science, Economics and Ecology, as discussed in the influential survey by Mossel and Tamuz (2014).

Tran and Vu (2019) considered the process on a Erdös-Rényi G(n, p) random graph and discovered the "Power of Few" phenomenon: Starting from a …

Frobenius And Commutative Pseudomonoids In The Bicategory Of Spans, Ivan Contreras, Rajan Amit Mehta, Walker H. Stern

Mathematics Sciences: Faculty Publications

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in Span correspond, respectively, to paracyclic sets and Γ-sets satisfying the 2-Segal conditions. These results connect closely with work of the third author on A∞ algebras in ∞-categories of spans, as well as the growing body of work on higher Segal objects. Because …

Searching For Cages In Graphs And Signed Graphs, Isaac Partee

Browse all Theses and Dissertations

The girth of a graph G is the minimum length of a cycle in G. A (k,g)-graph is a k-regular graph of girth g. A (k,g)-cage is a (k,g)-graph with the smallest possible number of vertices. For example, K4 is the unique (3,3)-cage, K3,3 is the unique (3,4)-cage, and the Petersen Graph is the unique (3,5)-cage. The search for cages particular values of (k,g) is an ongoing area of considerable research. For values of (k,g) where the number vertices in a (k,g)-cage is not known, covering graphs have recently been used for constructing progressively smaller and smaller (k,g)-graphs. The covering …

The Nested Dirichlet Process And Applications To Nonparametric Bayesian Inference, Evan C. Donald

Graduate Thesis and Dissertation post-2024

This body of research focuses on Bayesian nonparametric inference, in regards to an array of S-valued random variables. These random variables are assumed to take values in a complete and separable metric space and further have a particular symmetry property, row exchangeability.

The Nested Dirichlet Process is a particularly apt model for inference in such circumstances. However, calculating posterior distributions in this framework is exceedingly difficult, and become exponentially more complicated as sample size increases.

This research explores the method of using sequential imputation to calculate posterior distributions, providing a rigorous proof of the method’s suitability that had been missing …

A Bridge Too Low, John Adam

Mathematics & Statistics Faculty Publications

The article "A bridge too low" in the Physics Teacher journal discusses a low bridge near the River Greta in Keswick, England, with an arch shaped like a semiellipse. It presents questions about the maximum height a person of a certain height can walk under the bridge without hitting their head, the cross-sectional area of the arch, its eccentricity, and perimeter. The article also mentions the approximation by Indian mathematician Srinivasa Ramanujan for the perimeter of an ellipse and invites readers to find the answers online.

Bridging Theory And Practice: The Power Of A Mathematics Clinic In Teacher Preparation, Kelly A. Clark, Chrystal O. Dean, Josie Barnes, Hadley Seifert, Lauren Young

Journal of Case Learning & Exceptional Learners

The Council for Exceptional Children (CEC) and the National Council on Teaching Mathematics (NCTM) released a joint statement on the importance of preparing special education teachers to teach mathematics (NCTM & CEC, 2024). They provided implications for institutions of higher education with teacher preparation programs, one of which included requiring special education majors to take a minimum of one mathematics course with mathematics content directed specifically to PK-12 mathematics and include field-based learning opportunities within that course. Powell (2015) indicated that effective preparation in mathematics related to special education should include multiple facets, including planned field experiences. Planned field experiences …

A Qualitative Study: Supports Enabled By K-8 School Leaders To Bolster The Success Of Teachers In Their Mathematics Departments, Elizabeth A. Mercurio

Theses and Dissertations--Educational Leadership Studies

Despite decades of national policy efforts and reforms, mathematics achievement in public education in the United States, particularly in urban areas, remains persistently low, underscoring the critical need for effective leadership and targeted professional development to maximize student learning. There is limited research on the leadership approaches of building-level school leaders who have effectively strengthened mathematics departments and enhanced teacher capacity to improve student achievement. This qualitative study employed semi-structured interviews with building-level school leaders who have demonstrated success in fostering highly effective mathematics instruction to examine the leadership styles of effective urban school leaders and their use of professional …

Identifying The Types Of Collective Engagement Structures That Manifest In Middle Grade Mathematics Classrooms, Tye G. Campbell, Michael A. Lawson, Erin Rich, Sheunghyun Yeo, W. Boden Robertson

Teacher Education and Leadership Faculty Publications

While the potential of collaborative problem solving in mathematics is well known, facilitating effective group work is often challenging. One reason for this persistent difficulty educators face with collaborative problem solving can be traced to research and practice models that are not well-equipped to attend to the complexity of group work. The emerging construct of “collective engagement structures” represents one novel framework designed to attend to the complexity and variation of individual and group experience in collaborative problem solving. This study extends initial theorizing on collective engagement by offering descriptive accounts of the different types of collective engagement structures that …

School System And Student Performance: Examining Science And Mathematics Achievement In Pakistan, Sadia Muzaffar Bhutta, Sohail Ahmad, Aisha Naz Ansari

Institute for Educational Development, Karachi

Public and private school systems are the two largest providers of formal school education in Pakistan, differing significantly in their governance, resources, and practices. While previous studies showed differences in Student Learning Outcomes (SLOs), often favouring private schools, these comparisons remained largely descriptive. This paper compares elementary grade students’ performance in mathematics and science between public and private schools in Pakistan, accounting for factors such as grade level and geographical region. A cross-sectional survey was conducted with 15,391 students from grades 5, 6, and 8, selected through a multistage sampling technique from a total of­­ 153 public and private schools …

The Impacts And Interplay Of Adolescent Personal Contexts And Sense Of Belonging To Mathematics On Postsecondary Stem Field Interests And Intentions, Monica J. Weir

Dissertations, Theses, and Projects

This quantitative correlational research sought to examine the relationship between high school students’ sense of belonging to mathematics and their interests and intentions to pursue fields of study and careers in science, technology, engineering, and mathematics (STEM). With a theoretical framework based on the belongingness hypothesis and social cognitive career theory (SCCT), the author explicitly linked the universality of the phenomenon of sense of belonging to individual, contextual, environmental, and experiential factors that influence career-related interest development. A sample of 328 high school students attending one private, Catholic, college preparatory school in the Upper Midwest of the United States responded …

Existence And Uniqueness Of Solutions Of Fourth Order Iterative Boundary Value Problems, Zach Whaley

Theses and Dissertations

In this dissertation, we study the fourth-order iterative differential equation \begin{displaymath} x^{(4)}(t) = \f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)) \end{displaymath} where $x^{[2]}(t) = x(x(t))$ and $x^{[j]}(t) = x(x^{[j - 1]}(t))$ for $j > 2$. We consider the above equation with multiple sets of boundary conditions, and we state results on the existence and uniqueness of solutions for each set of boundary conditions. In Chapter 2, the boundary conditions are conjugate boundary conditions, \begin{align*} &x(-a) = -a, \ x'(-a) = b, \ x''(-a) = c, \ x(a) = a \\ &x(-a) = -a, \ x(a) = a, \ x'(a) = b, \ x''(a) …

Traveling Waves In Biological Population Models, Ahlam Alzahrani

Theses and Dissertations

In this thesis, we examine the existence of traveling waves in population models. We begin by exploring conditions under which traveling waves exist, even when the migration probability lacks a density function or, if the density function exists, it may be discontinuous. To address this, we impose certain conditions that ensure the monotonicity of the evolution operator, enabling the application of the Monotone Iteration Method. Next, we explore the existence of monotone traveling waves in a general class of integral-difference population models that depend on both the previous state and long-term memory, allowing for the consideration of multiple past states. …

Snap Recipients' Labor Supply In The Presence Of Children In Households, Sanae Tashiro, Jongsung Kim

Mathematics and Economics Faculty Journal Articles

No abstract provided.

Using Geogebra To Foster Imagination And Creativity To Build Mathematics Confidence In A Stem World, Joseph M. Furner

Transformations

Math teachers can better prepare all student for a STEM world using computer software like GeoGebra to better prepare our youth for such high-tech careers like that of which Einstein was part of while also emphasizing and fostering creativity in the learning of mathematics. It is critical we have young learners who are confident in their ability to do math and be creative. Teachers today need to check dispositions toward mathematics and address math anxiety while building math confidence in their students. In an ever-advancing world of technology, math teachers need to incorporate such emerging technologies like GeoGebra into the …

Advantages And Disadvantages Of Technology In The Mathematics Classroom, Francena D. Gargaro

Transformations

The integration of technology into teaching pedagogy, particularly in mathematics instruction, has been a subject of significant. This paper examines the impact of utilizing technology for mathematics education. It reviews multiple categories of technological tools available for classroom use, including hardware, software, and online resources, and discusses their potential benefits and drawbacks. Positive aspects of technology integration include personalized learning experiences, increased engagement and collaboration, and enhanced visualization and conceptual understanding. However, challenges such as dependency on technology, loss of conceptual understanding, distraction, and the widening achievement gap due to inequity are also addressed. The paper explores the role of …

Infographic: Timss 2023 Snapshot, Jo Earp

Teacher infographics

Results from the latest cycle of the Trends in International Mathematics and Science Study (TIMSS 2023) have just been released. Today’s infographic looks at the year 4 and year 8 mean scores in mathematics and science for students in Australia and selected participating countries.

An Examination Of Hilbert Spaces, Fnu Sumaiya

Theses and Dissertations

This independent study focuses on the exploration of Hilbert spaces, a cornerstone of modern analysis and a fundamental construct in mathematics. It starts by introducing vector space, metric and metric spaces and then Hilbert spaces and its properties.

This study includes good amount of mathematical proofs, theorems and examples. Through extensive research and analysis, this independent study aims to offer a comprehensive examination of Hilbert spaces, detailing their properties and their pivotal role in modern analysis.

A Study Of The Near-Surface Vertical Distribution And Chemistry Of Pollutants In Cold-Climate Urban Areas With The Novel Pact-1d Model, Jonas Kuhn, Jochen Stutz, Meeta Cesler-Maloney, William R. Simpson, Thorsten Bartels-Rausch, Tjarda J. Roberts, Jennie L. Thomas, Jack E. Dibb, Laura Heinlein, Michael O. Sunday, Cort Anastasio, Kathleen Fahey, James H. Flynn, Fangzhou Guo

Faculty Publications

Cold-climate urban areas often face severe air pollution events in wintertime because of residential heating and vehicle emissions into shallow surface inversion layers. Many state-of-the-art regional chemistry-transport models cannot capture the small spatio-temporal scale of the transport and chemical processes occurring in these environments.Here we introduce a new version of our one-dimensional atmospheric chemistry and transport model, PACT-1D that includes continuous exchange of atmospheric air with the interstitial air of a snow layer and a kinetic treatment of multi-phase chemical processes in air and snow. PACT-1D allows modeling and assessment of the interaction of transport, chemistry, and emissions on the …

Monte Carlo Error Analysis Of Lightning Interferometry With Lofar, Paulina Turekova, Brian Hare, Olaf Scholten, Steven Cummer, Joseph Dwyer, Ningyu Liu, Chris Sterpka, Sander Ter Veen

Faculty Publications

The LOFAR radio telescope works on a principle of radio interferometric imaging. It coherently sums the signal of hundreds of antennas in northern Netherlands, covering the 30-80 MHz window of the very high frequency (VHF) band of 30-300 MHz. We are using the TRI-D algorithm to extract 3-D polarization data of a lightning flash observed by LOFAR. TRI-D functions by coherently summing recorded voltages, accounting for the antenna function, polarization, and geometric time delay for each voxel. The result is split into time slices. A coherent intensity is calculated for each time slice, and the maximum of this value is …

A Study Of The Near-Surface Vertical Distribution And Chemistry Of Pollutants In Cold-Climate Urban Areas With The Novel Pact-1d Model, Jonas Kuhn, Jochen Stutz, Meeta Cesler-Maloney, William R. Simpson, Thorsten Bartels-Rausch, Tjarda J. Roberts, Jennie L. Thomas, Jack E. Dibb, Laura Heinlein, Michael O. Sunday, Cort Anastasio, Kathleen Fahey, James H. Flynn, Fangzhou Guo

Faculty Publications

Cold-climate urban areas often face severe air pollution events in wintertime because of residential heating and vehicle emissions into shallow surface inversion layers. Many state-of-the-art regional chemistry-transport models cannot capture the small spatio-temporal scale of the transport and chemical processes occurring in these environments.Here we introduce a new version of our one-dimensional atmospheric chemistry and transport model, PACT-1D that includes continuous exchange of atmospheric air with the interstitial air of a snow layer and a kinetic treatment of multi-phase chemical processes in air and snow. PACT-1D allows modeling and assessment of the interaction of transport, chemistry, and emissions on the …

Monte Carlo Error Analysis Of Lightning Interferometry With Lofar, Paulina Turekova, Brian Hare, Olaf Scholten, Steven Cummer, Joseph Dwyer, Ningyu Liu, Chris Sterpka, Sander Ter Veen

Faculty Publications

The LOFAR radio telescope works on a principle of radio interferometric imaging. It coherently sums the signal of hundreds of antennas in northern Netherlands, covering the 30-80 MHz window of the very high frequency (VHF) band of 30-300 MHz. We are using the TRI-D algorithm to extract 3-D polarization data of a lightning flash observed by LOFAR. TRI-D functions by coherently summing recorded voltages, accounting for the antenna function, polarization, and geometric time delay for each voxel. The result is split into time slices. A coherent intensity is calculated for each time slice, and the maximum of this value is …

Properties Of Relativistic Electron Precipitation: A Comparative Analysis Of Wave-Induced And Field Line Curvature Scattering Processes, Luisa Capannolo, Andrew Staff, Wen Li, Katharine Duderstadt, Nithin Sivadas, Joshua Pettit, Sadie Elliot, Murong Qin, Xiao-Chen Shen, Qianli Ma

Faculty Publications

We analyze the properties of relativistic (&gt700 keV) electron precipitation (REP) events measured by the low-Earth-orbit (LEO) POES/MetOp constellation of spacecraft from 2012 through 2023. Leveraging the different profiles of REP observed at LEO, we associate each event with its possible driver: waves or field line curvature scattering (FLCS). While waves typically precipitate electrons in a localized radial region within the outer radiation belt, FLCS drives energy-dependent precipitation at the edge of the belt. Wave-driven REP is detected at any MLT sector and L shell, with FLCS-driven REP occurring only over the nightside–a region where field line stretching is frequent. …

Properties Of Relativistic Electron Precipitation: A Comparative Analysis Of Wave-Induced And Field Line Curvature Scattering Processes, Luisa Capannolo, Andrew Staff, Wen Li, Katharine Duderstadt, Nithin Sivadas, Joshua Pettit, Sadie Elliot, Murong Qin, Xiao-Chen Shen, Qianli Ma

Faculty Publications

We analyze the properties of relativistic (&gt700 keV) electron precipitation (REP) events measured by the low-Earth-orbit (LEO) POES/MetOp constellation of spacecraft from 2012 through 2023. Leveraging the different profiles of REP observed at LEO, we associate each event with its possible driver: waves or field line curvature scattering (FLCS). While waves typically precipitate electrons in a localized radial region within the outer radiation belt, FLCS drives energy-dependent precipitation at the edge of the belt. Wave-driven REP is detected at any MLT sector and L shell, with FLCS-driven REP occurring only over the nightside–a region where field line stretching is frequent. …

Math Is Good, Elisa Benthem

Student Work

"There are endless ways we can explore creation, each revealing something wonderfully distinct."

Posting about ­­­­­­­­the beauty of mathematics from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Math is Good

Math From Every Angle, Jeff Ploegstra

Faculty Work Comprehensive List

"As an educator, I want students to understand, appreciate, and make decisions about reality in ways informed by quantities."

Posting about ­­­­­­­­the importance of mathematics in all areas of life from In All Things - an online hub that offers insight into maintaining a faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life—until all is made new.

Math from Every Angle

Exploring The Human Side Of Math, Mike Janssen

Faculty Work Comprehensive List

"Abstract mathematics is an important way to better understand and marvel at the beauty and intricacy of God’s creation."

Posting about ­­­­­­­­experiences in math research from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Exploring the Human Side of Math