Mathematics Commons^™

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

Structure Of Fine Selmer Groups In Abelian P-Adic Lie Extensions, Debanjana Kundu, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Sujatha Ramdorai

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This paper studies fine Selmer groups of elliptic curves in abelian p -adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Z p -extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

Constrained Quantization For The Cantor Distribution With A Family Of Constraints, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, for a given family of constraints and the classical Cantor distribution we determine the constrained optimal sets of n-points, nth constrained quantization errors for all positive integers n. We also calculate the constrained quantization dimension and the constrained quantization coefficient, and see that the constrained quantization dimension D(P) exists as a finite positive number, but the D(P)-dimensional constrained quantization coefficient does not exist.

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 and Statistical Sciences Faculty Publications and Presentations

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, …

Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius

Graduate College Dissertations and Theses

In this work, the current state of the field of edge-colored graphs is surveyed. Anew concept of unshrinkable edge colorings is introduced which is useful for rainbow subgraph problems and interesting in its own right. This concept is analyzed in some depth. Building upon the linear edge ordering described in a recent work from Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer, edge-ordering graphs with the cyclic group is introduced and some results are given on this and a related counting problem.

Computing The Canonical Ring Of Certain Stacks, Jesse Franklin

Graduate College Dissertations and Theses

We compute the canonical ring of some stacks. We first give a detailed account of what this problem means including several proofs of a famous historical example. The main body of work of this thesis expands on our article \cite{Franklin-geometry-Drinfeld-modular-forms} in describing the geometry of Drinfeld modular forms as sections of a specified line bundle on a certain stacky modular curve. As a consequence of that geometry, we provide a program: one can compute the (log) canonical ring of a stacky curve to determine generators and relations for an algebra of Drinfeld modular forms, answering a problem posed by Gekeler …

Complete Solution Of The Lady In The Lake Scenario, Alexander Von Moll, Meir Pachter

Faculty Publications

In the Lady in the Lake scenario, a mobile agent, L, is pitted against an agent, M, who is constrained to move along the perimeter of a circle. L is assumed to begin inside the circle and wishes to escape to the perimeter with some finite angular separation from M at the perimeter. This scenario has, in the past, been formulated as a zero-sum differential game wherein L seeks to maximize terminal separation and M seeks to minimize it. Its solution is well-known. However, there is a large portion of the state space for which the canonical solution does not …

Symmetries And Integrable Systems, Sen-Yue Lou, Bao-Feng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Symmetry plays key roles in modern physics especially in the study of integrable systems because of the existence of infinitely many local and nonlocal generalized symmetries. In addition to the fundamental role to find exact group invariant solutions via Lie point symmetries, some important new developments on symmetries and conservation laws are reviewed. The recursion operator method is important to find infinitely many local and nonlocal symmetries of (1+1)-dimensional integrable systems. In this paper, it is pointed out that a recursion operator may be obtained from one key symmetry, say, a residual symmetry. For (2+1)-dimensional integrable systems, the master-symmetry approach …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

Existence Of Solutions By Coincidence Degree Theory For Hadamard Fractional Differential Equations At Resonance, Martin Bohner, Alexander Domoshnitsky, Seshadev Padhi, Satyam Narayan Srivastava

Mathematics and Statistics Faculty Research & Creative Works

Using the Coincidence Degree Theory of Mawhin and Constructing Appropriate Operators, We Investigate the Existence of Solutions to Hadamard Fractional Differential Equations (FRDEs) at Resonance

A Spiral Workbook For Discrete Mathematics 2nd Edition, Harris Kwong

Milne Open Textbooks

This updated text covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions, relations, and elementary combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a different perspective or at a higher level of complexity. The goal is …

Analyzing A Smartphone Battle Using Bass Competition Model, Maila Hallare, Alireza Hosseinkhan, Hasala Senpathy K. Gallolu Kankanamalage

CODEE Journal

Many examples of 2x2 nonlinear systems in a first-course in ODE or a mathematical modeling class come from physics or biology. We present an example that comes from the business or management sciences, namely, the Bass diffusion model. We believe that students will appreciate this model because it does not require a lot of background material and it is used to analyze sales data and serve as a guide in pricing decisions for a single product. In this project, we create a 2x2 ODE system that is inspired by the Bass diffusion model; we call the resulting system the Bass …

Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

Journal of Nonprofit Innovation

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

How To Intercept A High-Speed Rocket With A Pair Of Compasses And A Straightedge?, Yagub N. Aliyev

CODEE Journal

In this paper a nonlinear differential equation arising from an elementary geometry problem is discussed. This geometry problem was inspired by one of the proofs of the first remarkable limit discussed in a typical first semester undergraduate Calculus course. It is known that the involved differential equation can be reduced to Abel’s differential equation of the first kind. In this paper the problem was solved using an approximate geometric method which constructs a piecewise linear solution approximation for the curve. The compass tool of GeoGebra was extensively used for these constructions. At the end of the paper, some generalizations are …

Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, Kengo Enami, Yumiko Ohno

Theory and Applications of Graphs

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.

For two triangulations $G$ and $G'$ on a surface, $\psi_3(G)$ may not be equal to $\psi_3(G')$ even if $G$ …

The Ricci Curvature On Simplicial Complexes, Taiki Yamada

Theory and Applications of Graphs

We define the Ricci curvature on simplicial complexes modifying the definition of the Ricci curvature on graphs, and prove upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial complexes by the Ricci curvature.

Social Justice Mathematics: Classroom Practices That Give Students Rigor While Building Agency, Emily Marquise

Masters Theses

The purpose of this study is to examine the impact of a social justice approach to mathematics instruction. While many students have math aversion, students in low socioeconomic communities exhibit this to a higher degree putting them at a disadvantage as they progress through their educational career. More than 3.4 million K-12 students in the United States come from families that earn less than the median income yet achieve scores in the top percentile (Wyner et al., 2007). This raises the question of why so many students in low-socioeconomic settings are not given rigorous content that will keep them competitive …

Building Community, Competency, And Creativity In Calculus 2: Summary Of A Pilot Year Of Project Implementation, Jennifer Beichman, Candice R. Price

Feminist Pedagogy

In light of the COVID-19 pandemic, instructional modes at our institution moved to fully online and remote, then fully online but on campus, and back to in-person learning in fall 2021. To combat perceived issues in student engagement, we piloted using group projects in place of exams at the natural content break points in Calculus 2.

Turing Patterns In A P-Adic Fitzhugh-Nagumo System On The Unit Ball, L. F. Chacón-Cortés, C. A. Garcia-Bibiano, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We introduce discrete and p-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional p-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the p-adic unit ball.