Mathematics Commons^™

Parametrization Of Formal Norm Compatible Sequences, Joseph Dicapua

Dissertations, Theses, and Capstone Projects

We give a classification of power series parametrizing Lubin-Tate trace compatible sequences. This proof answers a question posed in the literature by Berger and Fourquaux. Lubin-Tate trace compatible sequences are a generalization of norm compatible sequences, which arise in Iwasawa theory and local class field theory. The result we prove generalizes the interpolation theorem proved by Coleman in the classical norm compatible sequence case. We also, jointly with Victor Kolyvagin, give a method for finding such series explicitly in certain special cases.

Math 115: College Algebra Instructor Guide, Seth Lehman

Open Educational Resources

OER instructor guide for Math 115, College Algebra, Queens College

A Student Guide To Using Ai To Enhance Algebraic Understanding, Karan Puri

Open Educational Resources

This is a step-by-step guide that students can follow to use AI tools to check their understanding of concepts that have been tested in the introductory/college algebra classroom.

Project Title: Maximizing The Volume Of A Cardboard Box To Save Trees– An Application Of Polynomial Functions To Address Global Issues [Mathematics], Lucie Mingla

Open Educational Resources

MAT 115 College Algebra & Trigonometry/Precalculus

Project Title: Maximizing the Volume of a Cardboard Box to Save Trees– An Application of Polynomial Functions to Address Global Issues

Reflective Narrative:

This project was inspired by my participation in the "Designing and Implementation of STEM Co-Curricular Activities" CTL seminar in Spring 2023. I am grateful to Drs. Bukurie Gjoci, Daniel Gertner, Ingrid Veras, and Midas Tsai, along with fellow participants, for their invaluable feedback that helped shape its development. The project was implemented in two College Algebra and Trigonometry courses. I participated in two seminars to further develop this project. The Community …

Developing Mathematical Maturity By Solving A Given Quadratic Equation, Armando A. Amador, Nieves Angulo, Juan B. Lacay

Publications and Research

This article explores the instructional challenges faced by mathematics educators when addressing students’ limited background knowledge and underdeveloped mathematical maturity, two critical barriers to success in college-level algebra. Using the quadratic equation 2x2+x−3=0, which is reducible over the field of rational numbers ℚ ⊂ ℝ, we analyze how diverse solution methods—factoring, completing the square, and the quadratic formula—can support learning. An additional strategy, “Slide and Divide,” was introduced to promote procedural fluency and conceptual flexibility. To gain insight into students’ evolving mathematical maturity, an online survey was conducted to students at various levels of mathematics. The survey captured their perspectives …

Rings With Q-Torsionfree Canonical Modules, Naoki William Endo, Laura P. Ghezzi, Shiro S. Goto, Jooyoun Hong Phd, Shin-Ichiro Iai, Toshinori Kobayashi, Naoyuki Matsuoka, Ryo Takahashi

Publications and Research

Let A be a Noetherian local ring with canonical module K A . We characterize A when K A is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3 . If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K A is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein …

Mathematics In Contemporary Society, Patrick J. Wallach

Open Educational Resources

Mathematics in Contemporary Society is the textbook that corresponds to MA-321, the course of the same name. The course is designed to provide students with mathematical ideas and methods found in the social sciences, the arts, and in business. Topics will include fundamentals of statistics, scatterplots, graphics in the media, problem solving strategies, dimensional analysis, mathematics in music and art, and mathematical modeling. EXCEL is used to explore real world applications.

Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

Dissertations, Theses, and Capstone Projects

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, Michael R. Marinelli

Dissertations, Theses, and Capstone Projects

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was …

On Deformation Rings Of Residual Galois Representations With Three Jordan–Hölder Factors, Xiaoyu Huang

Dissertations, Theses, and Capstone Projects

In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan–Hölder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke algebra, we also prove an R = T theorem in the general context. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Assuming the Bloch-Kato conjecture, our result …

Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

Dissertations, Theses, and Capstone Projects

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

Mathematical Methods For Economics And Business Syllabus, Marian Melnyk

Open Educational Resources

This syllabus outlines an online asynchronous course introducing essential mathematical tools and techniques in economics. It spans 24 sessions covering Functions, Differentiation, Limits, Integration, and Multivariable Functions, grounded in college algebra and calculus. The course utilizes Open Educational Resources (OER) for all materials, available in text and video formats on Blackboard. Continuous assessment, including quizzes, homework, two 1-hour tests, and a final exam, ensures consistent progress. Weekly online office hours provide additional support. The primary objective is to equip students with the mathematical skills necessary for economic analysis and problem-solving.

Bi-Harmonic Maps' Existence And P-Balanced Energy And Connections To Harmonic Maps, Lina Wu

Publications and Research

We discover a bi-harmonic map’ existence, its energy growth, and its connections to a harmonic map. First, we prove the existence of a nontrivial bi-harmonic map in a unit sphere. Second, we investigate the p-balanced energy growth of biharmonic maps. As the most important energy technical breakthroughs, we propose an innovative energy algorithm called p-balanced energy technique to break the constraints of the existing L^q-energy technique in detecting L^q-energy growth towards boundlessness. The disadvantage of the finite L^q -energy technique in the L ^q spaces is not effective in dealing with infinite L^q- energy in Non-L^q spaces. Third, we study …

Me And Mathematics: “Doing What You’Re Talking About”: In Dialogue With My Family, Eden Morris

Dissertations, Theses, and Capstone Projects

This paper is a philosophically oriented accompaniment to my audio project (accessible through the following link: https://cuny.manifoldapp.org/projects/me-and-mathematics). Working together, the paper and audio collages form a call to action and a resource. My primary finding is the importance of doing what you’re talking about or exploring and implementing your ideas experientially. Doing what you’re talking about is important for effective teaching/learning and feeling in line with oneself. This working concept came to my attention during my research conversation with my oldest living relative, and then, again, with my youngest (non-baby) relative. This doing what you’re talking about is a way …

Parabolic And Non-Parabolic Surfaces With Small Or Large End Spaces Via Fenchel-Nielsen Parameters, Michael Antony Pandazis

Dissertations, Theses, and Capstone Projects

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each …

Higher Diffeology Theory, Emilio Minichiello

Dissertations, Theses, and Capstone Projects

Finite dimensional smooth manifolds have been studied for hundreds of years, and a massive theory has been built around them. However, modern mathematicians and physicists are commonly dealing with objects outside the purview of classical differential geometry, such as orbifolds and loop spaces. Diffeology is a new framework for dealing with such generalized smooth spaces. This theory (whose development started in earnest in the 1980s) has started to catch on amongst the wider mathematical community, thanks to its simplicity and power, but it is not the only approach to dealing with generalized smooth spaces. Higher topos theory is another such …

Bifurcations And Resultants For Rational Maps And Dynatomic Modular Curves In Positive Characteristic, Colette Lapointe

Dissertations, Theses, and Capstone Projects

No abstract provided.

Sigma_N-Correct Forcing Axioms, Benjamin P. Goodman

Dissertations, Theses, and Capstone Projects

I introduce a new family of axioms extending ZFC set theory, the Sigma_n-correct forcing axioms. These assert roughly that whenever a forcing name a' can be forced by a poset in some forcing class Gamma to have some Sigma_n property phi which is provably preserved by all further forcing in Gamma, then a' reflects to some small name such that there is already in V a filter which interprets that small name so that phi holds. Sigma_1-correct forcing axioms turn out to be equivalent to classical forcing axioms, while Sigma_2-correct forcing axioms for Sigma_2-definable forcing classes are consistent relative to …

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

Dissertations, Theses, and Capstone Projects

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, Brad Isaacson

Publications and Research

In a previous paper, we expressed three families of character sums by certain generalized Bernoulli functions which in turn were expressed by generalized Bernoulli numbers via a complicated and indirect process. In this paper, we generalize a theorem of Ibukiyama to directly express these generalized Bernoulli functions by generalized Bernoulli numbers. As a result, we can express the three families of character sums by generalized Bernoulli numbers in a more elegant fashion than was done before.

Syllabus For Oer Version Of Math 20100, Thea Pignataro

Open Educational Resources

Syllabus for OER version of first semester calculus with analytic geometry for physical sciences and engineering.

A Causal Inference Approach For Spike Train Interactions, Zach Saccomano

Dissertations, Theses, and Capstone Projects

Since the 1960s, neuroscientists have worked on the problem of estimating synaptic properties, such as connectivity and strength, from simultaneously recorded spike trains. Recent years have seen renewed interest in the problem coinciding with rapid advances in experimental technologies, including an approximate exponential increase in the number of neurons that can be recorded in parallel and perturbation techniques such as optogenetics that can be used to calibrate and validate causal hypotheses about functional connectivity. This thesis presents a mathematical examination of synaptic inference from two perspectives: (1) using in vivo data and biophysical models, we ask in what cases the …

New Algorithmic Support For The Fundamental Theorem Of Algebra, Vitaly Zaderman

Dissertations, Theses, and Capstone Projects

Univariate polynomial root-finding is a venerated subjects of Mathematics and Computational Mathematics studied for four millenia. In 1924 Herman Weyl published a seminal root-finder and called it an algorithmic proof of the Fundamental Theorem of Algebra. Steve Smale in 1981 and Arnold Schonhage in 1982 proposed to classify such algorithmic proofs in terms of their computational complexity. This prompted extensive research in 1980s and 1990s, culminated in a divide-and-conquer polynomial root-finder by Victor Pan at ACM STOC 1995, which used a near optimal number of bit-operations. The algorithm approximates all roots of a polynomial p almost as fast as one …

New Effective Transformational Computational Methods, Jun Zhang, Ruzong Fan, Fangyang Shen, Junyi Tu

Publications and Research

Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, …

On Cyclically 4-Connected Cubic Graphs, R. J. Kingan, S. R. Kingan

Publications and Research

A 3-connected cubic graph is cyclically 4-connected if it has at least n ≥ 8 vertices and when removal of a set of three edges results in a disconnected graph, only one component has cycles. By introducing the notion of cycle spread to quantify the distance between pairs of edges, we get a new characterization of cyclically 4-connected graphs. Let Q_n and V_n denote the ladder and Möbius ladder on n ≥ 8 vertices, respectively. We prove that a 3-connected cubic graph G is cyclically 4-connected if and only if G is either the Petersen graph, Q …

Time-Efficient Filtering Of Imaging Polarimetric Data By Checking Physical Realizability Of Experimental Mueller Matrices, Tatiana Novikova, Alexey Ovchinnikov, Gleb Pogudin, Jessica C. Ramella-Roman

Publications and Research

Motivation: Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing, and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filter out non-physical data, i.e. to test the positive semi-definiteness of the 4 × 4 Hermitian coherency matrix calculated from the elements of corresponding Mueller matrix pixel-wise. For this purpose, we compared the execution time for the calculations of (i) eigenvalues, (ii) Cholesky decomposition, (iii) Sylvester’s criterion, and (iv) coefficients of the characteristic …

Quotients Of The Curve Complex, Joseph Maher, Hidetoshi Masai, Saul Schleimer

Publications and Research

We consider three kinds of quotients of the curve complex, which are obtained by coning off uniformly quasiconvex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the mapping class group on those quotients are strongly weakly properly discontinuously (WPD), which implies that the actions are non-elementary and those quotients are of infinite diameter.

Are The Cans In The Store “Volume Optimized”? [Mathematics], Bukurie Gjoci

Open Educational Resources

This is one of LaGuardia’s Project Connexion STEM Team’s experiential learning activities. Project Connexion's purpose is to promote creative thinking on how to engage students in the classroom. As part of this, the STEM team developed Experiential/co-curricular activities that demonstrated to students how their work in class connects to the world around them. These activities were embedded into the syllabus to ensure the participation of all students. Each professor designed a Co-curricular activity for their courses, ensuring that the Co-curricular activity directly linked course material to the outside world.

This Calculus I Experiential Learning Project aligns with one of the …

Math 115: College Algebra For Pre-Calculus, Seth Lehman

Open Educational Resources

OER course syllabus for Math 115, College Algebra, at Queens College

Soundness And Completeness Results For The Logic Of Evidence Aggregation And Its Probability Semantics, Eoin Moore

Dissertations, Theses, and Capstone Projects

The Logic of Evidence Aggregation (LEA), introduced in 2020, offers a solution to the problem of evidence aggregation, but LEA is not complete with respect to the intended probability semantics. This left open the tasks to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. In this thesis we do both. We also develop the proof theory for some LEA-related logics and show surprising connections between LEA-related logics and Lax Logic.