Algebra Commons^™

Analysis Of A Group Of Automorphisms Of A Free Group As A Platform For Conjugacy-Based Group Cryptography, Pavel Shostak

Dissertations, Theses, and Capstone Projects

Let F be a finitely generated free group and Aut(F) its group of automorphisms.

In this monograph we discuss potential uses of Aut(F) in group-based cryptography.

Our main focus is on using Aut(F) as a platform group for the Anshel-Anshel-Goldfeld protocol, Ko-Lee protocol, and other protocols based on different versions of the conjugacy search problem or decomposition problem, such as Shpilrain-Ushakov protocol.

We attack the Anshel-Anshel-Goldfeld and Ko-Lee protocols by adapting the existing types of the length-based attack to the specifics of Aut(F). We also present our own version of the length-based attack that significantly increases the attack' success …

Understanding The Impact Of Peer-Led Workshops On Student Learning, Afolabi Ibitoye, Nadia Kennedy, Armando Cosme

Publications and Research

As students we often wonder why some subjects are easy to understand and requires not much effort in terms of re-reading the material, for us to grasp what it entails. One subject seems to remain elusive and uneasy for a vast majority of learners at all levels of education; that subject is Mathematics, it is one subject that most learners finds difficult even after doubling the amount of time spent on studying the material. My intention is to explore ways to make Mathematics easier for other students using feedback from students enrolled in NSF mathematics peer leading workshops, and use …

An Invitation To Linear Algebra, David N. Pham, Jonathon Funk, Wenjian Liu

Open Educational Resources

This is an OER textbook on linear algebra.

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these …

College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Dabkowska

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Masklanko, Ewa Dabkowska

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Inquiry Into Saving [Mathematics], Jeanne Funk

Open Educational Resources

‘Inquiry Into Saving’ is an assignment originally designed for MAT117, which is a course for students who have been placed in basic skills mathematics and who can apply a college level course in Algebra and Trigonometry to their program. These students should, ideally, be early in their LaGuardia career, though that is frequently not the case. All, however, are novices of mathematics. The assignment was vetted and revised based on feedback from the Inquiry and Problem Solving in STEM CTL seminar and a charrette not affiliated with the seminar. Revisions addressed connections to the Inquiry and Problem Solving/Written competency/ability pair, …

World Population Dynamics: Modeling Involving Polynomial Functions [Mathematics], Mangala Kothari

Open Educational Resources

In this Inquiry and Problem Solving Assignment students are expected to reflect on their analysis and compare their results with the actual population by conducting their own elementary level research such as searching databases, gathering information and interpreting. Students are expected to comment on the scope of the mathematical model and connect their learning in context to the real-world problem. The assignment includes open-ended questions such as: Write a paragraph about the dynamics of population for the world. What could be some of the possible parameters that contribute to the change in population size? Reflect on what you learned by …

Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of …

Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak

Dissertations, Theses, and Capstone Projects

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this dissertation, we seek to extend these techniques to finitely presented non-free groups, in particular to polycyclic and metabelian groups that are of interest to non-commutative cryptography.

As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. …

Arithmetic | Algebra Homework, Samar Elhitti, Ariane Masuda, Lin Zhou

Open Educational Resources

Arithmetic | Algebra Homework book is a static version of the WeBWork online homework assignments that accompany the textbook Arithmetic | Algebra for the developmental math courses MAT 0630 and MAT 0650 at New York City College of Technology, CUNY.

Study Guide For Cuny Elementary Algebra Final Exam (Ceafe), Olen Dias, William Baker, Amrit Singh

Open Educational Resources

No abstract provided.

College Algebra Through Problem Solving, Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Dabkowska

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Arithmetic | Algebra, Samar Elhitti, Marianna Bonanome, Holly Carley, Thomas Tradler, Lin Zhou

Open Educational Resources

Arithmetic | Algebra provides a customized open-source textbook for the math developmental students at New York City College of Technology. The book consists of short chapters, addressing essential concepts necessary to successfully proceed to credit-level math courses. Each chapter provides several solved examples and one unsolved “Exit Problem”. Each chapter is also supplemented by its own WeBWork online homework assignment. The book can be used in conjunction with WeBWork for homework (online) or with the Arithmetic | Algebra Homework handbook (traditional). The content in the book, WeBWork and the homework handbook are also aligned to prepare students for the CUNY …

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

Dissertations, Theses, and Capstone Projects

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form …

Some 2-Categorical Aspects In Physics, Arthur Parzygnat

Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description …

The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, Rachel A. Oppenheimer

Dissertations, Theses, and Capstone Projects

Nationwide, there is a crisis in math learning and math achievement at all levels of education. Upwards of 80% of students who enter the City University of New York’s community colleges from New York City’s Department of Education high schools fail to meet college level math proficiencies and as a result, are funneled into the system’s remedial math system. Once placed into pre-college remedial arithmetic, pre-algebra, and elementary algebra courses, students fail at alarming rates and research indicates that students’ failure in remedial math has negative ripple effects on their persistence and degree completion. CUNY is not alone in facing …

P-Adic L-Functions And The Geometry Of Hida Families, Joseph Kramer-Miller

Dissertations, Theses, and Capstone Projects

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using …

Supplemental Instruction For Developmental Mathematics: Two-Year Summary, Olen Dias, Alice W. Cunningham, Loreto Porte

Publications and Research

Supplemental instruction—using trained peer tutors to conduct additional class sessions in a group-work format—has been in use for over forty years. However, its success in developmental mathematics has been inconclusive. In the two years since institution of the strategy for developmental mathematics students at Hostos Community College, overall results (n = 5403 students) show significantly improved course pass rates to at least a 99% confidence level. Although no significant course retention differences have yet appeared, academic success itself promotes future retention. The program has proved beneficial for the College’s developmental mathematics students and is being expanded. Future research including …

Multiple Problem-Solving Strategies Provide Insight Into Students’ Understanding Of Open-Ended Linear Programming Problems, Marla A. Sole

Publications and Research

Open-ended questions that can be solved using different strategies help students learn and integrate content, and provide teachers with greater insights into students’ unique capabilities and levels of understanding. This article provides a problem that was modified to allow for multiple approaches. Students tended to employ high-powered, complex, familiar solution strategies rather than simpler, more intuitive strategies, which suggests that students might need more experience working with informal solution methods. During the semester, by incorporating open-ended questions, I gained valuable feedback, was able to better model real-world problems, challenge students with different abilities, and strengthen students’ problem solving skills.

My Math Gps: Elementary Algebra Guided Problem Solving (2016 Edition), Jonathan Cornick, G Michael Guy, Karan Puri

Open Educational Resources

My Math GPS: Elementary Algebra Guided Problem Solving is a textbook that aligns to the CUNY Elementary Algebra Learning Objectives that are tested on the CUNY Elementary Algebra Final Exam (CEAFE). This book contextualizes arithmetic skills into Elementary Algebra content using a problem-solving pedagogy. Classroom assessments and online homework are available from the authors.

My Math Gps: Elementary Algebra Guided Problem Solving, Jonathan Cornick, G Michael Guy, Karan Puri

Open Educational Resources

My Math GPS: Elementary Algebra Guided Problem Solving is a textbook that aligns to the CUNY Elementary Algebra Learning Objectives that are tested on the CUNY Elementary Algebra Final Exam (CEAFE). This book contextualizes arithmetic skills into Elementary Algebra content using a problem-solving pedagogy. Classroom assessments and online homework are available from the authors. In a paper to appear in PRIMUS, this textbook and accompanying pedagogy showed significant increases in learning outcomes.

Polynomials Occuring In Generating Function Identities For B-Ary Partitions, David Dakota Blair

Graduate Student Publications and Research

Let p_b(n) be the number of integer partitions of n whose parts are powers of b. For each m there is a generating function identity:

f_m(b,q)\sum_{n} p_b(n) q^n = (1-q)^m \sum_{n} p_b(b^m n q)q^n

where n ranges over all integer values. The proof of this identity appears in the doctoral thesis of the author. For more information see http://dakota.tensen.net/2015/rp/.

This dataset is a JSON object with keys m from 1 to 23 whose values are f_m(b,q).

A Primer For Mathematical Modeling, Marla A. Sole

Publications and Research

With the implementation of the National Council of Teachers of Mathematics recommendations and the adoption of the Common Core State Standards for Mathematics, modeling has moved to the forefront of K-12 education. Modeling activities not only reinforce purposeful problem-solving skills, they also connect the mathematics students learn in school with the mathematics they will use outside of school. Instructors have found mathematical modeling difficult to teach. To successfully incorporate modeling activities I believe that curricular changes should be accompanied by professional development for curriculum developers, classroom teachers, and higher education professionals. This article serves as an introduction to modeling by …

Noncomputable Functions In The Blub-Shub-Smale Model, Wesley Calvert, Ken Kramer, Russell Miller

Publications and Research

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more …

Iterated Aluthge Transforms: A Brief Survey, Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

Publications and Research

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then the Aluthge transform is defined by

∆(T) = |T|^1/2U|T|^1/2.

Let ∆ⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. ∆⁰(T) = T and ∆ⁿ(T) = ∆(∆ⁿ⁻¹(T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of …

On Groups Of Homological Dimension One, Jonathan Cornick

Publications and Research

It has been conjectured that the groups of homological dimension one are precisely the nontrivial locally free groups. Some algebraic, geometric and analytic properties of any potential counter example to the conjecture are discussed.