Physical Sciences and Mathematics Commons^™

The Mathematics And Applications Behind Image Warping And Morphing, Tanvir Prince, Maria Malik, Ildefonso Salva, Ariel Mazor, Sakhr Aldaylam

Publications and Research

This research is conducted in the summer of 2015 and is possible by the support of various agency, in particular, by the grant of Prof. Angulo Nieves and the New York City Research Initiative.

The purpose of this research is to reveal the mathematics and applications of the computer animation techniques of warping and morphing. A warp is a twist or distortion in the form of an object in an image while a morph is the smooth and gradual transformation of an object in one image into the object in another image. Linear algebra makes these computer animation techniques possible; …

Supporting Teachers’ Learning About Mathematical Modeling, June L. Gastón, Barbara A. Lawrence

Publications and Research

In the United States, one of the Standards for Mathematical Practice of the Common Core Curriculum (Common Core State Standards Initiative, 2010) is Model with mathematics. This standard requires that students be taught in a manner that will enable them to ―apply the mathematics they know to solve problems arising in everyday life, society, and the workplace‖ (p. 7). However many prospective and practicing teachers acquire a pedagogical style that does not support this standard. To promote higher levels of student thinking associated with mathematical modeling, teachers must thus be taught not only what mathematical modeling is, but how it …

Mathematics In Contemporary Society Chapter 1, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Mathematics In Contemporary Society Chapter 6, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Mathematics In Contemporary Society Chapter 7, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Mathematics In Contemporary Society Chapter 10, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Mathematics In Contemporary Society Chapter 9, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Mathematics In Contemporary Society Chapter 8, 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.

The textbook was posted in weekly installments:

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11

Equations Over Hyperbolic Groups, Alexander Kai-Chi Taam

Dissertations, Theses, and Capstone Projects

We show that the Diophantine problem, for quadratic equations over a non-elementary torsion-free hyperbolic group, is NP-complete. Furthermore, given a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a canonical $Hom$-diagram and a complete set of induced $\Gamma$-NTQ systems, for $\Gamma_{R(S)}$. Finally, the class of $\Gamma$-limit groups is the same as that of iterated generalized doubles over $\Gamma$.

Algorithmic Properties Of Poly-Z Groups And Secret Sharing Using Non-Commutative Groups, Bren Cavallo

Dissertations, Theses, and Capstone Projects

Computational aspects of polycyclic groups have been used to study cryptography since 2004 when Eick and Kahrobaei proposed polycyclic groups as a platform for conjugacy based cryptographic protocols.

In the first chapter we study the conjugacy problem in polycyclic groups and construct a family of torsion-free polycyclic groups where the uniform conjugacy problem over the entire family is at least as hard as the subset sum problem. We further show that the conjugacy problem in these groups is in NP, implying that the uniform conjugacy problem is NP-complete over these groups. This is joint work with Delaram Kahrobaei. We also …

On String Topology Operations And Algebraic Structures On Hochschild Complexes, Manuel Rivera

Dissertations, Theses, and Capstone Projects

The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan's observation of the fact that the intersection product on the homology of a smooth manifold $M$ can be combined with the concatenation product on the homology of the based loop space on $M$ to obtain a new product on the homology of $LM$, the space of free loops on $M$. Since then, a vast family of operations on the homology of $LM$ have been discovered.

In this thesis we focus our attention on …

Counting Restricted Integer Partitions, David Dakota Blair

Dissertations, Theses, and Capstone Projects

The first chapter examines $p_b(n)$, the number of partitions of $n$ into powers of $b$, along with a family of identities which can be deduced by iterating a recurrence satisfied by $p_b(n)$ in a suitable way. These identities can then be used to calculate $p_b(n)$ for large values of $n$.

The second chapter restricts these types of partitions even further, limiting the multiplicity of each part. Its object of study is $p_{b,d}(n)$, that is, the number of partitions of $n$ into powers of $b$ repeating each power at most $d$ times. The methods of the first chapter are applied, and …

The Moduli Space Of Rational Maps, Lloyd William West

Dissertations, Theses, and Capstone Projects

We construct the moduli space, M_d, of degree d rational maps on ℙ¹ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for M₃.

Using this construction, we give a method for solving the following problems: (1) explicitly construct, from a moduli point P ∈ M_d(k), a rational map ∅ with the given moduli; (2) find the field of definition of a rational map ∅. We work out the method in detail for the case d = 3.

Some Bernstein Type Results Of Graphical Self-Shrinkers With High Codimension In Euclidean Space, Hengyu Zhou

Dissertations, Theses, and Capstone Projects

A self-shrinker characterizes the type I singularity of the mean curvature flow. In this thesis we concern about some Bernstein type results of graphical self-shrinkers with high codimension in Euclidean space.

There are two main tools in our work. The first one is structure equations of graphical self-shriners in terms of parallel forms (Theorem 2.3.6). This is motivated by M.T.Wang's work ([Wan02]) on graphical mean curvature flows with arbitrary codimension in product manifolds. The second one is an integration technique (Lemma 2.4.5) based on the fact that every graphical graphical self-shrinker has the polynomial volume growth (Corollary 2.4.4). Because of …

The Geometry And Combinatorics Of Closed Geodesics On Hyperbolic Surfaces, Chris Arettines

Dissertations, Theses, and Capstone Projects

In this thesis, we obtain combinatorial algorithms that determine the minimal number of self-intersections necessary for a free homotopy class $[\gamma]$ on an orientable surface, using algebraic input. Using this same input, we describe another algorithm which determines whether or not a minimally intersecting curve in $[\gamma]$ is \textit{filling}, that is, whether or not the complement is a disjoint union of disks or punctured disks. Next, we use these algorithms as inspiration for proving the existence of filling curves which self-intersect $2g-1$ times, which is the minimal number of intersections possible. The combinatorial viewpoint that is developed can then be …

Precalculus, Thomas Tradler, Holly Carley

Open Educational Resources

From the preface, "These are notes for a course in precalculus, as it is taught at New York City College of Technology - CUNY (where it is offered under the course number MAT 1375). Our approach is calculator based. For this, we will use the currently standard TI-84 calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen …

What You Gotta Know To Play Good In The Iterated Prisoner’S Dilemma, Ethan Akin

Publications and Research

For the iterated Prisoner’s Dilemma there exist good strategies which solve the problem when we restrict attention to the long term average payoff. When used by both players, these assure the cooperative payoff for each of them. Neither player can benefit by moving unilaterally to any other strategy, i.e., these provide Nash equilibria. In addition, if a player uses instead an alternative which decreases the opponent’s payoff below the cooperative level, then his own payoff is decreased as well. Thus, if we limit attention to the long term payoff, these strategies effectively stabilize cooperative behavior. The existence of such strategies …

Geometric Separation And Packing Problems, Ivo Vigan

Dissertations, Theses, and Capstone Projects

The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the …

Force To Change Large Cardinal Strength, Erin Kathryn Carmody

Dissertations, Theses, and Capstone Projects

This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that $\kappa$ is still $\beta$-inaccessible in the extension, for every $\beta < \alpha$, but not $\alpha$-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo cardinals. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that for every $\beta < \alpha$, the cardinal $\kappa$ is still $\beta$-Mahlo in the extension, but not $\alpha$-Mahlo. I also show that a cardinal $\kappa$ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there. The thesis includes a collection of results which give forcing notions which change large cardinal strength from weakly compact to weakly measurable, including some earlier work by others that fit this theme. I consider in detail measurable cardinals and Mitchell rank. I show how to change a class of measurable cardinals by forcing to an extension where all measurable cardinals above some fixed ordinal $\alpha$ have Mitchell rank below $\alpha.$ Finally, I consider supercompact cardinals, and strongly compact cardinals. I show how to change the Mitchell rank for supercompactness for a class of cardinals.

New Results On Randomized Matrix Computations, Jesse Lowell Wolf

Dissertations, Theses, and Capstone Projects

The aim of this thesis is to present new results in randomized matrix computations. Specifically, and ultimately, we show how to modify, or preprocess an ill conditioned matrix having small numerical nullity (co-rank) into a nonsingular well conditioned matrix. This has intrinsic theoretical interest and we show a sample application to accurate solutions of nonsingular and ill conditioned linear systems. We discuss both multiplicative and additive preprocessing; in fact the multiplicative case assists in the derivation of the additive case. In the additive case, we approximate a nonsingular ill conditioned matrix by a singular well conditioned matrix which is then …

Some Applications Of Noncommutative Groups And Semigroups To Information Security, Lisa Bromberg

Dissertations, Theses, and Capstone Projects

We present evidence why the Burnside groups of exponent 3 could be a good candidate for a platform group for the HKKS semidirect product key exchange protocol. We also explore hashing with matrices over SL₂(F_p), and compute bounds on the girth of the Cayley graph of the subgroup of SL₂(F_p) for specific generators A, B. We demonstrate that even without optimization, these hashes have comparable performance to hashes in the SHA family.

The Holomorphic Couch Theorem, Maxime Fortier Bourque

Dissertations, Theses, and Capstone Projects

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.

The P -Royden And P -Harmonic Boundaries For Metric Measure Spaces, Marcello Lucia, Michael J. Puls

Publications and Research

Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfy certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

Discovering Geometric And Topological Properties Of Ellipsoids By Curvatures, Lina Wu, Shihshu Walter Wei, Jia Liu, Ye Li

Publications and Research

Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting.

Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics.

Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), …

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.

Generalized Least-Squares Regressions V: Multiple Variables, Nataniel Greene

Publications and Research

The multivariate theory of generalized least-squares is formulated here using the notion of generalized means. The multivariate generalized least-squares problem seeks an m dimensional hyperplane which minimizes the average generalized mean of the square deviations between the data and the hyperplane in m + 1 variables. The numerical examples presented suggest that a multivariate generalized least-squares method can be preferable to ordinary least-squares especially in situations where the data are ill- conditioned.

Explicit Solutions Of Imaginary Quadratic Norm Equations, Sandra Sze

Dissertations, Theses, and Capstone Projects

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic extension of $\mathbb{Q}$. Let $h$ be the class number and $D$ be the discriminant of the field $K$. Assume $p$ is a prime such that $\displaystyle\left(\frac{D}{p}\right)=1$. Then $p$ splits in $K$. The elements of the ring of integers $\mathcal{O}_K$ are of the form $x+\sqrt{-d}y$ if $d\equiv1,2\pmod{4}$ and $\displaystyle x+\frac{1+\sqrt{-d}}{2}y$ if $d\equiv3\pmod{4}$, where $x$ and $y\in \mathbb{Z}$. The norm \\$N_{K/\mathbb{Q}}(x+\sqrt{-d}y)=x^2+dy^2$ and $N_{K/\mathbb{Q}}\left(\displaystyle x+\frac{1+\sqrt{-d}}{2}y\right)=\displaystyle\frac{(2x+y)^2}{4}+\frac{dy^2}{4}$. In this thesis, we find the elements of norm $p^h$ explicitly. We also prove certain congruences for solutions of norm equations.

On Polynomial Roots Approximation Via Dominant Eigenspaces And Isolation Of Real Roots, Omar Ivan Retamoso Urbano

Dissertations, Theses, and Capstone Projects

Finding the roots of a given polynomial is a very old and noble problem in mathematics and computational mathematics. For about 4,000 years, various approaches had been proposed to solve this problem (see cite{FC99}). In 1824, Niels Abel showed that there existed polynomials of degree five, whose roots could not be expressed using radicals and arithmetic operations through their coefficients. Here is an example of such polynomials:$$x^5-4x-2.$$

Thus we must resort to iterative methods to approximate the roots of a polynomial given with its coefficients.

There are many algorithms that approximate the roots of a polynomial(see cite{B40}, cite{B68}, cite{MN93}, cite{MN97}, …

Calculus For Everyone, Sandra Kingan, Brooklyn College Library And Academic It

Open Educational Resources

Professor Kingan’s motivation for writing her free Calculus I textbook was to help address the departments high failure rates in Calculus. Along with another CUNY initiative to offer Calculus workshops in advance of taking the course, Kingan’s concise textbook in Calculus I offers students inside and outside of CUNY an opportunity to prepare for Calculus I at their own pace. She also believes that by providing free access to this material she could help to overcome some of the inequity students experience when Calculus is not offered in their high school. The textbook was written specifically for this pilot project. …

Elementary Algebra, Mangala Kothari

Open Educational Resources

Students use world population data to find the observed trend, the rate of population growth, build a model (linear equation) using the data, and predict the future world’s population size based on the model that they develop.