Mathematics Commons^™

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}, …

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).

Numeracy Infusion Course For Higher Education (Niche), 1: Teaching Faculty How To Improve Students' Quantitative Reasoning Skills Through Cognitive Illusions, Frank Wang, Esther I. Wilder

Publications and Research

We describe one of the eight units of a professional development program, the Numeracy Infusion Course for Higher Education (NICHE), which introduces research on cognition, including dual-processing theories, to university faculty. Under the dual-processing framework, System 1 (intuition) quickly proposes intuitive answers to judgment problems as they arise, while System 2 (deliberation) monitors the quality of these proposals, which it may endorse, correct, or override. We present several classic questions that demonstrate the pitfalls of overreliance on intuition without analytical thinking, then describe faculty participants’ responses to these questions and their ideas on how to apply cognitive illusion research to …

Non-Planar On-Shell Diagrams, Sebastían Franco, Daniele Galloni, Brenda Penante, Congkao Wen

Publications and Research

We initiate a systematic study of non-planar on-shell diagrams in N = 4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the d log form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in …

Mathematics In Contemporary Society Chapter 4, 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 …

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. …

Mathematics In Contemporary Society Chapter 2, 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 5, 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 3, 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

Mat 95: Problem Solving Process And Strategies, Salvatore Sommella

Open Educational Resources

Covers fundamentals and application of four steps of Problem Solving. Includes a problem solving strategy with a hands-on problems.

Math In The Modern World, Steven Cosares

Open Educational Resources

Students are asked to gather nutrition data regarding their favorite snacks and assess how well they contribute to their maintenance of a “healthy” diet, as defined through, articles, websites, and guidelines established by the US Department of Agriculture.

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.

Differentiability Of Correlations In Realistic Quantum Mechanics, Alejandro Cabrera, Edson De Faria, Enrique Pujals, Charles Tresser

Publications and Research

We prove a version of Bell’s theorem in which the locality assumption is weakened. We start by assuming theoretical quantum mechanics and weak forms of relativistic causality and of realism (essentially the fact that observable values are well defined independently of whether or not they are measured). Under these hypotheses, we show that only one of the correlation functions that can be formulated in the framework of the usual Bell theorem is unknown. We prove that this unknown function must be differentiable at certain angular configuration points that include the origin. We also prove that, if this correlation is assumed …

Semi-Fredholm Solvability In The Framework Of Singular Solutions For The (3+1)-D Protter-Morawetz Problem, Nedyu Popivanov, Todor Popov, Allen Tesdall

Publications and Research

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex �� of the boundary light characteristic cone and does not propagate along the bicharacteristics.The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to ��. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for …

Motivic Integration Over Nilpotent Structures, Andrew Ryan Stout

Dissertations, Theses, and Capstone Projects

This thesis concerns developing the notion of Motivic Integration in such a way that it captures infinitesimal information yet reduces to the classical notion of motivic integration for reduced schemes. Moreover, I extend the notion of Motivic Integration from a discrete valuation ring to any complete Noetherian ring with residue field $\kappa$, where $\kappa$ is any field. Schoutens' functorial approach (as opposed to the traditional model theoretic approach) allows for some very general notions of motivic integration. However, the central focus is on using this general framework to study generically smooth schemes, then non-reduced schemes, and then, finally, formal schemes. …

Self-Referentiality In Constructive Semantics Of Intuitionistic And Modal Logics, Junhua Yu

Dissertations, Theses, and Capstone Projects

This thesis explores self-referentiality in the framework of justification logic. In this framework initialed by Artemov, the language has formulas of the form t:F, which means "the term t is a justification of the formula F." Moreover, terms can occur inside formulas and hence it is legal to have t:F(t), which means "the term t is a justification of the formula F about t itself." Expressions like this is not only interesting in the semantics of justification logic, but also, as we will see, necessary in applications of justification logic in formalizing constructive contents implicitly carried by modal and intuitionistic …

An Algorithmic Approach To The Differential Galois Theory Of Second-Order Linear Differential Equations With Differential Parameters, Carlos Eduardo Arreche Aguayo

Dissertations, Theses, and Capstone Projects

We present algorithms to compute the differential Galois group G associated via the parameterized Picard-Vessiot theory to a parameterized second-order linear differential equation with respect to d/dx, with coefficients in the field of rational functions F(x) over a differential field F, where we think of the derivations on F as being derivations with respect to parameters. We build on an earlier procedure, developed by Dreyfus, that computes G when the equation is unimodular, assuming either that G is reductive, or else that its maximal reductive quotient is differentially constant. We first show how to modify the space of parametric derivations …

Combinatorial Properties Of Polyiamonds, Christopher Larson

Dissertations, Theses, and Capstone Projects

Polyiamonds are plane geometric figures constructed by pasting together equilateral triangles edge-to-edge. It is shown that a diophantine equation involving vertices of degrees 2, 3, 5 and 6 holds for all polyiamonds; then an Eberhard-type theorem is proved, showing that any four-tuple of non-negative integers that satisfies the diophantine equation can be realized geometrically by a polyiamond. Further combinatorial and graph-theoretic aspects of polyiamonds are discussed, including a characterization of those polyiamonds that are three-connected and so three-polytopal, a result on Hamiltonicity, and constructions that use minimal numbers of triangles in realizing four-vectors.

Martingales For Uniformly Quasisymmetric Circle Endomorphisms, Yunchun Hu

Dissertations, Theses, and Capstone Projects

The main subject studied in this thesis is the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Although many of our arguments work for any degree d≥2, our proof will be mainly written for degree 2 maps.

We will introduce a sequence of Markov partitions of the unit circle by using preimages of the fixed point of such circle endomorphism f. The uniform quasisymmetry condition is equivalent to the bounded nearby geometry condition of the Markov partitions. In Chapter 2 of this thesis, for each f, we use the Lebesgue invariant condition and the …

Generalized Least-Squares Regressions Iv: Theory And Classification Using Generalized Means, Nataniel Greene

Publications and Research

The theory of generalized least-squares is reformulated here using the notion of generalized means. The generalized least-squares problem seeks a line which minimizes the average generalized mean of the square deviations in x and y. The notion of a generalized mean is equivalent to the generating function concept of the previous papers but allows for a more robust understanding and has an already existing literature. Generalized means are applied to the task of constructing more examples, simplifying the theory, and further classifying generalized least-squares regressions.

Stable Commutator Length In Amalgamated Free Products, Timothy Susse

Dissertations, Theses, and Capstone Projects

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over Z^k, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. We then use the combinatorics of this algorithm to prove that for a word w in the (p, q)-torus knot complement, scl(w) is quasirational in p and q. Finally, we analyze central …

Asymptotic Invariants And Flatness Of Local Endomorphisms, Nikita Miasnikov

Dissertations, Theses, and Capstone Projects

For a local endomorphism of a noetherian local ring we introduce 3 asymptotic invariants one of which we call entropy. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite …

The Asymptotic Dirichlet Problems On Manifolds With Unbounded Negative Curvature, Ran Ji

Dissertations, Theses, and Capstone Projects

Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable if the curvature satisfies the condition $-C e^{(2-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition $-C e^{(2/3-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of $M$. As far as we know, this is the first result of this kind under unbounded curvature …

Exploring Platform (Semi)Groups For Non-Commutative Key-Exchange Protocols, Ha Lam

Dissertations, Theses, and Capstone Projects

In this work, my advisor Delaram Kahrobaei, our collaborator David Garber, and I explore polycyclic groups generated from number fields as platform for the AAG key-exchange protocol. This is done by implementing four different variations of the length-based attack, one of the major attacks for AAG, and submitting polycyclic groups to all four variations with a variety of tests. We note that this is the first time all four variations of the length-based attack are compared side by side. We conclude that high Hirsch length polycyclic groups generated from number fields are suitable for the AAG key-exchange protocol.

Delaram Kahrobaei …

Decoding Quantitative Language, Steven Cosares, Evelyn Burg

Open Educational Resources

We all need some “quantitative literacy” in order to communicate effectively. This means being able to read, write or speak intelligently in a medium where some quantitative information is a part the discussion, proof, or argument. The readers of these materials are expected to have sufficient wherewithal to understand this language so as to draw appropriate conclusions. Developing this type of literacy requires practice. It means using everyday language to describe the ways in which quantities relate to one another.