Physical Sciences and Mathematics Commons^™

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 …

Simulation As A Predictor In Probability, Xiaona Zhou

Publications and Research

In this study, we simulate bivariate normal data. We gain intuition about the bivariate normal distribution by comparing the generated data to the associated bivariate normal density surface. We also get results about covariance and correlation. We will use tools from linear algebra to discuss transformations of random normal vectors, and the use of contours.

Validation Of A Lottery, Xiaona Zhou

Publications and Research

The NY Pick 4 lottery consists of four "randomly" chosen digits from 0 to 9. For this to be fair, each digit should be equally likely to occur. To determine whether this is the case, a Chi-squared goodness of fit test will be applied to historical data. This provides a quantitative way of measuring how well the observed frequency of digits matches our expectations of a fair lottery. We also explore the "Lucky Sum", which is also a part of the Pick 4. We determine which sum is most likely to occur, and what the odds of winning are if …

On A Generalization Of A Theorem Of Ibukiyama, Brad Isaacson

Publications and Research

We generalize a theorem of Ibukiyama and express periodic generalized Bernoulli functions by generalized Bernoulli numbers. As a corollary, we obtain formulas expressing these character sums by generalized Bernoulli numbers using only elementary methods from algebra and number theory.

Teaching With Technology: Using A Virtual Learning Community And Peer Mentoring To Create An Interdisciplinary Intervention, Rebecca Mazumdar, Nadia Benakli, Pamela Brown

Publications and Research

This paper describes the development and implementation of engaging and supportive experiences to promote student engagement, persistence and success at a commuter, open enrollment, public, minority serving institution. Project components included faculty development at the SENCER Summer Institute (SSI) 2016, attended by a team comprised of an academic administrator, full-time faculty from English and math, and part-time faculty in chemistry; creation of a virtual learning community of freshmen enrolled in chemistry, English, and math linked by the specific theme of the environmental impacts of deicing roads with salt and the overarching theme of the impacts of human activities on the …

Evidence For Maxwell's Equations, Fields, Force Laws And Alternative Theories Of Classical Electrodynamics, Max Tran

Publications and Research

The set of equations known today as Maxwell's equations along with a few constitutive equations lie at the heart of classical electromagnetism. A common misconception held by many is that Maxwell's equations are essential, and that classical electromagnetic theory is settled science and is no longer an active field of investigations. We will review the four Maxwell's equations and related equations, their supporting experimental evidence, the field concept, and the Lorentz and Ritz force laws. We will give a brief outline of two approaches to classical electromagnetism which bypass Maxwell's equations, the propagated potential approach and the direct action approach …

Character Sums Of Lee And Weintraub, Brad Isaacson

Publications and Research

We prove two conjectures of Lee and Weintraub and one conjecture of Ibukiyama and Kaneko about character sums arising as fixed point contributions in the Atiyah–Singer holomorphic Lefshetz formula applied to finite group actions on the space of certain Siegel cusp forms.

R Program For Estimation Of Group Efficiency And Finding Its Gradient. Stochastic Data Envelopment Analysis With A Perfect Object Approach, Alexander Vaninsky

Publications and Research

The data presented here are related to the research article “Energy-environmental efficiency and optimal restructuring of the global economy” (Vaninsky, 2018) [1]. This article describes how the world economy can be restructured to become more energy-environmental efficient, while still increasing its growth potential. It demonstrates how available energy-environmental and economic information may support policy-making decisions on the atmosphere preservation and climate change prevention. This Data article presents a computer program in R language together with examples of input and output files that serve as a means of implementation of the novel approach suggested in publication[1]. The computer program utilizes stochastic …

Recent Trends In The Frequency And Duration Of Global Floods, Nasser Najibi, Naresh Devineni

Publications and Research

Frequency and duration of floods are analyzed using the global flood database of the Dartmouth Flood Observatory (DFO) to explore evidence of trends during 1985–2015 at global and latitudinal scales. Three classes of flood duration (i.e., short: 1–7, moderate: 8–20, and long: 21 days and above) are also considered for this analysis. The nonparametric Mann–Kendall trend analysis is used to evaluate three hypotheses addressing potential monotonic trends in the frequency of flood, moments of duration, and frequency of specific flood duration types. We also evaluated if trends could be related to large-scale atmospheric teleconnections using a generalized linear model framework. …

Higher Cluster Categories And Qft Dualities, Sebastián Franco, Gregg Musiker

Publications and Research

We introduce a unified mathematical framework that elegantly describes minimally supersymmetry gauge theories in even dimensions, ranging from six dimensions to zero dimensions, and their dualities. This approach combines and extends recent developments on graded quivers with potentials, higher Ginzburg algebras, and higher cluster categories (also known as m-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order-(m + 1) dualities of the gauge theories. Our work indicates that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities.

Validation Of A Lottery, Xiaona Zhou

Publications and Research

The NY Pick 4 lottery consists of four "randomly" chosen digits from 0 to 9. For this to be fair, each digit should be equally likely to occur. To determine whether this is the case, a Chi-squared goodness of fit test will be applied to historical data. This provides a quantitative way of measuring how well the observed frequency of digits matches our expectations of a fair lottery. The Chi-squared distribution gives us a number beyond which we reject fairness. However, there is another possibility. If the difference between the fair model and the observed frequency is too small, that …

Special Values Of Ibukiyama-Saito L-Functions, Brad Isaacson

Publications and Research

Following the method of Arakawa, we express the special values of an L-function originally introduced by Arakawa and Hashimoto and later generalized by Ibukiyama and Saito at non-positive integers by finite sums of products of Bernoulli polynomials. As a corollary, we prove an infinite family of identities expressing finite sums of products of Bernoulli polynomials by generalized Bernoulli numbers.

Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves, Nelson Carella

Publications and Research

Let $x \geq 1$ be a large number, let $f(n) \in \mathbb{Z}[x]$ be a prime producing polynomial of degree $\deg(f)=m$, and let \(u\neq \pm 1,v^2\) be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitive root $u$ is derived in this note. This asymptotic result has the form $$\pi_f(x)=\# \{ p=f(n)\leq x:\ord_p(u)=p-1 \}=\left (c(u,f)+ O\left (1/\log x )\right ) \right )x^{1/m}/\log x$$, where $c(u,f)$ is a constant depending on the polynomial and the fixed integer. Furthermore, new results for the asymptotic order of elliptic primes with …

Algorithmically Complex Residually Finite Groups, Olga Kharlampovich, Alexei Myanikov, Mark Sapir

Publications and Research

We construct the first examples of algorithmically complex finitely presented residually finite groups and the first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn functions, and arbitrarily large depth functions. The groups are solvable of class 3.

Case Study Of Undergraduate Research Projects In Vector Analysis, Alexander Vaninsky, Willy Baez Lara, Madieng Diao, Analilia Mendez

Publications and Research

This paper presents two examples of the undergraduate research projects in vector analysis conducted under the first author’s supervision at one of the community colleges that is an integral part of a large city university. The projects were accomplished by the students pursuing associated degrees in engineering, during their sophomore year. One project was to obtain an explicit formula for the curvature of a curve in plane defined implicitly in rectangular or polar coordinates. Another project was aimed to develop an alternative procedure for finding potential function for a vector field in space based on simultaneous integration. Participation in these …

A Novel Approach For Library Materials Acquisition Using Discrete Particle Swarm Optimization, Daniel A. Sabol

Publications and Research

The academic library materials acquisition problem is a challenge for librarian, since library cannot get enough funding from universities and the price of materials inflates greatly. In this paper, we analyze an integer mathematical model by considering the selection of acquired materials to maximize the average preference value as well as the budget execution rate under practical restrictions. The objective is to improve the Discrete Particle Swarm Optimization (DPSO) algorithm by adding a Simulate Annealing algorithm to reduce premature convergence. Furthermore, the algorithm is implemented in multiple threaded environment. The experimental results show the efficiency of this approach.

The History Of Algorithmic Complexity, Audrey A. Nasar

Publications and Research

This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. By providing a historical account of algorithmic complexity through a mathematical lens, this paper aims to equip mathematics educators with the necessary background and framework for incorporating the analysis of algorithmic complexity into mathematics courses as early on as algebra or pre-calculus.

Generalized Least-Powers Regressions I: Bivariate Regressions, Nataniel Greene

Publications and Research

The bivariate theory of generalized least-squares is extended here to least-powers. The bivariate generalized least-powers problem of order p seeks a line which minimizes the average generalized mean of the absolute pth power deviations between the data and the line. Least-squares regressions utilize second order moments of the data to construct the regression line whereas least-powers regressions use moments of order p to construct the line. The focus is on even values of p, since this case admits analytic solution methods for the regression coefficients. A numerical example shows generalized least-powers methods performing comparably to generalized least-squares methods, …

A P-Value Model For Theoretical Power Analysis And Its Applications In Multiple Testing Procedures, Fengqing Zhang, Jiangtao Gou

Publications and Research

Background: Power analysis is a critical aspect of the design of experiments to detect an effect of a given size. When multiple hypotheses are tested simultaneously, multiplicity adjustments to p-values should be taken into account in power analysis. There are a limited number of studies on power analysis in multiple testing procedures. For some methods, the theoretical analysis is difficult and extensive numerical simulations are often needed, while other methods oversimplify the information under the alternative hypothesis. To this end, this paper aims to develop a new statistical model for power analysis in multiple testing procedures.

Methods: We propose a …

Hyperplanes That Intersect Each Ray Of A Cone Once And A Banach Space Counterexample, Chris Mccarthy

Publications and Research

Suppose � is a cone contained in real vector space �. When does � contain a hyperplane � that intersects each of the 0-rays in �\{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question.We also present an example of a salient, closed Banach space cone � for which there does not exist a hyperplane that intersects each 0-ray in � \ {0} exactly once.

Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov

Publications and Research

We consider a complex representation of an arbitrary planar polygon P centered at the origin. Let P(1) be the normalized polygon obtained from P by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that P(1) is a normalized average of P. We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a complete and …

Set-Theoretic Mereology, Joel David Hamkins, Makoto Kikuchi

Publications and Research

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.

A Modularized Tablet-Based Approach To Preparation For Remedial Mathematics, Kenneth A. Parker

Publications and Research

Basic arithmetic forms the foundation of the math courses that students will face in their undergraduate careers. It is therefore crucial that students have a solid understanding of these fundamental concepts. At an open- access university offering both two-year and four-year degrees, incoming freshmen who were identified as lacking in basic arithmetic skills were engaged in an experimental technology-enhanced workshop designed to provide them with a deeper understanding of arithmetic prior to their initial remedial coursework. Customized online content was created specifically for this experiment, and the first implementation (n=27) yielded statistically significant improvement, not only from pretest to post- …

Review Paper: The Shape Of Phylogenetic Treespace, Katherine St. John

Publications and Research

Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and in running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths.

Experimental Demonstration Of Topological Effects In Bianisotropic Metamaterials, Alexey P. Slobozhanyuk, Alexander B. Khanikaev, Dmitry S. Filonov, Daria A. Smirnova, Andrey E. Miroshnichenko, Yuri S. Kivshar

Publications and Research

Existence of robust edge states at interfaces of topologically dissimilar systems is one of the most fascinating manifestations of a novel nontrivial state of matter, a topological insulator. Such nontrivial states were originally predicted and discovered in condensed matter physics, but they find their counterparts in other fields of physics, including the physics of classical waves and electromagnetism. Here, we present the first experimental realization of a topological insulator for electromagnetic waves based on engineered bianisotropic metamaterials. By employing the near-field scanning technique, we demonstrate experimentally the topologically robust propagation of electromagnetic waves around sharp corners without backscattering effects.

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.

Generalizing Liouville-Type Problems For Differential 1-Forms From Lq Spaces To Non-Lq Spaces, Lina Wu, Ye Li

Publications and Research

We obtain Liouville-type results for closed and p-pseudo-coclosed differential 1-forms ! with energy of lim inf r!1 1 r2 R B(x0;r) j!jqdv < 1 (that is, 2-finite growth), which extends finite q-energy ( R M j!jqdv < 1) in Lq spaces to infinite q-energy ( R M j!jqdv = 1) in non-Lq spaces. In particular, we recapture mathematicians' vanishing results of Liouville- type theorem for ! with finite q-energy in Lq spaces. Our method in this paper provides a successful way to work on Liouville-type problems for differential forms with a variety of energy conditions in broad spaces.

Locally Anisotropic Toposes, Jonathon Funk, Pieter Hofstra

Publications and Research

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.

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; …