Physical Sciences and Mathematics Commons^™

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.

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 …

Linear Progress With Exponential Decay In Weakly Hyperbolic Groups, Matthew H. Sunderland

Dissertations, Theses, and Capstone Projects

A random walk w_n on a separable, geodesic hyperbolic metric space X converges to the boundary ∂X with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on X is acylindrical. We extend exponential decay to the nonacylindrical case. We give an application to random Heegaard splittings.

The Philosophical Foundations Of Plen: A Protocol-Theoretic Logic Of Epistemic Norms, Ralph E. Jenkins

Dissertations, Theses, and Capstone Projects

In this dissertation, I defend the protocol-theoretic account of epistemic norms. The protocol-theoretic account amounts to three theses: (i) There are norms of epistemic rationality that are procedural; epistemic rationality is at least partially defined by rules that restrict the possible ways in which epistemic actions and processes can be sequenced, combined, or chosen among under varying conditions. (ii) Epistemic rationality is ineliminably defined by procedural norms; procedural restrictions provide an irreducible unifying structure for even apparently non-procedural prescriptions and normative expressions, and they are practically indispensable in our cognitive lives. (iii) These procedural epistemic norms are best analyzed in …

Studying The Space Of Almost Complex Structures On A Manifold Using De Rham Homotopy Theory, Bora Ferlengez

Dissertations, Theses, and Capstone Projects

In his seminal paper Infinitesimal Computations in Topology, Sullivan constructs algebraic models for spaces and then computes various invariants using them. In this thesis, we use those ideas to obtain a finiteness result for such an invariant (the de Rham homotopy type) for each connected component of the space of cross-sections of certain fibrations. We then apply this result to differential geometry and prove a finiteness theorem of the de Rham homotopy type for each connected component of the space of almost complex structures on a manifold. As a special case, we discuss the space of almost complex structures …

Mathematical Inquiry With Loops, Jeanne Funk

Open Educational Resources

Mathematical Inquiry with Loops is a fun activity designed to introduce students in LMF101 (First Year Seminar for Liberal Arts: Math and Science) to mathematical exploration and hypothesis formation via hands-on exploration with paper loops. It should make students understand that the “facts” that make up mathematics, are not arbitrary, but arise from observation and experience. It should also encourage students to view mathematics as a science rooted in exploration and problem solving, rather than an esoteric exercise in symbol manipulation.

This activity is designed to be very low stakes (ungraded) and to take one hour of class time. There …

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 …

Percent On & Off, Johannes Familton

Open Educational Resources

This PowerPoint was made for a Quantitative Reasoning, Liberal arts math course. This is an example of what one can do using only OER resources. This PowerPoint with the use of animation, animated gifs and a nice background keeps students engaged in the lesson. The contents of this lesson is taken from Math in Society, by David Lippman, Pierce College Ft Steilacoom. The text is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License. PPT Background: http://www.MyFreePPT.com Animated gifs: http://www.zingerbug.com

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. …

Elements Of Calculus 1, Nur Dean

Open Educational Resources

No abstract provided.

A Concise Workbook For College Algebra, Fei Ye

Open Educational Resources

This workbook is mainly based on the author’s worksheets for the college algebra course (MAD-119) at QCC of CUNY. It is intended as a concise introduction to college algebra at the intermediate level.

It contains 25 lessons. Each lesson, corresponding to roughly one class meeting, starts with a page on concepts, formulas and examples, and ends with a list of practice problems that students are expected to be able to solve and complete in class.

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams

Dissertations, Theses, and Capstone Projects

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve …

The Distribution Of Totally Positive Integers In Totally Real Number Fields, Tianyi Mao

Dissertations, Theses, and Capstone Projects

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the number field is quadratic, Beck also proved a mean value result using the continued fraction expansions of quadratic irrationals. We generalize Beck’s result to higher moments. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit …

Divergence Of Cat(0) Cube Complexes And Coxeter Groups, Ivan Levcovitz

Dissertations, Theses, and Capstone Projects

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we characterize right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. This characterization also has a direct application to the theory of random right-angled Coxeter groups. As another application of the divergence bounds obtained for cube complexes, we provide an inductive graph theoretic criterion on a right-angled Coxeter group's …

Coincidence Of Bargaining Solutions And Rationalizability In Epistemic Games, Todd Stambaugh

Dissertations, Theses, and Capstone Projects

Chapter 1: In 1950, John Nash proposed the Bargaining Problem, for which a solution is a function that assigns to each space of possible utility assignments a single point in the space, in some sense representing the ’fair’ deal for the agents involved. Nash provided a solution of his own, and several others have been presented since then, including a notable solution by Ehud Kalai and Meir Smorodinsky. In chapter 1, a complete account is given for the conditions under which the two solutions will coincide for two player bargaining scenarios.

Chapter 2: In the same year, Nash …

On Some Geometry Of Graphs, Zachary S. Mcguirk

Dissertations, Theses, and Capstone Projects

In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner's inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size of …

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices …

Geometry And Analysis Of Some Euler-Arnold Equations, Jae Min Lee

Dissertations, Theses, and Capstone Projects

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In this …

Mathematics In Contemporary Society - Chapter 2 (Spring 2018), 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.

Mathematics In Contemporary Society - Chapter 4 (Spring 2018), 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.

Mathematics In Contemporary Society - Chapter 1 (Spring 2018), 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.

Mathematics In Contemporary Society - Chapter 5 (Spring 2018), 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.

Mathematics In Contemporary Society - Chapter 3 (Spring 2018), 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.

Infinitely Many Solutions To Asymmetric, Polyharmonic Dirichlet Problems, Edger Sterjo

Dissertations, Theses, and Capstone Projects

In this dissertation we prove new results on the existence of infinitely many solutions to nonlinear partial differential equations that are perturbed from symmetry. Our main theorems focus on polyharmonic Dirichlet problems with exponential nonlinearities, and are now published in Topol. Methods Nonlinear Anal. Vol. 50, No.1, (2017), 27-63. In chapter 1 we give an introduction to the problem, its history, and the perturbation argument itself. In chapter 2 we prove the variational principle of Bolle on the behavior of critical values under perturbation, and the variational principle of Tanaka on the existence of critical points of large augmented Morse …

Gradient Estimation For Attractor Networks, Thomas Flynn

Dissertations, Theses, and Capstone Projects

It has been hypothesized that neural network models with cyclic connectivity may be more powerful than their feed-forward counterparts. This thesis investigates this hypothesis in several ways. We study the gradient estimation and optimization procedures for several variants of these networks. We show how the convergence of the gradient estimation procedures are related to the properties of the networks. Then we consider how to tune the relative rates of gradient estimation and parameter adaptation to ensure successful optimization in these models. We also derive new gradient estimators for stochastic models. First, we port the forward sensitivity analysis method to the …

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.

Models For Decision-Making, Steven Cosares

Open Educational Resources

No abstract provided.

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.