Mathematics Commons^™

Primes And Zeros: A Million Dollar Mystery, Brian Conrey

Dalrymple Lecture Series

A prime number is an integer greater than one whose only positive divisors are 1 and itself. In 1859 G. F. B. Riemann proposed a way to understand how the prime numbers are distributed among the natural numbers. More than 150 years later mathematicians still have not proven Riemann's Hypothesis. The stature of this problem has continued to rise so that today it is widely regarded as the most important unsolved problem in all of mathematics. In this talk I will describe some of the colorful history that surrounds this question.

The Fourier Coefficients Of Modular Forms, Kyle Pratt, Dr. Paul Jenkins

Journal of Undergraduate Research

Modular forms are complex analytic functions with remarkable properties. Modular forms possess interesting and surprising connections to many different branches of mathematics. For example, it is well-known that Andrew Wiles’ proof of Fermat’s Last Theorem, a conjecture that had been unresolved for more than three centuries, utilized modular forms in a crucial way.

Poincaré Disc: Hear The Whole Mind Sing, Laura Moss, John T. Shrontz

Research Horizons Day Posters

No abstract provided.

Stochastic Comparisons Of Weighted Sums Of Arrangement Increasing Random Variables, Xiaoqing Pan, Min Yuan, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

Assuming that the joint density of random variables X1,X2, . . . ,Xn is arrangement increasing (AI), we obtain some stochastic comparison results on weighted sums of Xi’s under some additional conditions. An application to optimal capital allocation is also given.

Multivariate Statistical Methodologies Applied In Biomedical Raman Spectroscopy: Assessing The Validity Of Partial Least Squares Regression Using Simulated Model Datasets., Mark Keating

Articles

Raman spectroscopy is fast becoming a valuable analytical tool in a number of biomedical scenarios, most notably disease diagnostics. Importantly, the technique has also shown increasing promise in the assessment of drug interactions on a cellular and subcellular level, particularly when coupled with multivariate statistical analysis. However, an important consideration, both with Raman spectroscopy and the associated statistical methodologies, is the accuracy of these techniques and more specifically the sensitivities which can be achieved and ultimately the limits of detection of the various methods. The purpose of this study is thus the construction of a model simulated data set with …

Review: On Pairs Of Generalized And Hypergeneralized Projections In A Hilbert Space, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Zeros Of Poincare Series Of Level 2, Andrew Haddock, Paul Jenkins

Journal of Undergraduate Research

Introduction Poincaré series are a certain type of modular form. Modular forms are complex-valued functions that satisfy certain symmetry properties. There are many different types of modular forms, and one way to classify modular forms is by their level, such as 1, 2, 3, etc. They are of much interest as a research subject because they are connected in surprising ways to many different fields in number theory—e.g., elliptic curves, quadratic forms, and partition functions, to name just a few. As we gain more insight into the various properties of modular forms, we gain more insight into how these various …

Does Being Bilingual Make You Better At Math?, Enxhi Elezi

Honors Projects in Mathematics

The purpose of this study is to examine if there is any relationship between being bilingual, defined as speaking your native language at home and another language in school, and your mathematical ability. Data from the National Longitudinal Study of Adolescent Health was used to compare the math grades of students who were not born in the US and speak English, Spanish, or Other at home. Also, data from the Bryant University first year students was used to test if students who speak a different language at home have a higher mathematical average than their monolingual peers. Results show that …

Mathematical Modeling Of Trending Topics On Twitter, Jonathan S. Skaza

Honors Projects in Mathematics

Created in 2006, Twitter is an online social networking service in which users share and read 140-character messages called Tweets. The site has approximately 288 million monthly active users who produce about 500 million Tweets per day. This study applies dynamical and statistical modeling strategies to quantify the spread of information on Twitter. Parameter estimates for the rates of infection and recovery are obtained using Bayesian Markov Chain Monte Carlo (MCMC) methods. The methodological strategy employed is an extension of techniques traditionally used in an epidemiological and biomedical context (particularly in the spread of infectious disease). This study, which addresses …

Analysis Of Random Metric Spaces Explains Emergence Phenomenon And Suggests Discreteness Of Physical Space, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, systems follow the pattern set by the second law of thermodynamics: they evolve from an organized inhomogeneous state into a homogeneous structure-free state. In many other practical situations, however, we observe the opposite emergence phenomenon: in an originally homogeneous structure-free state, an inhomogeneous structure spontaneously appears. In this paper, we show that the analysis of random metric spaces provides a possible explanation for this phenomenon. We also show that a similar analysis supports space-time models in which proper space is discrete.

Bytes Of Π, Spring 2015, Department Of Mathematics, Bridgewater State University

Department of Mathematics Newsletter

No abstract provided.

Topological Manifolds, Grant Wilson

Thinking Matters Symposium Archive

Topological Manifolds are abstract spaces that locally resemble Euclidean space. For example, consider a round globe and a flat map. The map is a 2-dimensional representation of a 3-dimensional space. Given any point on the globe we can find a corresponding position on the map, and vice versa. This correspondence is called a chart. With a sufficient number of charts, we can describe the whole space. Such a collection of charts is called an Atlas. It is possible to construct different Atlases for the same space, allowing us to move from one chart, to the space, to another chart. This …

Inside Out: Properties Of The Klein Bottle, Andrew Pogg, Jennifer Daigle, Deirdra Brown

Thinking Matters Symposium Archive

A Klein Bottle is a two-dimensional manifold in mathematics that, despite appearing like an ordinary bottle, is actually completely closed and completely open at the same time. The Klein Bottle, which can be represented in three dimensions with self-intersection, is a four dimensional object with no intersection of material. In this presentation we illustrate some topological properties of the Klein Bottle, use the Möbius Strip to help demonstrate the construction of the Klein Bottle, and use mathematical properties to show that the Klein Bottle intersection that appears in ℝ3 does not exist in ℝ4. Introduction: Topology

Parallel Symmetric Eigenvalue Problem Solvers, Alicia Marie Klinvex

Open Access Dissertations

Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, we describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, we discuss alternative methods for solving the saddle point …

Why Big-O And Little-O In Algorithm Complexity: A Pedagogical Remark, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the comparative analysis of different algorithm, O- and o-notions are frequently used. While their use is productive, most textbooks do not provide a convincing student-oriented explanation of why these particular notations are useful in algorithm analysis. In this note, we provide such an explanation.

Sometimes, It Is Beneficial To Process Different Types Of Uncertainty Separately, Chrysostomos D. Stylios, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we make predictions based on the measured and/or estimated values of different physical quantities. The accuracy of these predictions depends on the accuracy of the corresponding measurements and expert estimates. Often, for each quantity, there are several different sources of inaccuracy. Usually, to estimate the prediction accuracy, we first combine, for each input, inaccuracies from different sources into a single expression, and then use these expressions to estimate the prediction accuracy. In this paper, we show that it is often more computationally efficient to process different types of uncertainty separately, i.e., to estimate inaccuracies in the …

Symbolic Aggregate Approximation (Sax) Under Interval Uncertainty, Chrysostomos D. Stylios, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we monitor a system by continuously measuring the corresponding quantities, to make sure that an abnormal deviation is detected as early as possible. Often, we do not have ready algorithms to detect abnormality, so we need to use machine learning techniques. For these techniques to be efficient, we first need to compress the data. One of the most successful methods of data compression is the technique of Symbolic Aggregate approXimation (SAX). While this technique is motivated by measurement uncertainty, it does not explicitly take this uncertainty into account. In this paper, we show that we can …

A Simplified Explanation Of What It Means To Assign A Finite Value To An Infinite Sum, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Recently, a video made rounds that explained that it often makes sense to assign finite values to infinite sums. For example, it makes sense to claim that the sum of all natural numbers is equal to -1/12. This has picked up interested in media. However, judged by the viewers' and readers' comments, for many viewers and readers, neither the video, not the corresponding articles seem to explain the meaning of the above inequality clearly enough. One of the main stumbling blocks is the fact that the infinite sum is clearly divergent, so a natural value of the infinite sum is …

An Analysis Of Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia With Imatinib-Resistant Mutations, Dana Paquin, David Sacco, John Shamshoian

Mathematics

Chronic myelogenous leukemia (CML) is a cancer of the white blood cells that results from increased and uncontrolled growth of myeloid cells in the bone marrow and the accumulation of these cells in the blood. The most common form of treatment for CML is imatinib, a tyrosine kinase inhibitor. Although imatinib is an effective treatment for CML and most patients treated with imatinib do attain some form of remission, imatinib does not completely eradicate all leukemia cells, and if treatment is stopped, all patients eventually relapse (Cortes, 2005). In Kim (2008), the authors developed a mathematical model for the dynamics …

Mathematical Explorations Of Card Tricks, Timothy R. Weeks

Senior Honors Projects

In this project, we explore various mathematical topics as they apply to an assortment of card tricks. We will focus on an examination of theorems applied to the manipulation of cards in an attempt to prove why certain card tricks work. These theorems utilize abstract algebra, probability, number theory, and combinatorics. Many tricks can be explained this way, instead of singularly by sleight of hand or other “magical” methods. We will rigorously prove the theorems and principles that explain these concepts, focusing primarily on the card tricks and examples presented in Mathematical Card Magic by Colm Mulcahy (2013).

Hyperbolic Geometry And Mobius Transformations, Emmalee Stevens

Masters Essays

Imagine a world in which there are infinitely many lines through a single point that are all parallel to the same line; this world is referred to as the hyperbolic plane. It starts with a brief history of the development of hyperbolic geometry and the mathematicians who contributed their explorations of the hyperbolic plane. The reader will be reminded of some of the important aspects of Euclidean geometry and some of the basic properties that will later be adjusted to develop hyperbolic geometry. The hyperbolic plane will be constructed from the pseudosphere and the properties of the Poincare Upper Half …

Possible Bias In Asset Valuations: An Application Of The Fraud Risk Triangle To Divorce Cases, Jennifer Tomasetti

Honors Projects in Accounting

No abstract provided.

Applications Of Microlocal Analysis To Some Hyperbolic Inverse Problems, Andrew J Homan

Open Access Dissertations

This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when …

Math Quiz On The Radio, Darren B. Glass

Math Faculty Publications

What word, often spelled with an umlaut, is used to identify a point on a two-dimensional graph?

Many of you probably already figured out the answer is "coordinate." But that's because you are sitting comfortably in your dorm room rather than being on a stage with bright lights in front of a few hundred people being recorded for national broadcast on public radio. [excerpt]

Why Some Families Of Probability Distributions Are Practically Efficient: A Symmetry-Based Explanation, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Out of many possible families of probability distributions, some families turned out to be most efficient in practical situations. Why these particular families and not others? To explain this empirical success, we formulate the general problem of selecting a distribution with the largest possible utility under appropriate constraints. We then show that if we select the utility functional and the constraints which are invariant under natural symmetries -- shift and scaling corresponding to changing the starting point and the measuring unit for describing the corresponding quantity $x$. then the resulting optimal families of probability distributions indeed include most of the …

Transforming Precalculus Instruction: Evidence-Based Course Design, Wendy M. Smith

Discipline-Based Education Research Group Speaker Series

The UNL Mathematics Department has been focused on transforming precalculus instruction since 2012, with a goal of greater levels of student success. A short-term measure of student success is the passing rate (C or better), which has jumped from an average of 62% (2007-2011) to 80% for the past two falls. A longer-term measure of student success is recruiting and retaining undergraduates to STEM disciplines and careers. In this talk I will share specifics of the reform efforts (the who-what-when-where-why-and-how), and also share preliminary results from the research we have simultaneously been conducting into the reform efforts.

Once We Know That A Polynomial Mapping Is Rectifiable, We Can Algorithmically Find A Rectification, Julio Urenda, David Finston, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that some polynomial mappings φ: C^k --> Cⁿ are rectifiable in the sense that there exists a polynomial mapping α: Cⁿ --> Cⁿ whose inverse is also polynomial and for which α(φ(z₁, ...,z_k)) = (z₁, ...,z_k, 0, ..., 0) for all z₁, ...,z_k. In many cases, the existence of such a rectification is proven indirectly, without an explicit construction of the mapping α.

In this paper, we use Tarski-Seidenberg algorithm (for deciding the first order theory of real numbers) to design …

When Can We Simplify Data Processing: An Algorithmic Answer, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

In many real-life situations, we are interested in the values of physical quantities x₁, ..., x_n which are difficult (or even impossible) to measure directly. To estimate these values, we measure easier-to-measure quantities y₁, ..., y_m which are related to the desired quantities by a known relation, and use these measurement results to estimate x_i. The corresponding data processing algorithms are sometimes very complex and time-consuming, so a natural question is: are simpler (and, thus, faster) algorithms possible for solving this data processing problem? In this paper, we show that by using …

Sensitivity Of Mixed Models To Computational Algorithms Of Time Series Data, Gunaime Nevine

Doctoral Dissertations

Statistical analysis is influenced by implementation of the algorithms used to execute the computations associated with various statistical techniques. Over many years; very important criteria for model comparison has been studied and examined, and two algorithms on a single dataset have been performed numerous times. The goal of this research is not comparing two or more models on one dataset, but comparing models with numerical algorithms that have been used to solve them on the same dataset.

In this research, different models have been broadly applied in modeling and their contrasting which are affected by the numerical algorithms in different …

Root Cover Pebbling On Graphs, Claire A. Sonneborn

Honors Theses

Consider a graph, G, with pebbles on its vertices. A pebbling move is defined to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The cover pebbling number of a graph, γ(G), is the minimum number of pebbles such that, given any configuration of γ(G) pebbles on the vertices of G, pebbling moves can be used to place one pebble on each vertex of G. We define the root vertex of a graph and fix an initial configuration of pebbles on G where we place all pebbles on the root …