Mathematics Commons^™

Bhaskara’S Approximation For The Sine, Karel Stroethoff

The Mathematics Enthusiast

The 7th century Indian mathematician Bhaskara (c.600 – c.680) obtained a remarkable approximation for the sine function. Many subsequent ancient authors have given versions of this rule, but none provided a proof or described how the result was obtained. Grover [1] provides a possible explanation, but I think the rule can be explained more clearly. Rather than give the rule first, we will derive it, and then discuss its accuracy, and explore some alternative approximations. Our derivation is simply an exercise in modeling.

Difficulties In Solving Context-Based Pisa Mathematics Tasks: An Analysis Of Students’ Errors, Ariyadi Wijaya, Marja Van Den Heuvel-Panhuizen, Michiel Doorman, Alexander Robitzsch

The Mathematics Enthusiast

The intention of this study was to clarify students’ difficulties in solving context-based mathematics tasks as used in the Programme for International Student Assessment (PISA). The study was carried out with 362 Indonesian ninth- and tenth-grade students. In the study we used 34 released PISA mathematics tasks including three task types: reproduction, connection, and reflection. Students’ difficulties were identified by using Newman’s error categories, which were connected to the modeling process described by Blum and Leiss and to the PISA stages of mathematization, including (1) comprehending a task, (2) transforming the task into a mathematical problem, (3) processing mathematical procedures, …

Numerically Integrating Irregularly-Spaced (X, Y) Data, B. Cameron Reed

The Mathematics Enthusiast

This article describes a computer program for numerically integrating under a y(x) curve of experimental data where the abscissa values are not equally-spaced. This technique is based on fitting parabolas to successive groups of three data points, and may be regarded as a generalization of Simpson’s rule.

Math As A Tool Of Anti-Semitism, Jay Egenhoff

The Mathematics Enthusiast

At Moscow State University’s Department of Mathematics during the 1970’s and 1980’s, there was rampant discrimination against Jewish and other unwanted students. The professors at the math department made a strong effort to keep Jewish students out of the department. They designed "killer" or "coffin" problems and Jewish students had to answer them during an oral exam. These problems have simple solutions, but require a clever strategy to solve them. This paper explores some of the context of this episode and provides several problems with detailed solutions.

An Examination Of Pre-Service Secondary Mathematics Teachers’ Conceptions Of Angles, Melike Yigit

The Mathematics Enthusiast

The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students’ difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers’ (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process-Object-Schema …

Common Sense About The Common Core, Alan H. Schoenfeld

The Mathematics Enthusiast

Is the Common Core the best thing since sliced bread, or the work of the devil? Is it brand new, or a rehash of old ideas? Is it anything more than a brand name, or is there substance? Can it work, given the implementation challenges in our political and school systems? Opinions about the Common Core are everywhere, but the op-eds I’ve seen are often short on facts, and equally short on common sense. A mathematician by training, I’ve worked for nearly 40 years as an education researcher, curriculum materials developer, test developer, standards writer, and teacher. What follows is …

Scholastic Standards In The United States – The Discussion Concerning The ‘Common Core’, Alan H. Schoenfeld, Günter Törner

The Mathematics Enthusiast

Preface: This article has been developed based on a personal discussion between the German author Günter Törner and Alan Schoenfeld, who is an expert in the field of mathematical didactics. Basically there are three reasons for us to share our insights with the public:

(1) Readers, having subscribed to Jerry Becker’s e-mail information network, have received numerous messages over the past few months; what do we need to know about this fact in Germany?

(2) Scholastic standards – a keyword that sounds very familiar to us in terms of educational policy… But it is also a hot topic in other …

Book Review Of The Tower Of Hanoi: Myths And Maths (Birkhäuser), Cory Palmer

The Mathematics Enthusiast

As the title of the book suggests, the central topic of “The Tower of Hanoi – Myths and Maths” by Hinz, Klavˇzar, Milutinovi´c, and Petr is the famous puzzle of the same name. The classic Tower of Hanoi puzzle along with a number of its variations and related puzzles are examined in a rigorous mathematical framework.

Modeling Colony Collapse Disorder In Honeybees As A Contagion, Christopher Kribs, Christopher Mitchell

Mathematics Faculty Publications - Archive

Honeybee pollination accounts annually for over $14 billion in United States agriculture alone. Within the past decade there has been a mysterious mass die-off of honeybees, an estimated 10 million beehives and sometimes as much as 90% of an apiary. There is still no consensus on what causes this phenomenon, called Colony Collapse Disorder, or CCD. Several mathematical models have studied CCD by only focusing on infection dynamics. We created a model to account for both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection brought to the hive by foragers. The system of …

Development Of The Binary Number System And The Foundations Of Computer Science, Daniel R. Lande

The Mathematics Enthusiast

This paper discusses the formalization of the binary number system and the groundwork that was laid for the future of digital circuitry, computers, and the field of computer science. The goal of this paper is to show how Gottfried Leibniz formalized the binary number system and solidified his thoughts through an analysis of the Chinese I Ching. In addition, Leib-niz’s work in logic and with computing machines is presented. This work laid the foundation for Boolean algebra and digital circuitry which was continued by George Boole, Augustus De Mor-gan, and Claude Shannon in the centuries following. Some have coined Leibniz …

Reasoning-And-Proving Within Ireland’S Reform-Oriented National Syllabi, Jon Davis

The Mathematics Enthusiast

As educational systems around the world attempt to reform their mathematics programs to increase students’ opportunities to engage in processes central to the practice of mathematics such as proof, it is important to understand how this mathematical act is portrayed in national curriculum documents that drive that change. This study examined the presence of reasoning-and-proving (RP) in Ireland’s national reform-oriented secondary syllabi for junior cycle (ages 12-15) and senior cycle (ages 15-18) students. The analyses reveal that there were no differences among direct and indirect RP learning outcomes within each syllabus, but statistically significant differences did exist across syllabi in …

Tme Volume 11, Number 3

The Mathematics Enthusiast

No abstract provided.

Gradient-Based Compressive Sensing For Noise Image And Video Reconstruction, Huihuang Zhao, Yaonan Wang, Xiaojiang Peng, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this study, a fast gradient-based compressive sensing (FGB-CS) for noise image and video is proposed. Given a noise image or video, the authors first make it sparse by orthogonal transformation, and then reconstruct it by solving a convex optimisation problem with a novel gradient-based method. The main contribution is twofold. Firstly, they deal with the noise signal reconstruction as a convex minimisation problem, and propose a new compressive sensing based on gradient-based method for noise image and video. Secondly, to improve the computational efficiency of gradient-based compressive sensing, they formulate the convex optimisation of noise signal reconstruction under Lipschitz …

Metastable Behavior For Conservative Dynamics On A Finite Box With Open Boundary, Taizo Chiyonobu, Yusuke Takagi

Communications on Stochastic Analysis

No abstract provided.

An Entropy Approach To Bose-Einstein Condensation, Francesco De Vecchi, Stefania Ugolini

Communications on Stochastic Analysis

No abstract provided.

Absence Of Energy Level Crossing For The Ground State Energy Of The Rabi Model, Masao Hirokawa, Fumio Hiroshima

Communications on Stochastic Analysis

No abstract provided.

The Development Of Calculus In The Kerala School, Phoebe Webb

The Mathematics Enthusiast

The Kerala School of mathematics, founded by Madhava in Southern India, produced many great works in the area of trigonometry during the fifteenth through eighteenth centuries. This paper focuses on Madhava's derivation of the power series for sine and cosine, as well as a series similar to the well-known Taylor Series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in …

Truckload Shipment Planning And Procurement, Neo Nguyen

Graduate Theses and Dissertations

This dissertation presents three issues encountered by a shipper in the context of truckload transportation. In all of the studies, we utilize optimization techniques to model and solve the problems. Each study is inspired from the real world and much of the data used in the experiments is real data or representative of real data.

The first topic is about the freight consolidation in truckload transportation. We integrate it with a purchase incentive program to increase truckload utilization and maximize profit. The second topic is about supporting decision making collaboration among departments of a manufacturer. It is a bi-objective optimization …

Classical And Motivic Adams-Novikov Charts, Daniel C. Isaksen

Mathematics Research Reports

This document contains large-format Adams-Novikov charts that compute the classical 2-complete stable homotopy groups. The charts are essentially complete through the 59-stem. We believe that these are the most accurate and extensive charts of their kind. We also include a motivic Adams-Novikov E∞ chart.

Pursuing Coherence Among Proportionality, Linearity, And Similarity: Two Pathways From Preservice Teachers’ Geometric Representations, Hyung Sook Lee, Jaehoon Yim

The Mathematics Enthusiast

The importance of using multiple representations of a mathematical concept and connecting the representations has been discussed in learning and teaching mathematics. The Common Core State Standards further the discussion with an emphasis on focus and coherence in teaching mathematical concepts across grades. Preservice teachers in our problem solving class were asked to use geometric representations to solve a problem that required proportional reasoning. They were also asked to sequence the works of their peers as well as their own from a developmental perspective. Sequencing geometric representations with various levels was challenging because it required showing a coherent understanding of …

A Study On Malaysian Mathematicians’ Way Of Knowing, Lim Chap Sam

The Mathematics Enthusiast

“Can you name me a mathematician?” “Einstein??”

“Do you wish to become a mathematician one day?”

If these questions were asked to the public or any school students, the most likely answer to both questions might be a big ‘No!’. Why is this so?

The director of the Public Understanding of Mathematics Forum, Gene Kloz (1996) claims that the mathematics profession is the most misunderstood in all of academia. According to him, the public thinks that mathematicians contemplate ancient proofs and work as lonely recluses. Moreover, the most common public image of a mathematician has been furnished by a physicist …

Absolute Continuity Under Time Shift For Ornstein-Uhlenbeck Type Processes With Delay Or Anticipation, Jörg-Uwe Löbus

Communications on Stochastic Analysis

No abstract provided.

An Epsilon-Nash Equilibrium For Non-Linear Markov Games Of Mean-Field-Type On Finite Spaces, Rani Basna, Astrid Hilbert, Vassili N Kolokoltsov

Communications on Stochastic Analysis

No abstract provided.

Parameter Estimation From Occupation Times—A White Noise Approach, Wolfgang Bock, Thomas Götz, Martin Grothaus, Uditha Prabhath Liyanage

Communications on Stochastic Analysis

No abstract provided.

Higher Powers Of Quantum White Noise Derivatives, Aymen Ettaieb, Habib Ouerdiane, Hafedh Rguigui

Communications on Stochastic Analysis

No abstract provided.

Large Deviations Estimates For Some White Noise Distributions, Sonia Chaari, Achref Majid, Habib Ouerdiane

Communications on Stochastic Analysis

No abstract provided.

Prognostic Value Of Elevated Serum Ceruloplasmin Levels In Patients With Heart Failure, Muhammad Hammadah, Yiying Fan, Yuping Wu, Stanley L. Hazen, W.H. Wilson Tang

Mathematics and Statistics Faculty Publications

Background: Ceruloplasmin (Cp) is a copper-binding acute-phase protein that is increased in inflammatory states and deficient in Wilson's disease. Recent studies demonstrate that increased levels of Cp are associated with increased risk of developing heart failure. Our objective was to test the hypothesis that serum Cp provides incremental and independent prediction of survival in stable patients with heart failure. Methods and Results: We measured serum Cp levels in 890 patients with stable heart failure undergoing elective cardiac evaluation that included coronary angiography. We examined the role of Cp levels in predicting survival over 5 years of follow-up. Mean Cp level …

The Structure And Properties Of Clique Graphs Of Regular Graphs, Jan Burmeister

Master's Theses

In the following thesis, the structure and properties of G and its clique graph cl_t (G) are analyzed for graphs G that are non-complete, regular with degree δ , and where every edge of G is contained in a t -clique. In a clique graph cl_t (G), all cliques of order t of the original graph G become the clique graph’s vertices, and the vertices of the clique graph are adjacent if and only if the corresponding cliques in the original graph have at least 1 vertex in common. This thesis mainly investigates if …

The Use Of Variable-Bagging And The Cross-Validation Selector In The Prediction Of Alzheimer’S Using The Adni Database., Michael Wayne Godbey

Electronic Theses and Dissertations

Dimensionality plays a huge part in the modeling process. If there are more elements in a data set than variables in each element then there are very few restrictions in selection of an algorithm. Bagging, bootstrap aggregating (Breiman, 1994), may also be used to improve a model’s prediction capability. On the other hand, if there more variables in each observation than the number of observations in the dataset, the number of usable algorithms is greatly reduced. The recently developed algorithm, support vector machines, was designed for such situations, in comparison to algorithms such as logistic regression which have instability issues …

Mixed Variational Formulation For The Wellposedness And Numerical Approximation Of A Pde Model Arising In A 3-D Fluid-Structure Interaction, George Avalos, Tom Clark

Faculty Work Comprehensive List

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O coupled to a fourth order plate equation, possibly with rotational inertia parameter ρ>0. This plate PDE evolves on a flat portion Ω of the boundary of O. The coupling on Ω is implemented via the Dirichlet trace of the Stokes system …