Contributed Paper Session
Saturday Afternoon
Session I – UC 104
3:00-3:15 Euler and the Odd Perfect Numbers
Ed Sandifer, Western Connecticut State University
Abstract : We give Euler's proof that if N is an odd perfect number, then N has at least two distinct prime factors, one a prime that is congruent to 1 (mod 4) to a power that is also congruent to 1 (mod 4) and that all other prime factors (and there has to be at least one of them) are taken to even powers. It is still an open question whether or not there are any odd perfect numbers.
3:20 – 3:35 Medieval Islam as the birthplace for college algebra
Ezra Halleck, New York City College of Technology (CUNY)
Abstract : Many of the students at City Tech are from predominantly Muslim countries. I supplement the standard curriculum with writing assignments based on historically important contributions in mathematics made by\their ancestors. By doing so, I hope to raise their self-esteem as well as instill in all the students an appreciation for the Muslim world.
In my college algebra course, students read selections from the first surviving algebra text, written by Mohammed Ben Musa almost 1200 years ago while in the service of a Caliph in Iraq. In fact, algebra is derived from a word in the title of the book and algorithm derives from Al-Kharizmi, the name he is better known by, Khwarizm being the region on the Aral Sea from which he came, now part of Uzbekistan.
For the first assignment, students begin with the author's preface which explains who he is and his reverence for Allah, the prophet Mohammed, the investigators who came before him and the Caliph, without any of whom his text would not have been possible. Next, students read his method for solving a linear equation and then jump to an inheritance question that he uses to illustrate his method.
For the second assignment, students read a classification scheme for quadratic equations, study the algebraic solution of an example and follow with the geometric explanation for his method even today called "completing the square".
In the talk, I will present details of the excerpts and the corresponding assignments as well as show some sample student work.
3:40-3:55 A College Course for Math/Computer Science majors in a Community College
Zenaida Ramos, Quincy College
Abstract : In our community college, we teach College Algebra and Precalculus, where we follow the usual path of graphs, functions, zeroes of polynomials, exponential and log functions, and trigonometry.
However, we don’t have an Abstract Algebra course, a Number Theory course, or a course on proofs. We do have (besides the Calculus courses) Discrete Math and Linear Algebra courses. But I find that many students cannot understand these courses, and it seems to me that we need a course in preparation for them.
We consider the needs of students who plan to become math teachers at the middle or high school levels. Because they lack preparation at the high school level, didn’t have time to reflect on math or the role of proofs in math, it seems better for the students to learn about doing math (proving, analyzing, synthesizing, and evaluating specific examples) rather than simply being given a lot of information.
We will discuss the kind of curriculum needed for such a course and possible solutions to this problem.
Session II – UC 105
3:00 - 3:15 Clustering the Short Stories of Edgar Allan Poe with Formal Concept Theory
Roger Bilisoly, Central Connecticut State University
Abstract: Humans are experts at categorizing sets of objects and enjoy making such categorizations. Edgar Allan Poe wrote about seventy short stories in his lifetime, and these have been categorized in several different ways by literary critics. For example, his stories can be grouped into genres such as horror, detective, and science fiction. This talk will consider the challenge of having a computer find groups of related stories, where the groups make sense to a human. The approach combines two different techniques. First, we use term-document matrices, which were originally developed for the task of searching for documents in the field of information retrieval. Second, we use formal concept theory, which defines concepts through constructing lattices from object-attribute matrices. Finally, we will consider whether or not the stories grouped together do seem meaningful to a human.
3:20 – 3:35 Monte Carlo Simulation and Brownian motion in the Finance Industry
Xiaochuan (Frank) Wu, Norwalk, CT
Abstract : Monte Carlo simulation randomly generates values for uncertain variables over and over to simulate a model. It can be used to find the estimation of a very complicated multiple integrals which can not be solved by hand. It has been used widely in financial industry to simulate the price of the stock call options, estimate the Value at Risk of the portfolio, interest rate modeling etc.
The European call gives the call buyer the right that can exercise the call option at the strike price on the expiration date, so if the stock price is higher the strike price on the expiration date, then the stock buyer will get profits for buying the stock at the strike price and sell them immediately at the market price. This is called arbitrage.
Using the Monte Carlo simulation with the Nobel Prize-winning theory called “Black-Scholes Option Pricing theory”, a financial engineer can build a model to simulate the stock option prices and make proper investment strategies.
3:40-3:55 Dragon Folds and Turn Lists
Brian Kelly, Roger Williams University
Abstract: For the standard Dragon Curve the approaches of “paper folding” and “turn lists” both lead to the same curve. Other variations of the paper folding scheme lead to curves that are not represented by the most common method described for generating turn lists. We introduce a different method which represents all cases. We also characterize when the methods agree.
Session III – UC 106
3:00-3:15 A Report on a Class of Mechanical Problems
Larry Blaine, Plymouth State University
Abstract : Linear harmonic oscillators, simple or in coupled systems, are among the most-studied and best-understood systems in elementary mathematical mechanics. Textbook discussions usually postulate a frictional force continuously dependent on velocity and opposite in sign. In some situations, however, it is much more realistic to consider discontinuous frictional forces, and this introduces vast complexities into the analysis. The purpose of this talk is to explore some of these complexities and their implications in the real world.
3:20 – 3:35 Residuated semilattices and positive universal classes
Jeffrey S. Olson, Norwich University
Abstract: A commutative residuated semilattice, or CRS, is a meet-semilattice- ordered commutative monoid which possesses a residuation operation → satisfying x · y ≤ z ⇐⇒ y ≤ x → z . A CRS is called k–potent if it satisfies x k ≈ x k+ 1. A characterization of subdirectly irreducible k–potent CRSs was used previously to axiomatize varieties generated by positive universal classes of CRSs, i.e., those classes axiomatized by positive universal first-order sentences. We show that the axiomatization theorem can be significantly generalized. In particular, the condition of k–potence may be dropped. As an application, we show that the join of two finitely axiomatized varieties of CRSs is itself finitely axiomatized.
3:40 – 3:55 A Common Reflexive Basis for Studying the Degree of Certainty of Mathematical Models
Krassimir Tarkalanov, Quincy College
Abstract : Structure and domain-independence means constructing a model which solves the interpreted problem precisely, i.e. with an absolute degree of certainty at checking the result. Franklin and de Lapland (1999) discuss this topic by considering examples. This is not consistent with a philosophy of science approach. Hegel’s reflexion is an infinite series of consecutive irreversible negations. The author uses it for a philosophical substantiation of his pure mathematical research (1992) and of the development of the real world mathematical models (2000, 2001). The last substantiation disposes the models on one and the same or on different reflexive levels (the disposition will be illustrated pictorially). In the present paper we use parts of this approach for setting the indicated discussion on a common philosophical basis and its solution. A structure and domain-independence is impossible and checking the certainty can not be separated from modeling due to the irreversibility and infinity of the reflexion. We have shown simultaneously the necessity of a substantiation of the purposefulness of a mathematical research.