Department of Mathematics

One of the goals of the Math Club is to provide undergraduate students opportunities to participate in activities that are central to the mathematical experience. One such area is attendance and participation in conferences in mathematics, mathematics teaching, and mathematics education. Our Math Club has become increasingly active in their participation in local and national conferences. Local public events and colloquia that the Math Club helps organize and participates in are available at the page Public Events and Colloquia . Participation in conferences is detailed here.

• The Hudson River Undergraduate Mathematics Conference. The Westfield State College Math Club has participated in this wonderful annual conference in seven of its eight years.
  In each of the past three years the number of participants and the number of presenters from Westfield State College have been in the top 10 of all the schools represented.
  We have had over one-dozen student speakers and 100 participants over the past seven years of this annual conference!
• Regional meetings of the Mathematical Association of America
• Regional meetings of the National Council of Teachers of Mathematics

Abstracts to the talks are as follows:

Have You Seen the Golden Ratio Lately?
by Jonathan Eckard
Independent reading project supervised by Prof. Julian Fleron.
ABSTRACT: Throughout history, the Golden Ratio has been `found` in many places, including the Great Pyramid of Cheops, the paintings of Leonardo da Vinci, and even the UN`s Secretariat building. It has been reputed to be the first irrational number found, even before pi or the square root of two. The Golden Rectangle, a rectangle whose sides are in the Golden Ratio, was called `the most aesthetically pleasing of all shapes` by the ancient Greeks. With all of this attention directed toward it, then, it may come as a great surprise that many of the well-known `applications` of this ratio are really nothing more than fantasy. However, at the same time, the Golden Ratio does appear deeply within many advanced mathematical topics, including closed-term formulas for the nth Fibonacci number.

M⁴ - Mathematics, Music, Miles (Davis), and (Dave) Matthews
by Brandt Kronholm and Prof. Julian Fleron
ABSTRACT: In this talk we will share several of the striking connections between mathematics and music. For example: fret placement and 1/(2)^1/12: Eastern (15/13) vs. Western (12/8) rhythm structures: shapes of drums, Fourier series, and sound: Bach`s fractal fugues: Pythagorean semi-tones. Additionally, we will describe a new connection between mathematics and music, one implicit in the worlds of the Dave Matthews Band, guitar tablature, Miles Davis, and the Fender Stratocaster.

The Number "e"- It`s Origins, Equvalents, and Applications
by Lianne M. Smith
Independent reading project supervised by Prof. Philip K. Hotchkiss.
ABSTRACT: The number e is one of the most fundamental irrational numbers in all of mathematics. Beginning with it`s origin, the number e has appeared in an enormous amount of places, and has been defined several different ways. In this talk, we will examine the origins of e, these different definitions, and some of its diverse applications.

The Edge of Infinity as a Really Cool Subset of [0,1]
by Prof. Philip K. Hotchkiss
ABSTRACT: The boundary (or edge) of a finite object is easy to detect. The boundary of a disk (i.e. your dinner plate) is a circle, the boundary of the room you are in is (essentially) a cube. What happens if the object is infinite? For example, does the xy-plane have a boundary? It turns out that many "nice" objects, like the xy-plane, do have well defined boundaries. In this talk we will discuss a way of defining a boundary for "nice" infinite objects, and consider several examples. One of these will lead us to one of the most well-known and interesting subsets of the unit interval.

One group of students presented research on the explosion of the beaver population in the Commonwealth of Massachusetts. Their practice talk was attended by State Senator Michael Knapik and a story on their research , "Trap ban spells lots of beavers", appeared on page A1 of the Springfield Union News on 7 April, 2000.

Abstracts to the talks are as follows:

Busy Beavers in Massachusetts
by Inshirah Abdur-Ra'uf, Chantal Ayotte, Suzanne Gallagher, and Kathyrn Matras
Research project from the Honors Program course BIOL380/MATH380 - Population Modeling which was offered during the Fall 1999 semester and subsequent BIOL399 and MATH399 Independent Study Projects. Research supervised by Prof. Julian Fleron and Prof. Buzz Hoagland.
ABSTRACT: Over the past three years, the beaver population in Massachusetts has more than doubled: increasing from 24,000 to 52,000. The banning of leg-hold traps in 1996, which caused a dramatic decrease in the annual harvest, is suspected of being the key factor in this sudden population growth. The resulting overabundance of beavers directly impacts the environment: flooding streams, blocking roads, and flooding basements.
We created a mathematical model of the beaver population using spreadsheets and the Leslie Matrix method. This model incorporates beaver reproductive, harvest and survival rates. We used a sensitivity analysis to explore the effect of changing rates on the estimated population. Our model shows that several factors have a significant effect on the growth of the beaver population. Population models, similar to the one we created, are critical in estimating the growth of wildlife and determining effective wildlife management.

Controlling the Population of Black Bears in Wisconsin
by Christopher S. Adams, Suzanne Gallagher, Jessica Ferris
Research project from the Honors Program course BIOL380/MATH380 - Population Modeling which was offered during the Fall 1999 semester. Research supervised by Prof. Julian Fleron and Prof. Buzz Hoagland.
ABSTRACT: The population of black bears (Ursus americanus) in Wisconsin is on the rise resulting in an increase in the number of human-bear interactions. To minimize the potential for human injury from bears, wildlife managers are developing models for controlling bear populations at acceptable levels. These models generally include birth and mortality rates for specific age cohorts and harvest rates. In this project, a model was developed through the use of spreadsheets and population data to predict an optimal harvest rate to maintain the bear population at levels which minimize the danger to humans. Our model not only defines hunting levels, but identifies the important parameters that need to be regularly measured to assure that a sustainable bear population can be maintained.

A New Relationship Between Triangular And Square Numbers
by Brandt Kronholm
Presentation sponsored by Prof. Ronald Edwards.
ABSTRACT: Mathematicians since the time of Pythagoras have been aware of many relationships between triangular and square numbers. Often, these basic number forms serve as an introduction to number theory. While examining the properties of these numbers, I discovered a new relationship between them. Is this discovery truly unique, or has it been obscured since the times of the ancient Greeks?

The Sphere Packing Problem
by Rosann M. Tefts
Research project from the course MATH0390 - Senior Seminar in Mathematics and a subsequent MATH0399 Independent Study Project. Research supervised by Prof. Philip K. Hotchkiss.
ABSTRACT: Every supermarket runs into the problem of how to stack oranges, grapefruits, or any other spherical objects. The mathematician`s version of this is referred to as the sphere packing problem, or the Kepler Conjecture. In 1611, Johannes Kepler stated that the most efficient way to pack identical spheres is in the pyramid shape of the face-centered cubic lattice. Even though this conjecture has been around for several hundred years, the actual proof of it was not completed until 1998 by Thomas Hales of the University of Michigan. Hales announced his proof in 1998, but it is still under review. This presentation will outline the history of the Kepler Conjecture and some of the key ideas in Hales` proof.

Model Railroad Train Tracks, Tangles, Dominoes, and Tetris: Deep Mathematical Problems as the Progeny of Children's Toys
by Prof. Julian F. Fleron
ABSTRACT: We might think of model railroad train tracks, dominoes, the game Tetris, and the plastic Tangle toy (which has been popularized as a "cosmic guide" as well as "a folded protein model") as little more than recreational trinkets, games and hobbies. Yet they share critical mathematical connections. Exploring these connections leads us not only to wonderfully rich mathematical problems, but problems that are simultaneously deep and complex. These problems unite knot theory, topology, geometry, and combinatorics in wonderful ways. In this introductory talk we will illustrate some of these wonderful connections, introduce some of the mathematical problems that naturally arise from the study of these connections, and (hopefully) leave the audience Tangled in the wonderful web of mathematical intrigue woven together by these remarkable "children`s" "toys".

Cardboard Boxes, "Elliptic Curves", and the Quadratic Formula: It's Perfectly Rational
by Prof. Philip K. Hotchkiss
ABSTRACT: Anyone who has taken calculus is familiar with the Box Problem: "Suppose we wish to construct a box from a rectangular piece of cardboard, with side lengths a and b, by cutting out small congruent squares of side length s from each corner and then folding up the remaining flaps. What value of s will maximize the volume of the box?" The solution to this problem involves solving a quadratic equation and this generally requires the quadratic formula. However, picking rational values of a and b so that the quadratic factors nicely turns out to be harder than one would expect. In this talk we will show how these rational values can be parameterized: and surprisingly the answer lies parameterizing rational points on an ellipse.

Westfield State College participants in the conference were:
Rosann Tefts, Jenny Pawlishan, Adam Cardinal-Stakenas, Jason Gates, Debbie Palie, Nikali Benkert,, Chantal Ayotte, Katie Matras, Christopher Adams, Suzanne Gallagher, Brandt Kronholm, Inshirah Abdur-Rauf, Melissa Lathrop, Elizabeth Shea, Prof. Julian Fleron, Prof. Philip Hotchkiss, Prof. John Judge, Prof. Jim Robertson.

Westfield State College had 14 people participate in the conference, 11 students and 3 faculty. This group included 7 groups of speakers from our college, tied for third highest among all colleges and universities represented at the conference!

Abstracts to the talks are as follows:

(??? The mathematical symbols in this abstract need to be taken care of.???) Volumes, Areas, Perimeters, and Derivatives?
by Adam Cardinal-Stakenas
Presentation sponsored by Prof. Julian F. Fleron
ABSTRACT:An often-noticed relationship between the area and perimeter of a circle is that the derivative of the area function is equal to the circumference: A' = (pi r ^2)' = 2pi r = C. It is natural to wonder if this property holds for other familiar shapes. At first glance, though, the property does not even hold for squares: A' = (s ^ 2)' = 2s ??notequal?? P. However, if we consider the area with respect to the variable a = s/2 then A = (2a)^2 so A' = (4a^2)' = 8a = P. In fact, we show that by choosing an appropriate variable to calculate area, we can generalize this relationship to all "nice" two-dimensional shapes. In a similar manner, the derivative of the volume function of a sphere is equal to the surface area: V" = (4/3)(pi r^3)" = 4pi r^2 = SA. Again, we show how this relationship can be generalized to all "nice" 3-d shapes. This relationship, however, has limitation which we will show by giving examples where it does not hold and by demonstrating how similar properties arise in the solutions of certain optimization problems.

Rosencrantz and Guildenstern Are Dead (But Lucky!)
by Monique R. Despres
Presentation sponsored by Prof. Julian F. Fleron
ABSTRACT: What does Mathematics have to do with theater? for Rosencrantz and Guildenstern, some harmless coin tossing leads to a discussion of the laws of probability in this Tom Stoppard play. Come to this fun presentation to find out just how lucky these characters are. They didn't call it Theater of the Absurd for nothing.

Counting the Infinite
by Michael R. Nai
Presentation sponsored by Prof. Philip K. Hotchkiss.
ABSTRACT:Intuition tells us that there should be one size of infinity. Either soemthing is finite or it is not. During the 19th century Georg Cantor showed this is not the case. In fact, he showed there are different sizes of infinity. Int this talk we will consider Cantor's proofs of the countability or the rationals, the uncountability of the reals, as well as Cantor's Theorem which campares the cardinality of a set to the cardinality of its power set.

The Geometry, Calculus, and/or Topology of Model Railroad Train Tracks
by Prof. Julian F. Fleron, Jenny Pawlishen, and Rosann Tefts
Collaborative research project initiated by Prof. Fleron and future elementary school teachers in his course MATH0251 - Foundations: Geometry during the Fall semester of 1998. Ms. Pawlishen and Ms. Tefts followed up on this project via a subsequence MATH0360 - Current Trends in Mathematics and Mathematics Education Project which was supervised by Prof. Fleron.
ABSTRACT: Have you ever been playing with a model train track set and come up with this really awesome layout just to find when you get to the last piece, it will not connect to the other end? Either a small gap is formed, which cannot be compensated for by another piece of track, or a misalignment occurs. The frustration of this whole dilemma might cause you to dump the train set back in the box and stuff it away for a while. Our presentation explores our effots to find all feasible closed layouts of standard Lionel model train tracks. Some of our partial solutions include symmetry, counting arguments (connected possibly to Diophantine equations), and geometric reduction arguments (a' la Reidemeister moves from knot theory). So if you have ever had this problem, we have some informative ideas to help you keep that train from being thrown into the closet again before it is ever successfully put together.

Demonstration of a Dual Key Cryptographic System
by Frederick W. Ross
Presentation sponsored by Prof. Anne Pasquino.
ABSTRACT:In this presentation, we highlight the benefits and weaknesses inherent in a dual key system. A dual key system has the advantage of being judicially secure and conforms to at least two of the Payne criteria for a "perfect" crytographic system. We will demonstrate a working computer encryption program utilizing a dual key system.

The Fuzzy World of Fuzzy Set Theory and Fuzzy Logic
by Prof. Julian F. Fleron
ABSTRACT:Are you a Math Nerd? Do you always remember the +C when you indefintely integrate? Answers to these questions must certainly be matters of degree. Like most daily questions we face, they are not well suited to the definitive in/out nor true/false conclusions that characterize classical set theory and logic. Radical as it might seem, this relization should lead one to ask: Can classical set theory and classical logic be extended to cover the fuzzy memberships and fuzzy truth values that seem to characterize our varied daily lives? Promoters of the young areas of fuzzy set theory and fuzzy logic not only answer this question with a resounding yes, but they also claim that applications of their new fields are bot profound and revolutionary. Fuzzy systems, they claim, are revolutionizing technology systems that control high-speed trains, VCR's, camcorders, washing machines, and mony other things. In this expository talk we describe the alluring, yet elementary, way in which fuzzy theorists have fussified classical set theory and logic. Then, following the lead of one of the field's experts, we provide a typical application of fuzzy theory. Finally, we will briefly consider some reactions mathematicians and philosphers, including the speaker, have had to fuzzy theory.

A Fifth of Euclid
by Prof. Philip K. Hotchkiss
ABSTRACT:Did you know that there are triangles whose angle sum is less than 180'? Did you know that a straight line does not always give the shortest distance between two points? While these statements are false in Euclidean geometry, they are true in hyperbolic geometry. In his Elements, Euclid gives five axioms for geometry. His fifth axiom, commonly called the parallel postulate, which (essentially) states that given a line L and a point P not on L, there is only one line through P parallel to L, has drawn a lot of interest. For centuries, mathematicians were convinced that this axiom had to be a consequence of the other four axioms. In the 1820s several mathematicians, Gauss, Lobachevski and Bolyai showed that the parallel postulate was, in fact, indepedent of the other four. Hyperbolic geometry is a geometry that arises when the parallel postulate is replaced by the statement that given a line L and a point P not on L, there are infinitely many lines through P parallel to L. In this talk we will discuss the history of the parallel postulate and then look at a model of hyperbolic geometry and discuss some of the amazing differences between hyperbolic and Euclidean geometry.

Westfield State College participants in the conference were:
Rosann Tefts, Jenny Pawlishen, Adam Cardinal-Stakenas, Jason Gates, Debbie Palie, Monique Despres, Fred Ross, Mike Nai, Matt Collins, Mike Tirrell, Colleen Cooper, Prof. Julian Fleron, Prof. Philip Hotchkiss, Prof. Jim Robertson.

Abstracts from the talks are as follows:

Generalizing Fermat's Last Theorem
by Randall Vaill, jr.
Research project from the course MATH0390 - Senior Seminar in Mathematics and a subsequent MATH0360 - Current Trends in Mathematics and Mathematics Education independent study project. Research supervised by Prof. Julian F. Fleron.
ABSTRACT: By now it is widely known that Fermat's Last Theorem, the most famous problem in the history of mathematics, was proved by Andrew Wiles in 1994. However, this is not the end of the discussion about this problem. There are several interesting generalizations of this problem, including one discovered by the speaker and several fellow students which will be the focus of this talk.

An Introduction to Surreal Numbers
by Monique Despres
Independent research project pursued under MATH0399 - Independent Study in Mathematics and MATH0360 - Current Trends in Mathematics and Mathematics Education independent. Research supervised by Prof. Julian F. Fleron.
ABSTRACT: Surreal numbers, discovered by John H. Conway, are one of the newest and most exciting developments in mathematics. In Conway's own words, "Just as the real numbers fill in the gaps between the integers, the surreal numbers fill in the gaps between Cantor's [transfinite] ordinal numbers" (Conway, The Book of Numbers, 1996). What exactly is the infinitesimal 1/omega What are the relationships between the infinite numbers omega^2+1, and omega^omega? What does the Hackenbush game tell us about surreal numbers? This talk will explore these and other surreal dilemmas.

Pastures, Boxes, and Symmetry
by Adam Cardinal-Stakenas
Research project which grew out of a MA106 - Calculus I assignment. Pursued under MATH0360 - Current Trends in Mathematics and Mathematics Education under the supervision of Prof. Julian F. Fleron.
ABSTRACT: In calculus the two standard optimization problems in which one must maximize the area of a fenced in plot or maximize the volume of a topless box are specific examples of rectilinear isoparametric problems. We have discovered that entire families of these problems have an amazing and unique symmetry. In this presentation the speaker shall illustrate the symmetry in the classical examples, show how it extends to these large, related families, and attempt to shed light on what causes this symmetry.

Gabriel's Wedding Cake
by Prof. Julian F. Fleron
ABSTRACT: Calculus II students are often treated to the paradox of Gabriel's horn: an infinite solid which has infinite volume, but infinite surface area! This remarkable solid was actually discovered and analyzed by Torricelli before the advent of calculus. Recently the speaker has discovered a discrete analogue of Gabriel's horn, Gabriel's wedding cake, which can be shown to have the same remarkable properties as Gabriel's horn using rudimentary infinite series. Both of these examples and a smattering of their remarkable properties and histories will be considered in this talk.

When does 1/4=1/3=1/2?
by Jim Robertson
ABSTRACT: Given any circle, what is the probability that a random chord will be longer than the side of an inscribed equilateral triangle?
Intriguingly, three different solutions have been cogently argued to this problem. We will take a look at these solutions, whose explanations require only a minimal background in geometry. The question, "What constitutes a 'random' chord?" will be focal.

Kaleidoscopes
by Michelle M. Boussy
Independent research project pursued under MATH0399 - Independent Study in Mathematics and MATH0360 - Current Trends in Mathematics and Mathematics Education independent. Research supervised by Prof. Julian F. Fleron. Ms. Boussy's paper on this topic was published in The Best of Westfield State , Vol. 1, Spring 1997.
ABSTRACT: The kaleidoscope, invented by Sir David Brewster in 1781, has proven to be an instrument that utilizes physics and mathematics in order to capture the attention of people of all ages with its artistic images. We will consider the physics and optics involved in the kaleidoscope, including the number of mirrors incorporated in the tube, the angle at which they are placed, and the refraction of light which produces the images. We will use this discussion to focus on the mathematical structure behind the symmetrical images that can be seen through the kaleidoscope.

Constructible Numbers and the Problem of 'Doubling the Cube'
by John Baumann
Independent research project supervised by Prof. Julian F. Fleron.
ABSTRACT: Given a cube with a specific volume, is it possible using only a compass and unmarked straightedge to construct the edge of a cube that has twice the volume of the given cube? It is very interesting that not only was such a construction never accomplished but that the problem, namely that of "doubling cube", was actually proven to be unsolvable. We will look at what a constructible number is and what constitutes the set of constructible numbers, thereby showing that it is impossible to "double a cube".

Westfield State College participants in the conference were:
John Baumann ('96), Michelle Boussy ('97), Monique Despres ('99), Jason Kriacou ('97), Natalie Shaw ('96), Professor Maureen Bardwell, Professor Julian Fleron and Professor Jim Robertson.

The Opaque Square and Related Problems
by Prof. Julian F. Fleron
ABSTRACT:Related to Plateau's problem, minimal surfaces, and soap bubbles, is the so-called opaque square problem which can be stated as follows: Question: What is the shortest length of fencing that can block all lines of sight across a square plot of ground? The answer to this innoculous looking question is unknown. Intuitively evident configurations of fencing, together with a little calculus to fix the optimal lengths of segments in the configurations, readily lead to the "current world record." The holder of this record is a high scool mathematics teacher. By varying the question one obtains many interesting and surprising variants. Examples include: a totally disconnected opaque fractal fence, the impossibility of finding an analogue for the cube, a "wedding cake" candidate for the solution of the opaque sphere problem, and so forth. The variants are so plentiful and the results so sparse, that the interested party should be able to construct their own variant and become the "official world record holder" minutes after the presentation.

The Math Club at Westfield State College was on hiatus, was without an advisor, and there was no local organizer for the first, and soon to become annual, Hudson River Undergraduate mathematics Conference.

Regional Meetings of the American Mathematical Society

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