Excursions in Mathematics - Descriptions of Individual Faculty Members’ Courses

While it is the intent at this time of the faculty members cited here to present the material as listed, content and subject matter are subject to change.

L. Doty: Math 111

Graph Theory: This course is focused on the area of discrete mathematics known as graph theory.  A graph here is a mathematical model of a communication or transportation network.  Examples of such networks are street grids, bridge systems for cities, computer networks.  A good example of this kind of mathematical graph is any simplified, stylized mass transit map.  The distances are not to scale, and topographical features are not preserved.  Instead the order of the stations (or stops) and the interconnections of transit lines are emphasized.

S. Frank: Math 110

Mathematics of Light and Sound: This course will investigate how mathematics explains a variety of physical phenomena involving light and sound waves.  We will begin by investigating problems that led Newton to develop Calculus.  Attention will then focus on geometric reflection of light from flat surfaces and lenses.  The fascinating wave-particle nature of light will be discussed as well as the discovery of fiber optic technology.  Finally the characteristics of acoustic waves will be studied to understand why sonar is an effective tool for undersea detection.

J.  Helmreich: Math 111

Computational Probability and Statistics: The computer has revolutionized the field of statistics.  In this course we investigate questions in probability and statistics, and develop computer methods and models to aid in that investigation.  The technical term for what we do is called simulation and the bootstrap.  Students will learn a very simple and straightforward programming language, and write programs that will help answer various classical questions in probability and in statistics.

J.  Kirtland: Math 110

Check Digit Schemes: This course develops, from the ground up, the skills necessary to understand, apply and create check digits schemes. Check digits are digits that are appended to identification numbers (Universal Product Codes, ISBN numbers, credit card numbers, driver's license numbers, etc) to check for errors when the numbers are transmitted.  For example, they are used to catch forged credit card numbers, to make sure that grocery story scanners are reading the correct number (which gives the product price and other information), and to make sure that data is transmitted error free.

At the beginning of the semester, while studying basic number theory, the students learn about its application to simple cryptography and rudimentary check digit schemes.  As the student's mathematical skills grow to include the concepts of a function, permutation, and symmetry, applications to art and nature are discussed.  And toward the end of the semester, when group theory is introduced, applications to advanced check digit schemes are discussed.

K. P. Krog:  Math 111

Game Theory and Strategy: This course introduces students to the theory of two-person games.  A two-person game is any situation in which there are two players (a player may be an individual, a company, a nation, or even a biological species).  Each player has several strategies that he or she may choose to follow, and the strategies chosen by each player determine the outcome of the game.  With each possible outcome is associated a numerical value, or payoff, awarded to each player.  We will study both zero-sum games and non-zero-sum games.  Zero-sum games are games in which the sum of the payoffs for each outcome is zero.  Thus there is a winner and a loser each time the game is played.  In non-zero-sum games it is possible for both players to benefit simultaneously and so we must examine the relationship between competition and cooperation in such games.  In addition to studying the theory of two-person games, we will consider game-theoretical applications in business, political science, biology, and other disciplines as time permits.

R. McGovern:  Math 111

Voting Systems: In this section, we will attempt a mathematical analysis of various political institutions, principally: voting systems, proportional representation, the effect of coalitions on voting power and the possibility of fair voting systems.  If time permits, we may also study models of competitive behavior.

This course will attempt to give the student an experience in mathematical research in a supportive environment.  In the course of the semester you will work on approximately ten group research projects.  If you are uncomfortable working on open-ended problems, you may not be comfortable in this course.

You will be required to spend quite a bit of time working with your group both inside and outside of class.  Attendance will be part of your grade.  If you foresee that you will miss more than a few classes, you should not register for this class.  If it will be difficult for you to meet with your group outside of class, perhaps you should not register for this class.

You will have to write reports on your projects almost weekly.  If you find it difficult to write clear and grammatical English, this course may be difficult for you.  If you enjoy analytical writing, you will probably enjoy this course.

On the other hand, this course will require almost no algebraic skills nor will it develop any.  The emphasis will be on logical analysis of models, not calculation.  There are no mathematical prerequisites for this course beyond arithmetic.  You should not take this course as a preparation for other Mathematics courses.

No previous knowledge of political systems is required or expected.

T. McGrail: Math 110

Number Theory: This course studies subsets of the real numbers.  It typically begins with the study of the properties of a sequence of natural numbers called the Fibonacci sequence.  We look at connections between the Fibonacci numbers and various real-world phenomena.  We look for patterns in the Fibonacci sequence.  In the second part of this course, we discuss many of the issues surrounding infinite sets.  For instance, we consider the questions "Do any two infinite sets have the same size?" and "Are there more real numbers than integers?"  In addition, each student completes a research project and gives a presentation on his or her discoveries.

C. Vertullo: Math 110

Quantitative Reasoning: This a nontraditional mathematics course designed to develop better logical and mathematical thinking in students.  The approach of this course is from a quantitative reasoning perspective.  A significant amount of the classroom activities involve group work.  The mathematical topics studied include the methods of reasoning using inductive and deductive arguments, the process of problem solving and how to utilize tools that assist the process, and the concepts of percentages as applied to personal finance including savings plans, installment loans and retirement annuities.

Adjunct Instructors: Math 110

Mathematics as a Liberal Art: Topics are chosen to give the student an indication of the breadth and depth of current mathematical thought.   Topics will include three or four of the following: The mathematics of Voting, Weighted Voting Systems, Fair Division, The Mathematics of Apportionment, Euler Circuits, The Traveling-Salesman Problem, The Mathematics of Networks, The Mathematics of Scheduling.

Adjunct Instructors: Math 111

Mathematics as a Liberal Art: Topics are chosen to give the student an indication of the breadth and depth of current mathematical thought.   Topics will include three or four of the following:  Spiral Growth in Nature, The Mathematics of Population Growth, Symmetry, Fractal Geometry, Collecting Statistical Data, Descriptive Statistics, Chances, Probabilities and Odds, Normal Distributions.

 

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