UofL Math

Gazette

March 2000

The Newsletter of the Department of Mathematics

in the College of Arts and Sciences

of the University of Louisville

Math Club

Current Officers

Co-Presidents: Gena Boercker, Leanne Conaway

Treasurer : Sue Holt

Sponsored by the department and organized by graduate students, the club meets each Tuesday to eat pizza and discuss career opportunities, course work and general topics in mathematics. Guest speakers are invited to speak two or three times each semester.

Puzzle Page by Dr. Jeno Lehel

New Puzzle: Batting Average

Batter A has a higher batting average than batter B for every single day of the season. Does it follow that batter A has a better batting average than B for the whole season?

Old Puzzle: Traffic Light

At a certain corner, the traffic light is green for 30 seconds and then red for 30 seconds. On the average, how much time is lost at this corner?

Solution by Jim Twohey (MA '95)

The average time lost at the traffic light is 15/2 seconds. It is equal to the expected value of the time spent stopped at the light. The probability of being stopped at the light is 1/2. When not stopped by the light, the mean waiting time is zero. When stopped by the light, the mean waiting time is 15 seconds, because there is an equal probability of hitting the red light at any particular moment of the thirty seconds it is red. So 1/2 x 15 + 1/2 x 0 = 15/2 seconds is the mean waiting time.

-Also solved by Paul Buckthal (MA '76), Samuel Carter (BA '99), and Richard Farris (BA '72)

Mail or e-mail your solution to: Dr. Jeno Lehel lehel@louisville.edu , Math Dept, Louisville, KY 40292

Chairman's Corner

Dr. Michael Jacobson

This is shaping up to be a hallmark year for the department. Within the next six to nine months, I believe our Ph.D. proposal will finally get a review (hopefully positive) by the state Council on Post-Secondary Education. We expect that during this academic year to have awarded 14 M.A. degrees in Mathematics (an all-time high) and approximately 30 Baccalaureates. In spite of this, the department is under a microscope analyzing the efficiency of how we do our job. A significant reallocation review has been undertaken by the university in an effort to do an even better, more efficient job. I am optimistic that the department will fare well during this process and possibly see changes including a combining of Mathematics offerings across campus, and a true partnership in the training of mathematics teachers as we continue to work on an joint Center for Mathematics and Science Teacher Development, Education and Research in cooperation with the School of Education. These changes will help to strengthen the mathematical community in the area and help develop the scientific and mathematical literacy of the populace.

As always, I hope you find something interesting in this issue of the newsletter. Please feel free to contact me and let me know your thoughts - we appreciate all your support, and hope in some small way you are able to continue your mathematical experience.

Faculty Activities for 2000:

Michael Jacobson will be chairing a special session at the KYMAA meeting in March for statewide department chairs. He is also the featured speaker at the Fifth North Carolina Mini-conference on graph Theory, Combinatorics and Computing in April. He will give a Colloquium at Appalachian State University in April as well. In May he will be co-organizer of the 14'th Cumberland Conference to be held in Huntsville, Alabama.

Udayan Darji is currently visiting Caltech and will be working at the Institute for Defense Analysis Center in Communications Research in La Jolla, CA.

Marianne Korten one of our visiting professors, will give a talk at the Arkansas Spring Lecture 2000 in Fayetteville, Arkansas in March. She will also speak at the International Conference on Generalized Functions, Linear and Non-Linear Problems, Pointe-a-Pitre, Universite des Antilles et Guyane, Guadelope in April.

Richard Davitt will participate March 25-26, 2000 in the semiannual meeting of ORESME (a group of Kentucky, Ohio, and New York historians of mathematics who gather together periodically to research a designated seminal mathematical work and the life of the prominent mathematician who produced it) at Miami University in Oxford, Ohio. This session is extra special in that Professor John Fauvel, an internationally renowned historian of mathematics from the Open University of Great Britain, will lead the discussion of Sir Isaac Newton's fundamental text on integral calculus, De Analysi, and the accompanying investigation of the life and times of this most famous of mathematical scientists.

2000 Bullitt Lecture

Dr. Fernando Q. Gouvea

The Department of Mathematics is hosting the seventh annual Bullitt Lecture on Friday, April 7, 2000 at 7 p.m. in the Floyd Theatre located in the Swain Student Activities Center on UofL's Belknap Campus. This year's speaker is Professor Fernanco Q. Gouvea from Colby Collge. Dr. Gouvea's research interests are in number theory and arithmetic geometry, with a secondary interest in the history of mathematics. He is also the editor of FOCUS and MAA Online (the online newsletter of the Mathematical Association of America).

The talk will be a survey of the last 1000 years of mathematics focusing especially on how numbers moved from a peripheral position in mathematics to a central one, with some attention to how this has also influenced our way of thinking about the real world.

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Special Financial Lecture

For the last three years, our Department has hosted lectures by prominent scholars in insurance and mathematical economics, funded by the ARMS Financial Group. In 1997 our speaker was Dr. Jose Pinera, who discussed privatization of social insurance in Chile. In 1998, Dr. Stephen Ross presented the theoretical underpinnings of modern mathematical finance. In 1999, Dr. Hal Varian spoke about economic models of electronic commerce. Unfortunately, late in 1999, ARM Financial Group experienced financial difficulties and continued funding for the lecture became impossible.

This year, we will hold a special lecture in mathematical finance, without any external funding. It will take place on April 12, at 5:00 p.m. in Strickler Hall, Middleton Auditorium. The speaker is Dr. Richard Derrig, Senior Vice President of the Automobile Insurers Bureau of Massachusetts, and of the Insurance Fraud Bureau of Massachusetts. He will lecture on new mathematical methodologies of artificial intelligence helping in the discovery of insurance fraud.

Update On The Undergraduate Degree Programs

by Dr. Thomas Riedel

Based on a proposal worked out by the Undergraduate Studies Committee (Ewa Kubicka, Jenö Lehel, and Thomas Riedel) the Mathematics Department has made essential changes to its undergraduate degree program. Allowing students to specialize earlier than our previous programs permitted, our graduates will have more choices in the job market, or within their continuing studies.

We still offer the Bachelor of Arts and the Bachelor of Science degrees. The B.A. requirements have changed slightly; it still serves the purpose of getting a broad yet substantial background in Mathematics, while allowing the student to take a significant number of courses outside of Mathematics. Thus our B.A. program best serves students who intend to enter the field of education or continue their studies in other professional schools.

The B.S. provides a solid general background in Mathematics with an in depth study of one area together with its applications. This program serves students who enter into the technical job market or who wish to continue their studies of Mathematics in graduate school. The B.S. degree now contains a core curriculum including traditional courses like Analysis, Linear Algebra, Abstract Algebra, just to mention a few, and we offer four different concentration areas:

• Actuarial Mathematics
• Computational and Applied Mathematics
• Probability and Statistics
• Pure Mathematics

These changes in our programs will better serve our students' and our community's needs.

Faculty Highlight:

Manav Das

I received my Ph.D. in mathematics from The Ohio State University in 1996. After spending a year at OSU as a postdoc, I joined the University of Louisville as an Assistant Professor. I have taught courses in trigonometry, college algebra, honors calculus and linear algebra. Currently I am teaching a section of precalculus and a graduate level course in real analysis.

My main interest is in measure theory and its interplay with other areas of mathematics such as dynamical systems, probability theory and real analysis.

Roughly speaking, measure theory deals with the various techniques of measuring geometric objects. Calculating areas and volumes are the classical roots of this area. Historically therefore, it dates back to the Greeks. Integration theory, as developed by Newton and Leibnitz, provide a backdrop for many modern developments. Riemann, Lebesgue, Carathéodory and Hausdorff, aided by the solid foundations in set theory laid by Cantor and Dedekind, were able to generalize and extend the notion of integration by introducing novel methods of measuring sets.

For instance, consider a piece of metallic wire. One way of measuring this object would be its length. Another would be its mass. If the wire is made up of a single metal and is of uniform thickness, then its mass would be proportional to its length. This is not very interesting. But suppose two or more metals are combined into an alloy, then it is clear that the mass distribution should vary with the relative compositions of these metals. Different compositions would give rise to different mass distributions, or measures. Thus measure theory gives us a glimpse into the intrinsic geometry of sets and not just their classical shape.

In the beginning of the 1900s, great minds like Poincaré were trying to understand the notion of dimension. They thought inductively: a line (which is 1-dimensional) has boundaries that are points (which are zero-dimensional). A 2-dimensional object such as a square has boundaries that are 1-dimensional and so on. This dimension (termed topological dimension) was then connected with the notion of measure. To measure a 2-dimensional object A, length of A (denoted l(A)) is a bad choice because it is infinite. On the other hand the volume l3(A) is zero. Therefore area l2(A) is somehow just right. This crucial observation generalized the classical notions of length, area and volume. For each real number s, we can describe a measure which is like ls . For a set A there exists a unique number s¢ such that if we take any number t > s¢ , then lt (A) = 0 and if t < s¢ then lt (A) = infinity. This unique number s¢ may therefore be considered to be the dimension of the set A. This dimension is called Hausdorff dimension. If a set has Hausdorff dimension strictly larger than its topological dimension, the set is called a fractal.

Now imagine looking at an object under a magnifying glass. As we change the order of magnification, certain features will fall out of focus and certain others will become visible. This is simply because a given object could have finer structures at different scales. In order to understand the true geometry of this object, we would want to view it under all possible magnifications. Thus we would be compelled to study a multitude of fractals, each coming into focus at a different scale. This is multifractal theory.

The current research in multifractal theory is too vast for me to enumerate here.

There are numerous books on these topics. In particular there is a book by M. Schroeder titled Fractals, Chaos, Power Laws (W.H. Freeman, NY 1991) which gives a popular treatment of chaos and fractals. For a philosophical approach one can read B. Mandelbrot’s The Fractal Geometry of Nature (W. H. Freeman, NY 1983).

Please feel free to contact me if you have any questions. You can email me at manav@louisville.edu.

Department of Mathematics
College of Arts and Sciences
University of Louisville
Louisville, KY 40292

Tel. (502) 852-6826
Fax (502) 852-7132
E-mail: math@athena.louisville.edu

Chair of the Department:
Dr. Michael S. Jacobson
Professor of Mathematics
E-mail: mikej@louisville.edu
http://www.louisville.edu/a-s/math/msjaco01.html