Duke University Math News

Department of Mathematics
Box 90320, Durham, North Carolina 27708-0320
December 1995


Conference on Stochastic Processes

by Greg Lawler

The mathematics department will be host for the sixteenth annual Seminar on Stochastic Processes. These conferences began in 1981 as a means for established researchers, young specialists, and graduate students to discuss recent developments in all areas of stochastic processes. This is the first time the seminar has been held at Duke. In addition to invited lectures, there will be informal sessions for presenting open problems.

Among the five invited speakers is Priscilla E. Greenwood of the University of British Columbia who is a Duke alumna. Also scheduled are Harry Kesten (Cornell), Jean-François Le Gall (University of Paris), Yuval Peres (Berkeley), and Jay Rosen (City University of New York). In conjunction with the conference there will be a banquet in honor of Professor S. James Taylor of the University of Virginia, who is being recognized for his fundamental contributions in the areas of stochastic processes, in particular Hausdorff measure of random sets associated with Brownian motion and Levy processes.

The conference will be held March 14-16, 1996 and is open to all. There is no registration fee. More information can be obtained from Greg Lawler, jose@math.duke.edu.

And perhaps, posterity will thank me for having shown it that the ancients did not know everything.

-- Pierre de Fermat

The Mathematical Contest in Modeling

The MCM is for undergraduates in the area of mathematical modeling---the process of analyzing real world situations, making predictions, and drawing conclusions. Each school is permitted to have up to four teams of three people each.

There are two problems in each contest, one continuous and one discrete. Each team picks one of the problems to do during the contest.

The contest lasts from Friday morning until 5:00 the following Monday. During that time, the teams may consult books, use computer programs, and in general utilize any ``inanimate'' source to answer the questions posed in the problem. The teams must also write up a paper explaining the solution, which is turned in to the judges to be graded.

Each paper is given a rank of Successful Participant, Honorable Mention, Meritorious, or Outstanding. Those receiving Outstanding ratings will be published in The UMAP Journal, and some teams will be invited to present their papers at the annual meeting of the Society for Industrial Mathematics. Some papers may be given special awards by the sponsors of the contest.

This spring's contest will take place from Friday, February 10 to Monday, February 12. It is sponsored primarily by the National Security Agency.

Last spring, Duke entered two teams in the contest. Nate Bronson, Paul Dreyer, and Scott Harrington wrote a meritorious paper, and the team of Robert Schneck, Noam Shazeer, and Tung Tran received honorable mention.

Results of the VPI Contest

Congratulations are in order to Johanna Miller, a first year student at Duke. She was the top contestant in the Virginia Tech math contest, held earlier this fall, and will receive a check for $125. The contest is held annually in colleges and universities along the East Coast.

DUMU/SPS Semi-formal

The joint Math Union and Society of Physics Students semi-formal was a great success. About thirty undergraduates, graduate students, and faculty attended the pre-dance dinner at Café Parizade. After the dinner, about twenty people returned to the Von Cannon hall in the Bryan Center for three hours of dancing and company. Photos are available from Lester Chen, at lchen@math.duke.edu.

Undergraduate Lecture: The Fourth Dimension

In other DUMU news, Jeff Weeks will kick off the Undergraduate Lectures Series Friday, January 19th. His talk is titled ``Visualizing Four Dimensions,'' and will include applications to space-time and the ``myth of the passage of the time.''

Duke Math Contest

by Tung Tran

The Fourth Annual Duke University Mathematics Contest is being held March 2, 1996. More than fifty high schools from several states have been invited to send teams of five to compete in two divisions. The even is sponsored by the Duke University Math Union (DUMU), and students both organize and create questions for the meet. The format includes a proof-oriented power question, shorter team and individual questions, and the exciting relay round. No calculus background is assumed, only interest in challenging one's ingenuity and understanding of high school mathematics.

For more information on how to enter, or how to help out, contact DUMU co-chair Michael Rierson at rierson@math.duke.edu.


Smith Earns Educom Award

Three professors were awarded medals by Educom for their innovative efforts to improve and modernize undergraduate instruction using new information technology. The winners of the first Educom Medal Awards program were selected by the American Psychological Association, the American Chemical Society, and the Mathematical Association of America. David A. Smith, an associate professor of mathematics at Duke, received one for his work on redesigning the college calculus curriculum to take advantage of computers and calculators. He is co-director of Project CALC (Calculus As a Laboratory Course) which he has been working on since 1984. The program itself received recognition in 1991 by Educom.

Dr. Smith's award consisted of a silver medal, a bronze replica, and a check for $2,500.

New Course on Medical Mathematics for Spring '96

by David Schaeffer

Math 238 Mathematical Problems in Current Scientific and Engineering Research

This course will be conducted as an extended case study exploring mathematical problems in what is called ``shock-wave lithotripty,'' a medical treatment for breaking up kidney stones by shock waves without surgery (see discussion below).

The goals of the course will be to:

Develop through practice your ability to communicate with applied scientists,

Provide mathematical help for the research of a Duke scientist, and

Teach by example some specific techniques of applied mathematics.

To achieve these goals, the main emphasis in the course will be on student research, conducted in collaboration with Professor Laurens Howle (who proposed this problem area), his students, and Dr. Schaeffer.

In lithotripty, a shock wave generated outside the patient's body is focused on a kidney stone by an ellipsoidal reflector. The pressures involved are intense---as high as 1000 atmospheres---but very brief and localized in space---lasting only a few microseconds and extending over a region a few millimeters across. The positive-pressure pulse is followed by a slightly less intense, similarly short, negative-pressure pulse. Because of its short duration, the negative pulse does not tear the fluid apart; however, small bubbles (say a few microns in diameter) do expand greatly, increasing their volume by a factor of or more, after which they collapse violently. It is believed that this collapsing is what breaks the stones apart.

At least initially, research will focus on the response of a single, spherical bubble to a pressure pulse. (The full problem involves many bubbles deformed from spherical shape by their interaction with one another and their proximity to the boundary of the stone.) Specifically we will study the Gilmore-Akulichev description of bubble dynamics. This model consists of coupled integro-differential equations with time as the independent variable.

Specific issues for research are:

Nondimensionalize the equations and determine the appropriate scales; identify any small (dimensionless) parameters.

Develop intuition about the surprising behavior of the bubbles: viz., they expand a million fold, hold a nearly constant radius for many times the duration of the incident pulse, and then collapse almost instantaneously. (The theory predicts this, and it is observed experimentally.)

Write a computationally efficient numerical code to solve these equations.

Study the control problem: i.e., what input pulse is required to achieve a desired bubble response?

The class will be divided into different teams to work on these problems under the supervision of Dr. Schaeffer. The various teams will present their results both orally, in a lecture to the rest of the class, and in writing, in a short paper to be submitted at the end of the term. In addition to student research, Dr. Schaeffer will lecture on certain relevant techniques of applied mathematics.

Grades will be based on student lectures and research.

Prerequisites are multi-variable calculus and elementary PDE, say Math 114 or its equivalent.

Notes from the Department Chair

The Department has begun a program of Assistant Research Professors. These are three-year positions offered to new or recent Ph.D.'s in Mathematics. This program provide a rotating source of new Research and Teaching energies while offering at the same time a post-doctoral style extension of the training of young mathematicians.

One of these positions is of special significance to the Department because it is supported by a gift from William W. Elliott. Elliott was a member of the Mathematics Department from 1925 until his retirement in 1968. He served as chair from 1929 to 1937 and as Director of Undergraduate Studies from 1937 to 1949. The Elliott endowment was established in 1988, but the money only became available for use recently.

We are delighted to have Professor David Reed as the first William W. Elliott Assistant Research Professor. David got his B.A. from the University of Chicago in 1974 and his Ph.D. from Oxford in 1995. In between he had a career in investment banking in New York and London. His research is on Algebraic Geometry and Algebraic Number Theory. Specifically, he studies how the algebraic structure of a variety determines its topology and geometry.

We are also very happy that Professor Lester Coyle joined our faculty this fall on a two-year appointment as Lecturing Fellow. Professor Coyle received his B.A. in 1989 from Trinity College, Dublin and his Ph.D. in 1995 from the University of Michigan. His research is in the area of stochastic partial differential equations. Working with two of our graduate students, Mary Beth Fisher and Sharad Chaudray, and with Professor Greg Lawler, Professor Coyle is currently conducting a seminar in the Mathematics of Finance. He plans to develop two new courses in this area for next year, one at the undergraduate level and the second at the graduate level.

-- John Harer

Reminders from the DUS

Dear Mathematics Majors and Minors,

In my first semester as Director of Undergraduate Studies, it has been a great pleasure to meet many of you and to be of service---I hope---to all of you. Below I have listed some reminders that may be helpful.

-- Harold Layton


Capital One Needs Analysts

Capital One Financial Corporation seeks to hire business analysts with mathematics backgrounds, according to analyst Brent Baumbusch (Math AB '95). At Capital One, marketing analysts, working in a team-based, consultant-like framework, are responsible for charting product strategies and leading the project teams that implement those strategies. Qualifications for such positions include high grades in any analytical major, familiarity with modeling, and demonstrated leadership skills. The ability to interact with senior management is essential. Additional information is available in Room 217 Physics, from Mr. Baumbusch at baumbusc@aol.com, or from Capital One's College Recruiting Coordinator:

Constance Little
PO Box 85525, 11013 West Broad Street
Richmond, VA 23285-5525
phone: (804) 967-1181

A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.

-- Howard W. Eves


Furman Electronic Journal

Furman University has recently established an electronic journal of mathematical papers open to contributions by undergraduates. All kinds of papers are welcome, including technical, historical, and expository (although every paper must be approved by a faculty sponsor before it will be considered). Papers are submitted electronically, and files of any format or word processor are okay, but TeX, LaTeX, and PostScript are preferred.

For more information, contact the editor in chief, Mark Woodard, at:


The journal itself is at the URL:

-0.6ex126 mwoodard/fuejum/welcome.html

and is available through netscape or xmosaic.


Solutions from October

Solution to Problem 1:

Apply property 5 with z = 1 to get:

from (5)

from (3), which becomes:

from (4).

Combining these equations gives:

Solution to Problem 2:

To solve the equation

for positive integers x and y when z is some given number (20 in the problem), use the identity:

Then, substitute every possible combination of factors for a, b, and c such that abc = z. For example, a = 2, b = 10, c = 1 gives . Does this give all the solutions?

New Problems

Problem : Another Hole

Yes, it's another hole problem:

Let be a binary operator on the set of all non-negative integers with the following properties:

The problem is to find a general form for in terms of the elementary operations , and raising to powers.

Problem 2: For the D&D Fans

What is the volume of a regular icosahedron where all the edges are one unit long? For a real challenge, what is the volume of a dodecahedron with all the edges one unit long?


This newsletter is open to contributions from students, both graduate and undergraduate, and faculty. Information about the Mathematics Department, classes, degrees, and projects would all make good contributions. We are also looking for quotes, and of course problems and puzzles. Please send any ideas you might have to the editor, Garrett Mitchener: wgm2@acpub.duke.edu or talk to Harold Layton or Joan McLaughlin in the Mathematics Department.

Current and previous issues of Duke Math News are available online through the World Wide Web at: http://www.math.duke.edu


Garrett Mitchener wgm2@acpub.duke.edu


David Kraines dkrain@math.duke.edu


Joan McLaughlin joan@math.duke.edu

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Joan McLaughlin
Tue Dec 5 16:16:32 EST 1995