Scientific Computing at The Ohio State University

President Jennings (right) accepting the Ohio Eminent Scholars award from Governor Richard F. Celeste on May 11th

Research and teaching in the field of scientific computing will be part of the efforts of the Department of Mathematics. In May, the department received word that its proposal for such a program had been selected for funding by the State of Ohio. The allocation came through the Eminent Scholars Program established by the Ohio General Assembly. The purpose of that program is to foster "the growth of eminence in Ohio's academic programs while bringing educational resources to bear on compelling statewide problems."

The Center for Scientific Computation at The Ohio State University will provide a research base and shared access for all of Ohio to a national supercomputer network. Supercomputers are a departure from traditional sequential machines. Relying on new parallel processing architectures, these machines permit analysis of complex problems which previously have not yielded to solutions in reasonable periods of time. Computer-assisted design and manufacture, robotics, large-scale chemical processes, aerodynamical design - all of these use extensive computation. Power systems of the size and sophistication needed for contemporary applications could not be designed or controlled without large-scale computations validated by careful theoretical studies. Computer simulations have become crucial in all branches of industry since an accurate mathematical model may detect errors or suggest significant improvements; mathematical models are likely to be significantly cheaper than working prototypes. Indeed, in extensive studies of crisis situations, prototypes may be unavailable or too dangerous to attempt.

To exploit the capabilities of these machines, new algorithms must be conceived. This requires basic research in the areas of languages, algorithms, computational mathematics, and numerical analysis. Each Eminent Scholar Award provides funds to establish an Endowed Chair to attract a prominent researcher and teacher in the academic area. The individual filling the Chair in Scientific Computation will hold rank as a Professor in the Department of Mathematics and serve as Director of the Center for Scientific Computation. The Ohio General Assembly provided a $500,000 matching Grant to the University for establishing the Chair. In addition, the University has committed the necessary funds for acquiring computing equipment, space and building renovations.

It is also anticipated that the person serving as Director of the Center for Scientific Computation will have a beneficial effect on mathematical, scientific, and engineering research at Ohio State. A great deal of large-scale scientific computing is carried out here, and frequently scientists and engineers must delay work on their own projects to develop computational expertise which would most preferably be left to others. The existence of the Center for Scientific Computation would allow such people to concentrate their efforts on their own work and draw on the expertise of the Center.

The Center for Scientific Computation provides the necessary strong computing environment for the new Director to be able to recruit additional faculty and graduate students in the area of numerical analysis and scientific computing.

The academic need for strength in scientific computation is not limited to Ohio State, but affects colleges and universities throughout Ohio. Numerical analysis is a desirable part of the research effort and course pool at virtually all institutions of higher learning. The Center at The Ohio State University will be accessible to the various colleges and universities in Ohio and will provide a central location for training and research. In particular, it is expected that some visiting positions in the department will be earmarked for faculty from other Ohio institutions. A strong information campaign is planned to inform Ohio institutions of these possibilities and others related to the Center.

The Center for Scientific Computation is expected to contribute both to the revitalization of existing industry and to the attraction of new firms to Ohio. Specifically, the Center will be able to offer informal cooperation and advice, or more formal consulting assistance, both to large and small companies. More importantly, the Center will serve as a source for well-trained students available for temporary jobs or permanent recruitment. Ohio State President Edward Jennings has proposed the creation of a high-technology industrial and research park in the Columbus area. The Center will certainly contribute toward this goal. However, the benefits of the Center will not be restricted to Central Ohio. The existence of the Center for Scientific Computation will increase university-business cooperation throughout Ohio and serve as a magnet for business and industry on a national scale.

Woods' Log

Another academic year is upon us bringing with it clear signs of improvement in the scholastic preparation of incoming freshmen. The enrollment in calculus and upper-level sequences has increased. In particular, our honors sequences have significantly higher enrollments than in previous years. We offer two sequences of honors courses. Honors calculus is directed towards those most able students who need the full calculus sequence as a requirement in their major programs, mainly engineering. Mathematics honors is provided for students already familiar with the calculus and with the talent for and interest in developing the more theoretical aspects of the discipline. At present, class sizes average about 25 students in these honors sections and they are always taught by full faculty members. We are constantly on the lookout for good students at all levels. If you know of any, please suggest this department to them as a suitable place to continue their studies. I would be happy to describe our offerings and answer any questions either over the phone or in person. I can be reached at (614) 422-7173.

This fall we will be very busy recruiting to fill the chair in applied mathematics we were fortunate enough to be awarded in the recent Ohio Eminent Scholar competition. Provided we are successful, it is anticipated that the chosen scholar will lead a group in scientific computing and found a center for the subject here. This is one of the most exciting areas of applied mathematics today and the employment prospects for successful students in this field are virtually unlimited.

Although a public institution, we are relying more and more on contributions from the private sector to help in upgrading and initiating important services that do not naturally fall within areas supported by State subsidy. I would like to thank those of you who have contributed in the past. Incidentally, it is possible for donors to channel such contributions into particular programs of their choice.

If there is a departmental activity you would like to hear about, please don't hesitate to let me know. Until next time.

Alan Woods Chairman

An Invitation and a Challenge

The crisis in mathematics education is real! Funding for research in mathematics has dropped significantly in the last several years. In Ohio, as well as other states, funding for higher education has not kept pace with inflation nor with the requirements for maintaining quality programs. The Department of Mathematics needs additional resources to continue to provide an exciting and stimulating environment for students at all levels. This is an invitation and a challenge to you who have completed mathematics study at Ohio State. If you feel it is important that the standards of our programs be maintained at a high level, become a member of "Mathematics Motivators." Your contribution of $25.00 or more will enable us to establish scholarship support for talented students, an honors resources center, graduate student research funds, a visiting professor program, and, perhaps, an alumni special conference.

If you prefer, a direct contribution may also be made to any one of the following already established funds:

• John Riner Memorial Fund: for professional support, awards, and social activities for graduate students enrolled in the Master of Arts option for teachers.
• Mathematics Periodical Fund: for subscriptions to mathematics journals for the Mathematics Library.
• Notemin Fund: for use jointly by the department of Mathematics and the College of Education to search for better ways of teaching mathematics.
• Hortense Rickard Foundation: for fellowships and scholarships to students in mathematics.
• Bea Ross Memorial Fund: for scholarships and special lectures for the special summer program for highly talented high school students.

Contribution checks should be made payable to The Ohio State University Development Fund and mailed to James R. C. Leitzel, Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174. Remember all such contributions are tax deductible. You might also want to see if your firm has a "matching gifts" program that would double your support.

Department News

Henry Glover, together with Mark Feshbach of the University of Minnesota and Stuart Priddy of Northwestern University organized a one week conference this past summer. The conference, "On the Classifying Spaces of Groups," was held at the Institute for Mathematics and its Applications in Minneapolis. About 35 people participated in this conference. Funding for the program was provided by the Institute as a service to its member institutions in the midwest.

Every Spring Quarter, the department holds the Rasor-Bareis-Gordon Mathentatics Contests. Cash awards to the winners of these contests come from the correspondingly named endowment funds honoring Grace M. Bareis, Kate Deterly Gordon and S. E. Rasor. This year $1000 was distributed within three divisions. The winners of the Freshman division were Thomas Barrett, Tom Reischman and Mark Wolff. The Sophomore prizes were awarded to Voytek Zaleski and Carl Phillips. In the Junior-Senior division awards were made to Ben Bielefeld, Mark Hovey, Richard Toeniskoetter, and P. Scott Leonard.

Congratulations to all these young people on their excellent performance.

Fifty Years Ago

Fred Albert Lewis was born April 13, 1895 in Talladega, Alabama. He received his AB in 1915 and MA in 1916 from the University of Alabama. Dr. Lewis received his Ph.D. from Ohio State in August, 1934. After serving in World War I and teaching for a year at Texas A&M College, he returned, as an assistant professor, to the University of Alabama in 1920. He served on the faculty there until his retirement as professor in 1965. During his 45 years of distinguished service, he served as Head of the Department of Mathematics from 1945 to 1955. As a symbol of the respect and appreciation for the service given to the University of Alabama, the Fred A. Lewis Scholarship awards were named in his honor.

In the following letter, written by his wife Frances, he recalls some memories of his time in the department.

Dear Dr. Leitzel,

My husband, Dr. Fred A. Lewis, has asked me to write you in answer to your request of August 15th.

He was 89 years old last April 13th. His memory is bad, and so many, many events have taken place in the fifty years since 1934!! He thinks that you have a better chance of looking in Ohio State catalogues of the late twenties and thirties than anything he could tell you.

Professor H. W. Kuhn was Chairman of the Department. He had his Oral Examination for the Doctor of Philosophy at 2 p.m. on August 14, 1934, in room 314, University Hall. Dr. H. W. Kuhn was chairman, and members of the committee were Dr. C. C. MacDuffee, Dr. Tibor Radó, Dr. Henry Blumberg, Dr. F. R. Bamforth and Dr. D. R. Englis. His field of specialization is Group Theory.

There were five (5) Ph.D.'s in mathematics awarded on August 31st, 1934. They were to: Foster L. Brooks, Marjorie Leffler, Fred Albert Lewis, Robert L. Rinehart, and Wilbur J. Robinson.

Dr. Kuhn directed his thesis, and was a good friend. There was another Math Professor Fred liked very much -- Dr. Weaver, who was killed in an auto accident. We kept in touch with the Radós and the McDuffees for several years. We always went to the Association and Society meetings in August (or September) and often to the winter meetings. The last time I saw Mrs. Radó was in Kingston, Ontario, in August 1953.

Fred retired from the University of Alabama in 1965. We spent the next ten years traveling. We had our 59th wedding anniversary August 18th. His health is very bad now. We are Ohio State fans -- went to Rose Bowl with the football team twice!

I apologize for my "wandering"- I am 82!

Best wishes to the Math Department and to you on your newsletter.

Indonesia Bound

Jack Tull, associate professor of mathematics, will spend the next two years at Gadjah Mada University in Yogyakarta, Indonesia. He and his wife, Mary Nye, left Columbus on September 9th for their stay on the island of Java. Jack will be advising the mathematics faculty at the university on undergraduate training in analysis. He will also be advising on staff development and further mathematics training of the current staff.

Minimal Graphs Form A Finite Set

Finding the best method for laying down microscopic threads of metal, without intersections, on the surface of an integrated circuit chip is an enormously challenging problem. On some chips it is necessary to connect as many as a million circuit elements into an efficient electronic network. Graph theory, the branch of mathematics which studies arrays of points and their connections, has proved to be a useful tool in analyzing and designing such networks. Recently, Neil Robertson and Paul Seymour (now associated with Bell Communications Research, Inc. in Murray Hill, New Jersey) have achieved important results on so-called minimal graphs. These minimal graphs are the "obstructions" that prevent the drawing of a graph on a particular surface.

To join each of the five labelled points to every other point would require a total of ten lines. In the figure, it is not possible to join point 3 and point 5 without a crossing of lines in the diagram.

As early as 1930, mathematicians were aware that certain graphs could not be drawn in the plane without any of the lines crossing. In fact, the simplest example is to try to connect five points in the same plane so that lines join each point to every other point without any lines crossing in the diagram. There is no way to connect all pairs of these points by lines without introducing some crossing. The Polish mathematician, Kasimir Kuratowski, showed that this graph on five points and one other on six points were the minimal obstructing graphs for the plane. Any graph that contains either one of these two minimal graphs cannot be drawn on a plane, and any graph that does not contain either one of those can be drawn.

Not long after Kuratowski's result was announced, the Hungarian mathematician, Paul Erdos, posed the intriguing problem of determining whether the set of minimal graphs that cannot be drawn on other surfaces is a finite set. About 20 years ago, the German mathematician, Kurt Wagner, conjectured that for any well-described property of a graph, the list of minimal graphs is finite. The work of Robertson and Seymour deals with several special variants of Wagner's conjecture.

In 1980 progress was made on the Erdos conjecture by OSU department members Dan Archdeacon, Henry Glover, Phil Huneke and C. S. Wang. They verified by direct methods that, for the Mobius strip, there are 103 minimal graphs and for the torus, move than 800 minimal graphs exist. Further work by Huneke and Archdeacon in 1982 provided the result that the lists of minimal graphs not embeddable on unorientable surfaces are finite. This work gives a strong indication that Warner's conjecture might be true and feasible to prove.

The current results of Robertson and Seymour establish that for any fixed surface, the number of minimal graphs not drawable on that fixed surface is finite. This conclusion was achieved as a consequence of their work in developing a more general structure theory of graph inclusion relations. Define an antichain to be any set of pairwise unrelated graphs. The minimal members of any set of graphs form an antichain. Thus Wagner's conjecture is that all antichains of finite graphs under a certain inclusion relation are finite. Sets of graphs closed under the inclusion are called order ideals. The researchers have conjectured that certain structures are associated with these ideals in the sense that the ideal consists of those graphs which possess the given structure. Planar graphs are an example of this; they form an ideal under the inclusion, and the structure is given by their plane embeddings- Working with special cases of the inclusion property and results on tree structures, they were successful in showing that the set of graphs drawable on a fixed surface does not form infinite antichains under the special inclusion relation. The finiteness result follows from this property.

Since this is just a part of the more general structure they are developing, Robertson and Seymour are continuing their work. So far, eight long papers have resulted from their efforts. Seymour expects at least three more papers to come because, "There are a lot of details." Robertson states that a consequence of the structural and finiteness conditions proved or conjectured is that a computer program can be associated with each of the hereditary properties defined. That program would decide whether or not a graph possessed the property in a number of steps bounded by some fixed power of the number of vertices in the graph.

While their technique did not produce a systematic way of listing the minimal graphs, the contribution of Robertson and Seymour is a major advance in coming to a fuller understanding of the graph embedding problem.

Seed Grants Awarded

Paul Ponomarev, S. M. Tariq Rizvi (Lima), and David Trautman have received awards through the University's newly established Seed Grant Program. The program, administered through the Office of Research and Graduate Studies, supports faculty efforts to develop new research initiatives and other scholarly activities. Ponomarev's project is titled "Supersingular Elliptic Curves and Hecke's Conjecture." Rizvi will investigate the relationship between continuous and quasi-injective modules in his project "Theory of Types and Continuous Modules." Questions concerning non-locally convex function spaces will occupy the time of Trautman.

Ross Receives Honorary Degree

Arnold E. Ross (left) receiving his Honorary Degree from Denison University President Robert C. Good.

Arnold E. Ross, professor emeritus and former chairman of the department, received the honorary degree , Doctor of Science, from Denison University. The Honors Convocation was held April 27, 1984. Dr. Ross was recognized for his decades of work with high ability students. The citation reads in part, "... (a) distinguished mathematician and administrator, generous supporter of the research and scholarship of others, extraordinary discoverer of talent in young people in American cities and foreign and respected practitioner of international opportunities in the mathematical sciences..." Dr. Ross also addressed a faculty luncheon during the convocation week where he spoke on "Searching for Talent Among the Young: Problems and Opportunities."

Dr. Ross continued this summer, as he has for many years, his widely recognized program for high ability students. Fifty talented young people lived steeped in mathematics for an intense eight week session on the campus of Ohio Site. In addition to the study of number theory, the students also attended lectures on analysis, combinatorics and probability theory. Other members of the department working with the program this summer were Bogdan Baishanski, Ranko Bojanic, Barry Cipra, Tom Dowling, Joe Fiedler, Dan Shapiro and Gloria Woods. Antoine Brunel of the University of Paris, a visitor in the department this summer, gave the lectures on probability.

The program this year sadly missed the encouragement and support of Bea, Dr. Ross' wife, who passed away during the year. Her memory was honored through the establishment of a special lecture series, the Bertha Halley Ross series on Contemporary Issues in Mathematics Research and Education. Professor Felix Browder of the University of Chicago gave the inaugural lecture in that series. It was an exciting time for the interchange of ideas among the young students and Professor Browder.

The department extends warm congratulations to Dr. Ross for the honor he has received and for his continued work in successfully nurturing talented young students.

Alumnews

We received several responses from the first issue of Math Matrix Please, let us know about yourself. Take a minute and drop us a line.

Barbara S. Eberlin B.Sc. '59, M.Sc. Computer Science from Cleveland State University '64, currently Senior Software Engineer, Bailey Control Company, Cleveland

Sandy Hirschhorn M.S. '73, M.S. '75 in Computer Science, currently department manager -- design automation, CTE Research Laboratories, Waltham, Massachusetts.

Charles W. Johnson B.A. '46, retired engineer, General Electric Corp., presently involved in computer-oriented cryptanalysis of current short-wave intercepts.

Steve Kosciuk B.Sc. '76, MA. '77 and Ph.D. '82 from University of Wisconsin, Madison, currently data analyst for a cancer research project at the University of Wisconsin Medical School. Beginning August, 1985 expects to be involved in a training program for college mathematics faculty in Nicaragua.

Diane Matting B.A. '75, Comparative Literature, honors student in mathematics, now a Ph.D. candidate at Rutgers University in Comparative Literature. In Brazil next year studying on a Fulbright dissertation grant.

Jeffrey T. McLean M.S. '67, Ph.D. '73, associate professor, Department of Mathematics, College of St. Thomas, St. Paul, Minnesota. Summer of 1984 participated in an NEH Seminar on the History of Science at Harvard University; working on a manuscript or. discrete mathematics applied to computer science.

Agnes Sommer Merritt M.S. '22, retired educator.

Armin H. Meyer M.A. '41, theoretically retired, but serves as business consultant and adjunct professor in School of Foreign Service, Georgetown University, former U.S. Ambassador to Lebanon, Iran and Japan.

Harry Stebbins M.S. '71, M.Ed. '76 from University of Cincinnati, currently a mathematics teacher in Cincinnati.

Robert W. Votaw M.A. '39, retired from position as Manager, Guidance Mechanical and Sensor Engineering, Ordance Systems Department, General Electric Company, Massachusetts.

Roland Zielke M.S. '69, Ph.D. '71 from University of Konstanz, M.D. '83 from University of Muenster, West Germany, currently Professor of Applied Mathematics, University of Osnabrueck, West Germany.

Reader Reply

As a result of the article on public-key cryptography in the previous issue of Math Matrix, the Editor received the following note from Charles W. Johnson, 16 Thornwood Lane, Layetteville, New York 13066: "If any of your readers are cryptobuffs and would like to try their skills at the real thing instead of puzzle problems, suggest they write me and I will give them tips on how to get a hold of raw material."

Problem Corner

Here is your chance to renew or review your problem solving skills! Send your solutions to the Editor. The best (or most interesting) solutions will be included in the next issue of Math Matrix.

Problem 1. (from the Mathematical Magazine, January, 1882). Two men, A and B, hired a horse and carriage for $7, to go from Providence to Boston and back, the distance between the cities being 42 miles. At Attleboro', 12 miles from Providence, they took in C, agreeing to take him to Boston and Back to Attleboro' for his proportionate share of the expense. At Walpole, 24 miles from Providence they took in D, agreeing to take him to Boston and back to Walpole for his proportionate share of the expense.

What ought each person to pay?

Problem 2. A non-negative sequence u_n is defined by:   u₁=1/2, and 2u_n+l² = 1 + u_n (n = 1, 2,...). Show that

lim_n-> u_n = 1.

How fast does u_n coverge to 1? Evaluate

lim_n->4ⁿ(1-u_n)

Have you come across an interesting mathematical problem? Problems with an applied flavor are especially welcome. Submit your favorite candidate to the Editor for use in future issues.

[0]