CHAIR'S THOUGHTS

Dr. George Poole

This is our third issue of the mathematics newsletter. New personnel changes in the department include: Office Manager - Sherri Renfro, Graduate Coordinator - Dr. Jay Boland, Math Lab Director - Dr. Janice Huang, and statistician - Dr. Bob Price. Our number of graduate teaching assistantships has risen to 12 and has dramatically increased the activities in the mathematics graduate program.

Since the second issue of the Accent, the university has gained a new president. Dr. Paul Stanton, former Vice President for Health Affairs and Dean of the College of Medicine, became the eighth president of ETSU following five splendid years under the leadership of Dr. Roy Nicks. Among other important goals, President Stanton promises to establish greater ties between ETSU and the local community. Another important change at ETSU is the appointment of Dr. Don Johnson as the Dean of the College of Arts and Sciences following a two-year interim role. Finally, students from the first class of the Honors Program were graduated this past May.

Special faculty recognitions: Dr. Jay Boland received the 1996 University Outstanding Teaching Award, Dr. Teresa Haynes received the College of Arts and Sciences Outstanding Research Award, Dr. Bob Gardner was awarded promotion to Associate Professor, and Dr. Debra Knisley and Dr. Jeff Knisley were granted tenure.

Regarding "highlights," the department continues to be a leader in pedagogy and mathematical research. Departmental examinations are administered in College Algebra, Probability and Statistics, Precalculus and Calculus I. The Probability and Statistics course, Math 1080, continues to be the course of choice by students to satisfy the general education requirements (2200 students enroll each year). Because of its importance, this course is the best coordinated, most closely monitored multi-sectioned course at ETSU. Tod Jablonski has done a superb job in maintaining quality in the administration and instruction of Math 1080.

Changes in the undergraduate major program include: 2180, Foundations of Probability and Statistics, is a new required course; 3200, Differential Equations, is now required of all majors; and the major must select either Numerical Analysis or Numerical Linear Algebra to improve "number crunching" skills.

Recent changes in the graduate program include: Students are required to complete a thesis; the sequence requirement has been deleted; and students must orally present their proposed thesis problem to members of their committee at least three months before their anticipated thesis defense.

Jay Boland and Don Hong have proposed that the department develop an actuarial program at the graduate level. The discussion of such a program is ongoing.

The new library is going up fast and promises to be an imposing landmark. As mentioned in the second issue of the Accent, it should become the landmark most remembered by ETSU's graduates. How will the new library affect the Mathematics Department? In several ways, we hope. When the new library comes on line in fall 1998, the old Sherrod Library will be remodeled and backfilled with the College of Nursing and the Department of Computer and Information Science(C&IS). When C&IS vacates Gilbreath Hall, Mathematics hopes to gain some much-needed space for offices for GTAs, part-time instructors and a visiting instructor. A special office is needed to provide testing quarters for students needing a special testing environment. In addition, space for a conference and seminar room is needed.

I was delighted to hear from several of you regarding your current status and employment. When appropriate I shared those communications with the faculty. Please continue to communicate:

Chair: pooleg@access.etsu-tn.edu

Undergraduate Coordinator: kerleylm@etsu-tn.edu

Graduate Coordinator: boland@etsu-tn.edu

You should also visit the Mathematics Department's home page at:

http://www.etsu-tn.edu/~math/math.htm.

GRADUATE PROGRAM NEWS

Dr. Jay Boland

The graduate program has recently been the center of a whirlwind of activity. For the fall 1997 semester there are 18 graduate students enrolled full time! Twelve of these students are on paid assistantships and an additional two have full tuition waivers. Thanks go to Dr. Linda Lawson who piloted the graduate program through the last year of exceptional growth and change. Her excellent organization and management are especially appreciated by Dr. Jay Boland whose new position as graduate coordinator is greatly simplified.

Some activities related to our most recent graduates follow:

• Atif Abuida graduated in the fall of 1996, completing his thesis entitled "The Location of Zeros of Polynomial in the Complex Plane." His thesis advisor was Dr. Bob Gardner. Atif is now studying for his Ph.D. in mathematics at Auburn University.
• Bernadine Cochran completed her thesis "Matrix Factorizations" and graduated in August 1997. Her thesis advisor was Dr. George Poole. She is currently teaching mathematics at Pikeville College in Kentucky.
• Sherri Cockerham completed her degree requirements for the M.S. in Mathematical Sciences during summer 1997. Her thesis "f-Cycle Hybrid Triple Systems" was directed by Dr. Bob Gardner. Sherri has returned to teaching at Science Hill High School.
• Chris Hardin completed his master's degree in May 1997 under the direction of Dr. Paul Savarese. His thesis is "Regression Diagnostics for Comparison of Regression Models under Transformations."
• Lucas Van Der Merwe graduated in May 1997 having completed his thesis "Connectivity of a Generalized Cartesian Product Graph and a Generalized Prism over G and H" under the direction of Dr. Jay Boland. Lucas is currently teaching part time at ETSU while studying for his Ph.D. from the University of South Africa.
• Liu Ping completed his degree requirements in July 1997 under the co-direction of Dr. Teresa Haynes and Dr. Linda Lawson. His thesis is entitled "Domination In E-graphs." Liu is currently studying for a master's degree in Computer Science at ETSU.
• Dan Studer completed his degree in May 1997 under the co-direction of Dr. Teresa Haynes and Dr. Linda Lawson. His thesis is entitled "Induced-Paired Dominating Sets in Graphs." Dan is currently teaching in the Developmental Studies program at ETSU.

Recent accomplishments and activities of former graduate students include the following:

• Pam Torbett, Amy Weems and Bob Gardner had their paper "Decompositions of the Complete Digraph into each of the Orientations of a 4-Cycle which admit a Certain Automorphism"appear in Congressus Numerantium 120 (1996) 33-39.
• Pam Torbett, Jay Boland and Linda Lawson published "Eccentricity Stable Edges of Trees" in Congressus Numerantium 118 (1996) 203-208.
• Amy Weems and Bob Gardner submitted "A Bernstein Type L^p Inequality for a Certain Class of Polynomials" to Journal of Mathematics Analysis and its Applications.
• Atif Abuida and Bob Gardner submitted "Some Results on Location of Zeros of a Polynomial" to Approximation Theory and its Applications for possible publication.
• Coleen Huff, Bob Gardner and J. Kennedy submitted "Decompositions and Packings of Digraphs with Orientations of 4-Cycles" to Ars Combinatoria.
• Pam Torbett is currently teaching full time for us on the main campus on a one-year appointment.
• Coleen Huff was hired as an instructor to teach at the Kingsport center.
• Amy Weems received a fellowship from the University of Tennessee and is currently pursuing her Ph.D. in mathematics there.

HONORS

Christopher Wallace was recognized by the university during spring semester at Honors Day as being the most outstanding graduating senior in mathematics. Chris is presently enrolled as a graduate student in the M.S. program in mathematics at ETSU.

SCHOLARSHIP INFORMATION

Six deserving students were awarded scholarships for the fall 1997.

The Faber-Neal Scholarship which is designed to cover annual tuition costs to sophomore mathematics majors was awarded to Deborah Dye of Meadowview, Virginia. One week before his death (1995), Dr. Larry Neal, a distinguished professor of mathematics at East Tennessee State University, established a scholarship fund, the Faber-Neal Scholarship, to honor his very dear friend, Joe Faber, professor emeritus of East Tennessee State University.

The Wilson-Hartsell Scholarship which serves to help aspiring mathematics teachers was awarded to Katie Price from Kingsport, Tenn. The Wilson- Hartsell Scholarship was initiated in 1987 and named in honor of two former department chairs with long-term commitments to ETSU: Mr. Charles Wilson (retired, '86) and Dr. Lester Hartsell (retired, '87).

The Depew Scholarship which is designed to be awarded to students planning to major in mathematics was awarded to Ken Proffitt from Erwin, Tenn. Mrs. Rex Depew (Sara Margaret) established an endowed scholarship, the Depew Scholarship, in memory of her husband, who graduated from ETSU in 1940, having majored in mathematics. Dr. Depew (Ph.D. in mathematics at Vanderbilt University in 1963) served with the Air Force, taught at both Everett High School and North Alabama State University, and provided mathematics support and analysis for NASA Space Systems as an employee at the IBM Corporation. Mrs. Margaret Depew continues to reside in Maryville.

The Edward Stanley Scholarship is awarded to promising mathematics students planning to major in mathematics. Jerry Scott Wilson of Shawnee, Tenn, Kristin Waldo of Gate City, Va., and Jeffrey Bowles of Kingsport, Tenn were the recipients. The Edward Stanley Scholarship was endowed in 1986 from a generous gift from a former student, Dr. Ed Stanley, who completed an illustrious career teaching mathematics at Clemson University.

When you consider making contributions to the university, please consider these various funds.

NEW HIRES

Dr. Robert Price joined us in the fall 1997. He recently completed his Ph.D. in statistics at the University of Wyoming. His research interests are in nonparametric statistics and general linear models.

Sherri Renfro joined us as office manager in June 1997. She received her B.S. from ETSU in communications and has experience in human resources management in Arizona and Georgia.

Coleen Huff was hired as a full-time instructor at the Kingsport campus.

TEACHER EDUCATION PROGRAM

Dr. Janice Huang

We have recently revised the secondary mathematics education program to bring it more in line with the National Council of Teachers of Mathematics (NCTM) standards. The program now requires numerical analysis and a calculus-based course in statistics. Our Teaching Secondary Mathematics course continues to evolve and now includes a component on using technology in the classroom, as well as one on teaching mathematics on the block schedule. It is hoped that this new curriculum will receive accreditation from NCTM in the near future.

Our majors continue to do well after graduation. Lora Hopkins, outstanding math major in 1996, and Marty Johnson have been hired at Dobyns-Bennett High School in Kingsport. This fall (1997) one of our MAT students, Tony Donen, was hired at Science Hill High School in Johnson City.

We have been strengthening our relationship with area teachers of mathematics. The Upper East Tennessee Council of Teachers of Mathematics (UETCTM) meets on our campus regularly during the school year. In the summer of 1996, 25 high school teachers participated in a week-long workshop directed by Dr. Janice Huang entitled "Mathematical Modeling Using Statistics and CBL's."

FACULTY PROFESSIONAL ACTIVITIES

• Dr. George Poole was invited to participate in the sixth annual ATLAST workshop on software design in linear algebra held in San Diego, Calif., June 24-29. He was invited to participate in an army ROTC leadership assessment program held at Fort Knox, Ky. June 30 - July 3.
• Dr. Jay Boland received the 1996 University Outstanding Teaching Award.
• Dr. Bob Gardner's article "Optimal Packing and Coverings of the Complete Directed Graph with 3-Circuits and with Transitive Triples" will appear in Congressus Numerantium. He presented this paper at the 27^th Southeastern International Conference on Combinatorics, Graph Theory, and Computing at Boca Raton, Fla. (March 1997).
• Dr. Teresa W. Haynes received the 1997 College of Arts and Sciences Outstanding Research Award. She has eight papers accepted for publication in refereed journals and serves on the editorial board of the Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC). Dr. Haynes, S. T. Hedetniemi, and P. J. Slater have completed the following two books to be published by Marcel Dekker, N.Y., Dec. 1997. "Fundamentals of Domination in Graphs", and "Domination in Graphs : Advanced Topics."
• Dr. Don Hong recently presented a one-hour invited lecture at the sixth International Colloquium on Numerical Analysis and Computer Sciences with Applications held at Plovdiv, Bulgaria during August 11-17, 1997. His paper "Swapping edges of arbitrary triangulations to achieve the optimial order of approximation" (with Charles K. Chui at Texas A&M University) appeared in SIAM J. Numerical Analysis, No. 34, August 1997. He also has two other papers accepted for publication in Computers and Mathematics with Applications. These papers are "Optimal order approximation by mixed three-dimensional spline elements (with Ram N. Mohapara at University of Central Florida)" and "Some new formulations of smoothness conditions and conformality conditions for bivariate splines (with Huan-Wen Liu at University of Wollongong, Australia)."
• Dr. Janice Huang published an article "Counting Characters in Blocks of M ₂₂" in the Journal of Algebra in the May 1997 issue. Her research on the block schedule was outlined in The Mathematics Teacher in the Sept 1997 issue.
• Dr. Lyndell Kerley completed a three-year term as state director for Tennessee for the southeastern section of the Mathematics Association of America.
• Dr. Debra Knisley has the following three papers accepted for publication. "Neighborhood Unions, Generalized Degrees and Graphical Properties (with L. Lesniak)" in Graph Theory, Combinatorics, Algorithms and Applications, "Independence, Domination and Uniform Maximum Degree (with G. Domke, J. Dunbar, T. Haynes, and L. Markus)" in Graph Theory, Combinatorics, Algorithms and Applications, and "Independence, Domination and Generalized Maximum Degree (with G. Domke, J. Dunbar, T. Haynes, and L. Markus)" in Congressus Numeratium. She also gave an invited address entitled "Uniform Maximum Degree and Edge Density" at the special session on Graph Theory at University of Memphis (1997) at the southeast regional meeting of the American Mathematical Society.
• Dr. Rick Norwood has an article entitled "Why 2 + 2 = 2 x 2" accepted for publication in Mathematics Magazine.

Three faculty received ETSU Research and Development Grants:

• Bob Gardner, "Packing and Covering the Complete Directed Graph with Orientations of Cycles," $3230.
• George Poole, "Improving the Accuracy of the Computed Solutions of Large Linear systems in a Parallel Environment," $5880.
• Don Hong, "Bivariate Spline Functions and Applications," $5330.

NEW DISCOVERIES IN MATHEMATICS

Dr. Robert Gardner

The Navier-Stokes Equation and Turbulence

Over 150 years ago, the French engineer Claude Navier (1785-1836) and the Irish mathematician George Stokes (1819-1903) independently discovered the equations which describe the velocity and pressure of a fluid flowing around an object. The equations can be derived from Newton's Laws of Motion and expressed vectorially as

where the fluid is assumed to be incompressible , are vector functions, v=v(x,y,z,t) is the velocity of the flow, is the fluid density, p= p(x,y,z,t) is the fluid pressure, µ is the fluid viscosity coefficient, t is time, and f = f (x,y,z,t) is the body force (such as gravity). This partial differential equation (or PDE) must be supplemented with suitable boundary conditions in order for it to be well-defined (that is, there exists a unique solution which depends continuously on the boundary conditions). One of these boundary conditions is that there is no '' slip'' of the fluid at the surface of a body; that is, the fluid ''sticks'' to the body surface and has the same velocity as the surface itself.

The unfortunate property of the Navier-Stokes equation, as it is commonly called, is that it is nonlinear (a linear PDE consists of linear combinations of the derivatives of the dependent variables with respect to the independent variables - the nonlinearity of the Navier-Stokes equation can be seen in the term (v v ). There is a large body of knowledge on linear differential equations, however few methods exist to analytically solve nonlinear DEs.

A byproduct of the nonlinearity is turbulence. Turbulence, loosely speaking, is the chaotic motion displayed by a fluid as the speed exceeds a certain threshold. An example of this is the pattern displayed by smoke as it rises in still air from a cigarette. Just above the cigarette, the smoke displays a smooth, or laminar, flow. As the smoke rises, it displays a rolling, zigzagging motion reflecting the tiny eddies present in the air. Turbulence is usually thought of as a negative property of fluid flow. However, it is possible to take advantage of turbulence and to increase the efficiency of certain processes. For example, a golf ball is dimpled to increase the level of turbulence as it flies through the air (the ''fluid'' to which I refer is, in applications, most often air). As a smooth sphere flies through the air, a vacuum is created behind the sphere inducing a force resistant to its velocity. This force is a component of drag (along with the usual resistance to velocity produced by the air resistance which is proportional to cross sectional area). The dimpling of a golf ball allows turbulent air to get around the ball faster, thus reducing the vacuum behind the ball. This allows a golf ball to travel up to 250 per cent as far as an undimpled ball.

One reason turbulence may have such a bad reputation is the fact that it arises from this problem which is so difficult to solve mathematically. The first significant progress in the area of computational fluid dynamics was made in the late 1960s. Sufficient algorithms and computing technologies were present for the first time to solve numerically the Navier-Stokes equations for simple cases such as a slowly moving two dimensional fluid passing around an obstacle. This meant that the only way to test new aerodynamic designs was in the wind tunnel (requiring between 10 and 15 tests per design). Present day computational technology has now made it possible to reduce the number of wind tunnel tests to between two and four per design. This allows for quicker, cheaper tests of new designs.

This problem of turbulence is a fine example of a classical area of math in which there is still a flurry of research activity. One can expect this activity to continue in the future in both the applied and the theoretical realms of mathematics.

SELECTED REFERENCES

You might find the following publications of interest (especially the second one on which this article is loosely based).

• The Physics of Golf, T. P. Jorgensen, AIP Press (1994).
• Tackling Turbulence with Supercomputers, P. Moin and J. Kim, Scientific American 276(1) (1997), 62-68.
• Turbulent Times for Fluids, Tom Mullin in Exploring Chaos: A Guide to the New Science of Disorder, ed. N. Hall, W. W. Norton (1993).

MATHEMATICS LABORATORY

There were several changes in the Math Lab in the fall of 1997. Dr. Janice Huang assumed the Directorship of the laboratory and graduate student Becca Jablonski became the new Coordinator. Plans are evolving to make the operational structure of the Lab much more efficient for the benefit of the students. The new role of the graduate teaching assistants allows more tutoring of the students and less testing than in previous years. This is possible because the time required for testing in MATH 1010 and Math 1040 has been reduced considerably.

In September, a new MATH HOT LINE was established in the Lab, allowing students to call with questions. Meanwhile, tutoring sessions on probability and statistics, and the use of the HP48 graphing calculator are being given to the tutors. This enables the tutors to become more effective in assisting students with questions about such topics.

The hours during which the Lab is open this year are : Monday -- Thursday 11:30 a.m. - 7:30 p.m. and Sunday from 1:30 p.m. to 5:30 p.m. The HOT LINE is 423-439-8586.

KAPPA MU EPSILON MATHEMATICS HONOR SOCIETY

Kappa Mu Epsilon (KME) Mathematics Honor Society recently initiated the following 13 new members during spring 1997.

Stephanie Bauer, Katrina Blevins, Angie Dawn Greer, Michael R. Ivey, Dallas W. Lingerfelt, Verbie Jo Martin, Bradley Singletary, Deepak Taneja, Kristin D. Waldo, and Michael R. Young.

The 1996-1997 officers were

Ellie Alevritis - President

Corey Williams- Vice President

Brian Heaton - Secretary

Chris Wallace - Treasurer

WE NEED YOU NOW!

Please consider sending a contribution to the Mathematics Foundation Account so we might send a fall 1998 issue out to each of you. We also need additional dollars to cover expenses not normally covered by state revenues. In the past these monies have been used to purchase special research equipment, provide faculty release time to complete special departmental projects, and provide expenses for out-of-town speakers .

PLEASE SHARE

If you are an ETSU graduate with a major or minor in mathematics, please share with us your current type of work. More importantly, because we would like to improve our undergraduate and graduate programs, please share with us some things you liked about your experiences at ETSU and some things you would like to see changed. Please indicate your degree and the year you graduated from ETSU. See the form provided on page six of this newsletter or communicate electronically with George Poole, Chairman, Department of Mathematics, (POOLEG@ETSU.EDU).

Fall Newsletter Information

Currently our plans are to distribute a newsletter each fall. Please send information on your status to Dr. George Poole, Department of Mathematics, East Tennessee State University, Box 70663, Johnson City, TN 37614-0663.

Name --------------------------------------------------------------------

Year Graduated ---------------- Degree Obtained ---------------------------------

What are you doing now? -----------------------------------------------------------------------------------

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Preferred Mailing Address: ------------------------------------------------------------------------------

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I would like to contribute to the ETSU Department of Mathematics as follows:

Candidates for scholarships apply to the chair, Department of Mathematics.

----- Mathematics Foundation

For covering expenses not normally covered by state revenues, e.g. speakers, research equipment, etc.

----- Edward L. Stanley Scholarship

For meritorious students planning to major in mathematics. First-year students must have a minimum high school GPA of 3.2 and ACT composite of 25.

----- Wilson-Hartsell Scholarship

For students majoring in mathematics.

----- Faber-Neal Scholarship

For sophomores majoring in mathematics.

----- Depew Scholarship

For students majoring in mathematics.

Please make check or money order payable to:

Mathematics Foundation Account

East Tennessee State University Foundation

Box 70732

Johnson City, Tenn. 37614-0732