It's been 13 years since I joined the Mathematics Department at ETSU. Our mission remains pretty much the same, but many of our paradigms have changed dramatically. However, one thing hasn't changed: Our most important goal is to achieve "excellence" in teaching. Larry Neal, Jay Boland, and Bob Gardner won the university's Outstanding Teaching Award in 1990, 1996, and 1998, respectively. Jeff Knisley won the Arts and Sciences Outstanding Teaching Award in 1994 and is currently a nominee for the Outstanding Teaching Award for the Southeastern Section of the Mathematical Association of America. Written remarks by students on the Student Assessment of Instruction continue to support that teaching in mathematics steadily improves. Based on student input (or the lack thereof), students are accepting the fact that departmental final examinations are here to stay. Currently, departmental final examinations are administered for probability and statistics, college algebra, precalculus, calculus I and calculus II. Now that all students at ETSU are assigned an e-mail address and have access to the Internet through seven computer labs across campus, the paradigm of instruction is changing. Various faculty are beginning to use e-mail and web pages to communicate with and provide information for students. Electronic teaching and learning is "spoken here."

This October, our office manager, Sherri Renfro, took maternity leave to introduce her new daughter, Megan Alyse, to the world. Sherri plans to return in January 1999. Carolyn Peeples, former secretary in the Office of Student Affairs, joined us in a substitute role. So for November and December, Carolyn and the mathematics faculty were practicing and studying "Chaos Theory." As if Sherri's absence wasn't enough to test the faculty, this spring semester the department is short-handed due to the following faculty changes: Drs. Al Tirman and Vishwa Sakhare retired in December '98, Drs. Linda Lawson and Teresa Haynes are away on leave, and Dr. Bob Gardner is on a non-instructional assignment. Our goal is to supplement teaching power with part-time instructors who have served the department for several years: Tom Prior and Corlis Robe. Moreover, Michael Henning has joined us from South Africa to work with the graph theorists and is teaching one large section of Probability and Statistics.

Al Tirman retired December 1998. Al joined ETSU in 1970 as an assistant professor of Mathematics at the Kingsport Campus. By the early 90s, Al became a full-time faculty member on the main campus. Al became well-known on both campuses through his colorful memo writing and his exceptional teaching, often punctuated by expletives. I recall my first colorful memo from Al. After Al had been assigned two night classes, the first followed by an early morning class, I received a castigating memo which included (circa 1989): "Rather than give me a comparable teaching assignment, I would prefer that you take me out to the wood shed and rubber hose me!" Vish Sakhare retired in December 1998. Vish has always been the "quiet soul" of the department, but it never prevented him from speaking his mind at departmental meetings, or from slipping into my office to offer some free counsel. After joining the department in 1967, Vish journeyed to the "hill" at the University of Tennessee and completed his Ph.D. in 1973. Betty Lyle retires in May 1999 after a 37-year career at ETSU. Shortly after I joined the faculty in 1986, Betty began teaching our "new" statistics course, Math 1080. She, along with Tod Jablonski, were instrumental in getting this large program (1000 students per semester) off to a good start in the early 90s. As a department chair, I will sorely miss the presence of Al, Vish, and Betty.

We are attempting to increase the course offerings and enrollments at the satellite campuses of Kingsport, Bristol and Greeneville through increased closed-circuit TV courses and live instruction. Coleen Huff currently teaches courses at Kingsport. By appropriately scheduling TV and live instruction at Kingsport and Bristol, we hope to expand her responsibilities to include Bristol.

A "Vision Committee" considered some new goals for the department. Among these was one to reinforce the faculty's expertise in computational/applied mathematics or discrete mathematics. Consequently, a search is underway to fill two faculty positions with faculty whose backgrounds include these two areas. The research faculty continue to flourish. A new program in actuarial science is being developed by the department. The department will undergo an external review of its undergraduate and graduate programs during the spring.

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@etsu.edu

Undergraduate Coordinator: kerleylm@etsu.edu

Graduate Coordinator: knisleyd@etsu.edu

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

The ETSU mathematics graduate program remains strong. This year we welcome three new full-time graduate students: Suzanne Glass, Joby Kauffman, and Xiao Panrong. We also expect two additional graduate students to start in January.

Now for some news regarding recent graduate students.

• Pam Torbett took a position at the State University of West Georgia where she is a full-time instructor.

• Eric Carr left to pursue a doctorate at the University of Tennessee.

• Wang Xiaodong went to Purdue University to pursue a doctorate.

• Ben Phillips and Becca Jablonski are married and both are now in the mathematics doctoral program at the University of Alabama at Huntsville.

• Lucas Van Der Merwe is directing our math lab while trying valiantly to finish his Ph.D. research. His degree is to be granted by the University of Natal, South Africa.

• Coleen Huff continues to do an outstanding job in her full- time position with ETSU teaching at the Kingsport campus.

Activities by current students include:

• Brad Dyer, Anna Mu, and Kevin Ratliff are working on their theses under the direction of Dr. Don Hong.

• Tammy Carter is working with Dr. Bob Price on a thesis in statistics.

• Allison Coley is writing up her thesis on graph vulnerability under the direction of Dr. Jay Boland.

• Chris Wallace and Xiao Panrong have active research projects underway with Dr. Jay Boland on graph labeling schemes.

• Brian Heaton is beginning work on a problem in quantum mechanics. This work is likely to be jointly directed by Dr. Niall Shanks (Philosophy) and Dr. Bob Gardner.

A special fund has been established for the purpose of supporting mathematics graduate student travel and other expenses deemed suitable by the Math Department graduate coordinator.

The fund is named for former ETSU math graduate student Amy Weems. Amy was in our graduate program from 1994-1996. She received her M.S. under the direction of Dr. Robert Gardner in summer 1996. Her quality work while at ETSU resulted in three publications:

• Decompositions of the Complete Digraph into each of the Orientations of a 4-Cycle which admit a certain Automorphism, with R. Gardner and P. Torbett, Congressus Numerantium, 120, 33-39.

• Bicyclic Decompositions of the Complete Digraph into Orientations of a 4-Cycle, with G. Coker, R. Gardner, and P. Torbett, Journal of Combinatorial Mathematics and Combinatorial Computing, to appear.

• A Bernstein Type L^p Inequality for a Certain Class of Polynomials, with R. Gardner, Journal of Mathematical Analysis and its Applications, 219, 472-478.

In addition, the following manuscript is in preparation (and results from Amy's thesis work):

• "Some Results on the Rate of Growth of a Polynomial," with R. Gardner and N.K. Govil.
  
  Following her stay at ETSU, she started work on a Ph.D. at the University of Colorado, Boulder in fall 1996. Unfortunately health problems required a return to Tennessee. She again began work on a Ph.D. at the University of Tennessee in fall 1997. Tragically, Amy died of cancer during spring 1998. This fund has been established by the friends of Amy Weems in her memory.

For more information on this fund or anyone wishing to make contributions should contact the graduate coordinator.

Graduate Coordinator: knisleyd@etsu.edu

Kelly Diane Edmisten was recognized by the university during spring semester at Honors Day as being the most outstanding graduating senior in mathematics. She is the daughter of Mr. and Mrs. Mack Edmisten, Johnson City.

Three deserving students were awarded scholarships for the fall 1998.

The Wilson-Hartsell Scholarship, which serves to help aspiring mathematics teachers, was awarded to Jason Lewis of Johnson City, 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 Tiffiny Street of 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. Billie Jo Richardson of Morristown, Tenn. was the recipient. 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. Edith Seier joined us in the fall 1998. She recently completed her Ph.D. in statistics at the University of Wyoming.

Her present area of research is new measures of skewness and kurtosis, their applications and the use of computer-intensive methods. She also enjoys consulting, especially in the areas of modeling and time series analysis.

At the request of the College of Education, Dr. Huang has developed a new course for the new K-8 certification program. This course will add three hours of mathematics, for a total of nine, to the requirements for elementary education majors. The new course's title is "Number Concepts and Algebraic Structures." The other courses currently being required for these students are (1) "Introduction to Probability and Statistics" and (2) "Logic, Problem Solving, and Geometry."

Recent graduates of our secondary teacher education program are having no trouble finding positions. There is currently a shortage of mathematics teachers in Northeast Tennessee.

• Dr. Don Hong was invited by Dr. Carl de Boor, a member of the National Academy of Sciences, to give a lecture at the 98-Workshop on Wavelets and Approximation Theory at the University of Wisconsin, Madison, during May 28 - June 3, 1998. He also gave colloquium talks at Mississippi State University and Louisiana Tech University recently. He also has the following papers published or accepted to be published in research journals or proceedings recently. They are:

• On optimal order of approximation from bivariate spline spaces, in Proceedings of the Sixth International Colloquium on Numerical Analysis and Computers with Applications (D.D.Bainov and A.B. Dishliev Eds.), pp. 219-225, VSP, International Science Publishers, Utrecht, the Netherlands, 1998.
• Optimal triangulations and smoothness conditions for bivariate splines, in Approximation Theory IX (C.K. Chui and L.L. Schumaker Eds.), pp. 129-136, Vanderbilt University Press, Nashville, TN, 1998 (to appear).
• Recent progress on multivariate splines, in Approximation Theory: In Memory of A.K. Varma (N.K. Govil etc. Ed.), pp.265-291, Marcel Dekker, Inc., New York, NY, 1998.
• Some smoothness conditions and conformality conditions for quartic and quintic splines (with Dr. H.W. Liu at the University of Wollongong, Australia) has been accepted for publication by an Italian math research journal CALCOLO: A Quarterly on Numerical Analysis and Theory of Computation.

• Dr. Debra Knisley was invited to participate in DIMACS Institute of Graph Theory and Combinatorial Optimization workshop held at Rutgers University, July 20-July 26, 1998. The week-long workshop focused on cycles and paths in graphs.
• An invited paper, "Uniform Maximum Degree and Edge Density," by Dr. Debra Knisley, was published in a collection of works to commemorate Paul Erdos. The collected works was published by the Summer School and International Conference on Combinatorics and World Scientific Publishing Co.

• Teresa Haynes and Debra Knisley received an ETSU Research and Development Grant "Total Irredundance in Graphs," $6,000.
• Dr. Robert Gardner has had the following papers published or accepted in research journals:

• Bicyclic Directed Triple Systems, Ars Combinatoria, 49, 249-257
• Some Generalizations of the Enestrom-Kakeya Theorem, with N. K. Govil, Acta Mathematica Hungarica, 74(1-2),125-134.
• Functions of Exponential Type Not Vanishing in a Half-Plane," with N. K. Govil, Analysis, 17, 395-402.
• A Bernstein Type L^p Inequality for a Certain Class of Polynomials, with A. Weems, Journal of Mathematical Analysis and its Applications, 219, 472-478.
• Bicyclic Decompositions of Ku into copies of K3 U {e}, Utilitas Mathematica, (to appear).
• Bicyclic Decompositions of the Complete Digraph into Orientations of a 4-Cycle, with G. Coker, P. Torbett and A.Weems, Journal of Combinatorial Mathematics and Combinatorial Computing, (to appear).
• Triple Systems from Mixed Graphs, Bulletin of the Institute of Combinatorics and its Applications, (to appear).
• Dr. Gardner has also edited the book Logic, Probability and the Sciences; Poznan Studies in the Philosophy of Science, Book Series, with N. Shanks, Rodopi, Amsterdam/ Atlanta, 1998, which is to appear.

• Dr. Janice Huang gave a series of lectures at eight different universities in China on "Using the TI-92 in Teaching Mathematics" during July and August, 1998.

• She gave a presentation entitled "The Block Schedule: How Are We Doing?" at the National NCTM meeting in Washington, D.C., in April 1998. Her research results on how the block schedule is affecting the learning of mathematics will be published in the Winter Issue of the Virginia Mathematics Teacher.
• Janice attended a Mathematics Teacher Educator Short Course on Teachers Teaching with Technology (T3) at North Georgia College & State University in June of 1998.
• Dr. Huang is currently the President-Elect of the Tennessee Mathematics Teachers Association (TMTA) and will be installed as President on May 1, 1999.

• Dr. Teresa Haynes has had the following papers published or accepted in research journals:

• Efficient and excess domination in graphs, J. Combin. Math. Combin. Comput. 26 (1998) 83-96 (with Frank Harary and Peter J. Slater).
• Nordhaus-Gaddum type results for the domatic number of a graph, Graph Theory, Combinatorics, and Applications John Wiley & Sons, Inc. Vol. 1 (1998) 303-312 (with Jean E. Dunbar and Michael Henning).
• Bondage, Insensitiviy and Reinforcement, Domination in Graphs: Advanced Topics, Chapter 17, eds. Haynes, Hedetniemi, and Slater (1998) (with Jean Dunbar, Erich Teschner, and Lutz Volkmann). Book Chapter.
• The domatic number of a graph and its complement, Congr. Numer. 126 (1997) 53-63 (with Jean Dunbar and Michael Henning).
• Independence, domination, and uniform maximum degree, Graph theory, Combinatorics, and Applications, John Wiley & Sons, Inc. Vol. 1 (1998) 291-301 (with Jean Dunbar, Lisa Markus, Gayla Domke, and Debra Knisley).
• Double domination in graphs, to appear in Ars. Combin. (with Frank Harary).
• The codomatic number of a cubic graph, to appear in J. Combin. Math. Combin. Comput. (with Jean E. Dunbar and Michael Henning).
• Paired-domination in graphs, to appear in Networks (with Peter Slater).
• Domination critical graphs with respect to the relative complement, to appear in Austral. J. Combin. (with Michael Henning).
• Domatic numbers of factors of a graph, to appear in Ars Combin. (with Michael Henning).
• Perfect neighborhoods in graphs, to appear in Discrete Math. (with Gerd Fricke, Sandra Hedetniemi, Stephen Hedetniemi, and Michael Henning).
• Total domination edge critical graphs, to appear in Utilitas Math. (with Lucas van der Merwe and Christina Mynhardt).
• Criticality index of total domination, to appear in Congr. Numer. (with Lucas van der Merwe and Christina Mynhardt).
• Nordhaus-Gaddum bounds for domination, an invited paper to appear in A Commemorative Issue of SSICC for Professor Paul Erdös, World Scientific Publishing Co. (1998) (with Jean Dunbar and Stephen Hedetniemi).
• Induced-paired domination to appear in Ars Combin. (with Linda Lawson and Dan Studer).

Kappa Mu Epsilon (KME) Mathematics Honor Society recently initiated the following 15 new members during spring 1998:

Glenn Bigham, Jeff Bowles, Rachel Callahan, Robert Dautermann III, Rachel DeHart, Shannon Gosnell, Rebecca Hammons. Angela Harris, Susan Hosler, Justin Hyder, Jason Moore, Kenneth Proffitt, John Reynolds, Tara Templeton, and Scott Wilson.

The chapter currently has 20 members who are as follows: Robert Dautermann (President), Shannon Gosnell (Vice President) and Kim Tierney (Secretary-Treasurer), Trea Adams, Donny Belcher, David Bowles, Amy Crain, Rachel DeHart, Uriah Fletcher, Renee Gaunt, Angela Harris, Justin Hyder, Ronea James, Jobriath Kauffman, Jonathan Oakley, Kenneth Profitt, Michale Smith, Tabitha Taylor, Dewey Whaley, and Scott Wilson. The group meets twice a month, both on and off campus, to plan upcoming activities and enjoy the fellowship. Cici's Pizza was one of the more popular off-campus meeting places for spring of 1998.

Nine members attended the Southeast regional meeting of the MAA held at the College of Charleston in April of 1998. The chapter also was represented at the Mathfest held at Toronto, Canada by Kenny Proffitt. As a member of the chapter, Kenny was awarded a travel grant from the MAA. Kenny presented his research results from his summer research project under the direction of Dr. Jeff Knisley. The work was in fulfillment of his McNair research award.

Math Awareness Week was promoted by the members. Students from the chapter visited math classes at University School and Tennessee High School. They were also invited to speak to the Mathematics Club at Science Hill High School.

During the fall 1998 semester, the Chapter hosted the " Continuing Series Of Student Presentations ." Ken Proffitt was the September speaker and Jason Osborne was the October speaker. For the month of November, the chapter will help the department administer the Franklin Math Bowl to be held at ETSU in November.

Plans are being made to attend the regional meeting of the MAA which will be held at Rhodes College in March 1999.

An actuarial mathematics program is being developed in the Department of Mathematics. By emphasizing the mathematical topics of interest to Actuaries, this program will prepare students for careers in industries that use business modeling skills such as insurance companies, financial institutions, pension funds, governmental agencies, and consulting firms. Two courses, "Theory of Interest" and "Actuarial Mathematics I," have been developed and taught in 1997-98. A test center for exams of the Society of Actuaries has been arranged at ETSU since May 1998 and the exam center has been approved as a permanent one by the Society of Actuaries.

One student who graduated last May in the program received five interviews and two excellent job offers. Another student took four SOA exams last May and passed all of them with excellent scores. An SOA exam was administered on campus at the D. P. Culp center in November. The application deadline for the February SOA exams is January 15. The Actuarial Student Association (ASA) at ETSU has meetings on the first Monday of each month at Warf-Pickel 308 from 7:30 p.m. - 9 p.m.. The current president of ASA-ETSU is Renee Gaunt (zbrg5@etsu.edu).

For more information or recent activities of ETSU actuarial math program, please visit the web site www.etsu.edu/math/actuary

Dr. Janice Huang - Director

The MATH LAB has seen three different coordinators during the past two years. All three - Becca Jablonski, Chris Wallace, and Lucas van der Merwe - have done an outstanding job! The LAB increasingly is THE place to be while working on homework or studying for tests. It has become a focal point for students in mathematics courses. It also serves as a testing center for the department. The courses which give common tests, as well as several individual instructors, use the LAB to administer tests which have no time limit, thereby giving students with test anxiety a more relaxing atmosphere in which to take a test. The LAB is regularly involved in reaching out to students in the at-risk categories to help provide them with a support system and to encourage them in their efforts to be successful.

The graphics in the movies The Abyss and Terminator 2 are really a bit of mathematical magic: It is B-splines that derive the sophisticated computer graphics. In this note, we briefly discuss well-known Bernstein polynomials and properties, then Bèzier curves, and then B-splines.

The set of functions is also a basis of . It is called the Bernstein basis of the polynomial space.

For a function f defined on the closed interval [0, 1] the expression

is called the Bernstein polynomial of order n of the function f. The polynomial B_n(f, x) was introduced by S. Bernstein in 1912 to give an especially simple proof of the theorem of Weierstrass. For a continuous function f on [0, 1], B_n(f, x) converges to f(x) uniformly on [0, 1] as . However, the convergence rate of the Bernstein polynomials is very slow. Actually, if f is continuous on [a, b] and

is the modulus of continuity of f, then

On the other hand, Bernstein polynomials have very good shape preserving properties. For instance, if f is monotone non-decreasing, then . That is, B_n(f,x) is non-decreasing for any integer n > 0. If f is convex, then , which means B_n(f,x) is convex for any positive integer n, etc. In certain cases, B_n(f, x) may degenerate and become a polynomial of degree lower than n. It was shown in [7] that B_n(f, x) = B_n-1(f, x) if and only if f is a piecewise linear function. This result has been extended to a general s-dimensional setting in [1]. Because of their excellent geometric properties, Bernstein polynomials play an important role in Computer Aided Geometric Design (CAGD). Bèzier curves (surfaces), which are widely applied in curve or surface fitting in CAGD, are actually linear combinations of Bernstein basis functions. For given n+1 vectors b_k, k=0, 1, ..., n, the curve defined as
P(t) = for

is called a Bzier curve. Bzier curves have the following properties:

(1) P(t) interpolates two ending vectors, i.e., P(0) = b₀ and P(1) = b_n.

(2) Bèzier curves are bounded by the convex hull of the ''control vectors'' b_k, k=0, 1,..., n and preserve the convexity of the control polygon generated from the control vectors.

One can craft a Bèzier curve for any degree. However when fitting a curve through many points, one generally avoids high degree polynomials and instead uses some piecewise method of joining cubic pieces in order to produce curves that exhibit C¹ continuity.

Piecewise polynomials are called splines. The study of spline functions is one of the branches of approximation theory. Bernstein basis functions can be used to represent spline functions. Such a representation is called B-net of splines.

B-nets and some recent applications can be found in [5]. A spline with compact support usually is called a B-spline. Whenever curves and surfaces are represented mathematically, as they are in computer-aided design, analysis, and manufacturing, B-splines are the foundation of an efficient implementation. B-splines are especially important in the aircraft and automotive industries where shape is all important. Ray Sarraga of General Motors Research said '' Asking about the impact of B-splines in geometric design is like asking what the impact of the gasoline engine is for cars?'' Tom Grandine of the Boeing Company said '' No plane leaves Boeing without many billions of B-spline evaluations behind it. Splines demonstrate some of the good things that happen when you get the math right!'' (see [3]).

1. C. K. Chui, D. Hong, and S.T. Wu, On the degree of multivariate Bernstein polynomial operators, J. of Approx. Theory, 78(1994), 77-86. (MR 95j: 41023).

2. P.J. Davis, Interpolation and Approximation, Dover, New York, Second edition 1975.

3. Paul Davis, B-Splines and Geometric Design, SIAM News Volume 29/No. 5, 1996.

4. G.E. Farin, Curves and Surfaces for CAGD, Academic Press, Boston, Third edition 1992.

5. D. Hong, Recent progress on multivariate splines, in Approximation Theory: In Memory of A.K. Varma (N.K. Govil etc. Ed.), pp.265-291, Marcel Dekker, Inc, New York, NY, 1998.

6. G.G. Lorentz, Bernstein polynomials, Chelsea Publishing Company, New York, NY, Second edition 1986.

7. E. Passow, Deficient Bernstein polynomials, J. Approx. Theory 59 (1989), 282-285.

Please consider sending a contribution to the Mathematics Foundation Account so we might send a fall 1999 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 .

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 below or communicate electronically with George Poole, Chairman, Department of Mathematics, (POOLEG@ETSU.EDU).

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: ------------------------------------------------------------------------------

City: -----------------------------------------------------------------------------------------------------------

State: --------------------------------------------- Zip: -------------------------------

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.