I like to work on a variety of different problems and enjoy collaboration with other scientists. Currently I am interested in the use of nilpotents in real algebraic geometry and the algebraic theory of inequalities.

Born January 1, 1933 in Regina, Saskatchewan, Canada

U.S. citizen (American-born father, Marshall Stengle, born August 18, 1910 in Chicago, Cook County, Illinois)

Ph.D. (Mathematics) University of Wisconsin 1961

M.S. (Mathematics) University of Wisconsin 1959

B.E.P. (Engineering Physics) Cornell 1954

1999-- Professor, Lehigh University

1998-1999 Research Professor, Universidad Complutense de Madrid

1. A divergence theorem for Gaussian stochastic process expectations. Journal of Mathematical Analysis and Applications. Vol. 21, No. 3, 537-546, (1968)< /P>

2. (with Samir Khabbaz) An application of K-theory to the global reduction of matrix-valued functions and differential equations. Proceedings of the U.S.-Jap an seminar on functional equations. Benjamin (1968)

3. Uniform asymptotic solution of second order differential equations without turning varieties. Mathematics of Computation. Vol. 23, No. 105, 1-22 (1969)

4. (with Samir Khabbaz) An application of K-theory to the global analysis of matrix-valued functions. Math. Ann. , 179, 115-122 (1969)

5. A new basis for asymptotic solution of differential equations containing one or several parameters. Trans. AMS, Vol. 156, 1-43 (1971)

6. (with Samir Khabbaz) Normal forms for analytic matrix-valued functions. Proceedings of the Liverpool Singularities Symposium, Springer-Verlag, New York, 3 4-57 (1971)

7. (with J. Narayan) Uniform asymptotic splitting of linear differential equations. Analytic Theory of Differential Equations. Springer-Verlag, New York, 170 -175 (1971)

8. A nullstellensatz and a positivstellensatz in semi-algebraic geometry. Math. Ann. 207, 87-97 (1974)

9. Asymptotic estimates for the adiabatic invariance of a simple oscillator.SIAM J. Math. Anal. Vol. 8, No. 4, August, 640-654 (1977)

10. Integral solution of Hilbert’s seventeenth problem. Math. Ann. 246, 33-39 (1979)

11. A lower bound for the complexity of separating functions. Rocky Mountain Journal 14 (4) 927-29 (1984)

12. Transcendental estimates for the adiabatic variation of linear Hamiltonian systems. SIAM J. Math. Anal. Vol 16, No. 5, 932-940 (1985)

13. (with J. Narayan) A nonlinear scattering problem. Annales de L’Institut Henri Poincaré, Vol. 3, No. 1, 1-55 (1986)

14. (with W. Schiesser) Second derivative formulas for use in the numerical method of lines. In: Advances in Computer Methods for Partial Differential Equati ons: ed. R Vichnevetsky and R. R. Stepleman, 281-282, IMACS (1987)

15. A relation between formal and analytic equivalence for linear holomorphic systems containing a parameter. Resultate der Mathematik, Vol. 13, 173-187 (198 8)

16. Weierstrass preparation of quadratic differentials. Aequationes Mathematicae 35, 45-54 (1988)

17. Chebyshev interpolation with approximate nodes of unrestricted multiplicity. Journal of Approximation Theory, Vol. 57, No. 1, 1-13 (1989)

18. A measure for semialgebraic sets related to Boolean complexity. Math. Ann. 283, 203-209 (1989)

19. (with Terry Delph) An analysis of the Hui-Riedel equation. International Journal of Fracture, 40, 295-305 (1989)

20. Reduction of linear systems to Arnold canonical form. Asymptotic Analysis 2, 345-355 (1989)

21. (with J. Yukich) Some new Vapnik-Chervonenkis classes. Ann. of Stat., Vol. 17, No. 4, 1441-1446 (1989)

22. (with L. Bröcker) On the Mostowski number. Math. Zeitschrift 203, 629-633 (1990)

23. Numerical methods for systems with measurable coefficients. Applied Mathematics Letters, Vol. 3, No. 4, 25-29 (1990)

24. Constraints on the angular distribution of the zeros of a polynomial of low complexity. Discrete and Computational Geometry, Vol. 6, No. 2, 163-169 (1991 )

25. Algebraic theory of differential inequalities. Communications in Algebra, Vol. 19, No. 6, 1743-1764 (1991)

26. Applications of the real spectrum to differential algebra. Algebraic Theories for Systems of Differential Equations Rutgers (1991)

27. (with B. Eisenberg and G. Strang) The asymptotic probability of a tie for first place. Annals of Applied Probability, Aug 01, vol. 3, no. 1 ,731-745 (199 3)

28. Estimates for the nonuniformity in the representation of a definite polynomial as a sum of fourth powers. Recent Advances in Real Algebraic Geometry and Quadratic Forms, Contemporary Mathematics Vol. 155, AMS (1993) 243-246

29. (with Hasan Yousef) The spherical spectrum of a graded ring. Geometriae Dedicata (1994) 195-208

30. (with Rob Kusner) Computing the Mobius Energy of Torus Knots by Contour Integration. Experimental Mathematics, Vol. 2, No. 1 (1993) 4

31. Error analysis of a randomized numerical method, Numerische Mathematik, Vol. 27, (1995) 1- 15

32. (with Y. Baryshnikov and B. Eisenberg) A necessary and sufficient condition for the existence of the limiting probability of a tie for first place. Proba bility and Statistics Letters, 23 , 203-209 (1995)

33. (with B. Eisenberg) Minimizing the probability of a tie. Journal of Mathematical Analysis and Applications 198, 458-472 (1996)

34. Complexity estimates for the Schmudgen positivstellensatz. Journal of Complexity 12, 167-174 (1996)

35. (with J. McEnerney) The nonreduced order spectrum of a commutative ring. Revista Mathematica , Vol. 10, 251-275 (1997)

36. (with Pawel Hitczenko) Expected Number of distinct part sizes in arandom integer composition. to appear in Combinatorics, Probability and Computing.

Doctoral Students

Herbert Snyder, A Hypercomplex Function Theory Associated with Laplace'’s Equation

Gerard Cozzolino, Uniform Asymptotic Solution of Second Order Differential Eq uations with a Single Simple Turning Variety

Jack Narayan, Uniform Splitting of N-th Order Linear Dfifferential Equations Containing a Small Parameter

Steven Jeffrey Michaels, Probabilistic Investigation of Boundary Layer Phenomena in Singular Perturbation Problems

Donald Thompson, A Simultaneous Reduction and Comparison Technique for the Asymptotic Solution of a Third Order Differential Equation with Turning Varieties< /P>

Brian J. Kelley, An Analysis by Morse Theory of Admissible Domains for Higher Order Differential Equations

James McEnerney, Algebraic Theory of Semianalytic sets

Joseph Tanzini, Some Precision Asymptotics of Numerical Integration

Abdallah Badr, Gaussian Quadrature with Approximate Nodes of Unrestricted Multiplicity

Hasan Yousef, The Spherical Spectrum of a Graded Ring

Margaret Akata Dodson, Extended Runge-Kutta Monte-Carlo Methods

Elizabeth Kuehner Mauch, Representations of Schmuedgen Type for Semidefinite Functions