Plenary Lecture

Utilization of Fluctuationlessness Theorem in the Remainder Term of Taylor Polynomials

Professor N. A. Baykara
Marmara University, Mathematics Department
Istanbul, Turkey
E-mail: baki@be.itu.edu.tr

Abstract: The Fluctuationlessness approximation is based on a theorem which was conjectured and proven by M. Demiralp. This theorem states that the matrix representation of an algebraic operator which multiplies its argument by a scalar univariate function, is identical to the image of the independent variable's matrix representation over the same space via the same basis set, under that univariate function, when the uctuation terms are ignored. The same principle applies to multivariable functions too. In the first part of this lecture basic concepts of the Fluctuationlessness theorem are given and based on this theorem approximations to functions of a single variable as well as to multivariable functions are made by using the Taylor expansion with the remainder term expressed in integral form. Results are compared with those obtained from the corresponding Taylor series expansion without the error term. In the second part of the lecture, after using the Fluctuationlessness concept for approximations to univariate and multivariate functions by using the Taylor expansion with the explicit remainder term this approximate expression is integrated and a new quadrature-like numerical integration method is obtained. The results of numerical experiments are compared with the results obtained from the corresponding Taylor series expansion without the remainder term and errors are analyzed.

Brief Biography of the Speaker:
N. A. BAYKARA was born in Istanbul,Turkey on 29th July 1948. He received a B.Sc. degree in Chemistry from Bosphorous University in 1972. He obtained his PhD from Salford University, Greater Manchester, Lancashire,U.K. in 1977 with a thesis entitled "Studies in Self Consistent Field Molecular Orbital Theory". Between the years 1977-1981 and 1985-1990 he worked as a research scientist in the Applied Maths Department of The Scientific Research Council of Turkey. During the years 1981-1985 he did postdoctoral research in the Chemistry Department of Montreal University, Quebec, Canada. Since 1990 he is employed as a Staff member of Marmara University. He is now an Associate Professor of Applied Mathematics mainly teaching Numerical Analysis courses and is involved in HDMR research and is a member of Group for Science and Methods of Computing in Informatics Institute of Istanbul Technical University. Other research interests for him are "Density Functional Theory" and "Fluctuationlessness Theorem and its Applications" which he is actually involved in.