Plenary Lecture

Recent Developments In The Fluctuation Expansion Of Univariate Functions’ Matrix Representations

Professor Metin Demiralp
Informatics Institute
Istanbul Technical University
ITU Bilisim Enstitusu Ayazaga Yerleskesi
Maslak, 34469, Istanbul, Turkey

Abstract: The matrix representations of univariate or multivariate functions play important roles in many mathematical applications of sciences and in many engineering problems. They are mostly employed to truncate infinite dimensional problems for approximations. The residual terms can be, in principle, suppressed as long as there is convergence which depends on the choice of the basis set of the Hilbert space constructed for the problem under consideration. The best basis set choice, of course, is the one which diagonalizes the matrix representation. However it is the main difficulty in these problems. Hence, a new way is needed to reflect the omitted terms’ contributions to the truncated matrix representation. This has been done, at least, in one way which is called “Fluctuation Expansion”. A very important practical fact is revealed through this new concept: “A truncated matrix representation of a univariate function can be efficiently approximated, within quite high precision, by a matrix which is the image of the independent variable’s same type truncated matrix representation under the considered univariate function”. This is called Fluctuationless Approximation.

The fluctuationless approximation can also be improved by adding correction terms which contain certain type universal matrices, fluctuation matrices. The construction of these terms was quite cumbersome and containing infinite series which cause new trun- cation errors. Our recent efforts have changed this undesirable structures to compact analytical ones by using Cauchy theorem of complex analysis through certain appropriate operator argumented contour integrals. The geometric series expansion of the kernels of these integrals, and, the separation to rather simple matrix inverses and the going back via Cauchy theorem again enable us to get compact formulae for the fluctuation involving correction terms. This presentation focuses on certain level details of this procedure.

Brief Biography of the speaker:
Metin Demiralp was born in Turkey on 4 May 1948. His education from elementary school to university was all in Turkey. He got his BS, MS, and PhD from the same institution, Istanbul Technical University. He was originally chemical engineer, however, through theoretical chemistry, applied mathematics, and computational science years he is working on methodology for computational sciences. He has a group (Group for Science and Methods of Computing) in Informatics Institute of Istanbul Technical University (he is the founder of this institute).

He collaborated with the Prof. H. A. Rabitz's group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985--2003 after his 14 months long postdoctoral visit to same group in 1979--1980.

Metin Demiralp has roughly 70 papers in well known scientific journals and is the full member of Turkish Academy of Sciences ince 1994. He is also a member of European Mathematical Society and the chief--editor of WSEAS Transactions on Mathematics currently. He has also two important awards of Turkish scientific establishments.