Plenary Lecture
Singing in the Magic Empire of Stars: Probabilistic Evolution Approach to Celestial Mechanical Problems
Professor Metin Demiralp
Resigned Principal Member of Turkish Academy of Sciences
Istanbul Technical University
Informatics Institute
Istanbul, TURKEY
E-mail: metin.demiralp@be.itu.edu.tr
Abstract: Many problems of Celestial Mechanics are not analytically solvable until now even though there have been many attempts and developments for numerical solutions to this end. Perhaps the mathematically speaking the most important problem has been the three body problem whose certain solutions were given under certain specific restrictions while series solutions like Unsold’s expansion and perturbation expansion proposed by Poincare were made available some time ago. We are going to focus on the three particle problem in this talk. The issue is just to obtain the governing ODEs and initial conditions and then try to construct the solution. This is very well known and available almost everywhere. However, the governing equations are quite nonlinear and singular making the construction of an analytic solution almost impossible. The series solutions have slow numerical convergence problems and certain accelerating methods can be used to cure this situation. Nevertheless many well-developed approximating techniques are available.
The basic reason making the problem almost unsolvable is the nonlinearity. On the other hand we have recently developed a new method to solve any kind of explicit ODE or ODE system accompanied by certain compatible conditions by using a basis set expansion in the space where the unknown solution of the equations lie. We called this approach “Probabilistic Evolution Approach” which produces a denumerably infinite number of ODEs and accompanying conditions which are called “Probabilistic Evolution Equations (PEEs)”. PEEs can be written in a single first order linear and homogeneous infinite vector ODE whose denumerably-infinite-element coefficient matrix is constant (we call it “Evolution Matrix due to basic characteristics of the solution it possesses).
The important issues are autonomy, triangularity and the conicality in the Evolution Matrix and they may not arise unless the structure of the descriptive functions appearing as right hand side functions in the original ODE(s) permit. This nonexistence can be converted to existence by using space extension via new appropriate unknown definitions. These extensions work well in the three particle problem of celestial mechanics and new equations can be constructed such that the descriptive functions are conical and vanish at certain points of the extended space. Talk will focus on these issues within the time limitations.
Brief Biography of the Speaker: Metin Demiralp was born in T¨urkiye (Turkey) on 4 May 1948. His education from elementary school to university was entirely in Turkey. He got his BS, MS degrees and PhD from the same institution, Ë™Istanbul Technical University. He was originally chemical engineer, however, through theoretical chemistry, applied mathematics, and computational science years he was mostly working on methodology for computational sciences and he is continuing to do so. He has a group (Group for Science and Methods of Computing) in Informatics Institute of Ë™Istanbul Technical University (he is the founder of this institute).
He collaborated with the Prof. Herschel A. Rabitz’s group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985–2003 after his 14 month long postdoctoral visit to the same group in 1979–1980. He was also (and still is) in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in last three years (with certain publications in journals and proceedings).
Metin Demiralp has more than 90 papers in well known and prestigious scientific journals, and, more than 200 contributions to the proceedings of various international conferences. He gave many invited talks in various prestigious scientific meetings and academic institutions. He has a good scientific reputation in his country and he is one of the principal members of Turkish Academy of Sciences since 1994. He is also a member of European Mathematical Society. He has also two important awards of turkish scientific establishments.
The important recent foci in research areas of Metin Demiralp can be roughly listed as follows: Probabilistic Evolution Method in Explicit ODE Solutions and in Quantum and Liouville Mechanics, Fluctuation Expansions in Matrix Representations, High Dimensional Model Representations, Space Extension Methods, Data Processing via Multivariate Analytical Tools, Multivariate Numerical Integration via New Efficient Approaches, Matrix Decompositions, Multiway Array Decompositions, Enhanced Multivariate Product Representations, Quantum Optimal Control.