KeyNote Speaker Committee
  • Prof. Metin Demiralp
  • Prof. Metin Demiralp
  • Istanbul Technical University, Turkey
  • Title: Most Recent Status of Probabilistic Evolution (PET) Theory for Initial Value Problems of Explicit ODEs
  • Abstract:

    ODEs (Ordinary Differential Equations) are frequently used in many scientific and engineering problems arising in the modelling of practically encountered systems. The dynamic systems can be given as an example which finds a broad application area in this direction. The most concerned evolutions in the dynamic caharacterization of the systems appear to be the solution of explicit ODEs and accompanying initial conditions. The Probabilistic Evolution Theory (PET) has been developed basically in the last half decade through the studies of the Demiralp group (Group for Science and Methods of Computing). This theory constructs a denumerably infinite number of ODEs over the system vector’s (the vector composed of the unknown functions of the considered ODEs set) Kronecker powers. The accompanying initial values vector is replaced by a denumerably infinite vector whose vector blocks are the Kronecker powers of the initial vector. Thus, a homogeneous linear first order ODEs set with an infinite coefficient matrix composed of denumerably infinite number of rows and columns is constructed. We have called this infinite matrix Evolution Matrix because it produces all characteristics for the evolution of the system described by the unknown temporal functions of the given set of ODEs. All these mean that PET removes the nonlinearities from the ODEs even though they appear in the initial conditions. This theory pays the removal of the nonlinearities from the ODEs set by dealing with infinity. 


    The evolution matrix is in an upper block Hessenberg form and it can be replaced by an upper block triangular matrix form via extending the phase space (spanned by unknowns) of the system by deliberately adding new unknowns depending on the existing 1ones. Despite the triangularity in the evolution matrix facilitates the analysis pretty much it can be further simplified by conicalizing the system under consideration through further space extensions. Then the evolution matrix becomes a two adjacent upper block diagonal matrix. This can also be converted to a rather simple form where the blocks of the main diagonal become proportional to certain unit matrices. After having this form in our hands we can write the analytic solution of the infinite set of ODEs by using a two block term recursive ODE. The temporal structure of the result becomes quite simple and the vector nature of each summand is given as the image of a specified Kronecker power of the system’s initial vector under a specific rectangular matrix we call Telescope Matrix which are actually very sparse entities and their structures, which can be appeared awkward at the first glance, can be concisely processed via appropriate specific mathematical and algorithmic programming tricks.


    What we have stated until this point has been describing the recent status of PET and it was true until the beginning of October 2014. Since then there has been an important development to express the vector entities in summands of the expansion mentioned above in a more concise format. To this end the image of a Kronecker power of the initial vector under relevant telescope matrix could has been reexpressed as the image of the solely initial vector under a square matrix which is uniquely related to the telescope matrix under consideration. We call this most recent development Squarifying of Telescope Matrices. This is the most recent part of PET. The talk focuses on these issues as the time duration allows.

  • Biography: Metin Demiralp was born in Türkiye (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, ìstanbul 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 now he is continuing to do so. He has a group (Group for Science and Methods of Computing) in Informatics Institute of ìstanbul 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 in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in three years period 2010–2013 (with certain publications in journals and proceedings). Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 270 contributions together with various keynote, plenary, and, tutorial talks 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 was one of the principal members of Turkish Academy of Sciences since 1994. He has resigned on June 2012 because of the governmental decree changing the structure of the academy and putting politicial influence possibility by bringing a member assignation system. Metin Demiralp 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, and, Separate Nodes Ascending Derivatives Expansion.
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Conference:
April 14-16, 2015
Paper or Abstract Submission Due:
December 11, 2014  >> February 16, 2015
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