Keynote Lecture

Interpolative Aspects of Function Representations in Ascending Derivative Values

Professor Metin Demiralp
Istanbul Technical University
Informatics Institute

Abstract: Taylor series can be shown as a very well known example to function representations in ascending derivative values. Even though the most widely used ones are on univariate functions there exist very well designed formulae also on multivariate functions. The construction of Taylor series are based on the identity which dictates us that the integral of the derivative of a univariate function over an interval is equal to the difference between the function values evaluated on the greater and smaller interval endpoints respectively. This identity requires the continuity and the differentiability of the function at the focus, and, it seems to be sufficient to get the continuity and first order differentiability. However, this is not the case since this integral of derivative identity is not only required to be used for the target function but repetitiously for its all derivatives. This means that the basic requirement is not only continuity and differentiability but analyticity.
Taylor series are in fact the limiting form of the Taylor polynomials at infinite degree. These polynomials and a remainder integral defines the Taylor decomposition of a univariate function. In the case of multivariance a similar decomposition can be constructed between two points in a multidimensional Euclidean space. The convergence of the Taylor series is provided us when the integral remainder of the Taylor decomposition formula decreases towards zero as the polynomial degree grows unboundedly.
When exists a Taylor series is an infinite linear combination over an appropriate power basis set and the linear combination coefficients are the relevant function’s derivatives, evaluated at a common expansion point, divided by the factorial of the derivative order.
Quite recently, the presenter of this speech has used identity on the integral of the derivative of a function repetitiously but not on the same interval. Instead, the intervals whose one endpoints are at different nodal points while the other endpoints are located at the independent variable. This apparently changes everything. Each derivative value becomes evaluated at a different independent variable value while the basis set deviates from the power basis set to a polynomial basis set whose structure is completely determined by the nodal values. This approach has been called Separate Node Ascending Derivatives Expansion or as an acronym SNADE by Demiralp group. SNADE is now under an intense study in Demiralp’s so-called Group for Science and Methods of Computing. SNADE will be at one of the main foci of this presentation.
On the other hand, in a very recent attempt Demiralp has been able to show that Integral of Derivative Identity can be more generalized to cover more than one intervals by using interpolatory concepts. By having this more complicated identity it has been possible to use each derivative values at different set of nodes. This baby age approach has been called Separate Multinode Ascending Derivatives Expansion or in acronym format SMADE. A separate invited paper will also be presented in this conference to better explain SMADE.

Brief Biography of the Speaker: 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, ˙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 ˙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 100 papers in well known and prestigious scientific journals, and, more than 230 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.

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