**Keynote Lecture
Direct Power Series for Multivaried Functions: Welcome to the Charming World of Kronecker Product**

**Professor Metin Demiralp**

Istanbul Technical University

Informatics Institute

Istanbul, TURKEY

E-mail: metin.demiralp@gmail.com

**Abstract: ** Multivariance is plague of plethora for applied mathematical or linear algebraical applications. For multivariate functions which are analytic in a region of the Cartesian product of its variables’ complex planes, Taylor and/or Maclaurin series can be perhaps the most universal tools even though some other more generalized function series and the sums composed of finitely many terms like in high dimensional model representation (HDMR) or enhanced multivariance product representation (EMPR) can also be used to this end. However, the formulae in these types of approaches contain many indices and complicated structures for practical utilizations despite they may be considered worth of theoretical approaches like theorem proofs. Hence, to find a way for conciseness gains a lot of importance.

Perhaps the most prominent preference for the conciseness is the use of Kronecker power of vectors and/or matrices in ordinary linear algebra. Kronecker power is somehow a specific form a more general and rather abstract operation, direct product. We generally use the phrase “Direct Product” and “Kronecker Product” equivalently in linear algebra. Multiple products of a unique entity is called either “Kronecker Power”, or equivalently, “Direct Power”. Multivariate Taylor or Maclaurin series can be concisely represented in Kronecker powers of a state vector composed of independent variables with certain matrix coefficients. The matrix type of each term should be same even though the number of elements in state vector’s Kronecker powers changes from summand to summand. Hence the type of the each coefficient matrices from summand to summand should change to conserve the type of each summand.

We have recently developed a more general theory to solve the set of ODEs by using an infinite complete basis set of functions over unknowns. This led us to a denumerably infinite set of autonomous and homogenous ODEs. The infinite constant coefficient matrix is called “Evolution Matrix” because of its responsibility for the evolution of the system described by the original ODE set. We revealed that the initial infinite vector is composed of blocks which are indinvidually different Kronecker powers of a unique initial vector of finitely many elements. However, the constructed infinite ODE set does not need to have such a restrictive initial vector. It can be composed of completely different elements. This implies that the initial vector elements can be expressed of expectation values of certain Kronecker power blocks. Therefore initial vector definition may bring the probability concept to the theory. For this reason, we called this new concept “Probabilistic Evolution Approach (PEA)”.

PEA uses the Kronecker powers of certain vectors and their Kronecker products with some matrices and/or vectors. Despite the assistive properties of the Kronecker powers the formulae in PEA may necessitate further simplifications and shorthands to facilitate the analysis and its applications. To this end various properties of the Kronecker powers can be used. This talk focuses on this issue involving some original findings also.

**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, 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 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.