Abstract: The multilinear array decomposition is an intensely investigated area today. Although it is mostly used for three index arrays some other higher dimensional applications are also encountered. There are different approaches to this end although the most preferable one is the singular value decomposition’s multilinear counterpart. It aims to decompose the multilinear array under consideration to a sum over outer products composed of more than two factors. The construction is based on the suppression of the Euclidean distance between the approximant and the target array by finding optimal values for the proposed unknown entities. The decomposition attempts to additively represent the target array in terms of lower rank arrays. This type methods present quite nonlinear problems as long as the products appearing in additive representation contain more than two unknown factors. This happens when the multilinear dimensionality (the number of the indexes) is at least three. However, by keeping the number of each product’s unknown factors in the representation equal to two one can use the standard linear algebraic spectral tools to determine the unknowns. The author and Emre Demiralp (author’s son, PhD student in cognitive neuroscience program of the Psychology Department in University of Michigan at Ann Arbor) started to deal with the development of such decomposition methods in last two years. The purpose was and still is to find some ways which bypasses certain technical and sometimes conceptual difficulties encountered in the employment of the standing methods. Their inspiration resources were basically high dimensional model representation which was developed in last two decades and the fluctuation free matrix representation as a recently developed efficient approximation tool. Their efforts take the fruits in certain applications and now new openings seem to be appearing in the horizon.
High dimensional model representation (HDMR) and its quite new extension, Enhanced Multivariance Product Representation (EMPR) developed by the author can also be used as a decomposition method if the target function is considered as a data set given on the nodes of an orthonormal hyperprismatic grid and discrete geometry is utilized.
HDMR can be considered as a particular case of an additive representation over the single factor products. The terms are ordered in ascending multivariance. EMPR, on the other hand, uses products, each of which contains same number of univariate factors within a one-to-one relation to the independent variables (indexes in the discrete case). However, this increasing number of factors is balanced by keeping the number of unknowns in each product just as 1 for easy determination. The given factors have certain common properties also and we call them “supports” since one can control the approximation quality even in the very crude cases of constant or univariate level truncations.
Speech focuses on certain details of these issues by referring the original findings of the author and his group.

Brief Biography of the Speaker:
Metin Demiralp was born in Turkey on 4 May 1948. His education from elementary school to university was entirely 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 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 months long postdoctoral visit to the same group in 1979–1980.
Metin Demiralp has more than 70 papers in well known and prestigious scientific journals, and, more than 110 contributions to the proceedings of various international conferences. He has given many invited talks in various prestigious scientific meetings and academic institutions. He has a good scientific reputation in his country and he is the full member of Turkish Academy of Sciences since 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.
The important recent focii in research areas of Metin Demiralp can be roughly listed as follows: Fluctuation Free Matrix Representations, High Dimensional Model Representations, Space Extension Methods, Data Processing via Multivariate Analytical Tools, Multivariate Numerical Integration via New Efficient Approaches, Matrix Decompositions, Quantum Optimal Control.