Keynote Lecture

Optimisation Based Spectral Issues for Specific Rectangular Matrices: Monocular Matrices and Kronecker Power Spaces

Professor Metin Demiralp
Informatics Institute
Istanbul Technical University


Abstract: Recent years have brought the analytic solution possibility to explicit ODEs whose right hand side function vectors can be converted to very specific forms which are just proportional to Kronecker square of the system’s unknown vector with a known rectangular constant matrix coefficient we call “Kernel Matrix” or “Generator Matrix ”. This can be done by using space extension concept which basically based on the appropriate definitions of new extra unknowns as long as the right hand side of the original ODEs permit (as long as multinomially closedness under the Lie operator of the system enables to do so). For a set of ODEs on n unknown temporal functions the Kernel or Generator Matrix is of the type nxn
2. Hence it is a rectangular matrix with n number of nxn type (square) matrix block elements. If there is an n element vector which is proportional to its Kronecker square’s image under the generator, then we can call it a “Characteristic Vector” of the generator matrix. Then the proportionality coefficient is called “Characteristic Value”. Characteristic Vector can also be called “Eigenvector” under the impression from the ordinary linear algebra. However, it is just conventionally valid but not exactly true. A generator matrix can also be composed of n number of outer products whose left vectors are the characteristic vectors while the right factors are the transpose of these characteristic vectors’ Kronecker squares in very specific circumstances. Then the determination of the characteristic vectors gain great importance. Such outer products can be called “Spectral Outer Products” because they recall true spectral outer products of square matrices. We call the generator matrices which are composed by only spectral outer products “Spectrally Decomposable Generators”. Thus, the determination of the characteristic vectors takes a quite important place in Probabilistic Evolution Theory (PREVTH). Spectrally Decomposable Generators are quite rarely encountered. However, it is possible to approximate a given generator by an unknown spectrally decomposable generator whose spectral outer product vectors and the linear combination coefficients of the so-called spectral expansion (the linear combination of the unknown spectral outer products with unknown scalar coefficients). Then the norm square of the difference between the target generator and this unknown spectrally decomposable generator is desired to be suppressed. In this form this is a quite formidable task. Hence, instead of n additive terms including spectrally decomposable generator we intend to use a single unknown spectral outer product multiplied by an unknown scalar as a starting point. After solving this problem a recursive scheme can be established. The talk will focus on various essential points of this method and certain technicalities.

Brief Biography of the Speaker: Metin Demiralp was born in Turkiye (Turkey) on 4 May 1948. His education history can be found in the web site through above links. We do not intend to repeat those informations here. 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 around 2010 (with certain publications in journals and proceedings). Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 260 contributions to the proceedings of various international conferences. He gave many invited talks in various prestigious scientific meetings and academic institutions. For further information the above given web address can be visited. 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. Metin Demiralp has been officially retired on May 4th 2015 due to age limitation in Turkey state universities. However, he is still an active and productive scientist and occupying a post-retirement position like emeritus professor in his same institution at the moment. He has same group as before, PhD and MS students in the computational science and engineering program and teaching sophisticated graduate courses presently.
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