Keynote Lecture

Eigenspace Relations Amongst the Universal Matrices in Fluctuation Free Approximation Theory

Professor Metin Demiralp
Informatics Institute
Istanbul Technical University
ITU Bilisim Enstitusu Ayazaga Yerleskesi
Maslak, 34469, Istanbul, Turkey
E-mail: metin.demiralp@gmail.com

Abstract: Fluctuation free approximations are used everywhere matrix representation is directly or indirectly involved. The idea is simple: The matrix representation of a function operator, whose action on its operand is the multiplication by a function, is equivalent to the image of the matrix representations of the independent variables appearing in the argument of the function, under that function when all fluctuation terms are ignored. The fluctuation terms are matrix representations of certain operators involving a universal operator which is called the “fluctuation operator”. This operator, in fact, projects its operand, which should be taken from an appropriately defined Hilbert space of functions, to the complement of an appropriately chosen subspace of this Hilbert space. The basis functions spanning the subspace are the members of a subset of the full basis function set which spans the mother Hilbert space. The matrix representations of the independent variables are defined with respect to these functions and therefore their number of rows or columns is equivalent to the dimension of the subspace under consideration. The basis functions span a space of multivariate functions. Therefore there should be more than one independent variable operators each of which multiplies its operand by a different independent variable. The matrix representations of these operators are called “universal matrices” since they do not depend on the function mentioned in the fluctuation free approximation. The independent variable operators naturally commute by definition. However this may not imply the commutativity amongst their matrix representations. The commutativity serves us to find a unique eigenfunction set to spectrally decompose the matrix representation of each independent variable such that the decompositions’ projection matrices are constructed from this unique eigenfunction set while the linear combination coefficients vary from matrix representation to matrix representation. The case where the commutativity does not appear, one can conjecture that the norms of the commutators should decrease as the dimension of the subspace where the matrix representations are considered grows up to infinity. The fluctuation operator appears once or more than once in the structure of the commutators and tends to go to zero as the subspace dimension increases. This is the reason why the commutators should diminish as the dimension of the considered Hilbert subspace grows unboundedly. All these urge us to investigate the eigenspaces of the universal matrices. Each of these spaces is one dimensional if the corresponding eigenvalue has no multiplicity. Even the case of multiple eigenvalues does not prevent to orthogonally decompose the corresponding eigenspaces to one dimensional ones because of the symmetry in the universal matrices. Each of one dimensional spaces related to eigenspaces characterizes one axis in the considered subspace. The axes systems should be peculiar to the related universal matrix unless all of them commute. That is, they do not coincide to form a unique coordinate system. However, the angles between the coordinate axes corresponding to different universal matrices should diminish as the dimension of the considered Hilbert subspace grows unboundedly. This speech focuses on the issues roughly mentioned above in details as much as possible and tries to make comments and remarks on the possible pitfalls and misunderstandings. The talk sufficiently addresses to the related works emphasizing on the findings of the author’s and his group on this topics.

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.