Plenary Lecture

The Electrical Circuits Theory and Groups of Algebraic Topology

Professor Milnikov Alexander
Faculty of Computer Technologies and Engineering
International Black Sea University
Department of Informatics and Control Systems
Technical University of Georgia

Abstract: The work is devoted to substantiation of the tensor point of view on the electric circuit that was first introduced in the classical works of G. Kron. It is proved that: geometrical structure of a circuit naturally generates groups of homologies and cohomologies. The conception of homologically and cohomologically independence was introduced. It is proved that the ranks of one-dimensional homologic group and the rank of the one-dimensional group of coboundaries are equal to the number of homologically independent loops and to the number of cohomologically independent nodes pairs, respectively. To the groups two pairs of vector spaces are assigned isomorphically. Also it is proved that the invariance of the input and output powers turns out to be a natural consequence of the topological nature of the circuit, which enables one to construct a tensor model of electric circuits: currents and voltages of the pure-loop and pure-node circuits can be regarded as the contravariant and covariant vectors of the conjugate spaces, and the passage from the primitive circuit to the pure-loop (pure-node) one is done by transforming the bases of the homological and cohomological spaces; the matrices of inductances, capacities, and conductances are of tensor nature, i.e., are the coordinate representations of the covariant and countervariant circuit invariants, and the kinetic and potential energies of the circuit are represented as the corresponding bilinear functionals.

Brief Biography of the Speaker: Prof. A. Milnikov holds a Bs/Ms in Electronic Engineering, a PhD in Electric Engineering (1978) and a Doctor of Sciences in Applied Mathematics (2002) from the Technical University of Georgia.
He works as a full professor of: International Black Sea University (Tbilisi, Georgia) (1994-present), Technical University of Georgia (1999-present), Georgian Academy of Scineces, Institute of Applied Mechanics (1980-present), Academician (1980-1988), Leading Academician (1989-2005), Principal Academician (2005-present). Also he worked as a Dean of the Computer Technologies and Engineering Faculty at the International Black Sea University (1994-2008). His research interests include: the Electrical Circuits Theory, Modern Geometry (Differential Geometry and Tensorial Analysis), Statistics, Random Processes, Signals Theory and Digital Processing, Filters Design. He has more than 80 publications, qmong them 10 in impact journals, 15 proceedings in WSEAS and Other International Conferences.