Mathematical Sciences: Infinite Electrical Networks

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This project involves basic theoretical research on electrical networks having infinitely many branches. Presently there are far more unanswered questions than established results in this area. The theory of transfinite networks - networks obtained by connecting many infinite networks at their infinite extremities -is in its infancy and requires further development. A method for solving linear infinite grids with n-dimensional nonuniformities (n . 2) is needed. There is presently no efficient way of analyzing infinite nonlinear resistive grids. The same is true of infinite LRC networks, not to mention nonlinear ones. These and other problems, if solved, would contribute to the advancement of our meager knowledge of infinite networks, but will also have impact in several other fields, such as exterior boundary value problem, discrete harmonic analysis, classification of Riemann surfaces and operator theory.