Mathematical Sciences: Infinite Electrical Networks

本课题涉及无穷多支路电网的基础理论研究。目前，在这一领域，未解决的问题远远多于已取得的成果。超限网络理论——通过将许多无限网络连接到无限端点而得到的网络——尚处于起步阶段，需要进一步发展。一种求解n维非均匀性线性无限网格的方法。2)是必需的。目前还没有有效的方法来分析无限非线性电阻网格。无限LRC网络也是如此，更不用说非线性网络了。这些问题和其他问题，如果得到解决，将有助于提高我们对无限网络的贫乏知识，但也将对其他几个领域产生影响，如外边值问题，离散谐波分析，黎曼曲面的分类和算子理论。

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.