A Thomson’s principle for infinite, nonlinear resistive networks
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- by L. De Michele and P. M. Soardi PDF
- Proc. Amer. Math. Soc. 109 (1990), 461-468 Request permission
Abstract:
Suppose $\Gamma$ is an infinite resistive electrical network with resistors of the form (2). We prove an existence and uniqueness theorem for the current generated by external current sources by establishing the analogue of Thomson’s principle in suitable modular sequence spaces.References
- Vaclav Dolezal, Hilbert networks. I, SIAM J. Control 12 (1974), 755–778. MR 0363688, DOI 10.1137/0312059
- Vaclav Dolezal, Generalized Hilbert networks, SIAM J. Control Optim. 14 (1976), no. 1, 26–41. MR 484789, DOI 10.1137/0314003
- Vaclav Dolezal, Hilbert networks. I, SIAM J. Control 12 (1974), 755–778. MR 0363688, DOI 10.1137/0312059
- R. J. Duffin, Nonlinear networks. IIa, Bull. Amer. Math. Soc. 53 (1947), 963–971. MR 22960, DOI 10.1090/S0002-9904-1947-08917-5
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Harley Flanders, Infinite networks. I: Resistive networks, IEEE Trans. Circuit Theory CT-18 (1971), 326–331. MR 275998, DOI 10.1109/TCT.1971.1083286 M. G. Krasnoselskii and Y. B. Rutickii, Convex functions and Orlicz spaces, Nordhoff, Groningen, 1961.
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
- G. J. Minty, Monotone networks, Proc. Roy. Soc. London Ser. A 257 (1960), 194–212. MR 120163, DOI 10.1098/rspa.1960.0144 P. M. Soardi and W. Woess, Uniqueness of currents in infinite electrical networks, Università di Milano, 1988, Discrete Appl. Math., in print.
- Carsten Thomassen, Resistances and currents in infinite electrical networks, J. Combin. Theory Ser. B 49 (1990), no. 1, 87–102. MR 1056821, DOI 10.1016/0095-8956(90)90065-8
- M. M. Vaĭnberg, Variatsionnyĭ metod i metod monotonnykh operatorov v teorii nelineĭnykh uravneniĭ, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0467427
- Joseph Y. T. Woo, On modular sequence spaces, Studia Math. 48 (1973), 271–289. MR 358289, DOI 10.4064/sm-48-3-271-289
- Armen H. Zemanian, Infinite electrical networks, Proc. IEEE 64 (1976), no. 1, 6–17. Recent trends in system theory. MR 0453371, DOI 10.1109/PROC.1976.10062
- A. H. Zemanian, The complete behavior of certain infinite networks under Kirchhoff’s node and loop laws, SIAM J. Appl. Math. 30 (1976), no. 2, 278–295. MR 396113, DOI 10.1137/0130029
- A. H. Zemanian, Countably infinite nonlinear time-varying active electrical networks, SIAM J. Math. Anal. 10 (1979), no. 5, 944–960. MR 541093, DOI 10.1137/0510088
- A. H. Zemanian, Nonlinear, resistive, countably infinite electrical networks, Applicable Anal. 8 (1978/79), no. 2, 185–192. MR 523955, DOI 10.1080/00036817808839226
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 461-468
- MSC: Primary 94C05; Secondary 46N05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1014643-1
- MathSciNet review: 1014643