An application of algebraic topology: Kron’s method of tearing
Author:
J. Paul Roth
Journal:
Quart. Appl. Math. 17 (1959), 1-24
MSC:
Primary 65.00
DOI:
https://doi.org/10.1090/qam/104337
MathSciNet review:
104337
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H. B. Phillips and N. Wiener, Nets and the Dirichlet problem, J. Math. and Phys. 2, 105-124 (1923)
- L. A. Lyusternik, On electrical modelling of symmetric matrices, Uspehi Matem. Nauk (N.S.) 4 (1949), no. 2(30), 198–200 (Russian). MR 0031331
- Gabriel Kron, Numerical solution of ordinary and partial differential equations by means of equivalent circuits, J. Appl. Phys. 16 (1945), 172–186. MR 11808
- Gabriel Kron, Equivalent circuit of the field equations of Maxwell. I, Proc. I.R.E. 32 (1944), 289–299. MR 10695
- Gabriel Kron, Electric circuit models of the Schrödinger equation, Phys. Rev. (2) 67 (1945), 39–43. MR 11806
- Gabriel Kron, Equivalent circuits of the elastic field, J. Appl. Math. 11 (1944), A-149–A-161. MR 0011249
- Gabriel Kron, Equivalent circuits of compressible and incompressible fluid flow fields, J. Aeronaut. Sci. 12 (1945), 221–231. MR 11619
- Gabriel Kron, A set of principles to interconnect the solutions of physical systems, J. Appl. Phys. 24 (1953), 965–980. MR 60913
- Gabriel Kron, Solution of complex nonlinear plastic structures by the method of tearing, J. Aero. Sci. 23 (1956), 557–562. MR 81545
- R. v. Mises, On network methods in conformal mapping and in related problems, Construction and applications of conformal maps. Proceedings of a symposium, National Bureau of Standards, Appl. Math. Ser., No. 18, U. S. Government Printing Office, Washington, D. C., 1952, pp. 1–5. MR 0053616
- Garrett Birkhoff and J. B. Diaz, Non-linear network problems, Quart. Appl. Math. 13 (1956), 431–443. MR 77398, DOI https://doi.org/10.1090/S0033-569X-1956-77398-1
A. K. Oppenheim, Radiation analysis by the network method, Trans. ASME 78, 75-735 (1956)
- A. K. Oppenheim, The engineering radiation problem—an example of the interaction between engineering and mathematics, Z. Angew. Math. Mech. 36 (1956), 81–93 (English, with French, German and Russian summaries). MR 89707, DOI https://doi.org/10.1002/zamm.19560360302
F. H. Branin, Kron’s method of tearing and its applications, Proc. 2nd Midwest Symp. Circuit Theory, Mich. State U., 1956, p. 2.1-2.79
G. Kirchhoff, Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme führt wird, Poggendorf’s Ann. Physik u. Chemie 72, 497-508 (1847)
J. C. Maxwell, Electricity and Magnetism, vol. 1, p. 406, Oxford Univ. Press, 1892, 3rd edition
H. Weyl, Repartition de corriente et uno red conductora, Revista matematica, Hispano-Americana 5, 153-164 (1923)
- Beno Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex, Comment. Math. Helv. 17 (1945), 240–255 (German). MR 13318, DOI https://doi.org/10.1007/BF02566245
G. Kron, Tensor analysis of rotating machinery. I, Presented winter convention of AIEE (1933)
G. Kron, Non-Riemannian dynamics of rotating electrical machinery, J. Math, and Phys. 12, 103-194 (1934)
G. Kron, The application of tensors to rotating electrical machinery, General Electric Review, Schenectady, New York, 1938
G. Kron, Tensor analysis of networks, John Wiley & Sons, Inc., New York, 1939
G. Kron, Analyse tensorielle appliquée a l’Art de l’Ingénieur, Bulletin de l’Association des Ingénieurs Electriciens sortis de l’Institut Electrotechnique Montefiore, No. 9, Sept. 1936; No. 10, Oct. 1936; No. 1, Jan. 1937; No. 2, Feb. 1937
B. Hoffman, A mathematical interpretation of some work of Gabriel Kron, Mimeographed lectures delivered at Brown University, Nov. 12-13, 1943
- Banesh Hoffmann, Kron’s method of subspaces, Quart. Appl. Math. 2 (1944), 218–231. MR 11596, DOI https://doi.org/10.1090/S0033-569X-1944-11596-X
- Banesh Hoffmann, Kron’s non-Riemannian electrodynamics, Rev. Modern Physics 21 (1949), 535–540. MR 0034261, DOI https://doi.org/10.1103/revmodphys.21.535
- J. L. Synge, The fundamental theorem of electrical networks, Quart. Appl. Math. 9 (1951), 113–127. MR 42958, DOI https://doi.org/10.1090/S0033-569X-1951-42958-X
le Corbeiller, Matrix analysis of networks, John Wiley & Sons, Inc., New York, 1950
J. P. Roth, An application of algebraic topology to numerical analysis: Kron’s method I, Bull. Am. Math. Soc. 61, 183 (1955)
J. P. Roth, An application of algebraic topology to numerical analysis: Kron’s method II, Bull. Am. Math. Soc. 61, 183 (1955)
- J. P. Roth, An application of algebraic topology to numerical analysis: on the existence of a solution to the network problem, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 518–521. MR 74094, DOI https://doi.org/10.1073/pnas.41.7.518
- J. P. Roth, An application of algebraic topology to numerical analysis: on the existence of a solution to the network problem, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 518–521. MR 74094, DOI https://doi.org/10.1073/pnas.41.7.518
J. P. Roth, An application of algebraic topology to numerical analysis: III. Three theorems on networks, Bull. Am. Math. Soc. 62, 32 (1956)
H. Bode, Ein elektrisches Gerät zum Lösen von Systemen linearer Gleichungen, Z. angew. Math. u. Mech. 17, 213-223 (1937)
G. Kron, Electric circuit models for the vibration spectrum of polyatomic molecules, J. Chem. Phys. 14, 19-31 (1946)
- Gabriel Kron, Tearing and interconnecting as a form of transformation, Quart. Appl. Math. 13 (1955), 147–159. MR 73431, DOI https://doi.org/10.1090/S0033-569X-1955-73431-7
G. Kron, Electric-circuit models of the nuclear reactor, AIEE Trans. 1954
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952. MR 0050886
H. Cartan, Lectures on algebraic topology. Mimeographed lecture notes, Harvard University, 1949-1950
J. P. Roth, A method for finding the general solution to an arbitrary non-singular system of linear equations involving ${n^3}/2$ multiplications, The Institute for Advanced Study, ECP Tech. Rept. 56-01, Jan. 1956 (To appear in Math. tables and other aids to computation)
J. P. Roth, An application of algebraic topology to numerical analysis: IV. Tearing versus partitioning, Bull. Am. Math. Soc. 62, 45 (1956)
H. B. Phillips and N. Wiener, Nets and the Dirichlet problem, J. Math. and Phys. 2, 105-124 (1923)
L. A. Lyusternik, On electrical modelling of symmetric matrices, Uspekhi Matem. Nauk (N.S.) 41, 198-200 (1949)
G. Kron, Numerical solution of ordinary and partial differential equations by means of equivalent circuits, J. Appl. Mech. 16, 172-186 (1945)
G. Kron, Equivalent circuit of the field equations of Maxwell, Proc. IRE 32, 289-299 (1944)
G. Kron, Electric circuit models of the Schrödinger equation, Phys. Rev. 67, 39-43 (1945)
G. Kron, Equivalent circuits of the elastic field, J. Appl. Mech. 11, A149-161 (1944)
G. Kron, Equivalent circuits of compressible and incompressible fluid flow fields, J. Aero. Sci. 12, 221-231 (1945)
G. Kron, A set of principles to interconnect the solutions of physical systems, J. Appl. Phys. 24, 965-980 (1953)
G. Kron, Solution of complex non-linear plastic structures by the method of tearing, J. Aero. Sci. 23, 557-562 (1956)
R. von Mises, On network methods in conformal mapping and in related problems. Construction and application of conformal maps, U. S. Dept. of Commerce, Natl. Bur. Standards, Appl. Math. Ser. 18, 1-6 (1952)
G. D. Birkhoff and J. B. Diaz, Non-linear network problems, Quart. Appl. Math 13 (4) 431-443 (1956)
A. K. Oppenheim, Radiation analysis by the network method, Trans. ASME 78, 75-735 (1956)
A. K. Oppenheim, The engineering radiation problem—an example of the interaction between engineering and mathematics, Z. angew. Math. u. Mech. 36, 81-92 (1956)
F. H. Branin, Kron’s method of tearing and its applications, Proc. 2nd Midwest Symp. Circuit Theory, Mich. State U., 1956, p. 2.1-2.79
G. Kirchhoff, Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme führt wird, Poggendorf’s Ann. Physik u. Chemie 72, 497-508 (1847)
J. C. Maxwell, Electricity and Magnetism, vol. 1, p. 406, Oxford Univ. Press, 1892, 3rd edition
H. Weyl, Repartition de corriente et uno red conductora, Revista matematica, Hispano-Americana 5, 153-164 (1923)
B. Eckmann, Harmonische Funktionen und Randwert Aufgaben in einem Komplex, Comm. Math. Helvetici 17, 240-255 (1944-1945)
G. Kron, Tensor analysis of rotating machinery. I, Presented winter convention of AIEE (1933)
G. Kron, Non-Riemannian dynamics of rotating electrical machinery, J. Math, and Phys. 12, 103-194 (1934)
G. Kron, The application of tensors to rotating electrical machinery, General Electric Review, Schenectady, New York, 1938
G. Kron, Tensor analysis of networks, John Wiley & Sons, Inc., New York, 1939
G. Kron, Analyse tensorielle appliquée a l’Art de l’Ingénieur, Bulletin de l’Association des Ingénieurs Electriciens sortis de l’Institut Electrotechnique Montefiore, No. 9, Sept. 1936; No. 10, Oct. 1936; No. 1, Jan. 1937; No. 2, Feb. 1937
B. Hoffman, A mathematical interpretation of some work of Gabriel Kron, Mimeographed lectures delivered at Brown University, Nov. 12-13, 1943
B. Hoffman, Kron’s method of subspaces, Quart. Appl. Math. vol. 2, 218-231 (1944)
B. Hoffman, Kron’s non-Riemannian electrodynamics, Review Modern Physics 21, Commemorating Einstein’s 70th birthday 21, 535-540 (1949)
J. L. Synge, The fundamental theorem of electrical networks, Quart. Appl. Math. 9, 113-127 (1951)
le Corbeiller, Matrix analysis of networks, John Wiley & Sons, Inc., New York, 1950
J. P. Roth, An application of algebraic topology to numerical analysis: Kron’s method I, Bull. Am. Math. Soc. 61, 183 (1955)
J. P. Roth, An application of algebraic topology to numerical analysis: Kron’s method II, Bull. Am. Math. Soc. 61, 183 (1955)
J. P. Roth, An application of algebraic topology to numerical analysis: I. On the existence of a solution to the network problem, Proc. Natl. Acad. Sci. 41, 518-521 (1955)
J. P. Roth, An application of algebraic topology to numerical analysis: II. The validity of Kron’s method of tearing. Proc. Natl. Acad. Sci. 41, 599-600 (1955)
J. P. Roth, An application of algebraic topology to numerical analysis: III. Three theorems on networks, Bull. Am. Math. Soc. 62, 32 (1956)
H. Bode, Ein elektrisches Gerät zum Lösen von Systemen linearer Gleichungen, Z. angew. Math. u. Mech. 17, 213-223 (1937)
G. Kron, Electric circuit models for the vibration spectrum of polyatomic molecules, J. Chem. Phys. 14, 19-31 (1946)
G. Kron, Tearing and interconnecting as a form of transformation, Quart. Appl. Math. 13, 147-159 (1955)
G. Kron, Electric-circuit models of the nuclear reactor, AIEE Trans. 1954
S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N. J., 1952
H. Cartan, Lectures on algebraic topology. Mimeographed lecture notes, Harvard University, 1949-1950
J. P. Roth, A method for finding the general solution to an arbitrary non-singular system of linear equations involving ${n^3}/2$ multiplications, The Institute for Advanced Study, ECP Tech. Rept. 56-01, Jan. 1956 (To appear in Math. tables and other aids to computation)
J. P. Roth, An application of algebraic topology to numerical analysis: IV. Tearing versus partitioning, Bull. Am. Math. Soc. 62, 45 (1956)
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© Copyright 1959
American Mathematical Society