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Recursion-transform method and potential formulae of the m×n cobweb and fan networks |
Zhi-Zhong Tan(谭志中) |
Department of Physics, Nantong University, Nantong 226019, China |
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Abstract In this paper, we made a new breakthrough, which proposes a new Recursion-Transform (RT) method with potential parameters to evaluate the nodal potential in arbitrary resistor networks. For the first time, we found the exact potential formulae of arbitrary m×n cobweb and fan networks by the RT method, and the potential formulae of infinite and semi-infinite networks are derived. As applications, a series of interesting corollaries of potential formulae are given by using the general formula, the equivalent resistance formula is deduced by using the potential formula, and we find a new trigonometric identity by comparing two equivalence results with different forms.
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Received: 14 May 2017
Revised: 07 June 2017
Accepted manuscript online:
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PACS:
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05.50.+q
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(Lattice theory and statistics)
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84.30.Bv
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(Circuit theory)
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89.20.Ff
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(Computer science and technology)
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41.20.Cv
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(Electrostatics; Poisson and Laplace equations, boundary-value problems)
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Fund: Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20161278). |
Corresponding Authors:
Zhi-Zhong Tan
E-mail: tanz@ntu.edu.cn,tanzzh@163.com
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Cite this article:
Zhi-Zhong Tan(谭志中) Recursion-transform method and potential formulae of the m×n cobweb and fan networks 2017 Chin. Phys. B 26 090503
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[1] |
Kirchhoff G 1847 Ann. Phys. Chem. 148 497
|
[2] |
Kirkpatrick S 1973 Rev. Mod. Phys. 45 574
|
[3] |
Pennetta C, Alfinito E, Reggiani L, Fantini F, DeMunari I and Scorzoni A 2004 Phys. Rev. B 70 174305
|
[4] |
Redner S 2001 A Guide to First-Passage Processes (Cambridge: Cambridge University Press)
|
[5] |
Katsura S and Inawashiro S 1971 J. Math. Phys. 12 1622
|
[6] |
Doyle P G and Snell J L 1984 Random Walks and Electrical Networks, The Mathematical Association of America, Washington, DC
|
[7] |
Novak L and Gibbons A 2009 Hybrid Graph Theory and Network Analysis (Cambridge: Cambridge University Press)
|
[8] |
Goodrich C P, Liu A J and Nagel S R 2012 Phys. Rev. Lett. 109 095704
|
[9] |
Izmailian N S, Priezzhev V B, Philippe R and Hu C K 2005 Phys. Rev. Lett. 95 260602
|
[10] |
Dobrosavljević V and Kotliar G 1997 Phys. Rev. Lett. 78 3943
|
[11] |
Vollhardt D, Byczuk K and Kollar M 2011 Springer Series in Solid-State Sciences 171 203
|
[12] |
Caracciolo R, De Pace A, Feshbach H and Molinari A 1998 Ann. Phys. 262 105
|
[13] |
Weinan E and Wang X P 2000 SIAM J. Numer. Sci. Comp. 38 1647
|
[14] |
Garcia-Cervera C J, Gimbutas Z and E W 2003 J. Comput. Phys. 184 37
|
[15] |
Lai M C and Wang W C 2002 Numer. Methods Partial Differ. Equ. 18 56
|
[16] |
Fornberg B 1996 A Practical Guide to Pseudo Spectral Methods (Cambridge: Cambridge University Press) p. 17
|
[17] |
Houstis E, Lynch R and Rice J 1978 J. Comput. Phys. 27 323
|
[18] |
Borges L and Daripa P 2001 J. Comput. Phys. 169 151
|
[19] |
Kamrin K and Koval G 2012 Phys. Rev. Lett. 108 178301
|
[20] |
Cserti J 2000 Am. J. Phys. 68 896
|
[21] |
Giordano S 2005 Int. J. Circ. Theor. Appl. 33 519
|
[22] |
Wu F Y 2004 J. Phys. A: Math. Gen. 37 6653
|
[23] |
Tzeng W J and Wu F Y 2006 J. Phys.A: Math. Gen. 39 8579
|
[24] |
Izmailian N S and Kenna R and Wu F Y 2014 J. Phys. A: Math. Theor. 47 035003
|
[25] |
Izmailian N S and Kenna R 2014 J. Stat. Mech. E 09 1742
|
[26] |
Tan Z Z 2015 Chin. Phys. B 24 020503
|
[27] |
Tan Z Z 2015 Phys. Rev. E 91 052122
|
[28] |
Tan Z Z 2015 Sci. Rep. 5 11266
|
[29] |
Tan Z Z, Zhou L and Yang J H 2013 J. Phys. A: Math. Theor. 46 195202
|
[30] |
Tan Z Z, Essam J W and Wu F Y 2014 Phys. Rev. E 90 012130
|
[31] |
Essam J W, Tan Z Z and Wu F Y 2014 Phys. Rev. E 90 032130
|
[32] |
Tan Z Z 2015 Int. J. Circ. Theor. Appl. 43 1687
|
[33] |
Essam J W, Izmailian N S, Kenna R and Tan Z Z 2015 Roy. Soc. Open Sci. 2 140420
|
[34] |
Tan Z Z 2016 Chin. Phys. B. 25 050504
|
[35] |
Tan Z Z 2017 Commun. Theor. Phys. 67 280
|
[36] |
Tan Z Z and Zhang Q H 2017 Acta Phys. Sin. 66 070501 (in Chinese)
|
[37] |
He Z W, Zhan M, Wang J X and Yao C G 2017 Fron. Phys. 12 120701
|
[38] |
Tyson J J, Chen K and Novak B 2001 Nat. Rev. Mol. Cell Biol. 2 908
|
[39] |
Xia Q Z Liu L L, Ye W M and Hu G 2011 New J. Phys. 13 083002
|
[40] |
Davidich M I and Bornholdt S 2008 PLoS ONE 3 e1672
|
[41] |
Davidich M I and Bornholdt S 2013 PLoS ONE 8 e71786
|
[42] |
Bornholdt S and Rohlf T 2000 Phys. Rev. Lett. 84 6114
|
[43] |
Davidich M and Bornholdt S 2008 J. Theor. Biol. 255 269
|
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