An m x n cobweb network consists of n radial lines emanating from a center and connected by $m$ concentric n-sided polygons. A conjecture of Tan, Zhou and Yang for the resistance from center to perimeter of the cobweb is proved by extending the method used by the above authors to derive formulae for m = 1, 2 and 3 and general n. The resistance of an m x (s+t+1) fan network from the apex to a point on the boundary distant s from the corner is also found.
We study the corner-to-corner resistance of an M x N resistor network with resistors r and s in the two spatial directions, and obtain an asymptotic expansion of its exact expression for large M and N. For M = N, r = s =1, our result is R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6).
In light of the well-known fact that the $n$th divided difference of any polynomial of degree $m$ must be zero while $m<n$,the present paper proves the $(alpha,beta)$-inversion formula conjectured by Hsu and Ma [J. Math. Res. $&$ Exposition 25(4) (2005) 624]. As applications of $(alpha,beta)$-inversion, we not only recover some known matrix
We consider the problem of two-point resistance on an m x n cobweb network with a superconducting boundary, which is topologically equivalent to a geographic globe. We deduce a concise formula for the resistance between any two nodes on the globe using a method of direct summation pioneered by one of us [Z. Z. Tan, et al, J. Phys. A 46, 195202 (2013)]. This method contrasts the Laplacian matrix approach which is difficult to apply to the geometry of a globe. Our analysis gives the result directly as a single summation.
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}equiv (2q+2q^{-1}-1)[n]_{q^2}^4pmod{[n]_{q^2}^4Phi_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)cdots(1-aq^{k-1})$ for $kgeq 1$ and $Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.
J. W. Essam
,Zhi-Zhong Tan
,F. Y. Wu
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(2013)
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"Proof and extension of the resistance formula for an m x n cobweb network conjectured by Tan, Zhou and Yang"
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F. Y. Wu
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