For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-...For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.展开更多
This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness ...This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness and existence of solutions are proved by a new method.展开更多
文摘For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.
基金This research is supported by NSFC (No. 10471149)
文摘This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness and existence of solutions are proved by a new method.
