scholarly journals The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

2021 ◽  
Author(s):  
Hiroyuki Miyoshi ◽  
Darren Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Science Literature ◽  
Multiply Connected ◽  
Schottky Double ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Genus One ◽  
Physical Quantities ◽  
Prime Function ◽  
Van Der Pauw Method

AbstractThe van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

scholarly journals Fay meets van der Pauw: the trisecant identity and the resistivity of holey samples

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200796
Author(s):  
Hiroyuki Miyoshi ◽  
Darren G. Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Outer Boundary ◽  
Recent Interest ◽  
Schottky Double ◽  
Explicit Formulae ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Prime Function ◽  
Made In ◽  
Van Der Pauw Method

The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this ‘4-point probe’ method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. This paper introduces two new mathematical tools to this area of investigation—the prime function on the Schottky double of a planar domain and the Fay trisecant identity—and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. We show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of ‘envelopes’ observed in the case of holey samples by previous authors. We find explicit formulae for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. We describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

scholarly journals Secondary Schottky–Klein prime functions associated with multiply connected planar domains

2015 ◽  
Vol 471 (2173) ◽  
pp. 20140688 ◽  
Author(s):  
Giovani L. Vasconcelos ◽  
Jonathan S. Marshall ◽  
Darren G. Crowdy
Keyword(s):  
Mixed Type ◽  
Compact Riemann Surface ◽  
Mathematical Framework ◽  
Multiply Connected Domains ◽  
Additional Function ◽  
Multiply Connected ◽  
Planar Domains ◽  
Recent Interest ◽  
Schottky Double ◽  
Prime Function

In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky–Klein (S–K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S–K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.

scholarly journals A VISIBILITY-BASED PURSUIT-EVASION PROBLEM

1999 ◽  
Vol 09 (04n05) ◽  
pp. 471-493 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JEAN-CLAUDE LATOMBE ◽  
STEVEN M. LAVALLE ◽  
DAVID LIN ◽  
RAJEEV MOTWANI
Keyword(s):  
Finite Graph ◽  
Initial Position ◽  
Moving Target ◽  
Multiply Connected ◽  
Pursuit Evasion ◽  
Minimum Number ◽  
Evasion Problem ◽  
Free Spaces ◽  
Simply Connected ◽  
Motion Strategy

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ( lg  n). For multiply-connected free spaces, the bound is [Formula: see text] pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.

scholarly journals The Extension of the Van Der Pauw Method to Anisotropic Media

Informatica ◽  
2007 ◽  
Vol 18 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jonas Kleiza ◽  
Mifodijus Sapagovas ◽  
Vytautas Kleiza
Keyword(s):  
Anisotropic Media ◽  
Van Der Pauw ◽  
Van Der Pauw Method

CONNECTIVITY PROPERTIES OF SIERPIŃSKI RELATIVES

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 481-506 ◽  
Author(s):  
T. D. TAYLOR
Keyword(s):  
Fractal Dimension ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.

Probing Coulomb Interactions on Charge Transport in Few‐Layer Organic Crystalline Semiconductors by the Gated van der Pauw Method

2020 ◽  
Vol 6 (7) ◽  
pp. 2000136
Author(s):  
Sai Jiang ◽  
Jun Qian ◽  
Qijing Wang ◽  
Yiwei Duan ◽  
Jianhang Guo ◽  
...  
Keyword(s):  
Charge Transport ◽  
Coulomb Interactions ◽  
Van Der Pauw ◽  
Van Der Pauw Method
Sign in / Sign up

Export Citation Format

Share Document