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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Effective Properties of Doubly Periodic Media - Ri
chard Craster (Imperial College London)
DTSTART;TZID=Europe/London:20170303T115500
DTEND;TZID=Europe/London:20170303T121500
UID:TALK71230AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/71230
DESCRIPTION:In his seminal paper "A theorem on the conductivit
y of a composite medium"\, J. Math. Phys\, 5\, 196
4 Joe Keller in just two pages elucidated fundamen
tal properties for the effective conductivities of
composite media\; the setting was in electrostati
cs but the application is far broader than this.
This\, together with a parallel Russian literature
dominated by Dykhne'\;s 1971 result and others
\, has provided the bedrock of much of the theory
of composite media\, at least in the static settin
g. It seems only fitting that Joe Keller'\;s co
ntribution to this area be highlighted.

A remarkably simple result is that of the squ
are infinite checkerboard\, so a plane compose of
black and white squares with each phase having a d
ifferent conductivity\; the effective conductivity
turns out to be geometric mean of the conductivit
ies. Surprisingly for doubly periodic tilings of t
he plane there are few other known exact solutions
. One case is that of the four phase checkerboard
\, so a square subdivided into four equal squares
that are then of different colours\, that then til
es the plane\; or more generally rectangles. In 1
985 Mortola and Steffe conjectured the result and
it was\, for a while a little controversial indeed
it was alleged to be wrong by\, I think\, Kozlov
who promptly passed away without saying why. In 20
01 myself and Obnosov proved the result and simult
aneously using a very different approach by Milton
. This talk\, using the same overheads as I used
in a talk attended by Joe Keller in 2000\, will de
scribe the area and hopefully provide some insight
to the area\, Keller'\;s contribution and some
reminiscences of the discussion I had with him on
this.

This work was co-authored with Yuri
Obnosov

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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