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May 4, 1998, 3:00:00 AM5/4/98

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How have quasi-periodic tilings of the plane been studied so far? Can

anyone give me some useful references to consult?

Thanks,

Marco

May 4, 1998, 3:00:00 AM5/4/98

to

On Mon, 4 May 1998, Marco de Innocentis wrote:

> How have quasi-periodic tilings of the plane been studied so far? Can

> anyone give me some useful references to consult?

A good place to start is the recent undergraduate level book

Author: Senechal, Marjorie.

Title: Quasicrystals and geometry / Marjorie Senechal.

Pub. Info.: Cambridge ; New York : Cambridge University Press, 1995.

LC Subject: Quasicrystals.

Crystallography-Mathematical.

This book has voluminous references to the research literature and is

available in paperback :-)

An out-of-date (only up to to 1977--- there was a ten year delay in

publication) but still useful account of the very early work on Penrose

and Ammann tilings may be found in

Author: Grunbaum, Branko.

Title: Tilings and patterns / Branko Grunbaum, G.C. Shephard.

Pub. Info.: New York : W.H. Freeman, c1987.

LC Subject: Tiling-Mathematics.

Be careful, Shephard's account of the "essential holes theorem" of Conway

(who never published or, according to what he told me, even wrote down his

work on this) is in some respects misleading. Also, I believe that the

empires reported by Ammann for the Penrose tiling systematically missed

some tiles.

For a totally different take on the "language" of Sturmian shifts (one

dimensional analogs of Sturmian tilings, aka generalized Penrose tilings)

see "Markov and SFT Approximations" on my page

http://www.math.washington.edu/~hillman/postings.html

and "Empires in Sturmian Systems" on my page

http://www.math.washington.edu/~hillman/papers.html

My PhD thesis, when it is (knock wood) finished, will have lots more

information about various aspects of Sturmian tilings and the symbolic

(multidimensional) Sturm-Robinson shifts determined by them. I can send

you a postscript file of the introduction if you're interested.

Hope this helps!

Chris Hillman

May 5, 1998, 3:00:00 AM5/5/98

to

Marco de Innocentis wrote:

>

> Can anyone give me some useful references to consult?

Try this page. It's interactive, so you can make your own Penrose

tiling.

http://www.geom.umn.edu/apps/quasitiler/start.html

Dag Torvanger

May 5, 1998, 3:00:00 AM5/5/98

to

Suppose that the operator Tf(x) = \int K(x,y)f(y)dy is continuous

on L^1(0,1). Is it true that

ess sup \int |K(x,y)|dx

{y \in (0,1)}

is equal ||T||?

I would appreciate any help in solving the problem or

pointing to any reference.

Barbara Wolnik

May 8, 1998, 3:00:00 AM5/8/98

to

>Suppose that the operator Tf(x) = \int K(x,y)f(y)dy is continuous

>on L^1(0,1). Is it true that

>

> ess sup \int |K(x,y)|dx

> {y \in (0,1)}

>

>is equal ||T||?

I guess you additionally assume that K is product-measurable

(otherwise one may use the continuum hypothesis to construct

some K such that T=0 but K(.,y)=1 a.e. for fixed y, see e.g.

Rudin, "Real and complex analysis").

Under this additional assumption, the answer is positive:

Indeed, since the departure space is L^1(0,1) (and the

image space is a perfect Banach ideal space) the operator

T is order-bounded, and for the operator |T| we have

|| T || = || |T| || (see e.g. Theorem 2.2.16(a) in the book

S.S. Kutateladze (ed.) "Vector lattices and integral operators").

In particular, we may replace K by |K| without changing the norm of

the integral operator. Thus, without loss of generality, we may

assume that K is nonnegative. Now observe that

|| T || = sup \int \int K(x,y)f(y) dy dx.

|| f ||_1 \le 1

and f\ge 0

By Fubini-Tonelli, we may swap the order of integration and get

the statement, using the well-known formula

|| g ||_\infty = sup \int g(y)f(y) dy

|| f ||_1 \le 1

(for nonnegative g the supremum needs to be taken only over

nonnegative f, of course).

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