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Tag: apple

Conway’s puzzle M(13)

Recently, I’ve been playing with the idea of writing a book for the general public. Its title is still unclear to me (though an idea might be “The disposable science”, better suggestions are of course wellcome) but I’ve fixed the subtitle as “Mathematics’ puzzling fall from grace”. The book’s concept is simple : I would consider the mathematical puzzles creating an hype over the last three centuries : the 14-15 puzzle for the 19th century, Rubik’s cube for the 20th century and, of course, Sudoku for the present century.

For each puzzle, I would describe its origin, the mathematics involved and how it can be used to solve the puzzle and, finally, what the differing quality of these puzzles tells us about mathematics’ changing standing in society over the period. Needless to say, the subtitle already gives away my point of view. The final part of the book would then be more optimistic. What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?

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the cartographers’ groups (2)

Fortunately,
there is a drastic shortcut to the general tree-argument of last time, due to
Roger Alperin. Recall that the Moebius
transformations corresponding to u resp. v send z resp. to

1z and 11z

whence the Moebius transformation
corresponding to v1 send z to 11z.

Consider
the set P of all positive irrational real numbers and the
set N of all negative irrational real numbers and observe
that

u(P)N and
v±(N)P

We have to show
that no alternating word w=(u)v±uv±uv±(u) in
u and v± can be the identity in PSL2(Z).

If the
length of w is odd then either w(P)N or w(N)P depending on whether w starts with a u or with
a v± term. Either way, this proves that no odd-length word can
be the identity element in PSL2(Z).

If the length of
the word w is even we can assume that w=v±uv±uv±u (if necessary, after conjugating with u we get to this form).

There are two subcases, either w=v1uv±uv±u in which case w(P)v1(N)
and this latter set is contained in the set of all positive irrational
real numbers which are strictly larger than one .

Or, w=vuv±uv±u in which case
w(P)v(N) and this set is contained in
the set of all positive irrational real numbers strictly smaller than
one
.

Either way, this shows that w cannot be the identity
morphism on P so cannot be the identity element in
PSL2(Z).
Hence we have proved that

PSL2(Z)=C2C3=u,v:u2=1=v3

A
description of SL2(Z) in terms of generators and relations
follows

SL2(Z)=U,V:U4=1=V6,U2=V3

It is not true that SL2(Z) is the free
product C4C6 as there is the extra relation U2=V3.

This relation says that the cyclic groups C4=U
and C6=V share a common subgroup C2=U2=V3 and this extra condition is expressed by saying that
SL2(Z) is the amalgamated free product of C4 with
C6, amalgamated over the common subgroup C2 and denoted
as

SL2(Z)=C4C2C6

More
generally, if G and H are finite groups, then the free product GH consists of all words of the form  (g1)h1g2h2g3gnhn(gn1) (so alternating between non-identity
elements of G and H) and the group-law is induced by concatenation
of words (and group-laws in either G or H when end terms are
elements in the same group).

For example, take the dihedral groups D4=U,R:U4=1=R2,(RU)2=1 and D6=V,S:V6=1=S2,(SV)2=1 then the free product can be expressed
as

D4D6=U,V,R,S:U4=1=V6=R2=S2=(RV)2=(RU)2

This almost fits in with
our obtained description of
GL2(Z)

GL2(Z)=U,V,R:U4=1=V6=R2=(RU)2=(RV)2,U2=V3

except for the
extra relations R=S and U2=V3 which express the fact that we
demand that D4 and D6 have the same subgroup

D2=U2=V3,S=R

So, again we can express these relations by
saying that GL2(Z) is the amalgamated free product of
the subgroups D4=U,R and D6=V,R, amalgamated over the common subgroup D2=C2×C2=U2=V3,R. We write

GL2(Z)=D4D2D6

Similarly (but a bit easier) for
PGL2(Z) we have

$PGL_2(\mathbb{Z}) = \langle u,v,R
u^2=v^3=1=R^2 = (Ru)^2 = (Rv)^2 \rangle $

which can be seen as
the amalgamated free product of D2=u,R with D3=v,R, amalgamated over the common subgroup C2=R and therefore

PGL2(Z)=D2C2D3

Now let us turn to congruence subgroups of
the modular group
.
With Γ(n) one denotes the kernel of the natural
surjection

PSL2(Z)PSL2(Z/nZ)

that is all elements represented by a matrix

[abcd]

such that a=d=1 (mod n) and b=c=0
(mod n). On the other hand Γ0(n) consists of elements
represented by matrices such that only c=0 (mod n). Both are finite
index subgroups of PSL2(Z).

As we have seen that
PSL2(Z)=C2C3 it follows from general facts
on free products that any finite index subgroup is of the
form

C2C2C2C3C3C3CCC

that is the
free product of k copies of C2, l copies of C3 and m copies
of C where it should be noted that k,l and m are allowed
to be zero. There is an elegant way to calculate explicit generators of
these factors for congruence subgroups, due to Ravi S. Kulkarni (An
Arithmetic-Geometric Method in the Study of the Subgroups of the Modular
Group , American Journal of Mathematics, Vol. 113, No. 6. (Dec.,
1991), pp. 1053-1133) which deserves another (non-course) post.

Using this method one finds that Γ0(2) is generated by
the Moebius transformations corresponding to the
matrices

X=[1101] and
Y=[1121]

and that
generators for Γ(2) are given by the
matrices

A=[1021]
and B=[1223]

Next,
one has to write these generators in terms of the generating matrices
u and v of PSL2(Z) and as we know all relations between
u and v the relations of these congruence subgroups will follow.

We
will give the details for Γ0(2) and leave you to figure out
that Γ(2)=CC (that is that
there are no relations between the matrices A and
B).

Calculate that X=v2u and that Y=vuv2. Because the
only relations between u and v are v3=1=u2 we see that Y is an
element of order two as Y2=vuv3uv2=v3=1 and that no power of
X can be the identity transformation.

But then also none of the
elements  (Y)Xi1YXi2YYXin(Y) can be the identity
(write it out as a word in u and v) whence,
indeed

Γ0(2)=CC2

In fact,
the group Γ0(2) is staring you in the face whenever you come to
this site. I fear I’ve never added proper acknowledgements for the
beautiful header-picture

so I’d better do it now. The picture is due to Helena
Verrill
and she has a
page with
more pictures. The header-picture depicts a way to get a fundamental
domain for the action of Γ0(2) on the upper half plane. Such a
fundamental domain consists of any choice of 6 tiles with different
colours (note that there are two shades of blue and green). Helena also
has a
Java-applet
to draw fundamental domains of more congruence subgroups.

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bookmarks tuesday cleanup


Geeky Mom : Why am I blogging?
. Been there before. Sooner or later
all non-pseudonomenous bloggers are faced with the same dilemmas.
There’s really no answer or advice to give except : blog when you feel
like it, if not do something different, after all its just one of those
billion of blogs around.

Texmaker : another
LaTeX-frontend, possibly having a few extras such as : a structure-pane
including labels you gave to formulas, theorems etc. (click on them
brings you to them). Intend to use it now as I’m in another rewrite of
the never-ending-book..

Microformats : “Designed for
humans first and machines second, microformats are a set of simple, open
data formats built upon existing and widely adopted standards.” May
have another look.

Quicksilver : a recurring
link. At times when I feel learning key-strokes may save me a lot of
time I have (another) go at Quicksilver. Last week, Ive reinstalled this
blog more or less post by post and used keystrokes to send a line in the
SQL-file of the database dump of NEB as a clipping to Scrivener to
MultiMarkdown it further. I used the app Service Scrubber
to define my own key-strokes. Must have another go at Quicksilver soon.
Im sure it distinguishes ‚”power mac users” from the rest of
us.


List of GTDTools
: a good list of GTD-software. I’m probably just
too chaotic for GTD to improve my workflow but somehow I cannot resist
trying some of these things out.

LifeDEV : One of those sites that tells
me I should take GTD more seriously

DoIt : One of
these GTD-tools. It is said to go well with Quicksilver, so maybe, one
day.

Think
: Here a little seemingly completely useless tool which works well (at
least for me). No, it does not make you think, but at least it helps you
while you are thinking (or doing anything a bit focussed). Install it
and enjoy! The principle is that it just blocks out all other open
windows (and there are keystrokes (yes, again) to get you quickly in
and out.) Besides, it looks great. It’s in my dock and this says it
all

Thinkature :
a brainstorming tool. Dont know why I did bookmark this. Perhaps one
day, a few years from now

Stafford Talk :
a talk by Toby Stafford I came across by accident. Maybe there are other
interesting talks on the site?

Science Scouts : a great
idea! Give yourself badges for how well you do science (or talk/write
about science). Have to collect my badges soon. I’m sure this only
works for people with a scouting-history, but who
knows?

MacResearch : Here’s a site
that may become useful. MacResearch.org is an open and independent
community for scientists using Mac OS X and related hardware in their
research. It is the mission of this site to cultivate a knowledgeable
and vibrant community of researchers to exchange ideas and information,
and collectively escalate the prominence of Apple technologies in the
scientific research community. They have some interesting articles
and tutorials on e.g. DevonThink and BibDesk etc. Worth to
revisit.

Jennifer in love : well‚ should I say something about this?
probably best not.


Breakthrough CLI
: another pamphlet in favor of the Command Line! A
must read for those who perfer GUIs to CLIs.

<

p>CLI – the
site
: Rod is working hard on CLI-20. Whenever he releases version
2.0, neverendingbooks will be among the first sites to run it. I still
love the idea.

Why do I bother? : an n-category post I got briefly interested in,
but was somehow flooded by professional
math-philosophers

Newton Legacy Reviewed : just that, a first review
on the next bookmark.

the Newton
Legacy
: a free online book, a murder mystery with a physics touch.
Perhaps this is the best investment of time/energy : write a popular
science book rather than another paper. Read half way through it (sorry
but not the best prose Ive read so far), may continue but was held up
reading a (real) murder mystery Equinox featuring also Newton and
alchemy (must be in the air somehow), also not the best mystery read
so far

Stalking with Googleearth
: no comment

(to be continued)

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