neverendingbooks – Page 202

sudoku mania

Published June 26, 2005 by lievenlb

I never pay
much attention to the crossword-puzzle page of our regular newspaper DeMorgen. I did notice that they
started a new sort of puzzle a few weeks ago but figured it had to be
some bingo-like stupidity. It wasn’t until last friday that I had a
look at the simple set of rules and I was immediately addicted (as I am
mostly when the rules are simple enough!). One is given a 9×9 grid
filled with numbers from 1 to 9. You have to fill in the full grid
making sure that each number appears just once on each _horizontal
line_, on each _vertical line_ and in each
of the indicated 3×3 subgrids!

It is amazing how quickly one learns
the basic tricks to solve such _sudoku_s. At first, one plays by
the horizontal-vertical rule trying to find forbidden positions for
certain numbers but rapidly one fails to make more progress. Then, it
takes a while before you realize that the empty squares on a given line
in a 3×3 subgrid cannot be filled with any of the numbers already
present in the 3×3 subgrid. Easy enough, but it takes your
sudoku-experience to the next level. Anther simple trick I found useful
it to keep track how many times (from 0 to 9) you have already filled
out a given number. If it is 9, you may as well forget about this number
for elimination purposes and if it is 0 it will be hard to use it.
Optimal numbers to use are those that are already 4 to 6 times on the
board. And so on, and so on.

After having traced all back-copies
of the newspaper I ran out of sudokus but fortunately there is a
neverending (sic!) supply of them on the web. For example, try out the
archive of Daily
Sudoku, and there are plenty of similar sites as, no doubt, you’ll
find by Googling.

An intruiging fact I learned from my newspaper
is that there are exactly 6,670,903,752,021,072,936,960 different
filled-out Sudoku grids. You then think : this should be easy enough to
prove using some simple combi- and factorials until you give this number
to Mathematica to factor it and find that it is

$2^{20} \\times
3^{8} \\times 5 \\times 7 \\times 27704267971$

and hence has a
pretty big unexplained prime factor! This fact needed clarification, so
a little bit later I found this Sodoku
players forum page and shortly afterwards an excellent (really
excellent) Wikipedia on
Sudoku. There is enough material on that page to keep you interested
for a while (e.g. the fact that nxn sudoku is NP-complete).

neverending.sty

Published June 23, 2005 by lievenlb

You don’t
know how lucky you are to be able to begin a LaTeX-document with
something like \documentclass[a4]{amsart}. This simple line
tells the machine what the size of paper is it has to print to, which
part of that paper has to be filled with text, what the design of
titles, sections etc. is (and probably more). Just convince yourself
(for the sake of argument) that you want to produce a readable
mathematical book on paper of dimensions say 6 by 9 inch. How would you
do this? Simple : all you have to do is to adjust certain TeX-parameters
the way you want! Well, that’s exactly what the _neverendingbooks
design department_ has been doing for the last couple of weeks!

The package we would use was easy enough to find. A long time ago, Geert suggested that we
should use the
memoir package. The fun starts the moment you are foolish enough to
print the manual : 300 pages! After an inspiring account of
book-printing basics over the ages, you are told that you have total
freedom to set your _stock paper size_, how it needs to be
_trimmed_ to get the final result, how you should designs
everything from the title, over abstract, acknowledgement, thanks, table
of contents contents, dedications etc. down to chapter, section and page
styles. In short : ULTIMATE FREEDOM!!!

But, as some of you may
know from experience, there is nothing more frightening to the moderate
autist (and frankly, are there any other mathematicians?) than ultimate
freedom! So, we set up a task-force, had daily brain-storm sessions,
produced numerous trial-prints and eventually came out with something
that came very close to the _better designed book_. Let’s face it
: can you name me _one_ (yeah right, just 1) well-designed
mathematical book? If you don’t believe me, browse through the recent
mathematics-books on amazon (as our design-department
did for a whole week-end, deprived of beverages and other pleasures).
Found anything? Yes?? Please, please let us know! On the other hand, if
you browse through the Art, architecture and
photography section you will spot several extremely good-looking
books very soon. Well, after a week we succeeded in designing the
_arty-farty-fanshi-wanshi_ (as PD1 would name it, jealous that
she was not in on the fun) mathematical book! And, what did we do with
the labours of all this hard work? NOTHING! We simply dropped the whole
idea (if you are a graphics-designer trying to survive within a
mathematics publishing firm (not easy we know, you have all our
sympathy) and want to do something more fun, contact us and if we can
reach an adequate financial agreement we will be happy to send you our
ideas).

So, what went wrong? Nothing really, it just dawned on
us that _NeverEndingBooks.org_ should not go for the
better-designed mathematics book. All our (potential) authors can
publish as easily at Oxford University
Press, the European Mathematical
Society or, if they have no moral objection, at the AMS. So, why
would they choose us instead of these more established publishing
houses? Just because our books look slightly (well let’s face it : a
lot) better? Probably not. We, at neverendingbooks.org should not go for
the better-designed book, we should not go for the book concept at all,
we should invent something entirely NEW & SEXY & USABLE & DESIRING & (I
hope you get the drift!). Next time, I’ll let you in on the first ideas
of our design-department!

the Klein stack

Published June 15, 2005 by lievenlb

Klein’s
quartic $X$is the smooth plane projective curve defined by
$x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical
objects around. For example, it is a Hurwitz curve meaning that the
finite group of symmetries (when the genus is at least two this group
can have at most $84(g-1)$ elements) is as large as possible, which in
the case of the quartic is $168$ and the group itself is the unique
simple group of that order, $G = PSL_2(\mathbb{F}_7)$ also known as
Klein\’s group. John Baez has written a [beautiful page](http://math.ucr.edu/home/baez/klein.html) on the Klein quartic and
its symmetries. Another useful source of information is a paper by Noam
Elkies [The Klein quartic in number theory](www.msri.org/publications/books/Book35/files/elkies.pd).
The quotient map $X \rightarrow X/G \simeq \mathbb{P}^1$ has three
branch points of orders $2,3,7$ in the points on $\mathbb{P}^1$ with
coordinates $1728,0,\infty$. These points correspond to the three
non-free $G$-orbits consisting resp. of $84,56$ and $24$ points.
Now, remove from $X$ a couple of $G$-orbits to obtain an affine open
subset $Y$ such that $G$ acts on its cordinate ring $\mathbb{C}[Y]$ and
form the Klein stack (or hereditary order) $\mathbb{C}[Y] \bigstar G$,
the skew group algebra. In case the open subset $Y$ contains all
non-free orbits, the [one quiver](www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) of this
qurve has the following shape $\xymatrix{\vtx{} \ar@/^/[dd] \\
\\ \vtx{} \ar@/^/[uu]} $ $\xymatrix{& \vtx{} \ar[ddl] & \\
& & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]} $ $\xymatrix{& &
\vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{}
\ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur]
&} $ Here, the three components correspond to the three
non-free orbits and the vertices correspond to the isoclasses of simple
$\mathbb{C}[Y] \bigstar G$ of dimension smaller than $168$. There are
two such of dimension $84$, three of dimension $56$ and seven of
dimension $24$ which I gave the non-imaginative names \’twins\’,
\’trinity\’ and \’the dwarfs\’. As we want to spice up later this
Klein stack to a larger group, we need to know the structure of these
exceptional simples as $G$-representations. Surely, someone must have
written a paper on the general problem of finding the $G$-structure of
simples of skew-group algebras $A \bigstar G$, so if you know a
reference please let me know. I used an old paper by Idun Reiten and
Christine Riedtmann to do this case (which is easier as the stabilizer
subgroups are cyclic and hence the induced representations of their
one-dimensionals correspond to the exceptional simples).

neverendingbooks Posts

sudoku mania

neverending.sty

the Klein stack