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Stan # 88: How to be out…

August 2007

The answer
to the question: what are the odds of receiving four credit cards with the same
four digit PIN number is one in a trillion (approx). This is about 170 times
the number of people on the Planet.
Many of my
readers were obviously intrigued by the Lie group mentioned on my last Desk Top
and have demanded more information about the mysterious Lie Group 8. Well,
folks, here it is!
The result of the E_{8} calculation
is a matrix, or grid, with 453,060 rows and columns. There are 205,263,363,600
entries in the matrix, each of which is a polynomial. The largest entry in the
matrix is:
152
q^{22} + 3,472 q^{21} + 38,791 q^{20} + 293,021 q^{19}
+ 1,370,892 q^{18} + 4,067,059 q^{17} + 7,964,012 q^{16}
+ 11,159,003 q^{15} + 11,808,808 q^{14} + 9,859,915 q^{13}
+ 6,778,956 q^{12} + 3,964,369 q^{11} + 2,015,441 q^{10}
+ 906,567 q^{9} + 363,611 q^{8} + 129,820 q^{7} +
41,239 q^{6} + 11,426 q^{5} + 2,677 q^{4} + 492 q^{3}
+ 61 q^{2} + 3 q
If each
entry was written in a one inch square, then the entire matrix would measure
more than 7 miles on each side. Good job this question doesn’t turn up on an
exam paper. You wouldn’t have time to write the answer let alone solve the
problem.
Even with a
supercomputer it required very sophisticated mathematics and computer science
to carry out the calculation. The computation was completed on January 8, 2007.
Ultimately the computation took 77 hours of computer time, and 60 gigabytes to
store the answer in a highly compressed form.
This is a huge
amount of data. By way of comparison, a human genome can be stored in less than
one gigabyte. For a more down to earth comparison, 60 gigabytes is enough to
store 45 days of continuous music in MP3format which would probably result in
sore ears.
Some other facts about the answer:
Size of the
matrix: 453,060 Number of distinct polynomials: 1,181,642,979
Number of
coefficients in distinct polynomials: 13,721,641,221
Maximal
coefficient: 11,808,808
Polynomial
with the maximal coefficient: 152q^{22} + 3,472q^{21} + 38,791q^{20}
+ 293,021q^{19} + 1,370,892q^{18} + 4,067,059q^{17} +
7,964,012q^{16} + 11,159,003q^{15} + 11,808,808q^{14} +
9,859,915q^{13} + 6,778,956q^{12} + 3,964,369q^{11} +
2,015,441q^{10} + 906,567q^{9} + 363,611q^{8} +
129,820q^{7} + 41,239q^{6} + 11,426q^{5} + 2,677q^{4}
+ 492q^{3} + 61q^{2} + 3q
Value of
this polynomial at q=1: 60,779,787
Polynomial
with the largest value at 1 found so far: 1,583q^{22} + 18,668q^{21}
+ 127,878q^{20} + 604,872q^{19} + 2,040,844q^{18} +
4,880,797q^{17} + 8,470,080q^{16} + 11,143,777q^{15} +
11,467,297q^{14} + 9,503,114q^{13} + 6,554,446q^{12} +
3,862,269q^{11} + 1,979,443q^{10} + 896,537q^{9} +
361,489q^{8} + 129,510q^{7} + 41,211q^{6} + 11,425q^{5}
+ 2,677q^{4} + 492q^{3} + 61q^{2} + 3q
Value of
this polynomial at q=1: 62,098,473
If any
reader finds an error in my calculations, then please let me know.
To lighten
this Desk Top (for those who didn’t find the above light) there is a very
strange game called cricket which is played in Great Britain and a few of the
far flung corners of the old Empire such as Australia, South Africa and the
West Indies. There are ten ways to be ‘out’ which are: caught, bowled, leg
before wicket, stumped, run out, hit wicket, handling the ball, obstructing the
field, hit the ball twice and timed out. For those who are unfamiliar with
cricket, ‘out’ is not good if you are ‘in’.
There have
been some interesting ways of being out in recent years. The most recent
incident concerned Kevin Pieterson playing for England against the West Indies
who was hit on the head by a ball bowled by Dwayne Bravo or should I say hit on
the helmet which flew off his head and hit the wicket. To make things clear, it
was his helmet that came off and hit the wicket, not his head. Other strange
dismissals include Wayne Phillips playing in the Australia/England match in
1985 when he was caught by David Gower off Allan Lamb’s boot. The ball didn’t
hit the ground you see. Ian Botham playing for England against the West Indies
in 1991 attempted to pull a shot but fell backwards and tried in vain to step
over his stumps (three sticks of wood in the ground). He knocked off the bails
with a part of his anatomy which is a similar word which meant he was ‘out’
(not good).
Michael
Vaughan playing for England in 2001 missed a sweep shot, the ball struck his
pad and looped in the air. When it landed, he picked it up and was given out
for handling the ball. Muttiah Muralitharan playing for Sri Lanka against New
Zealand was given out when he stepped out of his crease to congratulate Kumar
Sangakkara on scoring a century and finally Graham Gooch (England against
Australia) defended a delivery from Merv Hughes which headed towards the stumps
and used his hand to brush it to safety but was given out (not good) because
the hand he used was not the one on the bat. Don’t these people know the rules?
Well, that’s just not cricket!
stan@adweb.co.uk