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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 E₈ 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 q²² + 3,472 q²¹ + 38,791 q²⁰ + 293,021 q¹⁹ + 1,370,892 q¹⁸ + 4,067,059 q¹⁷ + 7,964,012 q¹⁶ + 11,159,003 q¹⁵ + 11,808,808 q¹⁴ + 9,859,915 q¹³ + 6,778,956 q¹² + 3,964,369 q¹¹ + 2,015,441 q¹⁰ + 906,567 q⁹ + 363,611 q⁸ + 129,820 q⁷ + 41,239 q⁶ + 11,426 q⁵ + 2,677 q⁴ + 492 q³ + 61 q² + 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 MP3-format 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: 152q²² + 3,472q²¹ + 38,791q²⁰ + 293,021q¹⁹ + 1,370,892q¹⁸ + 4,067,059q¹⁷ + 7,964,012q¹⁶ + 11,159,003q¹⁵ + 11,808,808q¹⁴ + 9,859,915q¹³ + 6,778,956q¹² + 3,964,369q¹¹ + 2,015,441q¹⁰ + 906,567q⁹ + 363,611q⁸ + 129,820q⁷ + 41,239q⁶ + 11,426q⁵ + 2,677q⁴ + 492q³ + 61q² + 3q

Value of this polynomial at q=1: 60,779,787

Polynomial with the largest value at 1 found so far: 1,583q²² + 18,668q²¹ + 127,878q²⁰ + 604,872q¹⁹ + 2,040,844q¹⁸ + 4,880,797q¹⁷ + 8,470,080q¹⁶ + 11,143,777q¹⁵ + 11,467,297q¹⁴ + 9,503,114q¹³ + 6,554,446q¹² + 3,862,269q¹¹ + 1,979,443q¹⁰ + 896,537q⁹ + 361,489q⁸ + 129,510q⁷ + 41,211q⁶ + 11,425q⁵ + 2,677q⁴ + 492q³ + 61q² + 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