Chaitanya's Random Pages

October 31, 2010

Breaking the 10 second barrier for 100 metres

Filed under: sport — ckrao @ 12:45 pm

It has been achieved 482 times by 74 athletes (counting electronic times up to 21 Sep 2010). Here are the people who have done it the most times.

Asafa Powell 65
Maurice Greene 52
Ato Boldon 28
Frank Fredericks 27
Tyson Gay 26
Usain Bolt 20
Donovan Bailey 16
Carl Lewis 15
Dennis Mitchell 12
Justin Gatlin 11
John Drummond 10
Leroy Burrell 9
Linford Christie 9
Shawn Crawford 9

Note: several performances of Tim Montgomery and Justin Gatlin have been disqualified due to doping charges.

Update: The following graph shows how the frequency of such performances has increased in recent years, with as many as 53 sub-10 second times in 2008.

References:
http://www.iaaf.org/statistics/toplists/inout=O/age=N/season=0/sex=M/all=y/legal=A/disc=100/detail.html

http://en.wikipedia.org/wiki/10-second_barrier

http://www.oocities.com/sprintingelite/sub10chrono.html

October 24, 2010

My Six Favourite Formulas – #6

Filed under: mathematics — ckrao @ 11:02 am

This is the most elementary of my favourite formulas. I think I first saw this as a Year 8 student and have been enthralled by it ever since.

\displaystyle \left(\sum_{n=1}^N n \right)^2 = \sum_{n=1}^N n^3

In words, the sum of the first N cubes is the square of the sum of the first N squares. Furthermore, both sides are equal to the square of the binomial coefficient \displaystyle \binom{N+1}{2} = \frac{N(N+1)}{2}.

The formula was discovered by Aryabhata of Patna around 1500 years ago, but it may have been known before then. The most elementary proof is probably one by mathematical induction:

Let S be the set of positive integers N for which \left(\sum_{n=1}^N n \right)^2 = \sum_{n=1}^N n^3. We wish to show that S is the set of all positive integers \mathbb{N}.  Firstly 1 \in S since for N=1, \text{LHS} = 1^2 = 1^3 = \text{RHS} . Assume k \in S. That is,

\displaystyle \left(\sum_{n=1}^k n \right)^2 = \sum_{n=1}^k n^3.

For convenience denote by T the sum \sum_{n=1}^k n = k(k+1)/2. Then

\begin{array}{lcl} \sum_{n=1}^{k+1} n^3 &=& (k+1)^3 + \sum_{n=1}^k n^3\\&=& (k+1)(k+1)^2 + T^2\quad \text{(by the inductive assumption)}\\ &=& T^2 + (k+1)[k(k+1) + (k+1)]\\&=&T^2 + (k+1)[2T + (k+1)]\\ &=& T^2 + 2T(k+1) + (k+1)^2\\&=& (T + (k + 1))^2\\ &=& \left(\sum_{n=1}^{k+1} n \right)^2.\end{array}

This shows that if k \in S, then k+1 \in S. Hence by the principle of mathematical induction, S = \mathbb{N} and we are done.

Next I will show the nicest proof that I am aware of (see p126 of [1] which also contains a proof by picture). We form the grid of numbers of the form ij for i and j from 1 to N (as you would see in the multiplication tables) and sum the numbers of the grid in two ways. Below is an example for the case N=5.

1 2 3 4 5
2 4 6 8 10
3 6 9 12 15
4 8 12 16 20
5 10 15 20 25

The easier sum is simply \displaystyle \sum_{i=1}^N \sum_{j=1}^N ij = \left(\sum_{i=1}^N i \right) \left(\sum_{j=1}^N j \right) = \left(\sum_{i=1}^N i \right)^2, the left side of the formula.

Secondly, we sum along L-shapes, an example of which is shown in bold below.

1 2 3 4 5
2 4 6 8 10
3 6 9 12 15
4 8 12 16 20
5 10 15 20 25

Note that each number in such an L-shape is a multiple of max(i,j), which ranges from 1 to N. The sum of the numbers in an L-shape is then this multiple times the sum of 1, 2, …, max(i,j)-1, max(i,j), max(i,j) – 1, …, 2, 1, which can easily be shown to be \max(i,j)^2 (think of counting the points of a square grid along diagonals). Hence the total is

\displaystyle 1.1^2 + 2.2^2 + \ldots N.N^2 = \sum_{n=1}^N n^3.

In other words,

\begin{array}{lcl} \sum_{i=1}^N \sum_{j=1}^N ij &=& \sum_{i=1}^N \sum_{j=1}^N \max(i,j) \min(i,j)\\&=& \sum_{n=1}^N n \sum_{\max(i,j) = n} \min(i,j)\\&=& \sum_{n=1}^N n \left(\min(n,n) + \sum_{i=1}^{n-1} i + \sum_{j=1}^{n-1} j\right)\\&=& \sum_{n=1}^N n \left(n + \sum_{i=1}^{n-1} i + \sum_{i=1}^{n-1} (n-i)\right)\\&=&\sum_{n=1}^N n \left(n + \sum_{i=1}^{n-1} n\right)\\&=& \sum_{n=1}^N n(n + n(n-1)) \\&=& \sum_{n=1}^N n^3.\end{array}

Reference:
[1] C. Alsina and R. Nelsen, “An Invitation to Proofs Without Words”, Eur. J. Pure Appl. Math, 3 (2010), 118-127, available here.

October 21, 2010

Success of AFL/VFL Teams

Filed under: sport — ckrao @ 11:36 am

Not by win-loss records, but number of times reaching the finals, grand finals etc. Colllingwood, Carlton and Essendon have been the traditional powerhouses, while it hurts me to see that the Bulldogs have reached the grand final as many times as Port Adelaide, and have never topped the ladder at the end of the home and away season. Stats collected via the fabulous AFL Tables website.

Team Seasons Premier Runner Up Minor Premier Finalist Last
Collingwood 114 15 25 18 77 2
Carlton 114 16 13 17 66 3
Essendon 112 16 14 17 62 4
Richmond 103 10 12 8 34 6
Melbourne 111 12 5 9 38 12
Geelong 111 8 9 13 50 5
Hawthorn 86 10 5 7 28 11
Sydney/South Melbourne 113 4 10 7 36 11
Fitzroy 100 8 5 4 29 8
North Melbourne 86 4 5 4 28 13
St Kilda 112 1 6 3 25 26
West Coast 24 3 2 3 17 1
Brisbane Bears/Lions 24 3 1 0 10 3
Adelaide 20 2 0 1 11 0
Port Adelaide 14 1 1 3 7 0
Western Bulldogs 86 1 1 0 24 4
University 7 0 0 0 0 4
Fremantle 16 0 0 0 3 1

October 14, 2010

My Six Favourite Formulas – #5

Filed under: mathematics — ckrao @ 10:08 am

Back to my favourite formulas, we look now at the one I would use more frequently than the others, the Gaussian integral:

\displaystyle \int_{-\infty}^{\infty} e^{-x^2}\ dx = \sqrt{\pi}

This integral of a bell-shaped function was first found by Laplace in 1782. Early the next century Gauss used the  normal distribution (having density function of the form \displaystyle ke^{-x^2})  in the context that if measurement errors obey this distribution, then the arithmetic mean of measurements is the optimal estimator in a least squares sense.

The most popular proof of the Gaussian integral is to express its square as a double integral (method due to Poisson) and then change to polar coordinates. If I denotes the integral, we have

\displaystyle I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2 + y^2)}\ dxdy.

Now introduce the change of variables (x,y) = (r \cos \theta, r \sin \theta), r > 0, 0 < \theta < 2\pi and

\begin{array}{lcl} dx dy & = & d(r \cos \theta) \wedge d(r \sin \theta)\\&=& (dr \cos \theta - r \sin \theta \ d\theta) \wedge (dr \sin \theta + r \cos \theta \ d \theta)\\&=&r \cos^2 \theta \ dr \wedge d\theta - r \sin^2 \theta \ d\theta \wedge dr\\&=&r(\cos^2 \theta + \sin^2 \theta)dr \wedge d\theta\\&=& r dr d\theta.\end{array}

(Here we are using the language of differential forms to evaluate the Jacobian determinant when changing variables, also used in a previous post here and alluded to here when first explaining differential forms in the generalised Stokes Theorem.)

Then

\begin{array}{lcl}I^2 &=& \int_0^{2\pi} \int_0^{\infty} r e^{-r^2}\ dr d\theta\\&=& \int_0^{2\pi}\left[-\frac{1}{2}e^{-r^2} \right]_0^{\infty}\ d\theta\\&=&\int_0^{2\pi}\frac{1}{2}\ d\theta\\&=&\pi,\end{array}

leading to the desired result. I have skipped the details related to the integral being improper, but that can be easily treated through limits.

To this day I find this result and derivation remarkable. The \pi comes in the answer because the Gaussian distribution is rotationally invariant. This distribution pops up in many applications ranging from modelling errors and noise (due to the central limit theorem) to the solution of the heat equation in physics.

October 4, 2010

Unusual weather in Tokyo, San Diego, Los Angeles and Mumbai

Filed under: climate and weather,geography — ckrao @ 1:00 pm

Many parts of Japan have had their warmest summer on record. Tokyo for instance had its average August temperature range of 27.0–33.5°C (long-term average 23.1–30.5°C). This was the highest ever average minimum (previous record 26.1°C in 1995) and second highest average maximum (behind 33.7°C also in 1995). Add to this a relative humidity in excess of 60% and I can hardly imagine how difficult it must have been to cope with.

On the other hand Southern California had one of its coolest summers on record. San Diego had its third coolest summer, while Los Angeles International Airport had its second coolest summer:

We see that for close to four months the maximum temperature only poked its head up above the average mark a handful of times. After this remarkable cool spell, on the 27th of September LAX recorded 40.6°C (105°F) while downtown Los Angeles had its hottest temperature on record – 45°C (113°F)!

Finally, Mumbai has had one of its wettest monsoons on record. Below is the cumulative rain total for its weather station at Santacruz (copied from here).

Update (Oct 6) – More information about the unusual northern hemisphere summer at Christopher C. Burt’s blog entry here. It points out there that this was in fact Japan’s hottest summer since records began, plus the hottest July and August for China since 1961.

I also found out that Australia had its wettest September on record – see the rainfall maps here:
http://www.bom.gov.au/climate/current/month/aus/archive/201009.summary.shtml

Data for the graphs came from weather.msn.com, http://www.ogimet.com, accuweather.com and the Japan Meteorological Agency.

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