# Chaitanya's Random Pages

## October 31, 2012

### Kenenisa Bekele’s 10k runs

Filed under: sport — ckrao @ 12:11 pm

This year’s London Olympics was the first time the great Kenenisa Bekele did not win a 10,000m race in which he finished. The following is a list of his 10,000 track finals, taken from here (for anyone wondering, Haile Gebrselassie was undefeated from 1993 to 2001 in the 10,000m over 11 finals).

 Date Meet Venue Time Place 01/06/2002 Fanny Blankers-Koen Games Hengelo NED 26:53.7 1 IAAF World Outdoor Championships Paris FRA 26:49.6 1 (CR)* 08/06/2004 Golden Spike Ostrava CZE 26:20.3 1 (WR) 20/08/2004 XXVIII Olympic Games Athína GRE 27:05.1 1 (OR)** 29/05/2005 Fanny Blankers-Koen Games Hengelo NED 26:28.7 1 08/08/2005 IAAF World Outdoor Championships Helsinki FIN 27:08.3 1 Memorial Van Damme Bruxelles BEL 26:17.5 1 (WR) 27/08/2007 IAAF World Outdoor Championships Osaka JPN 27:05.9 1 14/09/2007 Memorial Van Damme Bruxelles BEL 26:46.2 1 Prefontaine Classic Eugene USA 26:26.0 1 17/08/2008 XXIX Olympic Games Beijing CHN 27:01.2 1 (OR) 17/08/2009 IAAF World Outdoor Championships Berlin GER 26:46.3 1 (CR) 28/08/2011 IAAF World Outdoor Championships Daegu KOR DNF – Memorial Van Damme Bruxelles BEL 26:43.2 1 UK Olympic Trials Birmingham GBR 27:02.6 1 04/08/2012 XXX Olympic Games London GBR 27:32.4 4

* he ran the last 5k in 12:57.24, raster than the winning time for the 5,000m at the same championships!

** 53.02s for the last lap!

According to this IAAF page he has four of the fastest six times in the event. In 2004-5, 2007-9 and 2011 he had the fastest time of the year. He also has the record for the most consecutive days holding the 5,000m and 10,000m records simultaneously (ref).

Watching his finishing kick has been a joy over the years (some of his final lap splits are here). Check out this instance in the 2007 World Championships where he was well behind with just over 200m remaining.

#### Reference

Unofficial Bekele

## October 30, 2012

### An interesting sum based on factorials

Filed under: mathematics — ckrao @ 11:40 am

This post is inspired by a question in this year’s University of Melbourne Maths Olympics. We wish to find sums of the following form.

$\displaystyle \frac{1}{1\times 2} + \frac{1}{2\times 3} + \frac{1}{3\times 4} + \ldots = \sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2)}$

$\displaystyle \frac{1}{1\times 2 \times 3} + \frac{1}{2\times 3 \times 4} + \frac{1}{3\times 4 \times 5} + \ldots = \sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2)(i+3)}$

More generally, how do we find

$\displaystyle \frac{1}{1 \times 2 \times \ldots \times k} + \frac{1}{2 \times 3 \times \ldots \times (k+1)} + \frac{1}{3 \times 4 \times \ldots \times (k+2)} + \ldots = \sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2) \ldots (i + k)}\text{?}$

The first sum is familiar enough to me – writing $\displaystyle \frac{1}{(i+1)(i+2)}$ as $\displaystyle \frac{1}{i+1} - \frac{1}{i+2}$ the sum telescopes:

\begin{aligned} \sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2)} &= \lim_{N \rightarrow \infty} \sum_{i=0}^N \frac{1}{(i+1)(i+2)}\\ &= \lim_{N \rightarrow \infty} \sum_{i=0}^N \left( \frac{1}{i+1} - \frac{1}{i+2}\right )\\ &= \lim_{N \rightarrow \infty} \sum_{i=0}^N \frac{1}{i+1} - \sum_{i=1}^{N+1} \frac{1}{i+1}\\ &= \lim_{N \rightarrow \infty} \frac{1}{2} - \frac{1}{N+2} \\ &= \frac{1}{2}.\end{aligned}

To do the next sum I thought one could use the partial fraction expansion

$\displaystyle \frac{1}{(i+1)(i+2)(i+3)} = \frac{1/2}{i+1} - \frac{1}{i+2} + \frac{1/2}{i+3}$

and continue as in the previous case. While this is doable, it is not easily generalisable. It is simpler to write

$\displaystyle \frac{1}{(i+1)(i+2)(i+3)} = \frac{1}{2}\left( \frac{1}{(i+1)(i+2)} - \frac{1}{(i+2)(i+3)} \right).$

In the general case, we write

\begin{aligned} \frac{1}{(i+1)(i+2) \ldots (i + k)} &= \frac{1}{(i+2)\ldots (i+k-1)} \left( \frac{1}{(i+1)(i+k)} \right)\\ &= \frac{1}{(i+2)\ldots (i+k-1)} \cdot \frac{1}{(k-1)}\left( \frac{1}{(i+1)} - \frac{1}{(i+k)} \right)\\ &= \frac{1}{k-1} \left( \frac{1}{(i+1)(i+2)\ldots (i+k-1)} - \frac{1}{(i+2)(i+3)\ldots (i+k)} \right), \end{aligned}

\begin{aligned} \sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2) \ldots (i + k)} &= \lim_{N \rightarrow \infty} \sum_{i=0}^N \frac{1}{(i+1)(i+2) \ldots (i + k)}\\ &= \lim_{N \rightarrow \infty} \frac{1}{k-1} \sum_{i=0}^N \left( \frac{1}{(i+1)(i+2)\ldots (i+k-1)} - \frac{1}{(i+2)(i+3)\ldots (i+k)} \right) \\ &= \lim_{N \rightarrow \infty} \frac{1}{k-1} \left( \sum_{i=0}^N \frac{1}{(i+1)(i+2) \ldots (i+k-1)} - \sum_{i=1}^{N+1} \frac{1}{(i+1)(i+2) \ldots (i + k - 1)} \right) \\ &= \lim_{N \rightarrow \infty} \frac{1}{k-1} \left( \frac{1}{1 \times 2 \times \ldots \times (k-1) } - \frac{1}{(N+2)(N+3) \ldots (N+k) } \right) \\ &= \frac{1}{(k-1) \times (k-1)!}. \end{aligned}

The reason for the title of the post is this sum can also be written as

$\displaystyle \sum_{i=0}^{\infty} \frac{i!}{(i+k)!} = \frac{1}{(k-1) \times (k-1)!}, \quad k \geq 2.$

Multiplying both sides by $k!$ we also obtain the following interesting infinite sum involving reciprocals of binomial coefficients:

$\displaystyle \sum_{i=0}^{\infty} \binom{i+k}{k} ^{-1} = \frac{k}{k-1}, \quad k \geq 2.$

For example,

$\displaystyle \binom{3}{3} ^{-1} + \binom{4}{3} ^ {-1} + \binom{5}{3} ^{-1} + \binom{6}{3} ^{-1} + \ldots = \frac{1}{1} + \frac{1}{4} + \frac{1}{10} + \frac{1}{20} + \ldots = \frac{3}{2}.$

## October 28, 2012

### The declining arctic sea ice

Filed under: climate and weather — ckrao @ 10:30 am

The following graphs are cause for concern. They show that

(1) the area of arctic ice at its minimum yearly extent (usually in September) was this year only a half of the average value of the 1980s.

(2) the volume of arctic ice at its minimum yearly extent  is on the decline with this year’s level just a fifth of that in 1979!

(3) Since the volume is decreasing at a faster rate than the area, this indicates that the mean ice thickness is also decreasing.

(Data from here and here.)

More on this at the following links.

Poles apart: A record-breaking summer and winter – Arctic Sea Ice News & Analysis, NSIDC

Ice records fall at both poles – New Scientist

Arctic sea ice graphs (Nevin Acropolis)

PIOMAS September 2012 (minimum) – Arctic Sea Ice in Arctic Sea Ice Blog (Nevin Acropolis)

## October 24, 2012

### How I remember the triple angle formulas

Filed under: mathematics — ckrao @ 5:04 am

The benefit of this post is for me to recall better the two following formulas:

$\displaystyle \cos 3x = 4\cos^3 x - 3 \cos x$

$\displaystyle \sin 3x = 3\sin x - 4\sin^3 x$

In the past, whenever I have had to find the cosine or sine of three times and angle (e.g. for some maths  contest problems), I have manually performed expansions based on the double angle formulas:

\begin{aligned} \cos 3x &= \cos (2x + x) \\&= \cos 2x \cos x - \sin 2x \sin x \\ &= (2 \cos^2 x - 1) \cos x - 2 \sin^2 x \cos x \\ &= (2 \cos^2 x - 1) \cos x - 2(1 - \cos^2 x) \cos x \\&= 4\cos^3 x - 3\cos x \end{aligned}

\begin{aligned} \sin 3x &= \sin (2x + x) \\ &= \sin 2x \cos x + \cos 2x \sin x \\&= 2 \sin x \cos^2 x + (1 -2 \sin^2 x) \sin x \\ &= 2\sin x (1 - \sin^2 x ) + (1 - 2\sin^2 s) \sin x \\ &= 3 \sin x - 4 \sin^3 x \end{aligned}

Now that I see these formulas together, I want a way of remembering them without having to write this many lines. Here is my aid.

• Both formulas are of the form $ay^3 + by$, where $y = \cos x$ for the cosine formula and $y = \sin x$ for the sine formula (I already knew this).
• The coefficients up to sign are 4 and 3 (I already knew this).
• The first coefficient is positive, the second negative (hence 4, -3 or 3, -4).
• The cubic term is paired up with 4. Alternatively, the cubic term is not paired up with 3

This last point comes about because the 2 from $2 \sin x \cos x$ is added to the 2 from $2\cos^2 x - 1 = 1 - 2 \sin^2 x$.

If one is unsure whether $\cos 3x$ is $4\cos ^3 x - 3 \cos x$ or $3 \cos x - 4\cos^3 x$, simply substitute $x = 0$ – the expression should evaluate to 1. The coefficients for $\sin 3x$ are then flipped in sign.

## October 18, 2012

### World population hot spots

Filed under: geography — ckrao @ 4:59 pm

Some time ago I displayed the world’s main large and dense population regions, shown below. By large, I mean significantly larger than a city.

Below are the places indicated. The January 2010 population estimates are my own based largely on:

Another interesting look at densely populated areas is this interactive map by Derek Watkins.

1. Ganges River Valley – 665 million people in $723,000 \text{km}^2$ ($920/\text{km}^2$), mostly rural

• 64m in $91,200\text{km}^2$ in Islamabad and much of Punjab (Pakistan)
• 27m in $50,400\text{km}^2$ in Punjab (India)
• 24m in $44,200\text{km}^2$ in Haryana
• 17m in $15,000\text{km}^2$ in Delhi
• 181m in $204,700\text{km}^2$ in most of Uttar Pradesh
• 96m in $94,200\text{km}^2$ in Bihar
• 90m in $88,800\text{km}^2$ in West Bengal
• 162m in $147,600\text{km}^2$ in Bangladesh

major cities: Islamabad, Lahore, Delhi, Kanpur, Kolkata, Dhaka

2. East China – 495 million people in $901,600 \text{km}^2$ ($550/\text{km}^2$)

• 95m in $153,300\text{km}^2$ in Shandong
• 95m in $167,000\text{km}^2$ in Henan
• 77m in $102,600\text{km}^2$ in Jiangsu
• 70m in $202,700\text{km}^2$ in Hebei
• 62m in $139,900\text{km}^2$ in Anhui
• 51m in $101,800\text{km}^2$ in Zhejiang
• 19m in $6,200\text{km}^2$ in Shanghai
• 16m in $16,800\text{km}^2$ in Beijing
• 11m in $11,300\text{km}^2$ in Tianjin

The area includes around 90m in $100,000\text{km}^2$ around the Yangtze River Delta, which may have the largest concentration of metropolitan areas in the world: Shanghai, Hangzhou, Suzhou, Ningbo and Nanjing.

3. Java – 135 million people in $132,200\text{km}^2$ ($1020/\text{km}^2$)

The most populated island on the planet:

• 41m in $38000\text{km}^2$ in Western Java
• 38m in $47900\text{km}^2$ in Eastern Java
• 34m in $34200\text{km}^2$ in Central Java
• 10m in $8200\text{km}^2$ in Banten
• 9m in $700\text{km}^2$ in Jakarta
• 4m in $3200\text{km}^2$ in Yogyakarta

4. Sichuan Basin – 96 million people in $217,000\text{km}^2$ ($441/\text{km}^2$)

• 73 million in $169,600\text{km}^2$ in eastern Sichuan
• 22 million in $47,500\text{km}^2$ in western Chongqing

5. Blue Banana – 94 million people in $189,000\text{km}^2$ ($500/\text{km}^2$)

• 26m in $34000\text{km}^2$ in UK (inc London, Birmingham, Manchester, Leeds, Sheffield, Liverpool)
• 8.8m in $17600\text{km}^2$ in Belgium
• 14m in $22800\text{km}^2$ in Netherlands
• 31.4m in $74500\text{km}^2$ in Germany (inc Essen-Dortmund, Frankfurt, Stuttgart)
• 4m in $20200\text{km}^2$ in Switzerland (inc Zurich)
• 8.6m in $19300\text{km}^2$ in Italy (inc Milan, Torino)

6. Nile Valley – 73 million people in $47,200\text{km}^2$ ($1540/\text{km}^2$)

95% of Egyptians live in less than 5% of the country’s land area.

7. Taiheiyō Belt – 78 million people in $60,700\text{km}^2$ ($1280/\text{km}^2$)

Includes Tokyo-Yokohama (35m), Osaka-Kobe (17m), Nagoya (9m), Fukuoka (2.5m)

8. Northeast Megalopolis – 49 million people in $85,800\text{km}^2$ ($570/\text{km}^2$)

Includes 5.8m in Boston, 22.2m in greater New York City, 6m in Philadelphia, 8.3m in Washington DC + Baltimore

9. Pearl River Delta – 48 million people in $32,000\text{km}^2$ ($1490/\text{km}^2$)

This is economically the fastest growing part of China and includes the following major centres:

• Guangzhou (10m)
• Shenzhen (8.6m)
• Hong Kong (7m)
• Dongguan (6.4m)
• Foshan (3.4m)
• Jiangmen (3.7m)
• Zhongshan (2.5m)
• Zhuhai (1.5m)
• Macau (550k people in just 28 square kilometres)

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