Chaitanya's Random Pages

March 26, 2011

A circle around mainland Australia

Filed under: geography — ckrao @ 7:18 am

Here is a circle of radius 2030km centred at 26S, 133.35E (on the Northern Territory/South Australian border of Australia). I used this tool to generate it.

This is a reasonably close approximation to the smallest circle containing all of mainland Australia. It is interesting that its centre should lie on the border. The three points closest to the circumference are the NSW/Vic border, Byron Bay (the easternmost point of mainland Australia) and about 75km SSW of Exmouth. There are also three other parts of coastal mainland Australia close to (within 75km of) the circumference, notably, near Dirk Hartog Island (WA), the south west coast and Cape York (Qld).

Hence one can almost say that 6 points of coastal mainland Australia form a cyclic hexagon whose circumcircle is the smallest circle containing mainland Australia! The largest distance I could find between two mainland points of Australia is 4048km, suggesting that Byron Bay to the point SSW of Exmouth is not too far from being a diameter of the circle. However due to spherical geometry, its midpoint is actually quite some distance south from the centre of the circle shown above.

Note: I tried a circle around Australia including Tasmania, but it wasn’t as interesting – only 3 points close to the circumference. 🙂

Ref: http://www.freemaptools.com/radius-around-point.htm

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Some Mathematical Coincidences

Filed under: mathematics — ckrao @ 3:42 am

Here are some mathematical coincidences I quite like. There are more at the references below.

  • \displaystyle 2^{10} = 1024 \approx 10^3 – this leads to such approximations as a doubling corresponding to a 3dB increase, or a kilobit of data being represented by 10 bits.
  • \displaystyle 3^2 + 4^2 = 5^2 and \displaystyle 3^3 + 4^3 + 5^3 = 6^3.
  • \displaystyle 10^2 + 11^2 + 12^2 = 13^2 + 14^2 (= 365, the number of days in a year)
  • \displaystyle 1.08^9 = 1.999004..., \displaystyle 1.02^{35} = 1.999889... so now you know the doubling time for investments at 8% or 2% return.
  • \pi \approx 22/7 to 0.04% or \pi \approx 355/113 to six decimal places
  • \displaystyle \left(\frac{2143}{22} \right)^{1/4}= 3.14159265258..., so \displaystyle \pi - \left(\frac{2143}{22} \right)^{1/4} < 10^{-9}
    This was found by Ramanujan, who knows how he came up with it!
  • \displaystyle e^{\pi}-\pi = 19.9990999792
  • 1 year is approximately \pi \times 10^7 seconds to 0.45% (1 year = 31,556,926 seconds)
  • Some involving 666: \phi(666) = 6.6.6, 666 is the sum of the squares of the first 7 primes, \sum_{i=1}^{6\times 6}i = 666
  • \displaystyle \int_0^{\infty}\prod_{i=1}^{\infty}\cos\left(\frac{x}{n}\right)\ dx \approx \frac{\pi}{8}, where the difference only occurs after the 42nd decimal place!

References

http://en.wikipedia.org/wiki/Mathematical_coincidences

http://en.wikipedia.org/wiki/Almost_integer

Weisstein, Eric W. “Almost Integer.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AlmostInteger.html

March 21, 2011

Most runs in n consecutive overs in ODIs

Filed under: cricket,sport — ckrao @ 11:13 am

After New Zealand’s amazing win against Pakistan in the group stages of the 2011 Cricket World Cup, I thought that surely no team had ever scored that quickly at the end of a one day international (ODI) innings. New Zealand in the last four overs scored 92 runs: 28, 15, 19 then a last over of 30! I did a little digging and found that while the Taylor-Oram partnership of 85 off 22 balls was easily the fastest 50+ partnership (thanks to this blog entry by Andy Zaltzman for the link), it was not quite the fastest end to an ODI innings, only just being outdone by the blitz by Astle and McMillan against the USA in 2004.

Below is an attempt at a list of the most runs scored in n consecutive overs of a one day international innings, for n ranging from 1 to 50. In the right column the team achieving the feat is mentioned first. I only have access to the ESPN Cricinfo scorecards along with their over-by-over comparisons, so corrections to this are most welcome. Hopefully over time the accuracy of this list will improve as I am made aware either of errors or other matches.

I found the following three matches have dominated the list so far:

SA vs Netherlands, ODI #2537:  This is to date the only one day international in which 6 sixes in an over were scored. In this 2007 World Cup match, heavy rain the previous night at St Kitts meant that the game was reduced to 40 overs per side. South Africa started slowly being just 4 runs for the loss of one wicket after 5 overs! After 29 overs the score was 2/178, before Herschelle Gibbs scored 36 off a Daan van Bunge over. An amazing 175 were scored off the last 11 overs, leading to a total of 4/353 from 40 overs.

New Zealand vs USA, ODI #2169: In this game at the 2004 ICC Champions Trophy, when McMillan joined Astle at the wicket the score was 4/211 after 42.2 overs with Astle on 87 off 130 balls. The pair then smashed 13 sixes and 6 fours in the last 46 balls, adding 136 in the process. Astle ended with 145*(151), McMillan 64*(27). The final six overs produced 12, 14, 27, 27, 16 and 26!

Australia vs South Africa, ODI #2349: Prior to this game no team had scored 400 in an ODI and it was achieved twice in this game! Australia batting first were 1/168 after 28 overs and scored 10 or more in 14 of the remaining 22 overs to finish at 4/434. South Africa paced the run chase remarkably well, with 111 from overs 20 to 30 to be at 2/279 after 30 overs. Then wickets fell at regular intervals while the lower order just managed to see off the target with one ball and one wicket to spare.

In addition there are matches for which I could not find over-by-over data, such as the world record effort by Sri Lanka against the Netherlands (more likely a steady run flow rather than a major burst at the end given the scorecard), or the 2008 game between New Zealand vs Ireland in which both openers scored 160+. I have checked the other 400+ innings games here. What other games may I have missed?

Edit (1/1/14): This game (ODI #3451) between New Zealand and West Indies on 1/1/14 has set new records for overs 7 through to 21 as shown below (game of previous record still shown).

Second edit (19/1/15): This game (ODI #3583) between South Africa and the West Indies on 18/1/15 set or equalled records for overs 8 to 17, 22 to 47 and 49 (game of earlier record still shown).

Third edit (6/9/16): ODI #3773 between England and Pakistan set records for overs 48-50.

n Most runs in n consecutive overs Game
1 36 ODI #2537 South Africa vs Netherlands, 2007
2 54 ODI #2169 New Zealand vs USA, 2004
3 78 ODI #3616 South Africa vs WI, 2015
4 96 ODI #2169 New Zealand vs USA, 2004
5 110 ODI #2169 New Zealand vs USA, 2004
6 122 ODI #2169 New Zealand vs USA, 2004
7 128 134 ODI #2169 New Zealand vs USA, 2004
8 148 ODI #3451, ODI #3583 New Zealand vs WI, 2004 and South Africa vs WI, 2015
9 160 ODI #3451, ODI #3583 New Zealand vs WI, 2014 and South Africa vs WI, 2015
10 151 171 182 ODI #3428 India vs Australia, 2013
11 175 179 197 ODI #2537 South Africa vs Netherlands, 2007
12 187 189 207 ODI #2537 South Africa vs Netherlands, 2007
13 198 200 220 ODI #2537 South Africa vs Netherlands, 2007
14 202 211 227 ODI #2537 South Africa vs Netherlands, 2007
15 203 219 236 ODI #2537 South Africa vs Netherlands, 2007
16 209 237 241 ODI #2537 South Africa vs Netherlands, 2007
17 251 ODI #3451, ODI #3583 New Zealand vs WI, 2014 and South Africa vs WI, 2015
18 225 262 ODI #2537 South Africa vs Netherlands, 2007
19 234 270 ODI #2537 South Africa vs Netherlands, 2007
20 237 278 ODI #2537 South Africa vs Netherlands, 2007
21 239 283 ODI #2537 South Africa vs Netherlands, 2007
22 243 280 ODI #2537 South Africa vs Netherlands, 2007
23 252 287 ODI #2349 Australia vs South Africa, 2006
24 266 294 ODI #2349 Australia vs South Africa, 2006
25 270 306 ODI #2349 Australia vs South Africa, 2006
26 278 311 ODI #2537 South Africa vs Netherlands, 2007
27 284 315 ODI #2349 Australia vs South Africa, 2006
28 292 323 ODI #2349 Australia vs South Africa, 2006
29 304 335 ODI #2349 Australia vs South Africa, 2006
30 311 341 ODI #2349 Australia vs South Africa, 2006
31 322 344 ODI #2537 South Africa vs Netherlands, 2007
32 327 350 ODI #2349 Australia vs South Africa, 2006
33 333 355 ODI #2537 South Africa vs Netherlands, 2007
34 342 364 ODI #2537 South Africa vs Netherlands, 2007
35 349 368 ODI #2537 South Africa vs Netherlands, 2007
36 350 371 ODI #2537 South Africa vs Netherlands, 2007
37 350 378 ODI #2537 South Africa vs Netherlands, 2007
38 354 381 ODI #2272 New Zealand vs Zimbabwe, 2005
39 365 385 ODI #2349 Australia vs South Africa, 2006
40 368 395 ODI #2349 Australia vs South Africa, 2006
41 375 398 ODI #2349 South Africa vs Australia, 2006
42 385 401 ODI #2349 South Africa vs Australia, 2006
43 392 408 ODI #2349 Australia vs South Africa, 2006
44 402 411 ODI #2349 Australia vs South Africa, 2006
45 411 418 ODI #2349 Australia vs South Africa, 2006
46 421 422 ODI #2349 Australia vs South Africa, 2006
47 428 ODI #2349, ODI #3583 Australia vs South Africa, 2006; South Africa vs WI 2015
48 433 434 ODI #3773 England vs Pakistan, 2016
49  436 440 ODI #3773 England vs Pakistan, 2016
50 443 444
ODI #3773 England vs Pakistan, 2016

March 20, 2011

The Isoperimetric Inequality

Filed under: mathematics — ckrao @ 1:50 am

The following result has been known since Ancient Greek times if not earlier, yet it was not until the 19th century that it was rigorously proven.

Among all regions in the plane with fixed perimeter, the circle encloses the greatest area.

I recently came across a solution to a special case of the isoperimetric inequality (smooth curves) using Fourier series which I shall outline here. This is based largely on reference [1] with support from [2-5]. Let the plane curve C be parametrised in the complex plane by \displaystyle z:[0,2\pi]\rightarrow \mathbb{C}, \displaystyle z(t) \in \mathbb{C}. We assume the curve is smooth, meaning its derivative z'(t) exists and is continuous. Also assume the curve is simple (z(t) is one-one on [0, 2\pi)) and closed (so that z(0) = z(\pi)). The length of the curve is given by

\displaystyle P = \int_0^{2\pi} |z'(t)|\ dt

and the area is

\displaystyle A = \frac{1}{2i} \int_C \overline{z} \ dz = \frac{1}{2i} \int_0^{2\pi}\overline{z(t)} z'(t)\ dt.

(These formulas for perimeter and area are beautiful in their own right! Feel free to pause to admire them if you have not seen them before. 🙂 )

Since we know that scaling the coordinates of the curve by k increases the arc length by k and its area enclosed by k^2 , we may assume that the perimeter is normalised to {P = 2\pi} and the curve is parametrised by arc length, so that {|z'(t)|=1}. Writing z(t) in terms of its Fourier series gives \displaystyle z(t) = \sum_{n \in \mathbb{Z}} c_n e^{int}.

By translating the curve by {c_0} if necessary (not changing the length or area enclosed) we may assume {c_0 = 0} and so

\displaystyle z(t) = \sum_{n \neq 0} c_n e^{int}.\quad ...(1)

Differentiating (1) term by term gives

\displaystyle z'(t) = \sum_{n\neq 0} c_n (in) e^{int} \quad ...(2)

(true by the smoothness assumption of z'(t)), and so

\begin{array}{lcl} A &=& \frac{1}{2i} \int_0^{2\pi}\overline{z(t)} z'(t)\ dt\\&=& \frac{1}{2i} \int_0^{2\pi} \sum_{m\neq 0} \overline{c_m} e^{-imt} \sum_{n \neq 0} c_n (in) e^{int} \ dt\\&=& \frac{1}{2}\sum_{m\neq 0}\sum_{n\neq 0}\overline{c_m}n c_n \int_0^{2\pi} e^{i(n-m)t}\ dt\\&=& \frac{1}{2}\sum_{m\neq 0}\sum_{n\neq 0}\overline{c_m}nc_n 2\pi \delta(n-m) \\&=& \pi\sum_{n\neq 0} n|c_n|^2. \quad ...(3)\end{array}

The interchange of summation and integration above is justified by the uniform convergence of the series. By Parseval’s relation on (2), we can write

\displaystyle \sum_{n \neq 0} n^2|c_n|^2 = \frac{1}{2\pi}\int_0^{2\pi} |z'(t)|^2\ dt = \frac{1}{2\pi}\int_0^{2\pi} 1\ dt = 1, \quad ...(4)

where the second equality follows from our assumption of z being parametrised by arc length. Using the elementary inequality {n \leq n^2} for {n \in \mathbb{Z}} from (3) and then (4) we have

\displaystyle A \leq \pi \sum_{n \neq 0} n^2 |c_n|^2 = \pi.

Since n is strictly less than n^2 unless n is 0 or 1, equality above holds if c_n = 0 for alln \neq 1, in which case z(t) has the Fourier series representation

\displaystyle z(t) = c_1 e^{it}.

Since |z'(t)| = 1 this means |c_1| = 1, so z(t) is the unit circle as expected.

By scaling by a factor of P/(2\pi), if z(t) has length P and encloses an area A, then A \leq \frac{P^2}{4 \pi}. This is the isoperimetric inequality.

References

[1] Alberto Candel, Notes on Fourier Series, available at http://www.csun.edu/~ac53971/courses/math650/fourier.pdf

[2] http://cornellmath.wordpress.com/2008/05/16/two-cute-proofs-of-the-isoperimetric-inequality/

[3] http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml

[4] Andrejs Treibergs, Inequalities that Imply the Isoperimetric Inequality, available at http://www.math.utah.edu/~treiberg/isoperim/isop.pdf

[5] Alan Siegel, A Historical Review of the Isoperimetric Theorem in 2-D, and its place in Elementary Plane Geometry, available at http://www.cs.nyu.edu/faculty/siegel/SCIAM.pdf

March 2, 2011

Short and interesting land borders

Filed under: geography — ckrao @ 11:54 am

Recently I noticed for the first time that North Korea shares a border with Russia. I had always thought it only borders China to the north and South Korea to the south. This made me look up some of the shorter borders between two countries. The following is a non-exhaustive list of short borders shared by two countries. I have deliberately omitted countries that themselves are small (e.g. Djibouti, Qatar, Swaziland, Luxembourg) that would more obviously have small borders.

Border Border Length (km)
Botswana-Zambia 2
Azerbaijan-Turkey 9
Morocco-Spain 17
North Korea-Russia 19
Croatia-Montenegro 25
Armenia-Iran 35
Afghanistan-China 76
Chad-Nigeria 87
Lithuania-Poland 91
Austria-Slovakia 91
Slovakia-Ukraine 97
Hungary-Slovenia 102
Hungary-Ukraine 103
Afghanistan-India 106

Notes

  • The Botswana-Zambia border is particularly interesting since it is the closest thing in the world we have to four countries meeting at a point (i.e. almost a quadripoint). Namibia and Zimbabwe are the other two countries in the area. If you look at a map, Namibia extends out east along a narrow finger of land called the Caprivi Strip that gives it access to the Zambezi River. Its border falls just short of Zimbabwe to the east, while Botswana to the South and Zambia to the north share the 2km stretch of Zambezi River in between.
  • Azerbaijan is one of the few countries to have more than one piece, and its western component shares a tiny border with Turkey.
  • How does Morocco border Spain? Spain has three tiny components on the African mainland (Ceuta, Melilla, Peñón de Vélez de la Gomera), all surrounded by Morocco, so the 17km border is the sum of three separate border lengths. The border with Peñón de Vélez de la Gomera is just 85 metres long, and only came into existence after a thunderstorm in 1934 made the former island part of the mainland!
  • Croatia is another country with more than one piece (giving the nation of Bosnia and Herzegovina a 26km coastline), and its smaller component has a short border with Montenegro. The same applies to Angola and its 201km border with Congo.

Apart from Azerbaijan, Angola, Croatia and Spain mentioned above, below is a non-exhaustive list of countries that have more than one mainland piece leading in some cases to interesting borders. An exclave is land which is not contiguous with another larger piece of the same country.

  • USA (Alaska)
  • Turkey (European and Asian sections, connected by two bridges)
  • Russia (Kaliningrad Oblast)
  • India and Bangladesh – there are 92 Bangladeshi exclaves within India and 106 Indian exclaves within Bangladesh, occupying less than 120 square kilometres in total. There exists within Bangladesh an Indian exclave with surrounds a Bangladeshi exclave, which itself surrounds an Indian exclave (whose area is less than a hectare)!
  • Belgium and the Netherlands have similarly small exclaves near their border.
  • The area between Kyrgyzstan, Uzbekistan and Tajikistan has a number of exclaves of each country. Have a look at how irregular the borders among these countries are if you have not already done so!
  • Oman has two exclaves within the UAE. One of these contains an exclave of the UAE.
  • Armenia also has an exclave in Azerbaijan.

Other countries with more than one piece (though not all continental) are the United Kingdom, Malaysia and Brunei.

A list of islands that are shared by more than one country is here – the largest ones are New Guinea, Borneo, Ireland, Hispaniola, Tierra del Fuego and Timor. I never knew about Sebatik Island off the east coast of Borneo, Usedom shared by Germany and Poland or the tiny ones listed there.

Other interesting borders are formed by panhandles (elongated pieces of land sticking out), some of which are listed here. Of these my most noteworthy would be in India (the seven sister states), north east Argentina, Cameroon, Namibia and southern Myanmar.

Finally, there are a number of countries currently having border disputes. I did not know there are as many as indicated here!

References

http://en.wikipedia.org/wiki/List_of_borders_by_length

http://en.wikipedia.org/wiki/List_of_enclaves_and_exclaves

http://en.wikipedia.org/wiki/Panhandle

http://en.wikipedia.org/wiki/List_of_divided_islands

http://en.wikipedia.org/wiki/List_of_territorial_disputes

Botswana Zambia 2
Azerbaijan Turkey 9
Morocco Spain 17
North Korea Russia 19
Croatia Montenegro 25
Armenia Iran 35
Afghanistan China 76
Chad Nigeria 87
Lithuania Poland 91
Slovakia Austria 91
Slovakia Ukraine 97
Hungary Slovenia 102
Hungary Ukraine 103
Mozambique Swaziland 105
Afghanistan India 106

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