gossip from the maths world

months ago, i posted something on the seven millennium problems, whose solutions will get be credited one million usd from clay institute. less than one decade into the new millennium, some progress has been made. i’m not posting any solutions or insights. but the following articles that “shed a different light” on the academia. yes. zero maths inside them.

the poincare conjecture: this one is almost confirmed solved by a “reclusive” russian, called perelman. Interesting thing is that he was awarded the fields medal (equivalent of nobel in maths) but he refused it on “moral grounds”. the following article details the so-called tussles over who solved it, but sheds a “bad light” over the politics in the academia. one of the authors of the article also wrote the book “the beautiful mind” and she tends to write only things (and perhaps skew them) in support of her view. to be fair to prof yau (and i feel that there’s sensationalisation on the part of the journalists), here is his defence.

the navier-stokes equation: prof penny smith on last sept posted her solution to this problem on the web. within two weeks, a major flaw was discovered and the paper was withdrawn. the interesting thing is that the error was not discovered through convention means. that is, a journal engages a referee to review the paper and hence point out mistakes. instead it was discovered through a forum discussion on a blog. and i remind you, it took less than two weeks of discussion to discover it. here’s another article entitled the proof is in the blogging.

last, the most famous of them all, the riemann hypothesis. the solution was posted online only on 19 mar 2007. no one has discovered any flaw yet. the interesting thing is that the solution was given by a 77 year old prof called Tribikram Pati from Allahabad University. it’s pretty extraordinary as most results are produced by young mathematicians. not being racist or nationalist or watever, the description here of prof pati makes him sound v unconventional and funny. in particular, i quote, “He is such a modest person that he gives credit for all his achievements to the divine revelations.”

hmmm... like that another weekend has passed, and a simple count puts it now to 2 years 9 months and 7 days = 36 months = 144 weeks = 1007 days...

1. last week went for this navy sprint challenge at east coast park. basically a very shortened version of a biathalon. 500m swim + 3 km run (actual bi = 1.5km swim + 10 km run). so not that siong lah... (heh... not implying that i'm fit or wat) but the "challenging" part is the swim. more like a big fight in the sea. nudge here kick there. when i was talking to my frens, they say that it's like that for the biathalons... some "veterans" intentionally pull pple to get forward in the real biathalon (can try imagine combining the pulling motion with the front crawl stroke... it's quite smooth actually). like ????!!! dunno true or not lah.

here's one photo of the huge mayhem. taken by my fren derek. can look at his other photos of the race (unfortunately... no hunks...)

2. yesterday went for first lesson for pilates with mika. it's at the civil service club at bukit batok. pulled mika along cos my fren told me that it's good for the core muscles... ie. abs. a bit disappointed with the instructor tho (perhaps cos she's a replacement) it just felt like slow abs + legs exercise in an aircon room with music. the instructor just said do this and this... inhale exhale... a bit rush and dun really know wat's happening.

mika complained that her tummy is aching... mine is still ok. i think bigger tummy needs more exercise. haha.

sian. life in the navy is so unpredictable. yesterday evening was so busy crafting and drafting a "opening address" for today's event (which the speaker capriciously chose not to use) and today am so boh liao to do the following calculations...

the numbers game - on channel 8, tues, 8pm.

in the final round (i only saw the final round, hee), there were nine cards labelled 1 to 9. the card '5' was picked out while the rest were covered and mixed up. the contestant was then to pick out one card and the aim was to guess correctly if the chosen card is bigger or smaller than ‘5’. if the guess is right, his winnings will be doubled. if not, the winnings will go to zero.

of cos, it was just that. then it'll be boring. so, there’s this guru who’ll thru some magical calculations and offer the contestant to open one of the other eight uncovered and unchosen cards for a certain amount of the contestant winnings.

example: for today, zhenrong chose a card (which of he didnt know before hand) and guessed that it was higher than ‘5’. then, the guru offered him $300 to open one of the eight unchosen cards. he took the offer and opened a ‘3’, which seem to imply that the card is more likely to be higher than ‘5’. Then the guru offered $350 for another card and he opened a ‘6’. Oops. Now it’s equally likely and he through some instinct decided to guess smaller than ‘5. Next, the guru offered $200 for another card and zhenrong decided to not open anymore. and fortunately the chosen card was ‘1’.

anyway... was boh liao and decided to do some calculations and figure out some strategy. made the following assumptions.

1. the contestant is not some gambler god who can memorise the positions of the cards. so the opening and choosing of the cards is totally random.

2. this one is debatable. i assumed the guru will make an offer such that the expected winnings without opening a card is the same as the expected winnings after opening a card. sounds complicated. for simple reasons, if the offer is too good, it’ll be to the disadvantage for the guru/banker. if the offer is too lousy, the contestant will not take the bait.

for example, if there are only two uncovered cards ‘1’ and ‘9’ left (one of which is the chosen one) and the contestant has only $100 left. then without opening, the expected winnings is (1/2)($200)+(1/2)($0)=$100. if the contestant opens the unchosen card, he’ll know the chosen one. so, my assumption is that the guru will offer $50. and the contestant will take back 2($100-$50)=$100.

as i said, this assumption is debatable. during the game, the guru never followed this offering system. and this assumption also needs a prerequisite that the contestants will do the necessary calculations. haha.


my conclusion.

case 1: the number of uncovered cards bigger than 5 = the number of uncovered cards smaller than 5. then open another card only if the offer the guru makes is less than (your winnings/number of uncovered cards). by right, the guru should make the offer = (your winnings/number of uncovered cards), following assumption 2.

case 2: the number of uncovered cards bigger than 5 and the number of uncovered cards smaller than 5 are not equal. this conclusion is quite surprising. do not take any offer as any positive offer will only lower the expected winnings. (of cos, if offer free, then take lor...)


disclaimer.

the expected winnings is defined in the statistics sense. that is, the strategy should be work in many many games. but, in the game, i saw, zhenrong nv followed the strategy, but he still won. just like any lottery thing lah. everyone is expected to lose, but there will always be the exception or outlier (which everyone hopes to be).

i also assumed the offer is to open one card. perhaps for two or more cards, the strategy may differ. and most likely, channel 8 will alter the "offer" contents to keep the audience interested.

my calculations may be totally wrong.... haha

I've recently assumed my first shipboard appointment as a navigating officer. in theory, i'm in charge of navigation, but in practice, it's much more than that. was on the train when i met an old fren from ri. when talking he asked, "so you're not doing office work." my answer was, "not supposed, but..." he gave the "i understand" nod. geez. if you think there's a job which is not desk bound, then think again.

but it's not a bad thing lah. i'm more conscious of my spelling, grammar, etc.. er... maybe not when i'm blogging.

more fascinating... i realised some powerful features of microsoft office. k... maybe not that power...

1. you actually can create your own animation schemes for your slides. i wanted to a sort of test for my friends, whereby you show a picture of a thing without the name and after u guess and then u click, the name appears. a bit leh cheh to do for like 100 plus slides. then there's no standard animation scheme in ppt that does this. but the good thing is that u can create it.

refer to http://skp.mvps.org/animxp.htm for more explanation.

2. the next one is more complicated and need some programming background. there are many functions in microsoft excel, like sum, product and stats functions. but there are also others which are not there. today i ran a search and found that one can actually create their own functions. advantages: 1. can cater to individual needs, 2. hide the formula and prevent others from accidentally editting the formula.

refer to http://office.microsoft.com/en-us/excel/HA011117011033.aspx for more explanation.

haha. this is such a geeky entry. never expected to become such a geek in the navy.