Mokka mit Schlag » Math

A quick proof of something that bothered me in basic topology. Assume the standard topology on ℝⁿ based on open balls. What about an open box? I.e. all points in ℝⁿ such that a₁ < x₁ < b₁; a₂ < x₂ < b₂;…;a_n < x_n < b_n. Is this an open set? I.e. can you build it up out of a union of open balls? Or, more colloquially, can you pack a square hole with round pegs without leaving any gaps?

Short answer: yes, if the balls can overlap and you have infinitely many of them. Long answer:
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C|Net accuses Apple of favoring iTunes songs over CD-ripped songs in iTunes random playlists. Unfortunately they don’t have the statistical chops to prove anything or do any real analysis:

It’s obviously difficult to tell whether back-room marketing deals or just dumb luck were responsible for the results we saw, but it appears that we can safely lend credence to the suspicions of myriad iPod users around the world. When it comes to choosing songs, ‘random’ clearly is relative.

Actually folks, it’s totally possible to figure out whether your results are random luck or not. For one thing, try repeating the experiment. But what you really need are better statistics. In particular try calculating the chance your results would occur by pure randomness. You haven’t published the raw data, so I can’t do it for you; but this should be well within the reach of anyone whose taken a couple of undergraduate courses in statistics. In fact, it would make a very nice final project for a statistics course. I don’t think it quite rises to the level of an undergraduate thesis though.
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After evaluating Chronosync for a month, the evaluation period is up and it’s time to make a decision. To buy or not to buy, that is the question. I think the answer is no. Chronosync is too slow and too complex to justify paying for.
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Did you watch the figure skating the other night? Did you notice almost every skater on the ice falling and making really embarrassing gaffes? Did you perhaps compare this to a typical Icecapades where falls almost never happen, or at least don’t happen nearly as frequently? There is a reason for this, and unlike what some TV commentators say it has little or nothing to do with pressure and the new scoring system and quite a lot to do with statistics and strategy.
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