Let be a measure on (the Borel sets of ) such that for each , exists. Show that is a constant.

A day later Well, has to be continuous. All I’ve got so far.

2 days later
Take , and let be such that is the cumulative distribution function of . Then since , we see using a taylor series expansion of that . So exists everywhere but is not constant. Assuming I haven’t made a mistake, this is a counterexample to the problem statement.

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After So You Think You Can Dance– the only reality show I watch diligently, because the contestants have to have um, … actual talent– I discovered Last Comic Standing, so apparently there’s another reality show worth watching. Fancy that.

Tonight was the semi-final round in Canada, and I was impressed by one comic in particular, DeAnne Smith, who has been doing comedy for only 2 years! She seems to go for brainy, dry humor. One of her jokes got me laughing harder than any other joke the entire show; here’s the gist of it:

Did you hear about the debate between evolutionary theory and intelligent design?
Is it really a debate, if they’re not speaking the same language? It’s like one guy says, “Nuclear reactions in the sun turn hydrogen into helium”, and the other guy says “Good point. But sunshine feels like a warm, warm hug. Mmmm… from Jesus.”

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I just now finished The Languages of Pao, by Jack Vance, a science fiction novel which explores the implications of the Sapir-Whorf hypothesis– the proposition that one’s language shapes and (in the strong version of the hypothesis) bounds one’s thoughts.

Apparently this is a famous book: see the wikipedia entry for the Sapir-Whorf hypothesis, and a lengthy Tenser, said the Tensor entry on the novel (written by a linguistist!).

Personally, I was impressed more by the fact that Vance was able to build an entire novel on a single linguistic hypothesis and nothing else, than by his writing skills.

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Ha ha! History can be so freaking amusing sometimes. I was shopping for some recreational reading at Amazon when I came across the term matelotage; having never heard it before, I Googled it and came back with an everything2 reference:

The pirates of the days of yore lived completely beyond the laws of society, and as such, occasionally had the opportunity to create societies of their own with new norms and institutions that shocked more conservative and law-abiding folks. Surviving documents such as pirate constitutions and pirate law codes were often ahead of their time with their emphasis on equalitarianism and basic rights for all.

One of the more interesting institutions developed by pirates in the 17th century was the practice of matelotage, a formal, often contractual permanent union between two consenting adult men. These men, known as each other’s matelot, jointly owned land and possessions, fought side by side, and nursed each other when ill. Matelots often drew up contracts stipulating that if one were to die, the other would inherit all his property, but even when contracts were not made, matelotage was such a prevalent practice that the surviving matelot was often awarded the property anyway.

In other words, matelotage was just like marriage, only between two men instead of a man and a woman.

And gay marriage is a lot older than some people think.

Matelotage is well documented in historical sources, and was well-known at the time. Matelotage was especially common on the pirate haven of Tortuga in the Caribbean, where one of the governors was so concerned about the practice that he imported hundreds of female prostitutes to try to lure men away from the arms of other men. But matelotage was also a common practice on several other islands where European women were hard to find, including Hispaniola and Jamaica.

The word matelot is a French term meaning “seaman” and matelotage means “seamanship” in French. But very early on the word was borrowed into English with the meaning of “buddy” or “comrade” and was eventually shortened to the word mate, so “matelotage” as an English word might best be glossed as “buddyship” or “comradeship.”

And I could have sworn that any sentence starting with with ‘gay pirates’ has a punchline…

On a more serious note, check out an article on gay pirates written by the author of the book.

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Let’s say we want to solve the PDE where and is a differential operator. Then we can use the fact that applying to is the same as multiplying by the polynomial in the frequency domain, so if a solution exists, it satisfies .

Does this give rise to any useful methods for numerically approximating solutions to PDEs? It seems like a good idea to pursue, but I’ve not seen anything based on it mentioned in my numerical methods for PDEs courses. On the other hand, something very fishy is going on here– will have poles unless zeros of just happen to coincide with those of – and in the case of the Laplace equation, but will be zero everywhere except at the origin, where it’s unclear what value it has. Seems like distributions might be called for…

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“Algorithms are sanity checks on lower bounds”– a quote of Michael Sipser’s that I saw at Computational Complexity. Definitely t-shirt worthy. The fact is, most of the algorithms I’ve seen (as a non-comp-sci specialist) do indeed theoretically accomplish the lower bounds on their performance. That’s why, for example, there are so many numerical linear algebra algorithms to solve for different special cases of . And if my numerical linear algebra prof sentiments are correct, as problem sizes increase, this quote will only become more accurate.

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You’ve probably seen the Towers of Hanoi already, but what about a probabilistic version? Namely, what is the probability of moving all the disks to a given peg in moves given that the initial position of each disk has uniform probability across the pegs? Assume that you follow the optimal strategy for moving the disks around.

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The ‘Alcubierre wave effect‘ was mentioned in Vacuum Diagrams– Stephen Baxter’s fictional history of the universe and the future of humanity, which I’m finding good reading so far. Looking it up on Wikipedia, I discovered that at least some serious physicists have given the topic of FTL serious consideration. It seems that a real warp-drive may be theoretically and practically impossible: most theories require the use of exotic matter to generate negative energy densities, and mind-boggling amounts of energy input, violate certain generalized uncertainty principles (the main sticking point), and don’t supply any hint of how one would apply them to the engineering of a working FTL propulsion system. It’s still exciting though! Enough to make me want to learn QM and GR.

Since we’re on the topic, I’d like to know what the science says on the feasibility of blackholes as methods of transport, ‘wormholes’. This is such a prevalent science fiction device that I often find myself nodding in familiarity when it’s mentioned in a book, but when I stop to think about it, I feel uncomfortable with their ubiquity. Even with my inferior popular science background in physics, I can identify some problems beside the obvious question of whether you can join two black holes (or a black hole and a white hole– do these things even theoretically exist?). For starters, there’re the tidal effects that would probably rip apart any vessel that entered the event horizon of the wormhole. Then there’s the question of how to stop the wormhole from eating everything in range. I wonder if there are any non-condescending popular science books on exotic applications of blackhole physics? Probably.

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