xkcd

MartianInvader

Ok, it's been a long time and no-one has gotten it. PaulT, do you know a solution? If so, perhaps it's time for a hint? This puzzle has been driving me crazy.

MartianInvader

The intro I like to talk about is gluing together edges of a square in order to make a sphere (all glued into a single point), a torus (opposite sides glued), a klein bottle (one pair of opposite sides glued with a flip), and real projective space (both opposite sides glued with flips).

MartianInvader

How are they not able to pass anything they want at all with a simple majority (51%) in both houses of Congress, though? Everybody's mentioned filibusters, but let's also remember the good ol' presidential veto. Doesn't that require 2/3 majority to overturn? And Bush has often flat-out stated "...

MartianInvader

So, just to make sure I'm understanding: If I have a set of points in the (n-1)-dimensional hyperplane plane through the origin given by x, and apply the rotation you have defined to them, their images will all be in the (n-1)-dimensional hyperplane through the origin defined by y? Yeah. You can se...

MartianInvader

Okay, so are you claiming this is a solution? I mean, if you slice the cake into two pieces and only one contains the center, then they're NOT congruent; this is easy to prove. I don't think you're allowed to cut up into lots of different pieces (since you did slice a couple of your congruent pieces...

MartianInvader

To put it algorithmically, get an orthonormal basis for your vector space so that the first two basis vectors span the plane generated by your two normal vectors. Say your vectors are at angle \theta apart. Then write the 2-dimensional rotation-by-\theta matrix in the upper-left corner of your n-by-...

MartianInvader

Hmm... Okay. There's something bothering me here. There's different standards being applied for what constitutes "sexist"... maybe not by the same person, but I thought it should be pointed out. The whole "bad mother" thing... this was being argued as sexist because one would nev...

MartianInvader

Yep, you're right, phlip. How about this? Let S_m be the number of sequences of length m so that no initial segment is repeated. Now we can obtain such a sequence of length m+1 by adding a new letter to such a sequence of length m. Then we have the following recurrence relations: If m is even, S_{m+...

MartianInvader

http://en.wikipedia.org/wiki/Riemann_sphere I think you might enjoy reading that. The circles which pass through the "point at infinity" are exactly the lines in the standard complex plane. Mobius transformations also give a nice feel for the similarities between lines and "infinite ...

MartianInvader

Isn't it as simple as (1/n) ^ x, where x ~ number of characters typed so far? Well, that's the chance of doing it on a certain step, but then you gotta sum it up. So you'd be looking at something like summing (1/n)^m from 1 to infinity, which would be 1/(n-1). Bu...

MartianInvader

That's a really nice development from the intuitive definition to the mathematical one.

MartianInvader

That proof strategy will probably work, though you have a few errors. You don't necessarily have a straight line to the center (maybe the pieces are all spiral-like), so you'll have to use a path. If you assume that all the "outer" edges correspond to each other, then you can assert that t...

MartianInvader

I like to view things topologically, so I just use the word "closed interval" and forget about this "extended real line" nonsense.

MartianInvader

Yeah, Token didn't subtract enough. I think we want {n \choose p}p^m - {n \choose p-1}(p-1)^m - {n \choose p-2}(p-2)^m - ... - {n \choose 1}(1)^m This might have some nice expression... I'm not sure. In the given case, that's the same as mine. I'm wrong because I missed some double counting, which ...

MartianInvader

WOW, this thread has gotten real crazy real fast. I wanna join in! I can't believe nobody has responded to this little tidbit yet- The blame for high gas prices in Europe falls on the metric system. Our government adds maybe $1 in taxes to every gallon. Europe adds about $1 per litre. I think in Fra...

MartianInvader

Man, reading that article just makes the lawsuit seem full of AWESOME. All the posts put in this thread talking about how this is wasting taxpayer money etc. are exactly the questions he was trying to get people to ask! If, as a result of this lawsuit, stricter measures get placed on what types of l...

MartianInvader

Yeah, Token didn't subtract enough. I think we want

{n \choose p}p^m - {n \choose p-1}(p-1)^m - {n \choose p-2}(p-2)^m - ... - {n \choose 1}(1)^m

This might have some nice expression... I'm not sure.

MartianInvader

Every time the McCain campaign has tried something like that, it's blown up in their face. I think they've learned their lesson by now.

MartianInvader

If I recall correctly, diamonds are only expensive now because of hoarding from the diamond companies... artificial diamonds, while cool, aren't exactly necessary yet. Would artificial diamonds be cost effective if natural diamonds were priced normally? I think the point is the companies are drivin...

MartianInvader

Belial wrote:The press's fabled "liberal bias" or "conservative bias" (depending on whom you ask) is extremely, ridiculously secondary to its laziness and sensationalism biases.

Oh man, I am so sigging that.

MartianInvader

I don't think he's full of shit, but it's just not that surprising. I mean, if I was at a rally for a candidate I supported, and someone I knew in the crowd yelled something like that, I might not approve of it, but I certainly wouldn't give his name to the FBI, especially if I thought he wasn't ser...

MartianInvader

Thanks for your input, weosdfl34! And thanks for those e-mails with the links to penis enlargement product sites! But seriously, we all know what this hexagon is. It's clearly some sort of underground alien complex, constructed by Saturnian life forms vastly different, and vastly superior to us. Our...

MartianInvader

Aw c'mon, now you're splitting hairs. Your (22/7) question clearly was phrased to try to discredit his (GreaterSteven's) original point by making it sound nigh-impossible to accomplish; he responded to this point by citing an example where it had been done in less time with less resources. Now you'r...

MartianInvader

I seem to recall that linear algebra works perfectly fine over division rings (i.e., fields without the communativity axiom for multiplication). So yeah, some of the axiomatic stuff is just thrown in there for comfort. But it probably wouldn't be a bad idea to just mentally note what axiom you're us...

MartianInvader

Yup. That's also why I was careful to say "linear in the size of the input". The size of the input would be the number of bits used to store it.

MartianInvader

Yeah, implicit and explicit differentiation are the same operation. The words "implicit" and "explicit" refer to what you're differentiating , not to how you're doing it. If you have a function defined explicitly, it means something like f(x) = x^3 +sin(x). A function defined imp...

MartianInvader

But doesn't everything that uses arbitrarily large input take at least linear time in the size of the input? I mean, it takes that much time just to read in the data.

MartianInvader

The differentials dx and dy are smooth sections of the cotangent bundle of R 2 . In the case you have a function f: R -> R , the graph of f is a one-dimensional submanifold, and dx and dy restrict to smooth sections of the cotangent bundle of graph(f). They are (locally) related to each other by dy...

MartianInvader

I understand what the problem is intended to be (and I'm working on it), but you may want to alter its set-up a smidgen. As it is, there's no basis for saying "we'll never know them", since there's nothing keeping A from just telling B what the sum of the numbers are and B from telling A t...

MartianInvader

Even worse - The function which is zero for x \leq 0 and e^{-1/x} for x > 0 is continuous and differentiable, but isn't even equal to its Taylor expansion around zero (indeed, every derivative at zero is zero, so the Taylor expansion would be the zero polynomial). Also, plenty of functions are once-...

MartianInvader

Sure! Your Russian-ness is just the average of the Russian-ness of all your ancestors at some fixed level (it's the average of your parents Russian-ness, which is the same as the average of your grandparents' Russian-ness, which is the same as the average of your great-grandparents Russian-ness, etc...

MartianInvader

Indeed. I count that as "doing something tricky." I purposely used the word "typically" when I said you get a k+m-n sphere. How about I phrase it this way: The intersection is is a k+m-n sphere 100% of the time

MartianInvader

What would one you get if you intersected two 3-spheres in a similar way? When intersecting spheres in this way, you should keep track of the ambient space. For examples, the intersection of two 1-spheres (a.k.a. circles) in two dimensions is a 0-sphere (a.k.a. two points), while their intersection...

MartianInvader

Yeah... at best, you're taking issue with McCain because he made a joke in poor taste.

At worst, you might as well talk about how McCain said "I will... destroy... America."

MartianInvader

You -could- do this in a calculus sort of way, but it seems like overkill: Let x be the amount of cake cut from the first cake (x in the interval [0, 1/2] ). Clearly, the cutter will want to cut the second cake either in half or into one big slice, depending on which cake the second person wants to ...

MartianInvader

No, it doesn't. Their result assumes that right-leaning members of Congress will use right-leaning sources and left-leaning members of Congress will use left-leaning sources. Well, I think it's possible to do a statistical study based on this assumption. If you were to take each source, look at the...

MartianInvader

In the 109th Congress (2005-2007) there were 41 more Republicans than Democrats (about 7 1/2 % difference). If anything, this metric should lead to a right-leaning press (if the press truly were central) since the Republicans had more seats, and therefore had more ability to write/push through bill...

MartianInvader

I can't think of a way to define it nicely in one equation, but I'll describe such a function for you: Let f(x) be zero for 0 <= x <= 1. Between x = 1 and x = 2, have it jump up and then down (in two straight lines if you don't mind it being nondifferentiable, or in a bump if you want it smooth), so...

MartianInvader

Why is Congress considered to be the arbiter of the mean? Probably because with Congress you're guaranteed to have a good deal of citation on both sides. You've got people who are very specifically political on both the right and left side of the spectrum making the citations quite regularly. Seems...

MartianInvader

Ordered pairs? Strings? Words?

I think maybe "Ordered n-tuple" is probably what you're looking for.

Search found 774 matches