xkcd

MartianInvader

The question as I see it is, if your best prmality test works in 1 millisecond for one number, is there any way to test 1000 random numbers in less than 1 second? And I think the answer is probably no. If you know something about the numbers (for instance if they're consecutive) then you can maybe ...

MartianInvader

Graph theory is a really big area.

You can study graphs without using any topology at all if you come at them from a combinatorial angle.

You can study graphs using almost nothing but topology if you come from, say, a geometric group theory angle.

MartianInvader

Great question. It's usually answered somewhere in the second half of second-semester calculus. You can determine the answer by breaking up the interval into lots of little pieces and approximating the function with straight lines on each piece (that is, connect the point on the graph at the start o...

MartianInvader

If the problem's in the code, that means a lot of groups of randomly wandering robot soldiers, who are essentially under the impression that they're still at war. They'll go through killing those perceived as enemies. If the groups make the size of an army, I'd call that genocide. Yes, they'd be con...

MartianInvader

Which means that when the robots finally do rise up and kill all humans, it'll be because of a software bug, not because they've suddenly become both self-aware and misanthropic. This! I mean, software bugs have caused deaths before , but never on a scale you'd see with robot soldiers. Think about ...

MartianInvader

Yeah, both are garbage. As uncivlengr pointed out, the sum isn't even "grammatically correct" because the upper limit is not an integer. The integral is "grammatically correct," but is a divergent improper integral due to the discontinuities of cosecant. Are you sure the integra...

MartianInvader

I don't understand what's being asked either. There's gotta be more to it than "minimize the total weight of the cycles", because then we can just split every vertex into a singleton, as Cauchy said. Or if we want there to BE a minimum, then it's just a matter of finding the least-weight c...

MartianInvader

22/7: "2 hours is enough time to get just the important stuff." Jack Saladin: "2 hours isn't enough time to get just the important stuff." Chill out. There's no misunderstanding, you guys just don't agree. I don't think there's any need to keep claiming the other person doesn't u...

MartianInvader

I just hope you're not one of those restless sleepers who wakes up and falls asleep again every half-hour... spend all night in bed with only a few total seconds of wakefulness, and by morning you'll have slept a fraction of a second.

MartianInvader

Since m^2 divides the norm, m must be among the real integer factors of a+bi (I'll leave you to figure out why). 5 is not a factor of 3+4i. Whoops! The statement only works if you replace a+bi with |a+bi|, and then (a/m) + (b/m)i isn't necessarily a Gaussian integer, so I guess the argument falls a...

MartianInvader

To do it without quoting any other results: First, a natural number is expressable as a sum of two squares iff its the norm of a gaussian integer: If N = a^2 + b^2, then N is the norm of a + bi and vice versa. Now if n isn't the sum of two squares, then if nm^2 were, then nm^2 would be the norm of s...

MartianInvader

Claiming blanket terms like "under God" don't show preference to a particular religion is a slippery slope. Lemme play Devil's advocate for a second: Catholicism is a relgion. Greek Orthodoxy is a religion. Evangelical Protestantism is a religion. Christianity isn't a religion, it's just a...

MartianInvader

Decepticons?

But seriously, year by year it does seem that technological warfare is playing a greater and greater role in politics, especially international politics.

MartianInvader

I am a Californian, who voted no via absentee. There, now 100% of our absentee voter sample voted no, well above the 60% we need! In seriousness, a large portion of those absentee ballots may be kids away at college, which I suspect would swing more to the "no" column. I don't know if it'l...

MartianInvader

Isn't it possible for this to be overturned as early as 2010? Another proposition could re-amend the constitution, right? And all the liberals won't be busy canvassing for Obama this time around and can do some campaigning. In an off-election year, it seems more likely to pass. I've got a couple fri...

MartianInvader

To put this blatently: "Good-guy/Bad guy" arguments aside, it's just not worth the return-on-investment for democrats to try to suppress votes. It's WAY easier to create long lines for voting in big cities (which tend to be more democratic); it's way easier to purge new voters than to try ...

MartianInvader

I don't read the "a" in that sentence as a quantifier at all. Compare to "a continuous function is differentiable", which I don't think anyone would interpret as "(at least) one continuous function is differentiable". *cough* a universal quantifier is still a quantifie...

MartianInvader

The reasons are mostly historical. Spherical geometry was researched because it was more interesting and simpler at the time. Today more general approaches are researched. For example, spherical geometry and euclidean geometry are special cases of Riemanian geometry of curved spaces. Hey, that's sl...

MartianInvader

No matter how corrupt the president, he can't just give billions to companies he has ties with without Congressional and public scrutiny. Um... he already did? Remember those no-bid contracts? And man, remember the original bailout plan? "Non-reviewable" decisions that the administration ...

MartianInvader

I'd say it IS still possible for McCain to win. If he takes Ohio, Pennsylvania, and Florida, then the election is his. The machines in Florida already have reports of switching votes from Obama to McCain. Officials claim they need to be recalibrated, which would leave them pretty vulnerable to tampe...

MartianInvader

There was an interesting blog post by Scott Adams on this subject recently that, summed up, said that Big War is a thing of the past after Iraq and Afghanistan as they have shown us the even "victory" is no longer economically viable. Countries quite simply can't afford to go to war while...

MartianInvader

Also, do not make the mistake of thinking that because someone is begging, they are not committing crimes, or that because someone is not begging they are committing crimes. It's not a binary system. You seem to be making the assumption that because they are begging, they are comminting crimes, whi...

MartianInvader

Ah yes, of course. I probably should have been able to think of that. Thanks a bunch.

MartianInvader

Some human rights organizations recently have attacked violent acts against women, standing against any type of violence... If a daughter or sister makes a mistake – especially a moral one – that negatively affects the entire family and its reputation, what’s the solution by such organizations? Wow...

MartianInvader

A question for all the people who like set-theory: I was wondering if there exists (under ZFC) a set S with the following property: \forall A \in S, \{f | f:[0,1] \rightarrow A\} \in S That is, if A is an element of S, then the set of paths in A (the set of functions from the interval [0,1] into A) ...

MartianInvader

Saying the answer is 50% is saying that you ALWAYS see 50% of each color. This is false, as looking dead-on at a blue section will cause more of that section to take up the circle. I believe the exact percentage will be something like \sqrt(2)/2. You can get this answer by noting that the radial lin...

MartianInvader

What really pisses me off about this kind of operation, regardless of whether it fits your definition of "terrorism", is that it has the opposite effect of what was intended. I mean, even if the strike was perfect and all of the dead were terrorists, how many civilians saw the US fly into ...

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.

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