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Re: HSC 2013 4U Marathon $Find the minimum possible value of $a^2+b^2$ where $a$ and $b$ are two real numbers such that the polynomial:\\ \\ $x^4+ax^3+bx^2+ax+1$\\ \\ has at least one real root.$

I bet I can find a countable and dense subset of your legs. (NB for those unfamiliar with topology: A topological space with a countable and dense subset is said to be separable.)

Not necessarily. Eg f(x)=x^{1/3}. Also functions that aren't continuous at the origin certainly won't be transformed into functions with stationary points at 0.

Its not stupidly simple, its just stupid. There are infinitely many such hyperbolas that pass through the point (15,16).

A few months into owning ultrasone pro 900's and zero regrets, they are amazing!

Re: HSC 2013 4U Marathon Chad O Dude has 1000 women interested in him. He knows that exactly one of them has a vicious, contagious, and fatal (if he catches it) case of herpes. He wants to sleep with all of the other 999 women, but needs to first determine which one is infected. The way he...

Incidentally, this is a great example of why we need rigorous definitions and proof in mathematical analysis. The high school notion of a limit is vague and imprecise, which makes false statements like this plausible.

The Mars Volta- Cassandra Gemini at 32 mins 32 secs. Really good song too. Am ignoring compilations here obviously.

Reiterating what others have said in a different way: Think of it as sort of like why we write \sqrt{1}=1 not \sqrt{1}=-1. We make an arbitrary choice although there are two numbers that could be considered a square root of 1. Analogously, there are infinitely many numbers...

Re: The p00n thread Being a douche has nothing to do with the presence or lack of brains. I am saying nothing about the latter. "Tend to" is indeed a generalisation but makes it sound like it is more common than not, extremely presumptuous. And I know that measurements of talent are...

Re: The p00n thread Haha wow, what a douche thing to say.

1. Yes. To prove this show that 2^{ab}-1 is divisible by 2^a-1 for positive integers a and b. 2. I guess. There are tests for primality, so this problem is a double application of them. We could for example use the ECPP primality test for p, and then the Lucas-Lehmer-Riesel test for 2^p-1. This...

As I expected. And in your expression now you have split up your alpha integral in an illegal way, your first integral is nonconvergent, and so must be your third. Its like writing: \int_0^1 1 \, dx = \int_0^1 \frac{1}{x}-\frac{1}{x} +1 \, dx= \int_0^1 \frac{1}{x}\, dx - \int_0^1...

Well yeah alpha is between 0 and pi always. Not sure what you mean by cases of cos2ntheta though. Whatever case argument you try be very careful with the inequalities and try not to skip steps. I am not that confident in it working but I don't have to time to try it myself right now.

But if alpha is greater than pi/2, then cos theta is negative on the interval of integration of the inside integral. So making the magnitude of that integrand bigger is making your inner integral smaller (more negative). Whereas if alpha is smaller than pi/2 then the opposite is true. Eg the...

Another quick visit to BoS. If you can convince me of page three you can convince me that your idea works, (after fixing up less important typos that may exist). From a glance right now I think the inequality part is dodgy. If you can explain each step of that part to me as if I knew no more...

Mates, if you are going to argue about mathematics at least argue about something of substance. Terminology varies and it always will. Personally I have seen the word piecewise used MUCH more than piecemeal, but pretty much everyone knows what is meant by the use of either term...so why make a...

Well the payoff corresponding to each action profile is a 2-vector of reals, saying that they are all distinct doesn't mean that they cannot still have one component equal, wherein lies the problem. (2,1)|(1,1) ----------- (2,2)|(1,2) for instance has all action profiles as pure strat...

No, convergence itself is not enough to justify the interchange of summation and integration. You need something stronger, like uniform convergence for example. Typically the sort of interchanges you are talking about are done via Fubini's theorem (wiki it). Note that summation is just a...

You edited your post so I assume you saw it but the things you were sandwiching your integral between weren't finite.