the tree

It's also helpful to use the forum's LaTeX feature. Click on the images to see how it's done. [math]| \tfrac{1}{10}(4 \hat i + 8 \hat j + n \hat k) | = 1[/math] [math]| \tfrac{1}{10}(4 \hat i + 8 \hat j + \sqrt {20} \hat k) | = 1 [/math]

Looks to me like the second form is seperable. Of course, I just misread what you were asking for. Sorry about that.

If you particuarly wanted people to read it, and someone had already expressed an interest in it, then I couldn't think of a better place for it to be hosted.

If it's only in the billions, then Wolfram Alpha tends to display numbers in a linear format (as well as whatever other format it feels like).

The plausable situations are that he has a daughter born on one of seven days, or a second son born on one of six days - since a second son born a tuesday would contradict the premise. So that is, out of 7+6=13 plausable situations (each with fairly equal probability), 6 that would result in the friend having two sons. So 6/13. Again with the slightly too easy.

Yes, there is a general approach. More than one less-than-general approach. You may wish to Google around for "(first order) inhomogenous linear ODEs". In this case, the important thing is that a sum of solutions to an ODE, is a solution to that ODE. So for the most general answer you will want to give. y(x) = u(x) + v(x) Where u(x) is the a y(x) that satifies LHS=0 (the implicit sol'n). And v(x) is a y(x) that satisfies y(x)=RHS and LHS=RHS (the particular sol'n).

Overall the universe is fairly homogeneous - on a (relatively speaking) small scale there are always going to be denser clusters and sparser areas. Since the formation of stars is largely dependent on pot luck, think of it as winning and loosing streaks on a fruit machine - in that they just happen. In terms of what to call that, I suppose the small bursts of heterogeneity in a homogeneous environment is essentially the study of entropy.

That is a hazard to consider with all objects. Remember kids: never leave your tequila stash next to the trampoline under the razor stalactites.

As the cost of LEDs goes down, that could have a very positive influence they may even be cheap enough to give away. I really don't see the concern about any danger from LEDs.

A simple proof by contradiction should at least show that there are no non-trivial linear solutions. (*) f(f(n-1))=f(n+1)-f(n) assume f to be linear f2(n-1)=f(1) f(n-1)=1 f=1 (*) 1=1-1 // so f is not linear

Oh I see, you're correct in that case.

Yes it is, some conventions allow 'closed form' to include all kinds of messy expression.

Depending on how you define 'closed form', transcendental numbers do not have a closed form.

You've really got to stop posting such embarrassingly easy 'puzzles'.

The thing is, you can change your mind. Even very dramatically, it's not unheard of for people to start with astrophysics and leave with a history degree. To pursue a CompSci degree you would do well to already have some programming under your belt (I think the resident CompSci guys would tell you to start learning with Python if you haven't done much before) and also some serious competence at discrete mathematics.

I think the mixing of tenses, is enough to call it nonsensical - so it cannot have a truth value anyway.

Seriously? A Word Document? Anyway, the entire document features nought but the following: The answer is of course that it is written in a needlessly obscured form, thus only really special in the Ralph Wigum sense of the word.

^^

Well, there probably wont be a 'transport of the future'. There's never going to be a need for a single form of transport. But there's no reason to suspect that pneumatic tubes will never become popular in some context - perhaps one where having an engine on every individual train becomes impractical (an underground equivalent to maglev, perhaps?).

Can you think of any reasons off of your own back? It's not exactly scientific, to pick a conclusion, and then find a way to get there. Using the underground is already more sensible than using a car in a densely populated city. It's in rural areas where mass transportation doesn't make sense, that personal cars are going to stick around.

Yes.

Because 1/4*100=2500%, and you're looking for 25%.