the tree

A pseudo-bot then: acting like a bot because he can't build one.

In what way? It appears to be fairly light, so it wouldn't deal much damage as a projectile. Being obviously difficult to shape it would make a very expensive blade - according to the story it can only be shaped while it is in it's liquid form so it's unclear how the razor edges of Wolverine's claws are ever created. The simplest military applications would be as shielding or armour.

For clarity, there really shouldn't be a definite article there. 'now' is local, there isn't just one.

To be clear, it's a bot - or a human acting very bot like. If you google the contents of any of his posts you'll find them on hundreds of forums. I'm fairly sure it's an attempt at humour. Remember the "how to tell if your son is a hacker" meme? I think it's an attempt at a forced version of that. Anyway: the reason you shouldn't respond to those posts is because there isn't any point.

Not really. For two n-dimensional vectors: [imath]\mathbf{u} = ( u_1 , u _2 , u_3 , \dots , u_n)[/imath] and [imath]\mathbf{v} = (v_1 , v_2 , v_3 , \dots , v_n)[/imath], their "dot product" is defined as: [math]\mathbf{u}\cdot\mathbf{v} = \sum_{i=1}^n u_{i}v_{i} = u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}\dots+u_{n}v_{n}[/math] The length of a vector is defined by it's "norm": [math]\left\|\mathbf{u}\right\| = \sqrt{\mathbf{u}\cdot\mathbf{u}} = \sqrt{u_1^2 + u _2^2 + u_3^2 + \dots + u_n^2 }[/math] And the cosine of the angle between two vectors is defined as: [math]\cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{\left\|\mathbf{u}\right\| \left\|\mathbf{v}\right\|}[/math] Are there any terms that you particular don't understand there? If you want to think of it more intuitively, then any two vectors are co-planar. That is, there exists a flat plane which they lie on and the angle between those two vectors can be seen on that plane in the 2-dimensional sense that you're used to. Almost surely a typo.

We're pretty strict on not giving straight up answers to homework questions, so it'd help if you could tell us whereabouts you're stuck?

That is again, just as undefined as it as ever it was. It is not. [math]\cot 90 = \frac{\cos 90 }{\sin {90}}[/math]or [math]\cot 90 = \lim_{\theta \to 90} \frac{1}{\tan \theta}[/math] with the condition of continuity.

Right, okay, whatever. So you're saying that [imath]5 \div 0=\infty[/imath] and that subsequently [imath]\infty \cdot 0 = 5[/imath]? Presumably you're also saying [imath]7 \div 0=\infty[/imath] and that subsequently [imath]\infty \cdot 0 = 7[/imath]? What else is [imath]\infty \cdot 0[/imath] equal to? Seriously though, there is no inverse operation to multiplication by zero. In other words, division by zero is just undefined. You're not going to find a way around that. No it isn't. Mirrors follow normal rules of geometry that can be described using well defined mathematics.

I'm noticing a lot of new users starting their own introduction threads - could we make this one more obvious somehow?

Division is the inverse operation to multiplication. Division of [imath]x[/imath] by zero is asking: "what number multiplied by zero gives [imath]x[/imath]?". It'd be [imath]\aleph_1[/imath] which, whilst infinite, is an entirely different beast to [imath]^{+} \infty[/imath].

The tangent function isn't defined at 90 degrees either - particularly in trigonometry there does not exist a Euclidean triangle with two right angles so the trigonometric definition tells you that the function isn't defined at that point.

No. No. Division by zero is undefined, likewise for infinity.

This. After one half-life, the probability of any individual isotope having decayed is 0.5. Strictly speaking it's only really called an atom when it has the right number of electrons to neutrons. A carbon-13 isotope has one too many neutrons which makes it unstable so it breaks apart. Exactly when this will happen is impossible to predict (like, literally impossible - quantum indeterminacy kicks in on this scale) which is why half-lives which are really just a statistical average are used to predict how many out of a large number isotopes will have decayed after a given amount of time.

Well, he was meant to resign as PM. Basically the old PM keeps being PM until the new government is formed (which only happened around the same time) so this is the right way of doing things. There's no big handover ceremony, the new PM goes to the Queen and requests the job*. Though what Gordon Brown has done is resign as leader of the Labour Party. There has been a general feeling since the campaigns season started that if Labour were to stay in power that it wouldn't be under Gordon Brown which is why a coalition between Labour and the Lib Dems never seemed likely. This has happened before, Tony Blair resigned/stepped/down/was quietly asked to leave in order for Gordon to get into power. When that happened, it was between general elections so the new PM was chosen within the party and there was a change of administration without a general election and not everyone was comfortable with that. Even though constitutionally speaking it was all legitimate, some people have been calling Gordon "our unelected PM" ever since. David Cameron, leader of the Conservatives, has now formed the new government. He could have tried to do this straight away but then he'd have been running a minority government which would have made him fairly useless. So the Conservatives, at about the same time (or at least within the same couple of hours) as Gordon Brown's resignation, have only just formed a new government in coallition with the Lib Dems. So we now have a majority government run by the conservatives but with Nick Clegg, leader of the Liberal Democrats, as deputy prime minister and other Lib Dems in other positions although exactly what is yet to become clear. The whole notion of this coalition is really awkward. Obviously Lib Dem voters and Conservative voters are pleased that their party is in government but the real die-hards aren't keen at all on working together. William Hague [conservative], new foreign secretary, has been on the Today programme saying there was a "true collective agreement" between the Tories and Lib Dems. This is a "realignment" of British politics, he said. "There are many things that the Lib Dems have had to swallow that are very difficult for them", he said, and the same was true of the Conservatives. He admitted "there will be people in both parties who will quietly wish it hadn't happened". --http://timesonline.typepad.com/election10/ Assuming the collation is going to be fairly strong, who gets to be the new leader of the Labour party isn't massively important although a really strong Labour party (something we haven't seen for a while) could impact the Con/Lib collation hard enough that it would break up before the next election**. Overall, everyone is feeling a little dazed and confused - we've had a Labour government for 13 years in which people have steadily either lost faith in them or joined their ranks, they've gone from the heroic trade unionists known as Labour who formed the National Health Service to "those-kind-of-centre-left-big-government-types" known as "New Labour", And now we've got a whole new government and we don't know what the hell it is. No-one, even within the parties, really knows what to expect of a Conservative/Liberal coalition. It's obviously not going to be the old Thatcherite Conservatives, nor are the Liberal Democrats going to have as much sway as they'd ideally like. So no-one knows what to expect in the next few years though whatever does it may be a welcome change. *this is one of those things were the Queen can veto it in theory but never has and never will. **which, incidentally, has already been announced. Cameron has introduced fixed terms of office where previously we've just had a maximum.

I say a collaborative effort would be in order. A Unified Application of String Theory to Pre-Cellular Rotoplexes in Retrograde Fourier Acids Abstract: In this paper we prove that... Anyone want to continue a la exquisite corpse?

What? No. In general processes aren't invertible. Particularly with explanations - the movement of shadows can be explained with the movement of the sun but not vice verser.

On that note, it's worth considering that a lot of the Bible is included to give the history of Christianity. "these are the stories that our ancestors told each other".

No. There may be more important things to do with stronger ethical drives than learning. I'm talking about people who have the luxury of having access to learning resources, and a spare hour or so in the day but don't chose to expand their horizons even if all that means is scanning through a wikipedia page. Yes, in most ethical systems not-going-around-killing-people is more important than tipping well - but tipping is still something to be considered even if it's not as important as the whole not-killing thing.

I was going for wilful ignorance - having the opportunity to learn and choosing not to. So the reason why learning is ethically "good" is just because of how it is practically empowering? What about learning about history or cosmology? This is kind of what I'm looking for. I more or less agree that people should learn - but I'm trying to form a coherent argument for why they should. I'm not very knowledgeable about Buddhism - do Buddhists seek enlightenment for themselves or both for themselves and for the "greater good"?

It could be read as a sort of coming-of-age story - with knowledge and awareness comes the human tendency to occasionally do bad things.

Given the opportunity (people living in poverty without access to learning resources can be exempt from this) is there an ethical drive beyond ones own personal gain to learn about the natural world, other cultures, their own historical roots etc? People who manage to 'live in a bubble' may be annoying, but are doing anything wrong?

Okay. Not specifically with motion but just in general: if you've got big numbers and long periods of time then it often makes sense to look at something as continuous even if it isn't really. Radioactive decay for instance - so long as the number of atoms involved is at least in the hundreds then exponential decay isn't a bad model. The "atto-fox" problem essentially just arises in boundary cases where a continuous model fails because the numbers are too small.

Would that really be continuous? Say a particle were moving horizontally for a while then there was a force applied that knocked it vertically so that it was travelling at a diagonal - wouldn't that point be a discontinuity? Well if you had a force being applied all the time (not really a collision) that force could change in an unpredictable manner. If you're not looking at a literal particle but were instead trying to model a stock market then some changes wouldn't be discrete events.

In your first sentence you mix up tenses a lot, which is fair enough since the first clause should be in the future tense (you wrote it in the present tense) and the rest should be in the past tense (whereas you wrote it in the conditional tense) In the second sentence you've accidentally used the conditional tense again, but that was only a minor error. You then went on say 'any thing', which in this context is spelled 'anything' but it seems that you mean 'everything'. (I'm not sure how to explain the difference, ask your teacher). Then you misspelled 'ambitions' when 'goals' would have been a more appropriate word. The next sentence is mostly correct apart from another superfluos 'had'. In the last sentence you misspelled 'beautiful', and you might want to consider other words in it's place such as 'enjoyable'. Overall it seems very good for a recent high-school graduate. What is your first language?

Actually they aren't all that cold, hence the need for no undies.