the tree

Here's a tip: try to ask actual questions that may have answers. Things like "isn't this cool?" or "freaky huh?" doesn't really lead anywhere. If you want to know something then "how does [X] work?" or "what is [Y]?" or even "how do we know about [Z]?" would be a good start. If you're not looking for anything specific, then try just reading existing threads.

I would definitely have to side with iNow on this one. The night sky as well, is even more impressive when you can appreciate how far away it is. The rainbow that you see and the rainbow that I see are created by different raindrops. Our eyes are in different places, so we detect different cones, produced by different drops. Rainbows are personal. Some people think that this kind of understanding "spoils" the emotional experience. I think this is rubbish. It demonstrates a depressing sort of aesthetic complacency. People who make such statements often like to pretend they are poetic types, wide open to the world's wonders, but in fact they suffer from a serious lack of curiosity: they refuse to believe the world is more wondrous than their own limited imaginations. Nature is always deeper, richer, and more interesting than you thought... - Ian stewart’s letters to a young mathematician. If you're going to appreciate something, at all, then you should learn about it. It's not very appreciative to say that you don't give a damn about anything other than it's superficial qualities.

Sets don't contain duplicates, so {1,1,1} = {1} and { {},{},{} }= { {} }. Correctly, 0 = {} the empty set 1 = { {} } the set containing the empty set 2 = { {} , {{}} } 3 = { {} , {{},{{}}} } et cetera The point is that you only need the definition of a set, the successor function as defined and one axiom 'there exists an empty set' to get a definition for all the natural numbers. Once you've got a solid definition like that, it's easier to define addition, multiplication etcetera. Well, you can eventually define the reals working this way, it just takes a very long time and isn't worth it.

This is, I dunno, it's not really representative of anything is it? They must know how rainbows work really, that's all kind of basic. There's a certain familiar edge of "I appreciate the world by actively avoiding learning about it" which I think is always the position of a severe minority. I don't think it's too worrying. If you ask that in the biology section I am certain that Moleke, our resident reptile representative, will be more than willing to give you a very detailed answer.

Does that really surprise you? Where did you think everything went? At least on earth it's all kind of big and there's a large delay between it leaving you and coming back - imagine how it works on the International Space Station. Are you sure? Alpha says that about 20% would be pushed off to the edge.

vordhosbn's joke gives a pretty good example. [math]1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots=2[/math] As for why, try sketching it.

Well, no. There is no universally agreed upon set of postulates. The Peano axioms are fairly standard for that type of stuff but they don't suit all situations and there are other just-as-okay alternatives. Realistically, the deepest you're going to go in matters of arithmetic are the field axioms which are just the definition of a field and no-one feels the need to justify that. Well, no. I mean, obviously there were lots of premises and conditions and whatnot that you left out, but still, ugh. And the line of thought was backwards - your argument started with your conclusion and went on to justify which just, ugh, you know better than that.

Well for a given value of S you could approximate theta via the Newton-Raphson method but you'd still need an initial guess and it's not exactly a speedy approach when looking for answers for different theta's.

Clicking links costs advertisers and directs money towards the costs of running SFN. Just saying.

I thought that'd apply in this, although I wasn't certain so I didn't want to say it outright. I'm still befuddled as to how it's connected to the expression presented in the OP.

But the LHS of equation 10 in that article is [imath]\nabla \cdot (W^\zeta \nabla W)[/imath].

We're talking axiomatics here, 'obvious' just doesn't cut it. The SQRT operator is usually taken to mean the positive square root but only by convention since any number with a strictly positive square root also has a negative one.

Argh, so many letters. I'm guessing the primes are for first derivatives, but with respect to what? And is e|phi| meant to read e|phi|? Knowing what path would certainly be helpful as well.

Sometimes I'm on a forum because I want help with something, or because I want something to do or because I just want to chat about something. An ideal forum would cater well to those at all levels - whether it's in a field I know like the back of my hand or something that I only found out about this morning. Moderation would somehow result in the right discussion winding up in the right place all the time. Banning would be okay because there's always people who just don't want to play nice and inclusion really does stop at the kid who only want to join in the game so he can kick the ball over the fence.

But it doesn't. Seriously, standard notation exists for a reason. [imath]x \times a[/imath] or [imath]x \cdot a[/imath] or even [imath]xa[/imath] aren't ambiguous but [imath]x(a)[/imath] definitely looks like [imath]x[/imath] refers to a function. Anyway, back to the question. The ratio for a:b:c is already set out as 1:1.5:2. There's no doubt that the smallest composite numbers matching that are going to be 4:6:8, and the value chosen for x doesn't affect the restriction so the smallest composite number, 4, may as well be chosen as Amir suggested So the smallest value I found for the expression to be minimised was 11 and I'd be pretty damn suspicious if any other values were given. (if the composite criterion were lifted, then 5, but that makes the question even more trivial) I guess what we're saying, is we don't believe you. Can you give any values for a,b,c,x that would satisfy the criteria you gave?

The human body doesn't come with an instruction manual, and there's no warranty to be voided if you use it "incorrectly". Ears might not be designed to have ear rings in them but you don't have a moral objection to that do you?

The difference in the derviatives of the denominator seem too big to ignore when you take n! to be constant.

Whatever, I use different computers to access the forums and on varied internet connections so I haven't noticed any change. I've never noticed any slowness in comparison with other websites though..

Damnit, I want a quiz. Even if we don't use those to filter participation, can we just have some for fun?

That's a good point. For really. really high [imath]u[/imath] it would approximate to: [math]\left( \frac{\mbox{d}u}{\mbox{d}x} \right)^2 = B e^{2 u} + C \sqrt{E e^{4 u}} [/math] [math]\left( \frac{\mbox{d}u}{\mbox{d}x} \right)^2 = (B + C \sqrt{E}) e^{2 u} [/math] [math] \frac{\mbox{d}u}{\mbox{d}x} = \left( \sqrt{B + C \sqrt{E}} \right) e^{u} [/math] [math]e^{u} = -x\sqrt{B + C \sqrt{E}}+F[/math]

What as of yet unanswered questions in oceanography would you realistically hope to see answers to in your lifetime? Are there famous unsolved problems for oceanographers like there are for physicists?

Not only has it bindun, it's been done musically. oQAnPFwOF2g

Biting the hand that feeds you not a good enough metaphor for you guys?

It doesn't. It was a coincidence that Shakes was making a joke of.

This is incredibly obvious if you think about it for a second, if you pick any x numbers whose mean is y then their sum will be xy from the definition of arithmetic mean. 9+10+11=30=3*10, for instance.