the tree

Okay, that's easy. If bodies with a large gravitational pull had a large magnetic force to go with them, then a priori it'd have a polarising effect on sheets of ferromagnetic material. As it happens, putting an oven tray on the floor doesn't turn that oven tray into a magnet. Therefore gravity and magnetism are entirely different beasts. (basic, primary school stuff that). No. A hypothesis must be testable - once the testing is done it stands a chance of being regarded as theory. It is idle speculation. What do you think the difference between flying pigs and an undetectable force from outer space is? Hint: one can be presented as a hypothesis, the other cannot. "if it were true then it'd work" hints at nothing more than internal consistency, which even idle speculation tends to live up to.

Really? If I feel like bursting into a spontaneous dance number in the street - while there are few people around so no-one is in danger, and without musical accompanyment so that I couldn't be breaking noise control levels - do I really have to find out if there's a law against dancing in the street?

It did? I rarely pay attention these things but do you have a citation on that? My understanding is that most anomolies on number stations can be put down to maintenance interuptions, and of course there is no telling their maitenance structure.

Okay, so a sphere is encased in a cube. What does this demonstrate? What are the positive and negative signs about? I'm guessing magnetic alignment, but then thier placement on the vertices seems a little strange since the vertices are as far away as you can get from the sphere, and subsequently not very close to the poles.

If anything, you are severly overestimating the goverment's involvement in the scientific community. Pseudoscience is a word used by scientists and philosophers to describe things that can be mistaken for science, but are not. Occasionally it refers to things maliciously disguised as science, but mostly just to misunderstandings.

The provision of efficient utilities in general would be what I'd want. Part of how energy will be produced in the future ties in with sensible ways of dealing with waste materials, and then there's stuff like clean drinking water and even public transport. None of these things exist independently from each other and an integrated approach is a must. Alongside all of that comes improvement scientific education - not just for producing a few super geniuses with wonderful ideas but also in order to have scientists and engineers all over the world to build and impliment all the things we want. You can build a hospital in a third world country, but you'd also need a doctor to but in it - similar applies for water supplies and modern power stations. Of course, all these problems, are everyone's problem and not just one for scientists. So some of the sollutions are going to have to come from goverments and buisinesses as well. For instance, there are many illnesses that we know how to eradicate - if we could just do that then a lot more time could be spent on dealing with the ones that we don't know much about.

You could probably get away without knowing series. None of the things you mention use them directly although the proofs of the boundary theorems require really getting how limits work - integration is essentially riemann sums which are basically infinite series. I think it would be worth flicking through the notes at least, you shouldn't need to know every little test for convergence - but there's really no point in not knowing Taylor series.

Often, a is chosen to be 0, for simplicity's sake.

What didn't you understand about it?

We're all aware that (1-1)=0, right? Or are we going for some crazy type of subtraction? Of course, trying to drag sense out of numerology is significantly harder than blood out of a stone.

To the best of my knowlege, all 'David does is post links to various lectures across the forum. I'm guessing he watches some of them as well, but I wouldn't hold your breath over getting a response.

Clearly, there is a secret magical football hidden inside, with the power to grant wishes or something.

I think freepowerboards.com (phpbb) and ipbfree.com (ivpb) might be a tad faster.

Yes, it is overkill but [ [A>B] => [A> B] ] is so trivial that you shouldn't even need to state it - just state what you've proven, and the conclusion that it implies. (I'm not sure why you'd have been asked to prove that when a slightly more powerful statement is provable, but there you go)

It's not easy, no, but I managed it in a few lines by taking a geometric view. This does suppose a basic knowledge of dot products, which I'm going to assume because it makes this proof nice and very, very general. (which is always good). Suppose we have two vectors of equal length in one vector space, |v|=|w|. Now it's trivial to prove that: v.v > v.w (use the cosine definition of dot products) Now, assign those vectors: v=(x2,y2,z2) w=(y2,z2,x2) You'll then have to show that |v|=|w| v.v=x2x2+y2y2+z2z2 v.w=x2y2+y2z2+z2x2 And then you'll have that x2x2+y2y2+z2z2> x2y2 + y2z2 + z2x2 Which should be all you need.

Oh I'm really sorry, I completely misread the OP. You shouldn't substitute in 90 since it's what you're trying to prove.

The LHS isn't a natural number (nor is the RHS), but really that's not the point. Basically, you know that x2,y2,z2,m and n are all strictly positive - that's all you should need. If you can prove that x2x2+y2y2+z2z2> x2y2 + y2z2 + z2x2 Then you will know that LHS>1 If you can prove that 2m + 3n < 3m + 3n Then you will know that RHS< 1 Since LHS>1>RHS it follows that LHS>RHS.

Well, the equation is symmetric about A, B and C for a start, so it doesn't matter which one you decide to be equal to 90. After that you'll need to look at various trig identities that you know.

This is particularly important, but perhaps not entirely obvious. The theoretical reason is that different aspects of shape respond to scale differently. For instance: while volume increases cubically, surface area only increases quadratically. So a significant drop in volume may represent only a small drop in surface area.

Definitely not. Look at the OP. m and n are to be natural numbers, both sides are strictly positive.Mr Skeptic is on the right track.

I'd go for finding the minimum value on the LHS, and maximum on the RHS. Induction may be appropriate for finding the maximum on the right, but probably not really necessary.

Sorry, I meant the derivative of it's inverse - what with its inverse not existing.

Do you mean applications for explaining stuff to old people or applications for esoteric algebra-like structures?

I did say that it'd work.

dy/dx is multivalued at nearly all points, and there exists points where dy/dx is 0 so it doesn't even have a reciprocal.But it can be chopped up so as to be piecewise differentiable and strictly monotonic which is why taking the reciprocal still works for most of the function. Same as if you were looking at sin(x), it works just fine but only over a small range.