Sorry, I was again unclear. I meant it would accelerate to "full sinking speed", whatever that is, independent of density.
I know about buoyancy, I'm just offering a way to see that independence of sinking speed wrt density leads to absurdity.
I didn't make it explicit enough, I guess, but it's supposed to be a thought experiment. If density didn't affect the speed, then something would go from floating to sinking at full speed as soon as a dust grain landed on it. Then it would go back to floating when the water washed the dust grain...
Hi, I'm giving a talk tomorrow morning, and I'd like to use the following fact: a path-connected subgroup of SO(3) consists of either a) only the identity, b) all the rotations about a single axis, or c) all of SO(3).
Unfortunately, I can't for the life of me find where I read it, and I'm not...
If we care only up to diffeomorphism then yes, since every surface has a "pants decomposition" and presumably there is an explicit parametrization of a pair of pants (though I don't know how to write one).
If we care about isometry (i.e. for every embedding into Euclidean space) then certainly...
In "regular flat space", i.e. not a sphere or something like that (the name for this is Euclidean space), two distinct points always uniquely determine a single straight line. This holds no matter how high the dimension, even in infinite dimensions.
In curved spaces it may not be true for all...
I suggest you just start doing it, and if you come across some terms you don't understand, look them up in the index, get a handle on them, then return to where you were. You may have to recurse a few levels. That's how everybody reads math books anyway.
Since you mentioned cross products and distance in higher dimensions, I'll expand a little on tiny-tim's answer. The norm |v| of a vector v in Euclidean space of any dimension is defined in terms of the dot product: |v|^2=v\cdot v. Then the distance between vectors is just the norm of their...
Instead of thinking about rotating lower-dimensional objects to get higher-dimensional ones, it's probably better to think about extrusion, as explained in the video posted to this thread.
The reason is that, qualitatively, rotations behave very differently in different dimensions. What...
I don't remember needing Zorn's Lemma when I had to do this exercise, but I think we were allowed to assume the boundary was compact.
Can you be more specific about what you mean by locally extending a retraction?
This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?
Just a quick hint: the points (T,100) and (T+100,100) tell you that the parabola is symmetric about the line x=T+50, and so your equation has the form y=c+a(x-(T+50-b))(x+(T+50+b)) for some real numbers a, b and c.
Edit: Here's maybe a better hint: the x-coordinate of the vertex of a parabola...
A metric is a locally-defined structure--that is, when defining one, we may consider regions as small as we like. So if one dimension is curled up into a circle of radius R, we could define our metric on balls of radius R/100000 for instance. At that scale, we can't even detect the curling...