Shadow

If I understand you correctly, I think you could solve that by designing the contraption so that the velocity of the side with the ball was high enough (when coming down) that you would get the effect seen on the video - or not?

https://www.facebook.com/383870055097958/videos/605245652960396/ The video is obviously a fake, my guess is that the two black stripes are in fact electromagnets, but frankly I don't really care. What I'm wondering about is, could something like this be constructed, in theory, using a mechanism like this? It goes without saying that it wouldn't display this "periodic" behaviour forever, just apparently so, eventually coming to equilibrium much the same way as a pendulum does (my gut tells me this operates on much the same principle as a pendulum does).

http://en.wikipedia.org/wiki/Linear_algebra#Quantum_mechanics

Hi there, I'm having trouble understanding a step in a textbook I'm currently reading (MIT Lectures on Dynamic Systems and Control); I just took a picture of the text to save myself the trouble of copying it over. W is a unitary matrix, [math]\sigma_i[/math] are the singular values of A. I haven't the slightest clue as to why this might be, and seeing as I've search the internet and found no refference to this and the author's don't elaborate I assume it's fairly simple and should be well within my reach, but for some reason I just don't get it. Thanks in advance for any help.

Say I have a sphere that's rotating with an angular velocity aproaching that of the speed of light. What would it look like to an outside observer?

Are you asking anything for anything specific or just want general information? If it's the latter case, wiki will probably give you a good start.

http://news.discovery.com/space/warp-drive-possible-nasa-tests-100yss-120917.html Comments? Is this the real deal, or media twisting the facts?

That's not true. The function [math]\frac{1}{(n+1)^2}[/math] is "infinitely long" and the area under it (from 0 to infinity) sums up to one. For an example of a function whose integral from negative to positive infinity (can one say "over the real numbers"?) is finite, have a look at the Gaussian integral.

Thank you for the book recommendations; they're certain to be my first stop after I've mastered complex analyses. Speaking of which, is there any book you might recommend on that topic?

From a formal point of view, yes, what PeterJ is saying is nonsense; it literally makes no sense. But we are human beings, not robots and from a purely intuitive point of view, which I believe is clearly implied in all of PeterJ's posts, I don't think he is very far off; it sounds very much like what is being said here, and while I have nothing to compare the information there with, I'd be very surprised to find out it was complete nonsense. So, DrRocket, from a intuitive, no-PhD-in-math point of view, is it correct? I would like to know myself; so far, I have looked at the Riemann hypothesis in the manner described in the aforementioned article. I would very much like to be alerted to the fact that it's all rubbish.

I know you didn't. I'm just saying that's the part that has me interested in this topic; I don't know what the set will look like, if you keep on iterating. Or, to make use of a limit metaphor, what the set will "approach" when the number of iterations goes to infinity. The fractal part was aimed solely at what DrRocket suggested.

Well, that can't be entirely true. As you said, there will always be plenty of space to put a new point, yet all the points in [math]\mathbb{R}^2[/math] can't possibly lie in [math]S[/math], to use the notation I used earlier. This in itself is enough for me to wonder what it would look like. It might just look like a black square, ie. it may appear that all the points in [math]\mathbb{R}^2[/math] are elements of [math]S[/math]. It might not. And DrRockets idea, if correct, would certainly yield an interesting fractal. I haven't gotten around to trying it out yet, I'll post here when I do. But thank you, and everyone else, for your inputs, they're much appreciated.

There's a homework section for that

Exactly. But the set can't ever contain all points in the plane. Thus, my question; what does it look like? EDIT: Ah, DrRocket replied while I was writing a reply. Thanks for the idea. EDIT2: That won't work; even after the first subdivision, every vertex will be colinear with two other vertices, for example the points {0}, {0, 1, 2}, {1, 2} in the following image:

I thought as much, but I was thinking of a different type of set, although I'm not even sure if it'll make sense. Imagine this process. Two points are given. You chose a third (at random) that's not colinear with the two and repeat the process. In general, given n points, you chose the (n+1)st so that no three of them are colinear. If repeated indefinitely, what set will I get? Can the process even be repeated indefinitely?

...no three of them lie on one line. Say in 2D euclidean space. What would it look like?

I should have been more clear, I realize that my mentioning real parts might have been confusing. Nevertheless, I really was interested in the real valued function. Complex functions are still beyond my understanding, from a formal point of view. I only understand what I've picked up around here and elsewhere. Thank you DrRocket; it never occurred to me, and should have, to look at the domains of the functions and your explanation clears things up nicely (and also offers a peek at complex functions, for which thanks are also in order). However, although I'm as far from wanting to participate in this argument as I could be, I would like to point out that DH has been of great help to me in different issues, and was clearly not deliberately misinforming me. It seems unfair to insult him, if that was your intention.

I'm not sure I understand. What I did, or at least attempted to do, was write out all the situations that could arise when evaluating an implication and what it evaluates to, ie. T->T (=T), T->F (=F), F->T (=T), F->F (=T) and tried to give an intuitive explanation of why it is evaluated in this way. The second statement could be viewed as a restatement of the first, but that wasn't the point. The same goes for the contrapositive.

Try thinking about it this way. If you start with a true statement and deduce something from it, then you can only get another true statement (or, to put it another way, it is possible to start with a true premise and arrive at a true consequence, ie. the implication is true). On the other hand, if you start with a true statement, you cannot deduce something untrue from it (or to again put it another way, it is not possible to start with a true premise and arrive at a false consequence, ie. the implication is false). It is now obvious why FT and FF are both true; it is perfectly possible to start with something untrue and arrive at something that is also untrue, ie. "Gravity repels massive objects which means that apples fall up", and it is equally possible to start with a false premise and arrive at a true consequence, ie. "The derivative of |x| evaluated at the point zero is equal to zero and thus zero is a local extreme of |x|".

If you take zero to be the identity element with respect to addition in some group, then just use the definitions of the identity element with respect to addition and of the additive inverse, and show that +0 and -0 must be the same.

Thanks. I'm completely unfamiliar with hyperbolic functions, thus the confusion.

Okay, this has confused me for a while now. Arctanh and arccotanh seem to be two different functions. Yet their derivatives are the same. What's more when I plot the two in WA, the plots overlap (at least the real parts). However, when I make WA solve the equation Re(arctanh(x)) = Re(arccotanh(x)), it gives me no real solutions, and a bunch of completely random complex solutions. So what's the deal?

I'm reading "A Working Excursion to Accompany Baby Rudin" (Evelyn M. Silvia, 1999), which is as far as I can tell is a "gentler" version of Rudin, and while I haven't gotten as far as calculus yet, it has been demanding enough for me to wonder if Rudin is the correct choice if one wants to understand the "logic" behind calculus. To understand calculus formally, sure, but if one seeks to develop his intuition, other books might be advisable. But then again, I have no sense whatsoever of what is out there, so maybe Rudin is a good choice after all.

Could you elaborate on how you got this result?

This might help answer a lot of your questions: http://plus.maths.org/content/music-primes