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May
19
comment Kinematics of rolling knots
The relevant material seems to be at the end of the talk, around 9:00? One thing to think about would be whether you want to ask these questions about knots with ideal shapes, or knots that are arbitrary fattened embeddings of the topological knot in 3-space. Ideal shapes are poorly characterized: arxiv.org/abs/1402.5760 . It's not obvious to me whether the dynamics are Hamiltonian. See, e.g., en.wikipedia.org/wiki/Chaplygin_sleigh .
May
19
comment Kinematics of rolling knots
possibly related: mathoverflow.net/questions/180194/…
May
19
comment Flat Riemannian manifold
An example that the OP might wish to think about would be a cylinder with a flat metric. That might help to clarify why the answer is framed in local terms.
May
18
awarded  Necromancer
May
18
comment Diffusion on a semi-Riemannian manifold?
if we lived in a space which was only semi-Riemannian Huh? We do live in such a space. There is no a priori guarantee that such a space is time-orientable, has Cauchy surfaces, or lacks closed, timelike curves. You may be interested in notions from general relativity such as global hyperbolicity. See, e.g., Hawking and Ellis, The Large Scale Structure of Space-Time, p. 206.
May
14
comment How to learn QFT from mathematical perspective?
We have a body of math that was created for the sole purpose of understanding topic X, and which is still practiced for almost the sole reason of applying it to X. It seems odd to me to be so insistent on refusing to learn about the application to X. This is not an example like group theory, where the original motivation was much more specific (permutation groups) but now constitutes only an insignificant portion of the whole landscape. If you learn an entire field of math without understanding a single application, I would question whether you understand the field.
May
12
comment Physics that needs “new” math
Physicist to string theorist: You've had this research program going on for 35 years, and it's not yet looking like a healthy physical theory. Why should anyone keep giving you funding? Standard answer from a string theorist: String theory is a piece of 21st-century mathematics that's fallen out of the sky into the 20th century, and it's going to require 22nd-century mathematics to solve it. Even if it's never going to be a correct Theory of Everything, keep funding us, because it's wonderful mathematics.
Feb
12
awarded  Yearling
Feb
12
revised space at the Planck scale
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Feb
12
answered space at the Planck scale
Jan
24
comment Are hyperreal numbers isomorphic to formal power series?
The product of two Laurent series need not be a Laurent series. If you want a non-archimedean field that has well-defined multiplication and division, a good candidate is the Levi-Civita field, but the LC field doesn't have $e^\omega$ where $\omega$ is infinite. It would also be odd if a structure as "natural" as the Laurent series were isomorphic to the hyperreals, since the hyperreals are non-unique in ZFC.
Dec
29
comment Which universities teach true infinitesimal calculus?
news.slashdot.org/comments.pl?sid=99232&cid=8500301
Dec
22
comment Why do we teach calculus students the derivative as a limit?
I've read through the whole dialog in comments about the sine function, and I'm still mystified. Among the mathematicians who got it wrong, what was the wrong answer that they gave? What was the reasoning that they offered?
Dec
10
comment Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
@JosephO'Rourke: I would bet you a nickel that the length before reaching a cul de sac has a distribution that drops off at least as fast as an exponential. As time goes on, the probability per unit time of being trapped should increase, because you've built up a trail that could help to trap you. Based on your simulation, I'd guess that the expected length is on the order of $10^4$.
Dec
8
revised Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
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Dec
8
revised Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
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Dec
8
revised Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
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Dec
8
revised Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
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Dec
8
answered Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon
Dec
4
comment Is non-existence of the hyperreals consistent with ZF?
Abraham Robinson suggested that ZF and ZFC were in some sense constructed exactly so as to allow us to do analysis on the reals. From that point of view, it's not surprising that in ZF(C) the reals exist and are unique, whereas ZF doesn't make the hyperreals exist, and ZFC doesn't make the hyperreals unique.