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 Yearling
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Mar
27
revised Introductory text on Riemannian geometry
change awkward title
Mar
25
revised Multiplicative infinitesimals in q-analogs?
clarify title
Mar
25
suggested approved edit on Multiplicative infinitesimals in q-analogs?
Mar
6
comment Is the Mendeleev table explained in quantum mechanics?
@SergeiAkbarov: What makes you think classical mechanics is an axiomatic system? Are you claiming that Newton's laws are an axiomatic system? I think that's clearly untrue. Or are you thinking of some restricted physical model within classical mechanics, which can be described axiomatically? That doesn't mean that classical mechanics in general is an axiomatic system. (I also don't know what you mean by the question about probability theory, since that isn't a physical theory.)
Feb
28
comment Electrons on a pancake ellipsoid
The continuous case has been solved for a more general class of shapes that includes oblate ellipsoids. See I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016 and math.stackexchange.com/questions/112662/… .
Feb
21
comment Surreal numbers vs. non-standard analysis
related: mathoverflow.net/questions/29300/whats-wrong-with-the-surreals
Feb
21
comment What's wrong with the surreals?
related: mathoverflow.net/questions/91646/…
Feb
19
comment Expanding disks lead to what packing of the plane?
This is also an interesting question if you substitute a sphere for the Euclidean plane. I did simulations when I was an undergrad to see what the resulting polytopes would look like.
Feb
12
awarded  Yearling
Feb
7
comment What is a foliation and why should I care?
In general relativity you see foliations over and over as ways of discussing Cauchy problems and the time evolution of systems. These are foliations using spacelike surfaces. Because GR doesn't have a preferred time coordinate or a preferred notion of simultaneity, a foliation in spacelike surfaces is the next best thing.
Jan
18
comment What is the exterior derivative intuitively?
The visualizations you refer to in Misner, Thorne, and Wheeler were originated by J.A. Schouten, and first presented in Ricci-Calculus: An Introduction to Tensor Analysis and its Geometrical Applications amazon.com/… . William L. Burke has given more concise and modern presentations, e.g., in Applied Differential Geometry.
Jan
8
comment Does a classical wave detect compact dimensions?
The question is about classical physics. When you refer to momentum, is that really interchangeable with wavenumber (inverse wavelength)?
Jan
8
comment What is the modern consensus on the difficulty of infinitesimals?
The motivation for the thread on math.SE seems to have been pedagogical: if one is teaching freshman calculus using an approach similar to Keisler's, how does one describe this issue to the students? In that context, I think the answer is clear. The students are learning a body of practices for manipulating infinitesimals, and these practices have been standardized and in use by scientists and engineers without interruption since Leibniz and Newton. In that body of standard practices, we never distinguish an individual infinitesimal.
Jan
6
awarded  Necromancer
Dec
30
awarded  Necromancer
Dec
27
comment Algorithmic complexity of formal proof verification?
The interpretation of this would seem to me to be that real-world proof assistants don't have the right theoretical properties to be good embodiments of whatever we mean theoretically by computationally effective proof. Similarly, a real-world computer doesn't have the right theoretical properties to be a good embodiment of a Turing machine. I could fill a room with monkeys banging on typewriters and set them the task of proving the Riemann hypothesis, with their output checked by grad students. This would be a proof assistant, albeit one with worse than polynomial performance.
Dec
23
comment Do mathematical objects disappear?
@Michael: I don't think we're in disagreement. Nothing in your comment contradicts anything in my answer. Quaternions used to be used for what we would nowadays do using div, grad, and curl. Nobody uses quaternions for that task anymore.
Dec
23
comment Do mathematical objects disappear?
@CarlMummert: Infinitesimals never disappeared from engineering and the sciences.
Dec
22
answered Do mathematical objects disappear?
Dec
22
comment Do mathematical objects disappear?
I don't think this is an accurate depiction of the history. Newton's notation and Leibniz's notation were isomorphic to each other. Newton's $o$ was equivalent to Leibniz's $dt$. (Newton had a convention about omitting the $o$ in certain contexts, which confused things somewhat. See Boyer, p. 201 ) Also, Leibniz did not wholeheartedly endorse the idea that his notation was a notation for infinitesimals. He sometimes preferred to describe differentials in the same way we would today describe differential forms (Boyer, p. 210). Leibniz notation won because it was more expressive.