3 ms·

I think you can get a generalisation of autodiff using this idea of "nonstandard real numbers": You just need a computable field with infinitesimals in it. The

I think you can get a generalisation of autodiff using this idea of "nonstandard real numbers": You just need a computable field with infinitesimals in it. The Levi-Civita field looks especially convenient because it's real-closed. You might be able to get an auto-limit algorithm from it by evaluating a program infinitely close to a limit. I'm not sure if there's a problem with numerical stability when something like division by infinitesimals gets done. Does this have something to do with how Mathematica and other CASes take limits of algebraic expressions? ----- Concerning the Dirac delta example: I think this is probably a pleasant way of using a sequence of better and better approximations to the Dirac delta. Terry Tao has some nice blog posts where he shows that a lot of NSA can be translated into sequences, either in a high-powered way using ultrafilters, or in an elementary way using passage to convergent subsequences where necessary. An interesting question is: What does distribution theory really accomplish? Why is it useful? I have an idea myself but I think it's an interesting question.

cjfd 2y ago
Distribution theory has lots of applications in physics. The charge density of a point particle is the delta function. Also when Fourier transforming over the whole real line (not just an interval where the function is periodic), one has identities that involve delta functions. E.g. \int dx e^(i * k1 * x) e^(-i * k2 * x) = 2 * pi * delta (k1 - k2).
- elcritch 2y ago
  That's fascinating about charge density of a particle being a dirac delta function. Is that a mathematical convenience or something deeper in the theory?
  - cjfd 2y ago
    Well, if we assume that a point particle is an infinitely small thing with all of its charge concentrated in one point, the dirac delta function is obviously the correct way to describe that. Of course, there is not really a way to find out whether that is true. Still, the delta function makes sense if something is so small that we do not know its size. This idealization has, however, led to problems in classical electrodynamics: https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force. Search for 'preacceleration' in this page. This particular problem was ultimately solved by realizing that in this context quantum electrodynamics is the theory to applies. But, then again, using point particles also causes problems, namely the need to renormalize the theory which is or is not a problem depending on your point of view.
- ogogmad 2y ago
  The article showed that Dirac deltas could be defined WITHOUT distributions. You ignored the article when answering my question. The question is why distribution theory is a particularly good approach to notions like the Dirac delta.
srean 2y ago
Thanks a bunch for pointing me towards Levi-Civita field. Where can I learn more ? Any pedagogic text ?
- yuppiemephisto 2y ago
  See my code at the end. The Wikipedia article is pretty good too. I can send you more if you like.
  - srean 2y ago
    Found it, thanks.
srean 2y ago
> I think this is probably a pleasant way of using a sequence of better and better approximations to the Dirac delta. That can give wrong answers because derivative of the limit is not always the limit of the derivative. When modeling phenomena with Dirac delta, I think the question becomes do I really need a discontinuity to have a useful model or can I get away with smoothening the discontinuity out.

cjfd 2y ago

Distribution theory has lots of applications in physics. The charge density of a point particle is the delta function. Also when Fourier transforming over the whole real line (not just an interval where the function is periodic), one has identities that involve delta functions. E.g. \int dx e^(i * k1 * x) e^(-i * k2 * x) = 2 * pi * delta (k1 - k2).

elcritch 2y ago
That's fascinating about charge density of a particle being a dirac delta function. Is that a mathematical convenience or something deeper in the theory?
- cjfd 2y ago
  Well, if we assume that a point particle is an infinitely small thing with all of its charge concentrated in one point, the dirac delta function is obviously the correct way to describe that. Of course, there is not really a way to find out whether that is true. Still, the delta function makes sense if something is so small that we do not know its size. This idealization has, however, led to problems in classical electrodynamics: https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force. Search for 'preacceleration' in this page. This particular problem was ultimately solved by realizing that in this context quantum electrodynamics is the theory to applies. But, then again, using point particles also causes problems, namely the need to renormalize the theory which is or is not a problem depending on your point of view.
ogogmad 2y ago
The article showed that Dirac deltas could be defined WITHOUT distributions. You ignored the article when answering my question. The question is why distribution theory is a particularly good approach to notions like the Dirac delta.

srean 2y ago

Thanks a bunch for pointing me towards Levi-Civita field. Where can I learn more ? Any pedagogic text ?

yuppiemephisto 2y ago
See my code at the end. The Wikipedia article is pretty good too. I can send you more if you like.
- srean 2y ago
  Found it, thanks.

srean 2y ago

> I think this is probably a pleasant way of using a sequence of better and better approximations to the Dirac delta. That can give wrong answers because derivative of the limit is not always the limit of the derivative. When modeling phenomena with Dirac delta, I think the question becomes do I really need a discontinuity to have a useful model or can I get away with smoothening the discontinuity out.