865 reputation
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bio website joshphysics.com
location Los Angeles
age 28
visits member for 1 year, 8 months
seen 11 hours ago

New project: phermi.com

Let me know if you know of any hard physics problems with clever solutions. (email listed to the left)

Personal website: joshphysics.com

Currently a visiting lecturer at the UCLA Department of Physics and Astronomy.

Ph.D. theoretical high energy physics from UCLA.

BA/BS in physics/math respectively from UC Berkeley.


Feb
8
comment Cochains: terminology
Thanks very much for this Olivier. Would you mind pointing me to a reference that discusses this stuff; I would very much appreciate it.
Feb
8
comment Cochains: terminology
@TedShifrin As a matter of terminology, would you happen to know if elements of $\mathrm{Hom}(M,\mathbb R)$ are ever referred to as cochains when $M$ is the $\mathbb Z$-module of chains? The $k$-form example you gave is precisely what I had in mind when I wrote this question; such objects arise naturally in physics (especially thermodynamics when modeling heat). By the way, I'm a fan of "multivariable mathematics," so I was, I must admit, somewhat star-struck when I saw your answer.
Feb
8
revised Cochains: terminology
edited body
Feb
8
comment Disprove a linear mapping
@SergioParreiras See the earlier version. For some reason, the OP removed the definition of $g$.
Feb
8
comment Disprove a linear mapping
@Slavica Yep. Sure thing.
Feb
8
comment Disprove a linear mapping
@Slavica What do you mean by a "general" disproof? This demonstrates that the statement "$g$ is linear" is false, because it is not the case that $g(\alpha x) = \alpha g(x)$ for all $\alpha$ and $x$. It's not as though "falsness" has multiple degrees.
Feb
8
answered Disprove a linear mapping
Feb
8
asked Cochains: terminology
Feb
7
comment Translations in two dimensions - Group theory
@user35952 Unfortunately I think not.
Feb
7
comment Translations in two dimensions - Group theory
@user35952 Yes it's a circle too. It's a circle with the same radius but whose center is at $(-a,-b)$. We could only have said that the equation for the circle was preserved in form (at least if we're using standard terminology) if there were some $R$, say, for which the transformed equation were $x^2+y^2 = R^2$; that equation has the same form. So, for example, a scaling $x\mapsto \alpha x, y\mapsto\alpha y$ would preserve the form of the equation for a circle at the origin because the new equation would be $x^2 + y^2 = (r/\alpha)^2$. Are thinking of a different notion of "preserve."
Feb
7
comment Translations in two dimensions - Group theory
@user35952 Well let's see. A circle is described by the equation $x^2 + y^2 = r^2$, and if we translate $x\mapsto x+a$, $y\mapsto y+b$, then we get $(x+a)^2 + (y+b)^2 = r^2$ which is certainly not an equation of the same form. In fact, the only such equation that will have the same form is the equation for any line in the direction of the translation $(a,b)$ itself.
Feb
7
comment Translations in two dimensions - Group theory
+1: I had forgotten about this way of packaging things; very slick.
Feb
7
comment Translations in two dimensions - Group theory
@user35952 I don't quite understand what you mean. Could you be more specific about what sorts of "equations" you're referring to?
Feb
7
answered Translations in two dimensions - Group theory
Feb
6
comment Intuition for chains and cochains
@AymanHourieh I actually have the first book; didn't think to look there. What I'd really like to read is some sort of exposition devoted entirely to the intuition. Perhaps something that does computations in the context of some physical applications of chains and cochains with lots of pictures etc. I feel like someone somewhere must have written something like this. Thanks for the references.
Feb
6
comment Intuition for chains and cochains
@AymanHourieh I don't know enough about cohomology to answer the first question, but I suspect I'd be interest in all of the above on some level. I've worked through most of Spivak's Calculus on Manifolds, I've learned a decent amount of differential geometry in the context of general relativity and high energy physics, I'm comfortable with differential forms, covariant derivatives, etc. (although I imagine understanding semi-Riemannian geometry won't really help much here)
Feb
6
comment Intuition for chains and cochains
Thanks. I'll check it out.
Feb
6
asked Intuition for chains and cochains
Feb
6
comment If differential 1-forms agree on chains with integer coefficients, are they equal?
Thanks; that's a nice argument.
Feb
5
comment If differential 1-forms agree on chains with integer coefficients, are they equal?
@BrunoJoyal Ah interesting ok. I need to learn what the universal cover is, but thanks.