henryforever14
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 Mar 4 comment Haar measure on the groups SO(n) and SO(n,m) for the nice integration formula displayed in the middle, is there a rigorous proof for it? It might be argued by using the uniqueness of Haar measure. It's certainly true that the RHS is left and right invariant. Normalization is not a problem. But how about regularity? Aug 6 comment Curvature of a level set If you want straight lines to be your answer, shouldn't you penalize for the sum of the absolute value of curvature of all points? Aug 1 comment Question about the definition of convergence in measure. The first equality in the second long equation, you are interchanging two limits. This is not always true. Jul 28 comment Are convex functions enough to determine a measure? @ByronSchmuland Thanks! It's a nice argument. Please change the very last measure to $\nu(dx)$. I will then accept your answer. Thanks a lot! Jul 28 comment Are convex functions enough to determine a measure? @ByronSchmuland I am sorry, but is that function still convex? For example, the function $f(x,y)=xy$ should not be convex. Jul 28 comment Are convex functions enough to determine a measure? @ByronSchmuland, what is the analogous functions for $f$ and $g$ in higher dimensions? Jul 28 comment Are convex functions enough to determine a measure? So it does work if the measures are compactly supported. Jul 27 comment Are convex functions enough to determine a measure? didn't realize compact domain would make such a huge difference. Thank you. For your last remark, suppose we are dealing with measures that have compact support, how may one prove convex functions are enough? I suppose it would require some approximation of continuous functions by convex functions that I am not aware of. Jul 25 comment Confusion about the definition of reflexive relation @BrianM.Scott Thanks! Edited. Jul 24 comment Is the boundedness necessary to extend harmonically? Why are the conditions different between harmonic functions and holomorphic function? Jul 24 comment Is the boundedness necessary to extend harmonically? So a priori, you don't need to know $v$ is bounded. As long as $v(z) \leq o(\log |z|)$, it will be still true that $v_\epsilon(z)$ will go to $-\infty$. It is the analysis that forced $v$ to be bounded. That means, the statement can be weakened as: "if $u$ is harmonic in the punctured disk and $u(z) \leq o(\log |z|)$, then $u$ can be extended harmonically at the origin." Jul 20 comment Define a relation — with functions and derivatives Let me just remark that in your case, a relation $D$ on $F$ means $D$ is a relation from $F$ to $F$, or, $D\subset F\times F$. Also, personally I would prefer not to use the words "domain" and "range". "domain" gives people a sense that you need to associate everything in it with something in the range, which is not the case in relation. It also might help if you just run some tests before you jump into the problem. For example, is $x^2 Dx$? Is $e^xDe^x+1$? Jun 30 comment second fundamental form and connection forms I guess what the author meant by "orthonormal frame" is orthonormal list of vector fields induced from local parametrization. Jun 24 comment why does Lie bracket of two coordinate vector fields always vanish? @JamesS.Cook, sure, that I agree. Thanks with the help. Jun 24 comment why does Lie bracket of two coordinate vector fields always vanish? @JamesS.Cook, "Let's see, if the commutator is nontrivial then I don't think that means it is not possible. I certainly can find vector fields on the plane which have nontrivial Lie Bracket." I meant vector fields with nontrivial commutator can't be coordinate derivations. Jun 24 comment why does Lie bracket of two coordinate vector fields always vanish? @JamesS.Cook, so this in a sense tells me if I start out with two vector fields such that the Lie bracket doesn't vanish, then there is no way I could find a coordinate system such that they happen to be coordinate derivations. Now, is the reverse true? That is, if I start out with two vector fields such that the Lie bracket DOES vanish, is it true that I can find a coordinate system with them being coordinate derivations? Thanks! Mar 25 comment borel measurable functions and measurable functions Thanks! Very neat construction! Mar 25 comment borel measurable functions and measurable functions @PhoemueX Thanks, that solves the problem! Feb 7 comment boundary of the support of a continuous function Thanks. That's a nice example! Nov 7 comment minimize p norm of f+c Never mind. I've already found a counter example. $e^x$ in $L^4$ doesn't follow this rule.