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reviewed Leave Closed Algebra with two variables
Jul
22
reviewed Leave Closed How to generalize $f = \frac{u}{v}$, $u,v$ scalars, for vectors?
Jul
16
reviewed No Action Needed Limits of systems of differential equations.
Jul
14
awarded  Good Answer
Jul
14
comment Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy
I agree with Giuseppe that this question is likely too general right now... are you solving an ODE or PDE? Of what type? Your analog computer acts as a good preconditioner for the low-frequency component of the solution; most numerical methods (the appropriate one depends on the type of problem) will be able to take advantage of this.
Jul
6
awarded  Nice Question
Jul
4
comment how much differential structure can we put on countable manifolds?
Very interesting! Thanks
Jul
4
awarded  Nice Question
Jul
1
answered How can you confirm that a problem is open?
Jun
30
comment how much differential structure can we put on countable manifolds?
@FredrikMeyer The subspace topology; though if something interesting can be said about other topologies I'm also interested.
Jun
29
asked how much differential structure can we put on countable manifolds?
Jun
25
revised Determining complexity of a 3D shape
added 80 characters in body
Jun
24
revised Determining complexity of a 3D shape
added 25 characters in body
Jun
24
answered Determining complexity of a 3D shape
Jun
18
asked gluing together real-analytic functions
Jun
17
answered Lagrange's Equation on a Manifold
Jun
17
comment Is the decimal notation the “right” notation for arithmetic?
It's not clear, though, that string length is the only important factor when it comes to "best" encoding. For instance if you only need to multiply and divide rational numbers, an encoding based on prime powers in the factorization will be far superior to positional encoding.
Jun
16
comment Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.
What is essential is that (1) the pushforward $A_*$ maps vectors in $T_pS^n$ to vectors in $T_{A(p)}S^n$ (this is true for any differentiable map, not just isometries) and (2) that for the antipodal map, this pushforward is just $-I$ in the ambient Euclidean coordinates.
Jun
16
comment Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.
It's true that $T_{A(p)}S^n = T_pS^n$ but focusing on this fact is a red herring: Amitai did not use it in his proof. (If you want to prove it, you can do so by finding a basis for the tangent space at $p$ and $-p$ and checking that they span the same $n-1$ dimensional hyperplane.)
Jun
16
comment Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.
I don't follow the proof in the document either, but you should be able to easily prove a more general fact: if $A$ is a smooth isometry of $\mathbb{R}^3$ it is also an isometry of any Riemmanian manifold embedded in $\mathbb{R}^3$ with the induced metric.