Willie Wong
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 5h answered What is difference between length of proof and length of its presentation in Peano Arithmetic? 5h comment Perturbations on SVD decompostions "So the discontinuity seems to be due to the fact that singular vectors are unique up to a factor of $\pm1$" That means you entirely didn't read/understand the reference I gave you. From the link I sent, read (really read) Robert Israel's answer. 5h comment Inverse trace theorem for partial trace Define $f(\vec{x},y) = g(\vec{x}) \phi(y)$ where $g$ is the function on the submanifold, $y$ is the transversal coordinate in the tubular, and $\phi$ is a smooth cut-off function. $f$ is as smooth and as integrable as $g$ is. 5h comment Algorithm: Computing the intersection of two conics For numerical stability, the website scicomp.stackexchange may be a better place. 8h comment What is difference between length of proof and length of its presentation in Peano Arithmetic? I skimmed through Section 3 of the article, and didn't quite see what you are referring. Can you edit in a page reference in your question, and better yet a quote of the relevant sentences from the article, so it is more easily identifiable? 8h comment Algorithm: Computing the intersection of two conics The algorithm for computing the intersection of two conics are easily found in many places. Here's a Wikipedia link. For help converting this to code or pseudo-code, please ask on StackOverflow instead. 8h reviewed Close Algorithm: Computing the intersection of two conics 8h comment Inverse trace theorem for partial trace What I don't understand is your notation: if $\partial\Omega'$ is an arbitrary subset, for example, a discrete subset of $\partial\Omega$, how do you intend to define $W^{1-1/p,p}(\partial\Omega')$? Similarly if $\partial\Omega$ is a submanifold of positive codimension? Your question may admit a good technical answer, but first you have to specify what you mean. 8h comment Inverse trace theorem for partial trace In terms of simply extension theorems for Sobolev spaces, the recent works of Fefferman, Israel, and Luli (in some combination) come to mind. If your $\partial\Omega'$ is sufficiently nice you can extend to $\partial\Omega$ and apply the standard results. One of the problems however is with the definition of $W^{1 - 1/p,p}(\partial\Omega')$. If $\partial\Omega'$ is a submanifold of $\partial\Omega$ and the Sobolev space is the intrinsic one, then by taking a tubular nbhd the extension is trivial. 11h comment Dynamics of pairwise distances in the $n$-body problem The number of pairwise distances is $n(n-1)/2$. Including the pairwise speeds you have $n(n-1)$ degrees of freedom captured by just $D$. (Note, the actual number should be smaller since when $n > 4$ there are non-trivial relationships between the $D_{ij}$ to be admissible as actual distances between masses. ) For $n \leq 5$ we have $6n-9 > n(n-1)$ so you definitely will be missing degrees of freedom: your equations would not fully capture the classical mechanics. For $n \geq 6$ I suspect after factoring in the aforementioned nontrivial relationships you would still be short, but I am not 100%. 11h comment Dynamics of pairwise distances in the $n$-body problem Basically: you can think of this in terms of symmetry reductions. The "general" system has $6n$ degrees of freedom (position and velocity of each mass). If you demand that they interact only through relative displacement and not from absolute position, then you have translation invariance which can kill 3 degrees. rotational invariance (in the Newtonian case) kills another 3. Galilean invariance (linear change of reference frames) kills yet 3 more. So the equations should number $6n-9$. 11h comment Dynamics of pairwise distances in the $n$-body problem If you want the equations to just depend on $D$ and $U$, then no. With just $D$ and $U$ you cannot distinguish between 2 bodies orbiting each other (constant $D$) and 2 bodies crashing into each other. 11h comment Inverse trace theorem for partial trace This will be true as soon you have an extension operator $W^{1-1/p,p}(\partial\Omega') \to W^{1-1/p,p}(\partial\Omega)$. Do you assume more on $\partial\Omega'$ than just "arbitrary subset"? 1d revised Nice way to express $H^{-1}(\mathbb{S}^1)$ edited tags 1d comment Perturbations on SVD decompostions 1d comment Perturbations on SVD decompostions Defining a function $f$ is not that hard. But presumably you want some properties of $f$. You may want to read Kato's Perturbation Theory for Linear Operators. Basically: eigenvalues and eigenprojections are continuous, but eigenvectors not necessarily. 2d comment Connection between cardiac equations and untangling knots? If you ignore the "connection" to heart muscles, and take the particular reaction diffusion equation as given, the mathematical heuristics is actually quite well explained in the article. Roughly speaking if you take the PDE and solve it in 2D, it supports vortex-like solutions (things that look like it is spinning around about a center). Plugging in data with two different vortices it looks like that the two vortices will repel each other. By a "vortex string" the authors mean a solution of the 3D equation that in a tubular nbhd of a embedded circle looks like the 2D vortex times the circle. 2d comment Connection between cardiac equations and untangling knots? The FitzHugh-Nagumo model is usually presented as a two dimensional dynamical system. This differs slightly from the version used in the pre-print you cited (which has a diffusion term thrown in making it a PDE instead of ODE system). Not being an expert in mathematical biology, I am not sure how much the diffusion term changes things (but being somewhat of an expert in PDEs, I would expect the diffusion term, in the presence of decaying boundary conditions at infinity, to make things quite different). 2d comment Using Headpose Vector and 2D Point to Compute Distances Your question seems to be about computer vision. You should try dsp.stackexchange.com/questions/tagged/computer-vision or cs.stackexchange.com/questions/tagged/computer-vision or scicomp.stackexchange.com/questions/tagged/computer-vision 2d comment Compressing a hypersurface on the sphere I am not sure which one Ben McKay means specifically, but you can do his construction in either stereographic projection or orthographic projection.