Marek
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 Jan 28 comment Embedded surface in $\mathbb{R}^3$ Do you understand why $\epsilon$ can't be arbitrary in general (i.e. must be small for certain surfaces)? Understanding this point will take you half-way to the answer. Jan 26 comment Calculating the limit $\lim \limits_{n \to \infty} \sqrt[n]{4n + \sin \sqrt{n} + \cos (\frac{1}{n^2}) + 17}$ Do you know how to compute the limit for $n^{1/n}$? Jan 26 comment Differential of the exponential map on the sphere Correct. Glad I could help. Jan 26 answered Differential of the exponential map on the sphere Jan 25 answered Computation of Laplace-Beltrami operator in a conformally equivalent metric Jan 25 comment Is Goedel term (in incomleteness theorem) both true and unproveable? @Suzan: in logic you have two sides: syntactic and semantic. Syntactic is what you can prove formally, while semantic is what is really true (in some model). Ideal theory for a given model should be both sound (you can only prove syntactically what is true semantically) and complete (you can prove syntactically everything that is true semantically). So by adding $\neg G$ you obtain a theory that is not sound anymore (relative to the model for $T$ you were working with previously). Jan 25 answered Is Goedel term (in incomleteness theorem) both true and unproveable? Jan 25 comment Is Goedel term (in incomleteness theorem) both true and unproveable? @Berci: but it is also important to note that extending by $\not \phi$ produces a theory that is not sound since we already know that $\phi$ is true (semantically). Jan 24 comment $Sp(V)$ acts transitively on $V^*-\{0\}$ where $\Omega$ here is symplectic 2 form @rschwieb: all of the stuff mentioned in the question is standard and natural. Still, since $\Omega$ is non-degenerate, we can translate all questions about forms to equivalent questions about vectors by $v \mapsto \Omega(v, \cdot)$ and its inverse, so it's a bit puzzling that the author insists on working with $V^*$. Jan 24 answered normalize subgroup Jan 24 answered Tangent cone to a subset of $\mathbb{R}^3$ Jan 24 comment Tangent cone to a subset of $\mathbb{R}^3$ Well, one way would be to notice that you can set $z=0$ and work in the $(x,y)$-plane. There the cone will be just two lines and returning back to three dimensions, you obtain the full cone by revolving those two lines around their axis (since $y^2 + z^2 = r^2$ is an equation of a circle in the $(y,z)$-plane). Jan 24 comment Given any ring $A$, how to obtain $A$ from a ring $B$, in which p is invertible? Have you heard about localization and are just looking for alternate descriptions? If not, it is possible that looking up information on localization is all you need. In particular, if $R$ is an integral domain and $p \neq 0$ then the ring you are interested in can be obtained as a subring of the field of fractions $K(R)$ of $R$. If $R$ is not an integral domain then one needs to kill the elements $q$ for which $pq = 0$ since otherwise we'd get $q = p^{-1}pq = 0$. I suppose this killing might also be described using standard operations. Jan 24 answered What is the homology group of the sphere with an annular ring? Jan 24 answered Minimal Connected Set containing a Closed Connected Set in a Compact Space Jan 24 comment Minimal Connected Set containing a Closed Connected Set in a Compact Space @K.Stm.: that doesn't work. Intersection of connected subsets need not be connected. E.g. take $X = S^1$ and $A$ the union two points in $S^1$. Then two arcs containing these points as boundaries are closed and connected but their intersection is just $A$. Jan 24 comment problem on continuous and differentiable function Also, as you said, it is not hard to prove this for analytic $f$ (since $f$ behaves like $x^k$ and $f'$ like $x^{k-1}$ near a zero of $f$). I think this generalizes to arbitrary differentiable functions but I can't really come up with the proof. Any ideas? Jan 24 comment problem on continuous and differentiable function Looks fine to me. Also, I think it might be possible to show $g(x)$ is actually continuous since otherwise there would need to be an essential discontinuity for $f'$, which means that it would be oscillating a lot and this violates $f'(x) \geq f(x)$ immediately. Jan 23 comment Show that minimum exists (direct method) Well, you can generalize this to functional that computes distance along a curve in the plane $\sqrt{1 + (y')^2}dx$ or even further to a functional for a geodesic in a Riemannian manifold or a functional for motion of a point in a potential. The latter is just integral of the Lagrangian $L = T - V$ where $T = v'/2$ is kinetic energy and $V$ is the potential. Your functional corresponds to a free particle in 1D and the solution here is that it stands still. Jan 22 comment Why does a circle cut a torus into an annulus? I think you assume that $\phi$ is not contractible in $T^2$ for otherwise, $T^2 \setminus \phi(S^1)$ has two components whereas $T^2 \setminus \iota(S^1)$ has just one and so can't be homeomorphic. But $\psi$ would induce such a homeomorphism.