User kallikanzarid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T04:09:28Z http://mathoverflow.net/feeds/user/9435 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45154/riemannian-metric-induced-by-a-metric Riemannian metric induced by a metric Kallikanzarid 2010-11-07T10:07:13Z 2010-11-07T14:39:31Z <p>Let $M$ be a smooth manifold, $\rho(p, q)$ — a differentiable metric on $M$. Can we construct Riemannian metric $g(X,Y)$ on $TM$ that induces $\rho(p, q)$? Under what conditions?</p> <p>I'm sure this question has been dealt with, I just didn't find it in the quick survey of literature :)</p> http://mathoverflow.net/questions/39676/which-bianchi-identity-is-due-to-bianchi-or-not-since-it-might-be-due-to-ricci/42379#42379 Answer by Kallikanzarid for Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa? Kallikanzarid 2010-10-16T10:01:26Z 2010-10-16T10:01:26Z <p>The first one is due to Ricci, and the second one is due to Bianchi.</p> http://mathoverflow.net/questions/40082/why-do-we-teach-calculus-students-the-derivative-as-a-limit/40102#40102 Answer by Kallikanzarid for Why do we teach calculus students the derivative as a limit? Kallikanzarid 2010-09-27T07:21:28Z 2010-09-27T07:29:48Z <p>$\frac{\sin x}{x}$ at $x = 0$ should be a good example.</p> <p>P.S.: Talking about esoteric definitions, if you can introduce stationary point without derivatives, you can then introduce derivatives using sheaves, like you would introduce vectors on a smooth manifold. It would broaden the consciousness of your freshmen, he-he ^_^</p> http://mathoverflow.net/questions/39579/studying-non-linear-pdes-with-manifolds Studying non-linear PDEs with manifolds Kallikanzarid 2010-09-22T05:11:55Z 2010-09-22T17:32:04Z <p>I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a non-linear PDE on <code>$\mathbb{R}^n$</code> to a distribution or a 'good' PDE on a smooth manifold? It seems to me like a natural step, but I don't know anything about it yet :(</p> http://mathoverflow.net/questions/50048/non-linear-fourier-analysis Comment by Kallikanzarid Kallikanzarid 2010-12-22T13:20:36Z 2010-12-22T13:20:36Z If some family of these beasts forms a frame, you can. Otherwise, you can't. http://mathoverflow.net/questions/45778/philosophical-consistency-proof-for-set-theory Comment by Kallikanzarid Kallikanzarid 2010-11-12T03:40:12Z 2010-11-12T03:40:12Z I'm just a grad student in geometry, though :) http://mathoverflow.net/questions/45778/philosophical-consistency-proof-for-set-theory Comment by Kallikanzarid Kallikanzarid 2010-11-12T03:39:13Z 2010-11-12T03:39:13Z PP seems like assuming a lot about space-time, and it disagrees with my intuitive understanding of mathematics as studying relationships in formal systems, not events. http://mathoverflow.net/questions/45530/approach-to-solving-a-differential-functional-equation/45533#45533 Comment by Kallikanzarid Kallikanzarid 2010-11-10T17:29:31Z 2010-11-10T17:29:31Z Would you please stop spamming threads? http://mathoverflow.net/questions/45277/how-close-are-we-to-extending-tetration-to-the-real-and-complex-numbers Comment by Kallikanzarid Kallikanzarid 2010-11-09T05:09:28Z 2010-11-09T05:09:28Z Mathematics has gone a long way since tetration was first studied. I'm afraid very few professional mathematicians will study it as something more than a hobby. However, if you want to generate a sustainable interest of mathematical community even at this level, the best way to do it would be to 1) dig up and organize for easy access previous professional works on the subject 2) rework amateur papers into a monograph, increasing their rigor and providing a consistent and spartan language and notation. I'm just a graduate student myself, though, so take my words with a grain of salt :) http://mathoverflow.net/questions/45154/riemannian-metric-induced-by-a-metric/45172#45172 Comment by Kallikanzarid Kallikanzarid 2010-11-07T15:30:33Z 2010-11-07T15:30:33Z Upd: I think the problem in Igor's comment and the one in my previous comment are equivalent, because $g \mapsto \rho_g$ is injective, and so it is bijective to it's total image. http://mathoverflow.net/questions/45154/riemannian-metric-induced-by-a-metric/45172#45172 Comment by Kallikanzarid Kallikanzarid 2010-11-07T15:26:44Z 2010-11-07T15:26:44Z I was interested in finding the conditions, under which for $\rho(p,q)$ there is a $g(X,Y)$ such that $\rho_g(p,q) \equiv \rho(p,q)$, where $\rho_g(p,q)$ is constructed from $g$ as usual, and how can such $g$ be constructed from $\rho$. Thanks for the links :) http://mathoverflow.net/questions/45154/riemannian-metric-induced-by-a-metric Comment by Kallikanzarid Kallikanzarid 2010-11-07T10:51:06Z 2010-11-07T10:51:06Z @Suvrit I doubt it. @Christian Good point. I feel a bit fishy about it, though, I'll dig into it deeper :) @Paolo Yeah, I was thinking in this line, too. http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism Comment by Kallikanzarid Kallikanzarid 2010-10-30T12:37:11Z 2010-10-30T12:37:11Z Nelson doesn't look too much sane to me ;) http://mathoverflow.net/questions/42929/suggestions-for-good-notation/42941#42941 Comment by Kallikanzarid Kallikanzarid 2010-10-21T03:45:08Z 2010-10-21T03:45:08Z I like $i = \overline{1, n}$ http://mathoverflow.net/questions/42929/suggestions-for-good-notation/42945#42945 Comment by Kallikanzarid Kallikanzarid 2010-10-21T03:42:48Z 2010-10-21T03:42:48Z $\mathbf{1}_X$ is conventionally used for characteristic function of a set. http://mathoverflow.net/questions/42929/suggestions-for-good-notation Comment by Kallikanzarid Kallikanzarid 2010-10-21T03:38:28Z 2010-10-21T03:38:28Z Isn't $x \mapsto f(x)$ commonplace? As for homomorphisms, they are not simply maps, and $\mathrm{Hom}(A, B)$ denotes the whole class, while $A \to B$ denotes a single mapping. http://mathoverflow.net/questions/13320/cool-problems-to-impress-students-with-group-theory/13329#13329 Comment by Kallikanzarid Kallikanzarid 2010-10-06T14:00:39Z 2010-10-06T14:00:39Z Yakov Perelman demonstrated the solution without explicitly introducing any math above elementary school level at all :) http://mathoverflow.net/questions/39579/studying-non-linear-pdes-with-manifolds Comment by Kallikanzarid Kallikanzarid 2010-09-22T13:48:50Z 2010-09-22T13:48:50Z By 'good' PDE I meant a PDE which preserves a certain operation on manifold just like linear PDEs on <code>$\mathbf{R}^n$</code> preserve linear combination of the solution. I'm afraid I cannot formulate this with more rigor, I'm only starting to seriously study manifolds, and I'm not very experienced with PDEs either.