Sam Lisi
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 May 31 comment Horn and spindle tori Can you define what the spindle torus and horn torus are? May 29 comment Chern class of tautological line bundle After you have done the computation this way, I think it would be instructive to see how Milnor-Stasheff do it. I learned a lot from working through that section of their book. May 25 awarded Enthusiast May 24 revised Does this IVP have a unique solution for all $x \in \mathbb R$ added a sentence for clarity May 24 comment Does this IVP have a unique solution for all $x \in \mathbb R$ @Tunococ: if you've cited a local uniqueness theorem, you've done all the necessary work to answer the question (see Artem's solution below). Ron, if you look at the example I give in my answer (with $y' = y^{2/3}$), you will see why I am somewhat concerned about how to provide a complete argument when you cross the singular points in your separation of variables. I'd think it was really cool if you could indeed provide the missing details in your argument to show global uniqueness -- if it worked, it would be a lower tech solution to the problem than any I know. May 24 answered Does this IVP have a unique solution for all $x \in \mathbb R$ May 23 comment Does this IVP have a unique solution for all $x \in \mathbb R$ OK, perhaps this is a question of taste -- nothing you wrote is false. I do, however, feel this is misleading because the uniqueness of global solutions to this IVP has nothing to do with its separability. This is why I asked above what theorems the OP knew about existence/uniqueness. May 23 comment Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.? To tack an example on to this idea, quite often one gets a set with some structure and then want to show it has even more structure. For instance, I may have a set of solutions to some equation, but then, exploiting something about the equation, I want to show that it is also a metric space. I guess I could say, "I will now find a metric space structure" on this set, but it is nicer to say "I will now find a metric". Of course, if I wanted to show that this set had a group structure... I'd have to say "group structure". May 23 comment Does this IVP have a unique solution for all $x \in \mathbb R$ Indeed you are correct about the integral, sorry about my mistake there. I don't see how what you wrote addresses uniqueness fully. You are saying that if a solution exists for which $\sin(y)$ is never $0$ and for which $\tan(y/2)$ is never $0$ or undefined, it must satisfy the last displayed equation. Now, this displayed equation does not have unique solutions, so it isn't clear why the uniqueness of the solution to the IVP follows from this. One now needs an argument to say that the $y$ has to stay in the same branch as $Y$ (and deal with the edge cases of $y=0$ etc). May 23 comment Does this IVP have a unique solution for all $x \in \mathbb R$ There are two problems with this: (1) This doesn't really answer the question asked. The OP wants to know about existence/uniqueness. This doesn't address that. (2) you seem to have a mistake in $\int \frac{1}{sin(y)}\, dy$. May 23 comment Does this IVP have a unique solution for all $x \in \mathbb R$ Do you know any theorems about existence/uniqueness of solutions for IVP? Some version of : en.wikipedia.org/wiki/Picard–Lindelöf_theorem May 23 comment Operations on vector spaces This question is very open-ended... I could write you a linear algebra book and an elementary differential geometry book as an answer. Can you focus the scope a bit? Otherwise, as a starting point for reading up on tangent vectors, differentials, covectors etc, I recommend looking at Spivak's Calculus on Manifolds. May 18 awarded Constituent May 14 comment I just don't see what I do wrong - number of surjections seems higher than number of functions. If you have a problem with Maple like this, you might want to contact Maple technical support. At the very least, they should be grateful for you finding a bug. May 14 comment Every manifold admits a vector field with only finitely many zeros Obtaining a vector field with transverse zeros is (slightly) easier than obtaining the existence of a Morse function. As Ted explained, you want to use Sard's theorem. The idea is much more general -- it allows you to say that you can always arrange the section of a vector bundle to be transverse to the 0 section. Any book on differential topology should explain this -- for instance Hirsch does it. May 12 revised An alternative description of the first Stiefel-Whitney class fixed latex display problem May 10 comment Determining the embedding space: What do you mean by embedding a usual triangle (or circle) in a sphere? In the case of the circle, you can mean that you take the set of points that are the same (spherical) distance from a given point. I don't know off the top of my head what these geometries are called. A Riemannian geometer should be able to answer this. May 9 comment Does Differential Topology or Differential Geometry play a larger role in Chaos Theory? I think that the answer depends a lot on the circumstances special to your situation, e.g. what exactly the courses will cover, what your faculty supervisor [assuming you have one] expects you to know, etc. I think you should talk to your faculty supervisor about this question and/or to the instructors of the two courses in question. May 7 comment Is there a relation between Super Riemannian manifolds and Kähler manifolds? @Trimok: why do you think there should be a relationship? did someone mention this in a seminar? do you have some examples in which there is some strange relationship you'd like to generalize? I don't know any physics, so I don't think I can help at all, but I don't even understand where the question is coming from. May 7 answered Determining the embedding space: