4,109 reputation
821
bio website shapetales.wordpress.com
location Prague, Czech Republic
age 28
visits member for 3 years, 9 months
seen yesterday

Math PhD. student and functional programming enthusiast.


Jan
10
comment Fundamental group of CW-Complex only depends on 2-Skeleton
math.cornell.edu/~hatcher/AT/AT.pdf Corollary 4.12 in the Hatcher's book should do the trick.
Jan
10
comment Fundamental group of CW-Complex only depends on 2-Skeleton
The intuition seems correct to me. Of course, it's sweeping few pages of technical details under the rug, as you mention yourself :)
Jan
10
comment Motivation for introducing algebraic topology?
@KCd: good points but your comment is quite imprecise. All ${\bf R}^n$ have homotopy type of a point, so that their homology and actually all algebraic topological properties coincide. What's different is e.g. the homology of ${\bf R}^n \setminus {\rm pt}$ which has the homotopy type of a sphere. Another way would be using compactly supported cohomology but these invariants are of course not preserved by homotopy.
Jan
10
comment Motivation for introducing algebraic topology?
Even more than that, maps between Riemann surfaces (and even branched covers) were understood almost a century before algebraic topology appeared. I think the invariance of domain might be a better example (although I'm not sure it doesn't have a simple proof without using AT).
Jan
10
comment Motivation for introducing algebraic topology?
Very good answer, I agree with most of what you have said. But your example $S^2 \to T^2$ is actually pretty simple. Just write $S^2$ as union of hemispheres. Image of the equator must be contractible, so that image of one of the hemispheres is then a disk, while the image of the other one contains two great circles (generators of homology of $T^2$) and wouldn't be contractible.
Jan
9
comment Orthogonal basis for infinite-dimensional vector spaces
@Ketty: the standard definition of basis for inner product spaces is not the same as the one for linear algebra, (see e.g. here en.wikipedia.org/wiki/Inner_product_space#Orthonormal_sequences). I'm voting to close because of your uncooperativeness and refusal to make yourself and your question clear even after much discussion with your peers.
Jan
9
comment Calculating the lie algebra of $SO(2,1)$
The equation here is then $A = - \eta A^t \eta$. It's quite simple to solve but as a baby step you can try $SO(1,1)$ with $\eta = {\rm diag}(1, -1)$ first.
Jan
9
comment Calculating the lie algebra of $SO(2,1)$
Your equation should have $0$ on the right hand side since $\eta$ doesn't depend on $t$. Otherwise it's correct. Note that if instead of $\eta$ we would have identity matrix (group $SO(3)$), this equation would be solved by anti-symmetric matrices.
Jan
9
comment Orthogonal basis for infinite-dimensional vector spaces
@Ketty: we are talking about which basis is better since the answer depends on which kind of basis (Hamel, Schauder or perhaps something else) you want. Also, you want to talk about the inner product. Every vector space with compatible inner product is a pre-Hilbert space and its completion is a Hilbert space. So if you know the answer for Hilbert spaces, that's all there is to know.
Jan
9
comment Orthogonal basis for infinite-dimensional vector spaces
TL;DR: if you want an inner product space that is useful, then we're talking about Hilbert spaces. If you explicitly want to learn about incomplete inner product spaces, please state that in your question.
Jan
9
comment Orthogonal basis for infinite-dimensional vector spaces
The standard definition of basis from linear algebra isn't very useful in infinite-dimensional vector spaces. There is a notion of a Hamel basis but such a basis must be necessarily uncountable. It is much better to weaken the notion of linear independence by using infinite summation, which leads to the Schauder basis. But to be able to do summation you need to have norm on your vector space and preferrably a complete one, so this leads directly to Banach spaces. If you are interested in orthogonality as well then you need inner product and such a space is actually a Hilbert space.
Jan
8
awarded  Custodian
Jan
8
reviewed Reject suggested edit on Distribution of stochastic integral in small time
Jan
8
comment Nets and Convergence: Why directed indices?
Is this really supposed to be an answer? If so, then you know the answer, so why did you ask the question in the first place. If not, why is this "answer" here? And in any case, these observations really belong to the question itself. Only now you made it obvious that it's relations (such as partial order) that you are interested in. Which completely changed the question. Sigh...
Jan
8
comment Nets and Convergence: Why directed indices?
@Alex: 1) I'm not sure what kind of answer you expect. Convergence is a phenomenon of getting near to something. So having sense of a direction is useful. I'm not sure if you can have more intuition than this since bad things happen in pathological spaces. 2) I've explained why it converges both formally and informally. What more do you need? For nets, the concept of direction is more complex than for sequences. Here, the net goes up and right and so for convergence issues only top matters and the net is constant there. So it's eventually close to $0$. I'm not sure what more do you want.
Jan
8
answered Nets and Convergence: Why directed indices?
Jan
8
comment Hilbert's syzygy theorem in the analytic setting
I don't know much about this topic but I think you mean finite resolutions. Infinite ones should always exist. Anyway, nice question.
Jan
8
comment Finding conjugacy classes of $D_{10}$
$(g\tau g^{-1})^2 = g\tau g^{-1}g\tau g^{-1} = g \tau^2 g^{-1} = e$.
Jan
7
comment What's the explanation for why n^2+1 is never divisible by 3?
This is the best answer given yet in my opinion. Not necessarily for the OP, but certainly for me because it strikes right at the heart of the problem.
Jan
7
comment What's the explanation for why n^2+1 is never divisible by 3?
@fretty: it seems to me OP understands the proof perfectly. But knowledge of a topic doesn't come from perfect understanding of one proof but from knowing multiple proofs and how the topic connects with rest of mathematics. As the many answers show, there is much more going on here than meets the eye. Kudos to OP for asking this question.