Marek
Reputation
4,397
Top tag
Next privilege 5,000 Rep.
Approve tag wiki edits
 Jan 14 answered How do I maximize $|t-e^z|$, for $z\in D$, the unit disk? Jan 14 comment How do I maximize $|t-e^z|$, for $z\in D$, the unit disk? There are obviously other solutions even for $y_0 = 0$ because also $\phi = \pi$ works and moreover normal to the curve at every point intersects real axis somewhere. Of course, one needs to check which of the these three candidate points is the true maximizer of the distance. Jan 13 comment Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$ @janmarqz: it's standard definition for the space of smooth functions defined on the interval $(0, T)$. Jan 13 comment Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$ @janmarqz: $g$ doesn't it anything. It's a generalization of $f$ that takes values in general Banach space $X$ rather than in $\bf R$. Otherwise it's the same. Jan 13 comment Properties of $\pi_n$ from a category theoretical point of view Aren't the structures one is looking for essentially $n$-groupoids? E.g. crossed module is a $2$-group and generally I'd expect $n$-type to be captured by $n$-groupoid that should be an $n$-truncation of $\infty$-groupoid, which should reflect the full homotopy type. But your last paragraph suggests that the situation is not very well understood. Does the problem lie in finding nice algebraic definition for $n$-groupoids? Jan 11 answered Field of fractions Jan 10 comment — Cartan matrix for an exotic type of Lie algebra -- @Idear: are you sure that this is the same matrix? At first sight it seems to be quite a different object just accidentally having a same name (I'm sure Cartan has looked at more than one matrix during his life...). Also, I think you are doing neither modular representation theory nor working with associative algebras, right? 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 Distribution of stochastic integral in small time