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Dec
22
comment Why is ${n-1 \choose -1} = 0$ when (-1)! is undefined?
You also can't choose k things from $-5$ or $3/2$ object and yet binomial coefficients $\binom{-5}k$ and $\binom{3/2}k$ are [usually defined to be] non-zero.
Dec
22
comment How to prove that the sum of squared binomials equals $\binom{2n}{n}$
or of math.stackexchange.com/questions/373122/… or of math.stackexchange.com/questions/320348/… or of math.stackexchange.com/questions/404715/… ...
Dec
22
comment Understanding the Definition of the Tensor Product of Chain Complexes
...details can be found in most algebraic topology textbook (e.g. in Hatcher's). And IMHO it's very useful in general to study (some) algebraic topology in parallel with homological algebra.
Dec
22
comment Understanding the Definition of the Tensor Product of Chain Complexes
@Exterior For cellular homology this is [not some complicated quasiisomorphism but] an almost tautological isomorphism: on the level of elements that's just $\mathbb R^n\times\mathbb R^m=\mathbb R^{n+m}$ — and for the differential that's exactly what motivates the definition of the differential in $C\otimes D$. As for the Kunneth formula, aforementioned lemma allows one apply Kunneth formuma from homological algebra (computing $H(C\otimes D)$ in terms of $H(C)$ and $H(D)$) to compute $H(X\times Y)$.
Dec
22
comment Understanding the Definition of the Tensor Product of Chain Complexes
@Exterior Details depend on the version of (co)homology you're using — but if, say, $X$ and $Y$ are CW-spaces then $C(X\times Y)\cong C(X)\otimes C(Y)$ where $C(-)$ is the cellular chain complex.
Dec
22
comment Understanding the Definition of the Tensor Product of Chain Complexes
If you're computing (co)homology of topological spaces, tensor product of complexes corresponds to the usual product of spaces.
Dec
22
comment Counting subsets with r mod 5 elements
Dear Adriano, that edit a) was quite unnecessary; b) title now looks quite ugly. I'm rolling back.
Dec
21
comment An example where $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is the number of ways of counting something?
(For me the interpretation is combinatorial enough, but the proof is not. It would be nice, of course, to have a combinatorial proof of the fact that $S(m,n)$ indeed counts these paths.)
Dec
21
comment An example where $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is the number of ways of counting something?
Related: math.stackexchange.com/q/9935
Dec
21
comment primes of the form $4k+3$ and sums of squares
@WillJagy Oops, indeed say math.stackexchange.com/q/105034 is a better duplicate
Dec
21
comment Evaluating Combination Sum $\sum{n+k\choose 2k} 2^{n-k}$
Also answer essentially coincides with math.stackexchange.com/q/918 — I wonder if there exists a bijection...
Dec
21
comment Evaluating Combination Sum $\sum{n+k\choose 2k} 2^{n-k}$
Related: math.stackexchange.com/q/121407
Dec
21
comment How to calculate variant of geometric series based on sequences of Catalan numbers?
en.wikipedia.org/wiki/Catalan_number#First_proof
Dec
21
comment Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?
Related: Polynomial bijection from ℚ×ℚ to ℚ? @ MO
Dec
20
comment At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?
(Re: «It sounds like basically any such metric space is actually a manifold.») No. A manifold is something locally isomorphic to $\mathbb R^n$, not just embeddable in $\mathbb R^n$. Arbitrary subsets of $\mathbb R^n$ can be very complicated.
Dec
19
comment Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases
(Tangentially) related: uniform proof of the fact that 3 altitudes of a triangle are concurrent
Dec
17
comment Definition of bordism - gluing manifolds with structure
Related: math.stackexchange.com/q/410917
Dec
16
comment Reference Request: Thom Spectrum of a virtual vector bundle
AFAIR, 1) For Thom spaces of vector bundles $\operatorname{Th}(\xi+1)=\Sigma\operatorname{Th}(\xi)$. 2) Any virtual bundle is of the form $\xi+n$, where $\xi$ is a vector bundle and $n$ is an integer. 3) For spectra suspension map is invertible.
Dec
15
comment Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases
@Willemien Are you asking if the law of sines (in this form) true in hyperbolic geometry? Certainly — see e.g. en.wikipedia.org/wiki/Law_of_sines#Unified_formulation
Dec
14
comment Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$
Thank you. I need to think about it for some time.