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 Apr 27 comment A binomial-related inequality I remember this was a good exercise for using Chebyshev's inequality. Apr 27 comment Finding the sum of a complex series Can you sum $\sum_{n=1}^\infty z^n$? If so, notice that that your series is exactly $z \cdot \frac{d}{dz} (\sum_{n=1}^\infty z^n)$. Apr 22 comment Do we care about singularities on the path when using the Residue theorem? (And other theorems in complex analysis) The integral $\int_C f(z) dz$ is not even defined when $f$ is not defined on the whole curve $C$. So, it doesn't make sense to even talk about the integral of $1/(z-1)$ around the unit circle, as it isn't defined there. Apr 22 comment How can we write (2,5) in the countable family of disjoint open intervals? Expanding on @Travis point, finite union is also countable, so you can just express $(2, 5)$ as a countable union of a single interval, that is, itself. In fact, it can be proven that $(2, 5)$ cannot be written as a union of more than one disjoint open interval -- that what the whole notion of connectedness is all about. Apr 22 comment Algebraic Topology; Hatcher 2.23 Let's even consider a special case: consider a continuous map $f: \Delta^1 \to \Delta^1$. Let $e_0, e_1$ be the vertices of $\Delta^1$. Then, viewing $f$ as a member of $C_1(\Delta^1)$, the boundary $\partial f \in C_0(\Delta^1)$, $\partial f = a -b$ where $a, b: \Delta_0 \to \Delta_1$, $a(e_0) = f(e_1)$, $b(e_0) = f(e_0)$. We see thus that $f$ is a cycle iff $\partial f = 0$ iff $a = b$ iff $f(e_0) = f(e_1)$, that is, $f$ is a loop. Apr 22 comment Algebraic Topology; Hatcher 2.23 I still don't understand your point. Every cycle $C(\Delta^n)$ is a boundary (because $\Delta^n$ is contractible), but this is no longer true for relative chains $C(\Delta^n, \partial \Delta^n)$. Apr 22 comment Compute the distribution function of the random variable $Y:=-ln(F(X))$ You confused $F$ with the cumulative distribution function $F_X$. Apr 22 comment Modulus of roots of polynomial tend to infinity Yeah, that's true, but I think it's a good exercise to generalize the proof above for the case of any nonzero entire function. Apr 22 answered Modulus of roots of polynomial tend to infinity Apr 22 comment Algebraic Topology; Hatcher 2.23 I don't really understand your less precise way :) Less precisely, I'd just say that a chain $s \in C(X, A)$ is a cycle precisely when its boundary lies inside $A$. Usually it's not easy to see instantly what's a boundary of a chain, but in our case it's pretty trivial. Apr 22 answered Algebraic Topology; Hatcher 2.23 Apr 21 comment Polynomial equation: $P(\sin t) = P(\cos t)$ Ah. It's OK if you ask me, but if I were you, I wouldn't be surprised if this question was closed because of this -- the rules on stackexchange websites can be weird sometimes... Apr 21 answered Polynomial equation: $P(\sin t) = P(\cos t)$ Apr 21 comment Why is the following expectation inequality true? Actually one must use $|\int f| \leq \int |f|$, which is less trivial than just $\int f \leq \int |f|$. Apr 21 answered How to compute $\int \sqrt{x}\sin{\sqrt{x}}dx$? Apr 20 comment Derivative of map $f: S^n \to \mathbb{R}P^n$ is an isomorphism $T_x \pi$ has no chance to be injective, as the dimension is reduced. Apr 20 revised Ring Extension: Mapping: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q$ added 453 characters in body Apr 20 answered Ring Extension: Mapping: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q$ Apr 17 answered How do I show that $\gcd(n,\frac{n}{k})=\frac{n}{k}$? Apr 15 accepted Series of inverses of binomial coefficients