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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