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475160
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location Buenos Aires, Argentina
age 21
visits member for 2 years, 7 months
seen 9 hours ago

(My avatar is a piece by artist Pollock named "Number 8".)

Some interesting questions, with great answers:

  1. The so-called Axiom of Choice

  2. Real numbers and sets

  3. How discontinuous can a derivative be?

  4. Why is summation by parts important? This is one example.

  5. Amazing work


2d
comment The integral $\int_{\varepsilon}^1 r^n(1-r)^{k-n}\,dr $
I was joking. I take it you want to estimate the value of the integral asymptotically as $\varepsilon \to 0$?
2d
comment The integral $\int_{\varepsilon}^1 r^n(1-r)^{k-n}\,dr $
Approximately $\int_0^1 r^n(1-r)^{k-n}\,dr = \frac{1}{(k+1)\dbinom k n}.$.
Aug
23
comment The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer
In general, if you have a monic polynomial $x^2+ax+b$ with integer coefficients and (possibly complex) roots $\alpha,\beta$, $\alpha^n+\beta^n$ is an integer for any $n$, from $\alpha+\beta=-a$, $\alpha\beta=b$ integers and $\alpha^{n+1}+\beta^{n+1}=(\alpha^n+\beta^n)(\alpha+\beta)-\alpha\beta(\alpha^{n‌​-1}+\beta^{n-1})$.
Aug
22
comment Is $\frac{\mathrm d}{\mathrm dx} \sin x/x = \cos x/x - \sin x/x^2$ Lebesgue integrable?
What did you try...?
Aug
20
comment If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?
Cardinality. ${}{}{}$
Aug
20
comment If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?
Since $T$ is uncountable, this means by definition that $\# T>\aleph_0$. But since the union is disjoint, $\# \bigcup_{i=1}^n C_i =\sum_{i=1}^n \# C_i=n\aleph_0=\aleph_0$. But of course $\aleph_0>\aleph_0$ is contradictory. One can prove that the finite union of countable sets is countable with no AOC, but probably someone will comment that countable union of sets countable or that every infinite set has a countable subset are AOC-dependent.
Aug
20
comment $(a, b) = (b, c) = (a, c) = 1$ implies $(c^2, ab) = (ab, a^n - b^n) = (c^2, a^n - b^n) = 1$?
It is true that $(a,bc)=1$ follows from $(a,b),(a,c)=1$. It is also true that $(a,a^n-b^n)=(a,-b^n)=(a,b^n)=(a,b)=1$ since $a^n-b^n=-b^n\mod a$, the same with $(bc,a^n-b^n)$. In general, if $b=b'\mod a$, $(a,b)=(a,b')$.
Aug
19
revised The Sum of ${11^{th}}$ power of the roots of the equation ${x^5+5x+1=0}$
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Aug
19
comment The Sum of ${11^{th}}$ power of the roots of the equation ${x^5+5x+1=0}$
What's with the boldface...?
Aug
18
comment No. of homomorphisms from $\mathbb Z_n$ to $\mathbb Q$
(I'm not sure if the OP is thinking $\Bbb Q$ as an additive of multiplicative group. In the former case the only torsion elt is $0$, in the latter, there are two torsion elts.)
Aug
18
answered No. of homomorphisms from $\mathbb Z_n$ to $\mathbb Q$
Aug
17
comment What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?
Yes, and the formula is proven using inclusion-exclusion by counting surjections.
Aug
17
comment What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?
You might be thinking about the inclusion-exclusion principle.
Aug
17
comment Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.
(Given this argument fails for $2$ instead of $3$, I'm tempted to find a counterexample in the case of $\Bbb Z_2$, or an improved argument).
Aug
16
comment If $R$ is nonunital, is it true that $R/A$ is a field iff $A$ is a maximal ideal?
@JonasMeyer True, I missed that. =)
Aug
16
answered Is $\{f \in End_{\mathbb R}(\mathbb R^n) : d(f(x),f(y))=(x,y) \space \forall x,y \in \mathbb R^n\}$ a group?
Aug
16
comment If $R$ is nonunital, is it true that $R/A$ is a field iff $A$ is a maximal ideal?
A commutative ring is a field iff its only ideals are $(0)$ and $(1)$. If $R$ is any ring and $\mathfrak a$ an ideal, there is a correspondence between the ideals of $R/\mathfrak a$ and the ideals lying over $\mathfrak a$. A maximal ideal $\mathfrak m$ is such that the only ideals lying over $\mathfrak m$ are $\mathfrak m$ and $(1)$.
Aug
13
revised Use Sylow's theorem to show that $G = HN_G(P)$
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Aug
13
answered Use Sylow's theorem to show that $G = HN_G(P)$
Aug
13
revised Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?
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