99,228 reputation
887150
bio website
location
age
visits member for 1 year, 6 months
seen 3 hours ago

A long time ago, I studied mathematics. I used to know three or four things about complex analysis, topological vector spaces, and point set topology. But I've forgotten at least five of them.


6h
comment Why $x\in C_k$ implies $x/3\in C_{k+1}$.
Induction would be the method of choice, I think. You could write $$C_k = \bigcup_{n=0}^{2^k-1} \frac{1}{3^k}\cdot [a_n^{(k)}, a_n^{(k)}+1],$$ then look how the $a_n^{(k)}$ develop. Looking at the $a_n^{(k)}$ in base three for $k = 0,1,2,3$ should give you an idea how the $a_n^{(k)}$ look, which you would also prove via induction. That's a little tedious, however (but not difficult once you've seen what goes on). I may be overlooking something simpler.
6h
comment Why $x\in C_k$ implies $x/3\in C_{k+1}$.
You could prove that that is equivalent to the construction above.
6h
comment Why $x\in C_k$ implies $x/3\in C_{k+1}$.
The argument depends on how exactly $C_{k+1}$ is defined. If you define $C_{k+1} = \frac{1}{3}\cdot C_k \cup \bigl( 1 - \frac{1}{3}\cdot C_k\bigr)$, it is immediate. If you use a different (but of course equivalent) definition, you need a different argument.
6h
comment Determining a radius convergence of a power series
What would count as immediate?
7h
awarded  Constituent
10h
comment Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $
The sequence $(g_k)$ is monotonic.
10h
comment Is $\mathbb{Z}_p$ a Finite Field?
Many people prefer blackboard bold also on paper, there's nothing wrong with that. I understood your point was the subscript, but my point was, the subscript notation is also standard for the ring of integers modulo $n$ or $p$, nothing wrong with that either.
11h
comment Is $\mathbb{Z}_p$ a Finite Field?
$\mathbf{Z}_n$ or $\mathbb{Z}_n$ is also a widely used standard notation for the ring of integers modulo $n$. Usually, if $\mathbf{Z}_p$ is used, it is clear from the context whether the ring of integers modulo $p$ or the $p$-adic integers are meant. If not, one needs to explicitly say which is meant.
12h
reviewed Close Linear regression, analyze of correlation, remove a variable
13h
comment What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
Yes. I'd have taken $x = 0$, but yours works too.
13h
comment Theorem: the first positive number to have 500 divisors has to be even.
It may help to try proving the stronger "Let $d > 1$. Then the smallest positive integer having $d$ divisors is even."
13h
comment Simply connected and connected in complex analysis
There ought to be a "... the set is connected and ..." (or something equivalent) in there. It may be that the book's author(s) forgot it, that's an erratum then, or you may have missed it. Check again carefully to see which is the case.
13h
comment What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
That's easy to check. Suppose $0 < p < 1$. Is there an $x\in\mathbb{R}$ with $\{x\} \neq \{x+p\}$?
14h
reviewed Close advice for self studying
14h
reviewed Looks OK An example of finite, connected topological group
14h
comment An example of finite, connected topological group
The question asks for a connected topological group.
1d
comment An abelian subgroup of symmetric group
If $p$ is a prime, what does an element of $S_n$ of order $p$ look like?
1d
comment Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions
The characteristic function of a suitable Borel set works. What kind of Borel set might work?
1d
reviewed Close Prove that ${\frac{d}{{dt}}_{t = 0}}(\det ({e^{tX}})) = Tr(X)?$
2d
comment Solving this complicated integral using the Residue Theorem
No, if $m$ is not an integer, then your integrand doesn't have a pole at $z^\ast$ but a branch-point, and you must have a branch-cut from $z^\ast$ to $\infty$. Sometimes, one can use branch-cuts and the residue theorem to compute integrals (things like $\int_0^\infty \frac{\log x}{1+x^2}\,dx$), but I don't see a way to do that here for non-integer $m$.