@Sawarnik @user63477 Now prove a generalization : $\tau(k)$ be the number of divisors of $k$. prove that $$\sum_{k | n} \tau(k)^3 = \left ( \sum_{k|n} \tau(k) \right )^2$$
Well, I have a follow up that you can try : Can you find a multiset with sum of cube of elements equal to square of their sum, not belonging to the divisor class above? @MartinSleziak
@BalarkaSen by that condition $f$ is monotone increasing, therefore differentiable almost everywhere. However, by the condition given it is nowhere differentiable.
@MartinSleziak can we say its bacause ie $(-\infty,5)\cup(6,\infty)$ and $(-3,\infty)\cup(-\infty,7)$ are in in $\tau_1 \cup \tau_2$ but intersection of these two subset $(-3,5)\cup (6,7)$ isnt in $\tau_1 \cup \tau_2$
I initially ended up in 'upper room' by mistake, a stack exchange chat room where the discussions are regarding the difference between evangelicals and protestants...
@AntonioVargas Can I ask an analysis question...Do you know what kind of mapping there exists between R^n and its dual, I know that every member of the dual maps to a row vector in R^n but I'm not sure what the properties of this mapping is, do you have any idea?
@AntonioVargas I'm just saying that any function from the dual space (R^n)' can be mapped to some row vector [a_1,...,a_n] in R^{n}. So I was just interested as to what properties this mapping between R^n and and its dual R^n' has...
@JohnDoe Ohh I see what you mean now. Yes the mapping should be bijective and linear, i.e. if $\lambda$ is the mapping then $\lambda(a_1 \mathrm{e}^1 + a_2 \mathrm{e}^2) = a_1 \lambda(\mathrm{e}^1) + a_2 \lambda(\mathrm{e}^2)$
But I don't really know much more than that. It's not my field.
@lyme What I meant was that $\{1\}=(0,1]\cap[1,2)$ is an intersection of two sets from $\tau_1\cup \tau_2$, but it does not belong to $\tau_1\cup \tau_2$ itself.
@r9m My research led me to some discoveries done by Lewin. So, here is a nice result $$\int_0^{\pi/2} \int_0^{\pi/2} \operatorname{Li}_2(-y^2 \tan^2(x)) \ dx \ dy=\pi^2\log\left(1+\frac{\pi}{2}\right)+2\pi \log(2+\pi)+\pi^2 \operatorname{Li}_2\left(-\frac{\pi}{2}\right)-2\pi\log(2)-\pi^2 $$
@seaturtles Suppose we're given $n$ integers $a_1,\ldots,a_n$. Then there exists $1\leqslant k\leqslant \ell\leqslant n$ such that $a_k+a_{k+1}+\ldots+a_\ell$ is divisible by $n$.
Suppose we have finite sets $A,B$, and let $R$ be a relation on $A\times B$. There are is $x\in A$ such that $\#{\rm cl}(a)=\#\{b\in B:a\sim b\}\geqslant \# R/\#A$.
@PedroTamaroff Yes... Rather than ask a question either state they are or tell them to show that they are "show that the $a_k$ are $n^\text{th}$ roots of unity"
