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4h
comment Evaluating integral involving product of cosine inverse
I'm truly not trying to be a jerk here ... but you should really master integration by parts in the single-integral setting before trying to do double integrals at all.
13h
comment Evaluating integral involving product of cosine inverse
Maybe start by switching the order of integration and integrate by parts to get rid of the second arccos?
13h
revised Evaluating integral involving product of cosine inverse
added 6 characters in body; edited title
20h
answered $P(n)$ is product of all digits of $n$. Find all $n$ such that $P(n)$ = $n^2āˆ’10nāˆ’22$.
1d
comment On multiplicity representations of integer partitions of fixed length
I think that sort of truncated statistic is much harder to access with generating functions; I'd be inclined to try to understand it by induction on $L$. As for a source for generating functions, I recommend the excellent and capitvating book "Concrete Mathematics" by Graham, Knuth, and Patashnik.
1d
comment Find the dimension of a subspace by find a basis for the null space.
Isn't the set in question precisely the null space of $[2\;{-3}\;1]$?
2d
comment Find the dimension of a subspace by find a basis for the null space.
The question says "by exhibiting a basis for the null space". The null space of what? I worry that it's asking you to consider that set,z not as the image of some matrix (as you've done), but as the null space of the matrix $[2\; {-3}\; 1]$. (Also note that you dropped a negative sign in the second column of your first matrix.)
2d
answered alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?
2d
comment A and B are sets. Prove that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$
Your first step doesn't give $x\in A$; it gives $X\in A$.
2d
answered On multiplicity representations of integer partitions of fixed length
Sep
19
comment How to find if an integral is possible to compute: Failing to solve integral for quadratic functional
What does $dt\, dt$ mean?
Sep
19
comment A bit stuck on proving a function, $ f : [-1,1] \rightarrow \mathbb{R} $ is differentiable from the definition.
At $x=0$ it should be easy. For $x>0$ you can ignore the absolute values in the definition of $f_n$, since the limit considers only $c$ near $x$; similarly for $x<0$. Don't know if that helps you....
Sep
18
revised Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$
edited title
Sep
18
comment Cauchy product $\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}a_{n-k}b_k$
This is essentially Fubini's theorem on the measure space $\Bbb Z^2$ with the counting measure. The proof of this case will differ from the proof in the question you linked to only in the bookkeeping of indices.
Sep
18
comment Number of integer lattice points within a circle
The supposed formula $N(n)-N(n-1) = 4[n\sqrt2]$ is really, really false. For one thing, the asymptotic size of the difference is $\pi n^2 - \pi(n-1)^2 \sim 2\pi n$, not $4\sqrt2 n$. Surely your computations revealed that?
Sep
17
comment How can I find out when a system of linear equations have a non-trivial solution?
In that case, I'd first recommend that you pick a specific value of $a$ like $6$ and understand how to use both augmented matrices and determinants to answer the question for that $a$.
Sep
17
comment Vectors and polyhedra: a surprising fact
You can decompose a polyhedron into the union of tetrahedra whose interiors are disjoint and which meet in full faces, like you did for $n=6$. (This is the three-dimensional analogue of a triangulation of a polygon.) That's probably what you're looking for. By the way, don't forget $n=5$!
Sep
17
revised Evaluate the integral $\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$
edited title
Sep
17
comment Is this proof correct? concerning abelian groups
Ok, well any two countable sets can be put in bijection with $\Bbb N$ and hence with each other. But your map is not a bijection. Consider the case $r=0$. You're claiming that there's a bijection between the finite abelian group $C(\Bbb Q)^{tors}$ and the free abelian group $(T,+)$, but the latter set is infinite.
Sep
16
comment quesiton about residuals and poles
That's a valid worry, but it turns out not to happen. The last term "blows up the fastest", swamping all other terms. One way to see this is to write $f(z) = (z-z_0)^{-n} \cdot \big\{ (z-z_0)^n f(z) \big\}$; the second factor approaches $a_{-n}$ as $z\to z_0$, while the first factor blows up, so the product $f(z)$ must blow up.