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Apr
20
awarded  Yearling
Jan
9
comment How important are automorphic representations among admissible ones?
Thanks for the answer!
Jan
2
comment How prove this $XY=YX$
If $X$ and $Y$ commute with $A$ then they also commute with $A^{-1}$...
Oct
6
revised Help solving this Linear First order ODE
There were no "{}"-brackets for exponentials, making the formulae unreadable. I added them where it seemed the case, the OP should check.
Oct
6
suggested approved edit on Help solving this Linear First order ODE
Sep
17
asked Analytic Continuation of a nowhere existing Mellin Transform
Sep
16
comment How to calculate the following double integral
Do you have a proof for the $a=c=0$ case? I think that the knowledge of the right integral representation of $K_\nu$ could unlock the problem...
Sep
16
comment How to calculate the following double integral
There is a small difference between your expression and the one of the Normal Product Distribution you cite, i.e. in the denominators of the exponents you should take $2b^2$ and $2d^2$ instead of $2b$ and $2d$. Is it just a typo or you want to compute exactly what you wrote?
Jul
10
revised Turning a Line Integral into a Contour one
added 136 characters in body
Jul
10
comment Turning a Line Integral into a Contour one
@DanielFischer Yes, this agrees with my computation. I think I could try to compute the additional pole (and relative residue) by brute force, but it's gonna be long and not elegant. While the observation of the authors of the paper on the possibility of turning it into a contour integral with only one pole is very tantalizing. Do you think it is possible to do that? On the other hand, according to the suggestions of Avitus I was trying to play a bit with different contours to exclude this additional pole. Thanks for your interest!
Jul
10
revised Turning a Line Integral into a Contour one
added 1 characters in body
Jul
10
comment Turning a Line Integral into a Contour one
Problems arise if I try some "classic" transformation. $z:=e^{i\theta}$ gives rise to a function holomorphic in $\mathbb{C}\setminus\{0\}$, but the integration path is not closed (it is the right semi-circle). For the transformation $z:=e^{2i\theta}$ the integration path is closed (the unit circle) but $i\sin(\theta)= \frac{1}{2}(\sqrt{z} -\frac{1}{\sqrt{z}})$ which is not holomorphic on the whole unit circle. I think those are just two expressions of the same pathology, i.e. the original integrand is not $\pi$-periodic. What do you think? Thanks again!
Jul
10
comment Turning a Line Integral into a Contour one
@Avitus Thanks for your response! Actually we are supposed to consider both $B$ and $\lambda$ as parameters. As you say the imaginary part of the denominator is killed by $\theta=k\pi$, but the real part is always $\neq0$ for any $\theta$ (excluding at most 2 "annoying" choices of $\lambda$, but we are allowed to do it). In specific the integrand, as a function of $\theta$, is non singular.
Jul
10
asked Turning a Line Integral into a Contour one
Jul
1
comment Finding the integer solutions of $246x + 217y = 3$
Ooops, Clearly $7+1+2\neq 9$ then! I apologize for the mistake... ...in this case we simply observe that $3/3$ and $3/246$ implies that $3/y$ for any solution $y$. So all the solutions are of the form $(x_0,3y_0)$ where $(x_0,y_0)$ is a solution of $\frac{246}{3}x +217y =\frac{3}{3}$. The strategy of the comment above applies to this equation. PS @Tomas, thanks for pointing it out!
Jul
1
comment Finding the integer solutions of $246x + 217y = 3$
You can simplify the equation observing that all the coefficients are divisible by 3. Then apply the Euclidean algorithm en.wikipedia.org/wiki/Extended_Euclidean_algorithm to compute a solution, and finally use Bezout's identity en.wikipedia.org/wiki/Bézout's_identity to compute all of them.
Jun
25
comment How important are automorphic representations among admissible ones?
Thanks! This is the kind of answer I was looking for! In specific I find extremely enlightening the observation about "factorization of automorphic forms" vs "factorization of automorphic representation", I completely oversaw it while studying!
Jun
25
accepted How important are automorphic representations among admissible ones?
Jun
20
asked How important are automorphic representations among admissible ones?
Jun
20
comment Definition of Automorphic Representation
What is the definition of weakly contained in $L^2(G_F\backslash G_A)$?