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Nov
28
awarded  Tumbleweed
Nov
21
asked Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function
Oct
23
comment Inverse Mellin of the exponential of the digamma function
and also the signs do not match up: $\mathcal{M}\{\exp(\pi^2/6)\delta(t-\exp(\pi^2/6))\}(s) = \exp((\pi^2/6)s)$ the first term in the product expansion. I do not know if it is possible to do this for $k> 1$ and then perhaps also this wouldn't give you the answer you want even if it does work.
Oct
23
comment Inverse Mellin of the exponential of the digamma function
I mean $m_k$ in $\mathbb{R}$.
Oct
23
comment Inverse Mellin of the exponential of the digamma function
I have a naive idea, but I got stuck with it. I am posting it as a comment in case someone can use it or show it won't work. First, find the inverse Mellin transform of $\mbox{exp}(m_k(-1)^ks^k)$ for each $k>0$ and any $m_k\in\mathbb{Q}$. Use expansion $\psi(s+1) = -\gamma- \sum_{k=1}^{\infty}\zeta(k+1)(-s)^k$ and multiplicative convolution for Mellin transform to arrive at a product $\mbox{exp}(-\gamma)\prod_{k=1}^{\infty}\mbox{exp}(-\zeta(k+1)(-z)^k) = \mbox{exp}(\psi(s+1))$. For example, the Mellin transform of $\mbox{exp}(-\pi^2/6)\delta(t-\mbox{exp}(-\pi^2/6))$ is $\exp(-(\pi^2/6)s)$.
Oct
11
asked quasi-ordinary singularities on a versal deformation?
May
25
awarded  Nice Answer
May
24
revised John Nash's Mathematical Legacy
added 3 characters in body
May
24
revised John Nash's Mathematical Legacy
added 3 characters in body
May
24
answered John Nash's Mathematical Legacy
May
8
comment When is the Hom-scheme connected?
oh, I thought of a trivial example: $Hom_k(spec(k), spec(k) \sqcup spec(k)) $. I think $A$ and $B$ should local artinian rings with residue field $k$ (or at least a finite extension of $k$). Then, I truly do not know of example having worked with such things for a few years.
May
7
comment When is the Hom-scheme connected?
I like this question. When A and B are (truncated) infinitesimal neighborhoods of a smooth point, then $Hom$ will be connected (in fact, it will be iso to some $\mathbb{A}^M$). But, at non-singular points, explicit computations show that the space will often not be irreducible (yet remain connected). Off the top of my head at the moment, I cannot think of an example where $Hom$ is disconnected. The same question may be asked for non finite algebras, if you are willing to work with formal schemes.
Jan
16
revised Why do we need localization by Leftschetz motive?
edited body
Jan
16
answered Why do we need localization by Leftschetz motive?
Jan
16
comment Why do we need localization by Leftschetz motive?
Just to nit-pick, in any commutative ring $R$ and given any element $f$ of $R$, $R_f=$ the localization with respect to the stably multiplicative subset $\{1,f, f^2,\ldots\}$ is a ring. In particular $f^{-1}$ always exists in $R_f$ and is nonzero whenever $f$ is not nilpotent.
Jan
3
comment Why is the CM closure of $\mathbb{Q}$ the “ultimate” coefficient field for motives?
This is a guess: it probably has something to do with periods. From section 3 of maths.ed.ac.uk/~aar/papers/kontzagi.pdf, if $f$ is a modular form of positive weight $k$ and $z_0\in\mathbb{H}$ is a CM point, then $\pi^kf(z_0)$ is a period. Among other things, there is a connection between nori motives and periods.
Oct
20
comment Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$
Actually, this way is very easy also $\sum_{j=0}^{k-1} j^m = \frac{1}{m}(B_m(k) - B_m)$ which will immediately imply the result which you proved. The Bernoulli numbers were a red-herring.
Oct
20
accepted Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$
Oct
20
comment Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$
ah, yes, now the other hint makes sense too. Thanks for your help. Your answer implies that $(B_m(k)-B_m)/mn^m$ goes to zero as $m$ approaches infinity provided $n>1$. I thought that maybe there is a way to prove this second fact directly.
Oct
20
comment Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$
Thanks for your hint. Yes, it is true, but the events $E_i$ given by $\{(l_j) \mid l_i = n\}$ are not mutually exclusive. I should say that neither probability theory nor analysis are my specialty so perhaps I missed the hint.