All Questions

10 views

Residue fields of attached to coefficients of modular forms

Let $f = \sum_n a_n q^n$ be a cuspidal newform of some weight and level. Here I want to view the $a_n$ of $p$-adic numbers (by embedding $\overline{{\bf Q}}_p$ in ${\bf C}$ in so …
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Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig ident …
31 views

A double grading of catalan numbers

This is something I found in trying to work on Vince Vatter's excellent question. I have no solution, but a much more precise conjecture. Recall that a rooted planar tree is a roo …
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How can this fail if my pairing is nondegenerate?

I have two infinite-dimensional (Frechet) linear spaces $X$ and $Y$ over $\mathbb R$, and a nondegenerate pairing $\langle\cdot,\cdot\rangle:X\times Y\to \mathbb R$. Neither $X$ no …
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Existence of particular embeddings in euclidean spaces for non compact manifolds

Let $M$ be a $n$-dimensional smooth connected not compact manifold s.t. groups (singular cohomology) $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a suff …
243 views

Where is the belly button of the Universe?

It's fine and nice and wonderful when a part of learning mathematics is chaotic, ad hoc, spontaneous, social, ... However it would be perhaps of fundamental value to know a very c …
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What is the inverse of this kind of integral transform?

Let $$\hat{f}(\lambda):= \int_0^{+\infty}K(x,\lambda)\ f(x)\ dx, \text{where } K(x,\lambda)=\sqrt{\frac{2}{\pi}}\frac{\lambda \cos(\lambda x)+h\sin(\lambda x)}{\lambda^2+h^2}$$ be …
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Closure of one relation w.r.t other

I have two relations R and R' satisfying following property - R(a,b) & R'(a,a') & R(a',b') => R(a,b') Pictorially, it looks like this - a --(R)-> b | \ (R') \ (* n …
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Why this synchronization error dynamic for Krasovskii-Lyapunov?

I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front. The problem is …
113 views

Why did Bourbaki not use universal algebra?

I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra? Well, universal algebra is not much …
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How many trees can be constructed from k vertices using an LCA operator?

Consider the class of rooted trees and suppose I have at my disposal a lowest common ancestor (LCA) operator given by  \textrm{lca}(u,v) = \text{the lowest common ancestor of $u$ …
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Extension of equivalent norms

Let $(X,||\cdot||_1)$ be a normed space and $Y$ a linear subspace of $X$. Let $||\cdot||_2$ be a norm on $X$ which is equivalent to $||\cdot||_1$ on $Y$. Does there exist a norm on …
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Why does Grothendieck’s period conjecture imply Hodge’s conjecture ?

Hello to all of you : I would like to know if it is true that the Grothendieck period conjecture implies the Hodge conjecture in the case of non-singular complex algebraic varietie …
Let $A(t)$ and $B(t)$ be matrices with each element in $L^\infty(0,T).$ Let $A(t)$ have an inverse. I know nothing else about this inverse. Let $c(t)$ be a vector in $L^2(0,T).$ …