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If I understand right the character table of an association scheme is the table of irreducible representations of the Bose-Mesner algebra. If we can recover this algebra, and the adjaceny matrices wi …

Here are two relevant counterexamples: Let $R$ be a ring and $I$ an ideal, then the morphism $\operatorname{Spec} R[x]/(x-x^2,Ix) \to \operatorname{Spec} R$ has connected fibers outside $V(I)$ but di …

Can we just do this the hard way? First we construct the set of points of the etalification $\bar{Z}$. A point of the etalification is a point $P$ of $X$ plus a map from the spectrum $T$ of the etal …

A corrected version of this argument is contained in Section 4 of Six Lonely Runners by Bohman, Holzman, and Kleitman. Their argument shows, using equidistribution, that the case of the lonely runner …

Let $G$ be a complex, not necessarily connected reductive group and let $K$ be a maximal compact subgroup. There is a natural function from algebraic representations of $G$ to continuous representatio …

I think the limit of the red cone as the green point approaches the base is a quadrant, in agreement with The Masked Avenger's earlier impressions but not his later ones. Imagine viewing the circle $ …

Let's use the notation $\{ x\}$ for the fractional part of a number $x$. Assume $u, v$, and $u/v$ are all irrational. Then, $\{un\}$ and $\{vn\}$ behave as independent uniform random variables. (Thi …

Let $X$ be the total number of plays and let $Y$ be the number of plays for the track with the most plays. I think a good strategy is to estimate $$E [ Y- X | X = X_0 ]$$ It seems like this invol …

Based on Christian Remling's answer, we assume $b-a<1$. We know that $f$ has a pole on $b$. I don't know whether such an $f$ exists but I would at least like to understand the order of growth at this …

They are not always independent. There are two problematic cases: the trivial case $a=b$, and the trinomial case when $c_1x^n + c_2x^m +c_3 =0$ for rational $c_1,c_2,c_3$, $a= x^n$, and $ b= x^m$, so …

Given a metric space $X$ with $\operatorname{cov}(X,\epsilon) \leq N(\epsilon)$, there exist $N(\epsilon_1)$ points such that each point is within $\epsilon_1$ of each of them. That set of $N(\epsilon …

To see when this is possible, we can consider each orbit of $G \times H$ separately. An orbit of $G \times H$ corresponds to a subgroup of $G \times H$. A subgroup of $G \times H$ corresponds to a tr …

The key point is that unitary matrices have orthogonal eigenvectors, thus you can form an orthonormal basis of eigenvectors, which is the same thing as a unitary matrix $A$ with the property you descr …

If $G=\mathbb Z^d \rtimes H$ for some group $H$ that acts on $\mathbb Z^d$, there is indeed a natural bijection $G/H \cong \mathbb Z^d$. As Qiaochu says, what $H$ would be depends on the structure yo …

One justification for this is the Euler product expression. To find the Euler product expression for the Hasse-Weil function, you have to ask yourself what the appropriate analogue of a prime is. It t …