# All Questions

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### Existence of $\sqrt{2}$ in a finite group algebra over $\mathbb{Q}$

I cannot find a finite group $G$ such that $\exists x\in \mathbb{Q}[G]$ with $x^2=2e$, where $\mathbb{Q}[G]$ is the group algebra of $G$ over $\mathbb{Q}$. I also could not prove it does not exist. ...
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### How to simplify this homotopy totalization coming from an arc-cover into a pullback?

My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know ...
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### Digit sum of a prime number

Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime? e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)...
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### Inclusion-exclusion in a set of multivariables

I want to determine the risk of a multi-asset portfolio, in a way different from previous attempts. It is because current methods focus on just the correlation coefficient of two variables, while in a ...
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### Cluster expansion, Mayer expansion and perturbative renormalization group

This is a second part of my previous question, which I decided to split into two parts not to mix up different topics at one giant question. Again, according to V. Rivasseau (section 1.5 of ...
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### What is a large field problem?

I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify. On page 2, Rivasseau talks about the large field problem and, if I understood it ...
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### Let $V$ be a variety. A point $P \in V$ is nonsingular iff $\dim_k(M_P/M_{P}^{2})=\dim(V)$ [closed]

First of all, we consider $k$ to be an algebraically closed field, and by $M_P$ I denote the maximal ideal of the coordinate ring $k[V]$ at $P$. As for the statement, I have managed to understand how ...
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### Possible characterisation of compactly generated weakly Hausdorff spaces

Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of ...
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### When does the null-cone consist entirely of eigenvectors?

Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$. For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
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### Examples of homology sphere that bound a nonsmoothable contractible 4-manifold

Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible ...
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### A $p$-adic homotopy theory for non-simply connected spaces?

I'm looking to understand the state of the art for $p$-adic (unstable) homotopy theory of non-simply connected (non-nilpotent!) spaces. Ideally, I'd also like integral versions, e.g. things like ...
The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a ...