# All Questions

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### Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
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Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
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### concentration inequalities for quadratic forms of correlated random vectors

Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...
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### Purely inseparable field extensions of degree p

Take a field $k$. If $k'/k$ is a field extension of degree $p$, it is known that there are many possibilities for the isomorphism class of $k'$. See ...
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### Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
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### Divisibility among discriminants

Let $f(x)$ be an algebraic function over the field $\mathcal F$ of algebraic numbers over $\mathbb{Q}$. Suppose that $r \in \mathcal F$. Does the discriminant of $f(r)$ divide the discriminant of ...
A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...