# All Questions

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### Does exterior product commute functor Hom?

Let $M$ be an module over the commutative ring $R$. I'd like to ask do we have the following isomorphism? $$Hom_R(\wedge^n_RM,R)\simeq \wedge^n_R Hom_R(M,R)$$ We can obviously see it's true for the ...
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### Solve a PDE related to free boundary problem

I would like to solve the following system for my problem: $$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$ where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
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### Results true in a dimension and false for higher dimensions

Some theorems are true in vector spaces for a given dimension $n$ but become false in higher dimensions. Here are two examples: A positive polynomial not reaching its minimum. Impossible in ...
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### Almost periodic sequence

Say that a real sequence ${(x_k)}_{k \in \mathbb{Z}}$ is almost periodic if the set of all its shifted sequences ${\left\{{(x_k)}_{k+n \in \mathbb{Z}}\right\}}_{n \in \mathbb{Z}}$ is relatively ...
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### Minimizing deep holes in sphere packings

What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I ...
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### Optimize the distribution if it is left unsmoothed

I have a question about distribution. Let see my problem The paper said that the distributions p and q are left unsmoothed, so we can ignore Kernel density. But I don't understand what is left ...
### On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements. We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...
In more detail, let $F$ be any field and $A$ a set of matrices, $F\cdot Id_n \subseteq A\subseteq GL_n(F)$, closed under addition and product, which is irreducible: If \$\{0\}\subsetneq V \subsetneq ...