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Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$\begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases}$$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
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Is $LIS(\pi)+LIS(\sigma)+LIS(\sigma\pi^{-1})$ lower bounded?

In the title, $LIS$ stands for the length of longest increasing subsequence and Greek letters stand for permutations from symmetric group $S_n$. Considering some cases such as ...
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What is the full subcategory of $\it{Cat}$ consisting of categories $\mathbb{C}$ where $\mathsf{ob}(\mathbb{C}) = \mathsf{arr}(\mathbb{C})$ hold?

Let $\it{Cat}'$ be the full subcategory of $\it{Cat}$ such that $\mathsf{ob}(\it{Cat}') = \{ \mathbb{C} \in \it{Cat} \mid \mathsf{ob}(\mathbb{C}) = \mathsf{arr}(\mathbb{C}) \}$. The questions are as ...
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Some calculus in $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...