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### Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...
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### Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain. The question is all in the title: is there ...
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### Gabber's original proof of his purity theorem

Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$. It is remarkable because it ...
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### Simple decomposition of $K_{2n}-I$ into hamiltonian cycles

http://mathworld.wolfram.com/HamiltonDecomposition.html In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...
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### Is there an elementary proof of the polar factorization theorem for vector-valued function?

I have recently learned the polar factorization theorem for vector-valued functions due to Brenier. Namely, given a probability space $(X,\mu)$ and a bounded domain $\Omega\subset \mathbb{R}^n$ with ...
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### What's the relationship between the different versions of the BBD decomposition theorem?

I have a few questions relating to the BBD decomposition theorem. I have come across the following two versions of the decomposition theorem. Version 1. Let $f : X \to Y$ be a proper map of ...
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### complexity of eigenvalue decomposition

what is the computational complexity of eigenvalue decomposition for a unitary matrix? is O(n^3) a correct answer?
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### Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space. Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
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### When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...
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### A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a connected finite dimensional Lie group $G$ so that ...
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### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values. K is a sparse banded matrix ...
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### Trace of multiplied positive definite matrices

I have to compute $Tr(K^{-1}\Sigma)$ where both $K$ and $\Sigma$ are symmetric positive definite matrices. Question is considering that I have computed the Cholesky, $L_1$ of $K$ previously, is there ...
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### Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer? More precisely: $F \subset S$ is a proper connected (usually reducible) half-...
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### How to do Computations Using the Decomposition Theorem for Perverse Sheaves

This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general. My question is how does one use the ...