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0
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1answer
69 views

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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0answers
16 views

Is it possible to design an algorithm an optimization problem with non-convex and coupled constraints, using decomposition methods?

Recently, I am working on an optimization problem in wireless networks. I formulate the problem to be an optimization problem with non-convex and coupled non-linear constraints. And I want to design a ...
5
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1answer
1k views

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 ...
2
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1answer
186 views

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 ...
1
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1answer
82 views

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 ...
3
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1answer
114 views

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 ...
5
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0answers
133 views

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) ...
1
vote
1answer
444 views

Eigenfunctions and eigenvalues of the product of two exponential kernels

Consider the following exponential kernel: $k(x_1, x_2) = \exp\left(\frac{|x_1 - x_2|}{L}\right)$, which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have $k(x_1, x_2) ...
1
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1answer
254 views

Do signed measures on sigma-rings always have a Hahn decomposition?

Let $X$ be a set. Let $\mathcal{R}$ be a set of subsets of $X$ such that $\{\} \in \mathcal{R}$ and For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} ...
2
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3answers
628 views

Decomposing a discrete signal into a sum of rectangle functions

Hello mathoverflow community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
7
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4answers
6k views

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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0answers
471 views

Which statement do people usually call the Decomposition Theorem, and what is the precise reference for it?

Which statement is usually called the Decomposition Theorem (for perverse sheaves)? Is this (roughly): a proper pushforward of an intersection complex could be decomposed into a direct sum of ...
8
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2answers
2k views

Iwasawa Decomposition & Polar Decomposition related how ?

In an earlier post (Use Lie Sub-Groups of GL(3, R) for elastic deformation ? here), I mentioned polar decompositions as in F = RU where R in SO(3) & U in symmetric positive-semidefinite matrices. ...
5
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1answer
583 views

Easy special cases of the decomposition theorem?

The decomposition theorem states roughly, that the pushforward of an IC complex, along a proper map decomposes into a direct sum of shifted IC complexes. Are there special cases for the decomposition ...
4
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1answer
452 views

Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
16
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3answers
2k views

Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
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3answers
2k views

Examples for Decomposition Theorem

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne. Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...
8
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4answers
2k views

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 ...
5
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2answers
667 views

Explicit Direct Summands in the Decomposition Theorem

Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...