# Questions tagged [decomposition-theorem]

In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.

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### Adding valid cuts for integer feasibility problem under Benders decomposition framework?

Traditional infeasibility cut is derived under the assumption that the feasibility problem is LP instead of ILP and relies on the dual form of the LP.
Is there a systematic way of adding valid cuts ...

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### Decomposition theorem over more general base schemes

The BBDG decomposition theorem says that if $f\colon X \to Y$ is a projective morphism of finite type $\mathbf{C}$-schemes and $X$ is smooth of (pure) dimension $d$ then $\mathbf{R}f_*\mathbf{Q}_\ell[...

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### Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them

I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...

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### Decomposition of direct image of a smooth morphism, Deligne's theorem, motives

Let $f : X \to Y$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $X$ and $Y$, so use classical topology for simplicity.
I can take the ...

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### Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?

I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...

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### Etale cohomology of a nodal (cuspidal) curve

Let $k$ be a separably closed field, and $X/k$ be a curve (not necessarily complete) with a single singularity, a simple node $x$. Suppose $\ell$ is a prime number invertible in $k$, how do we compute ...

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### Intersection homology of toric resolutions

I'm interested in the intersection homology of toric varieties associated to a polytope $P$ with proper faces F, and a subdivision $P'$ of P. Let $X_P$ be the toric variety associated to the polytope $...

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### What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?

Let $f\colon X\to Y$ be a surjective morphism of smooth projective varieties. If the decomposition theorem for $f$ is given by $$Rf_*\mathbb{C} \simeq \bigoplus_i R^if_*\mathbb{C}[-i],$$ what are the ...

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### Is it possible to prove the Jordan decomposition starting from Schur's decomposition?

Schur's decomposition says any matrix $A$ is similar to a upper triangular matrix $U$ i.e., there exists unitary $Q$ such that $A = Q^{-1}UQ$. If we split $U$ as $D+N$ where $D$ is the diagonal part ...

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### Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is ...

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### What is the elementary proof of Weil's polynomial theorem of decomposition?

André Weil sometimes glosses his Theorem of Decomposition in a simplified polynomial form:
If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically
prime to each other, with integer ...

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### Being additive map using spectral decomposition theorem

Let $(\mathcal{A},\tau_\mathcal{A})$ and $(\mathcal{B},\tau_\mathcal{B})$ be semifinite von Neumann algebras with normal semifinite faithful traces $\tau_\mathcal{A}$ and $\tau_\mathcal{B}$. I defined ...

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### Efficient scissors congruence between efficiently describable convex polytopes and simplex?

Is there a convex polytope in $\mathbb R^n$ describable by only $O(poly(\log n))$ half-plane inequalities with positive volume (so at least $n+1$ vertices) such that the standard simplex has a ...

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### Connections between eigenvectors after matrix multiplication

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...

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### Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n}$. $D$ is a positive diagonal matrix and $m > n$.
Is there any equality or inequality over $|B|$, $|AA^...

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### Find a minimum set of paths that cover all pairs of dependent vertices

Let $G=(V,A)$ be a simple directed acyclic graph. A set consisting of two vertices is called dependent if there is a directed path from one of the vertices to the other. The question is to find ...

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### What is this matrix decomposition called and does it exist always? - II

Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...

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### What is this matrix decomposition called and does it exist always?

Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds?
...

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### Singular Value decomposition of huge dimensional matrix

i would like to consider singular value decomposition of such type of matrix
creation of matrix from small sample is not big issue, i have ready code for this
...

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### Decomposition of Henstock-Kurzweil-integrable functions

Let $f:[a,b]\to\mathbb R$ be a Henstock-Kurzweil-integrable function (short: HK-integrable).
Can $f$ always be written as a sum of a Lebesgue-integrable function and a function which has a ...

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### The closure of cyclic modules under direct sums and direct summands

Cohen and Kaplansky have proven that a commutative ring $R$ has the property
C: Every $R$-module is a direct sum of cyclic $R$-modules.
if and only if $R$ is an Artinian principal ideal ring. Can ...

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### How to find $K,W,S$ in the Mostow decomposition theorem?

The Mostow decomposition theorem states:
Let $Z$ a nonsingular complex matrix, then $Z$ can be factored as:
$$Z=We^{iK}e^S$$ where: $W$ is unitary, $K$ is real and skew symmetric and $S$ is real and ...

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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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### 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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### 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) =...

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### 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 $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$...

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### 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 (rectangular)...

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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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### 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 (shifted)...

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### 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. ...

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### 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 ...

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### 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) ...

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### 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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### 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 ...

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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 ...

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### 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 ...