29 votes
Accepted

Gabber's original proof of his purity theorem

It's different, but it also uses Weil II. See Purity for intersection cohomology after Deligne-Gabber for my translation of the original.
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16 votes
Accepted

When does a perverse sheaf occur in the decomposition theorem?

In general it is a difficult problem. For example, the core of Ngô's proof of the fundamental lemma is his support theorem which implies that in the context of the Hitchin fibration, all simple ...
11 votes
Accepted

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

If $P(x,y),Q(x,y)$ are relatively prime, then so are the one-variable polynomials $p(x)=P(x,1),q(x)=Q(x,1)$ (since we can homogenize any common factor of $p,q$ to a common factor of $P,Q$). It follows ...
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7 votes

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

I don't know how to characterize such morphisms, which I think is your first question. However, this can certainly happen, even if $f$ is not smooth. (By the way, your comment about absence of local ...
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5 votes
Accepted

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

No, you can't always find two such matrices. Let $n \gt 2$. Choose a $2$-dimensional subspace $V$ of $\mathbb{Q}^n$ that does not intersect the positive orthant except at $\vec 0$. Let $M$ be a ...
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4 votes

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

Yes, there is, using exterior product as a functor: Let $e_i$ be the basis of $\mathbb R^n$. Then \begin{align} \det(A.D.A^\top)e_1\wedge\dots\wedge e_n &= \Lambda^n(A.D.A^\top)(e_1\wedge\dots\...
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4 votes

Singular Value decomposition of huge dimensional matrix

This problem of updating the singular value decomposition of a matrix upon repeatedly appending a row or a column to the matrix is discussed in A stable and fast algorithm for updating the singular ...
4 votes
Accepted

Decomposition of Henstock-Kurzweil-integrable functions

The answer is negative. Consider the space $\Delta'=\Delta'([a,b])$ of functions, which are classical derivatives of functions $[a,b]\to\mathbb R$. Assume the claim was true. Then any HK-integrable $f:...
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4 votes
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The closure of cyclic modules under direct sums and direct summands

The introduction of Warfield, R.B.jun, Rings whose modules have nice decompositions, Math. Z. 125, 187-192 (1972). ZBL0218.13012. says It is also shown that if $R$ is a commutative ring and there ...
2 votes

complexity of eigenvalue decomposition

I think the other answers are wrong. I periodically look up this problem and I believe it to be open. I will summarize my opinion: The symmetric eigenvalue problems is "solved". Wilkinson was able to ...
2 votes
Accepted

Simple decomposition of $K_{2n}-I$ into hamiltonian cycles

In a 1-factor decomposition of $K_{2n}$ which you describe (an edge from the center of a regular $(2n-1)$-gon to a vertex and all chords perpendicular to it), two matchings which correspond to `almost ...
2 votes

Is there an elementary proof of the polar factorization theorem for vector-valued function?

Here's an elementary type argument I learned, based on unique splitting $T=S+A$ of linear operators $T: \mathbb{R}^n \to \mathbb{R}^n$. (i) every linear operator $T$ can be uniquely decomposed $T=S+A$ ...
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2 votes
Accepted

Connections between eigenvectors after matrix multiplication

$U_s$ is not recoverable from $H$ and $U_A$. Consider the following examples, one each for $M < N, M > N, M = N$. In each example, there are 2 different $R$'s, $R1$ and $R2$, having different $...
2 votes
Accepted

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

I am not sure I get what you mean by "brought to Jordan form", but if you don't consider the structure of $D$ while changing basis for $N$ then it won't work. Example: $$ U = \begin{bmatrix}...
2 votes
Accepted

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

Since $\sigma_1^2=\lambda_+$ and $\sigma_2^2=\lambda_-$ are the two eigenvalues of the symmetric matrix product $MM^t$, we have $\lambda_++\lambda_-={\rm tr}\,MM^t=\|m_1\|^2+\|m_2\|^2$. Hence we may ...
2 votes
Accepted

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

We can give many counterexamples to semisimplicity in positive characteristic using the observation that if the canonical morphism $k_Y\rightarrow f_*k_X$ in $D^b_c(Y,k)$ is split, then the induced ...
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2 votes
Accepted

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

Here is an example where $f$ is not smooth but $Rf_* \mathbb{C}$ behaves as if it were: Let $X$ be a hyperelliptic surface and $f$ the natural morphism to $Y \cong\mathbb{P}^1$. All reduced fibres of $...
1 vote

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

Yes. Search for combinatorial Benders decomposition or logic-based Benders decomposition. In particular, Benders feasibility cuts for a binary master problem are “no-good” cuts of the form $$\sum_{j\...
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1 vote

Decomposition theorem over more general base schemes

If $X$ is any excellent scheme with equicodimensional irreducible components, there is a perverse t-structure on $D^b_c(X,\mathbf{Q}_\ell)$ with all expected properties ($\ell$ any prime invertible on ...
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1 vote
Accepted

Generate a two-variable polynomial from its "roots

There a number of very well researched techniques that can be used to solve your problem in a practical sense. Most of these have come from computer graphics and computer vision where taking a set of ...
  • 4,492
1 vote

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

An answer via hypergraph theory There exists an obvious reduction to the minimum edge cover problem in hypergraph theory (also known as the set cover problem). I will describe this reduction. A ...
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1 vote

A Krull-Schmidt Theorem for Lie groups?

I see that you are restricting to (connected) compact Lie groups. This means that $G$ is a product of a semi-simple group and a central torus. The semi-simple piece decomposes uniquely (up to ...
1 vote
Accepted

Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us write your matrix equation as the following block format, $$\begin{bmatrix}A & B\\ C & D(x,y) \end{bmatrix}\begin{bmatrix}x\\ y \end{bmatrix}=\begin{bmatrix}b\\ c \end{bmatrix},$$ in ...
1 vote

Examples for Decomposition Theorem

A nice example of the decomposition theorem for a proper morphism of smooth varieties is the blowup of the points $p_i$ lying at the intersection of two plane curves; that is, let $f, g \in \mathbb{C}[...
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