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# 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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### Primary decomposition thm proof [migrated]

I'm studying the proof of primary decomposition thm in linear algebra, hoffman and kunze. Page 221, in the process of proving that minimal polynimial for T_i is (p_i)^r_i, I don't understand why g(T_i)...
63 views

### A question to the Wedderburn-Mal’cev decomposition

Excuse me, I saw the result on the Wedderburn-Mal’cev decomposition of unital compact rings which M.I. Ursul and A. Tripe introduced in the attached file. However, I cannot contact them because ...
1 vote
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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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### 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 ...
1 vote
261 views

### 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 ...
1 vote
380 views

### 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 ...
367 views

### 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 ...
1 vote
62 views

### 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 ...
1 vote
44 views

### 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 ...
138 views

### 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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### 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? ...
176 views

### 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 ...
221 views

### 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 ...
256 views

### 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 ...
215 views

### 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 ...
181 views

### 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 ...
1 vote
526 views

### 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 ...
2k views

### 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 ...
531 views

### 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 ...
600 views

### 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 ...
514 views

### 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 ...
149 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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### 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 ...
386 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 vote
219 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 ...
330 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 ...
183 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) half-...
1 vote
989 views