All Questions
Tagged with decomposition-theorem not perverse-sheaves
35 questions
0
votes
0
answers
99
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 ...
- 141
1
vote
1
answer
175
views
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 ...
2
votes
1
answer
178
views
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 ...
- 45
2
votes
0
answers
108
views
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 ...
4
votes
0
answers
581
views
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 ...
- 547
12
votes
2
answers
554
views
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 ...
- 452
1
vote
1
answer
378
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 ...
- 15
1
vote
1
answer
463
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 ...
- 113
5
votes
1
answer
387
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 ...
- 11.1k
1
vote
0
answers
64
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 ...
- 11
1
vote
0
answers
48
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 ...
- 1,826
3
votes
1
answer
145
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 $R_s = U_s\Lambda_sU_s^*$. ...
- 33
2
votes
1
answer
111
views
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^...
- 39
2
votes
1
answer
961
views
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 ...
- 31
6
votes
0
answers
99
views
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_+-...
- 13.9k
4
votes
1
answer
294
views
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?
...
- 13.9k
2
votes
1
answer
217
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
...
- 143
9
votes
1
answer
266
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 ...
- 231
5
votes
1
answer
296
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 ...
- 63.1k
0
votes
1
answer
257
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 ...
- 399
4
votes
0
answers
184
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 ...
- 15.4k
1
vote
0
answers
681
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 ...
- 1,959
2
votes
1
answer
606
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 ...
- 400
4
votes
1
answer
723
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 ...
- 1,881
0
votes
1
answer
158
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 ...
- 2,436
5
votes
2
answers
424
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 ...
- 3,017
1
vote
1
answer
254
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
votes
1
answer
549
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 ...
- 171
7
votes
0
answers
188
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-...
- 6,162
1
vote
1
answer
1k
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) =...
- 113
1
vote
1
answer
394
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 $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$...
user5810
2
votes
3
answers
1k
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 (rectangular)...
- 129
23
votes
4
answers
52k
views
complexity of eigenvalue decomposition
what is the computational complexity of eigenvalue decomposition for a unitary matrix?
is O(n^3) a correct answer?
- 231
10
votes
2
answers
5k
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. ...
- 157
6
votes
2
answers
1k
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 ...
- 8,866