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21939
bio website fph.altervista.org
location Pisa, Italy
age 31
visits member for 4 years, 5 months
seen 2 hours ago

Numerical linear algebra. Mainly matrix equations and their applications (control theory, applied probability).


2h
comment How to transform matrix to this form by unitary transformation?
"Notice that UMV still has spectral radius $m_1$" - why? The singular values are preserved, but the spectrum may change.
Apr
15
comment Trace of Inverse matrix from Cholesky
Oh, and for a SPD matrix, essentially the eigendecomposition is the SVD.
Apr
15
comment Trace of Inverse matrix from Cholesky
What about computing an upper triangular Cholesky factor instead? Then, from the shape of $\Sigma^{-1}=U^{-T}U^{-1}$, you get that the diagonal entries are the squares of the inverse of the diagonal of $U$.
Apr
8
reviewed Reject suggested edit on Generalized Radon transform (Relaxed sufficient condition for invertibility)
Apr
5
comment Approximate the square root of (1-X) efficiently through (nested) products
@DehuaCheng I see. Don't you have the same problem in your example with $(I-X)^{-1}$, then? If you expand out all the products, it becomes $2^d$ operations again.
Apr
4
answered Approximate the square root of (1-X) efficiently through (nested) products
Apr
3
comment Complexity of turning a d-degree polynomial to 2-degree polynomial
Also, your "general polynomial" has a very special form. For that polynomial, you can get rid of the exponent $d$ in the minimum problem, for instance.
Apr
2
comment Complexity of turning a d-degree polynomial to 2-degree polynomial
Your question is not well defined. As it is, one can just set $z:=p(x)-x^2$ to get $p(x)=x^2+z$, which has degree 2.
Apr
2
comment What areas of pure mathematics research are best for a post-PhD transition to industry?
I wouldn't define machine learning and mathematical biology "pure mathematics".
Apr
2
comment NP-hard problems in linear algebra and real analysis
Computing the tensor rank is also NP-hard.
Mar
30
comment Does $Mv$ converge to i.i.d in some sense?
There are several different questions here; one is "is each of the $u_i$ separately $Bin(n,1/4)$" (yes), another is "are the $u_i$ independent from each other?" (no, not always, I guess), and a third (if I read between the lines) is "does some result such as a LLN/CLT hold as if they were independent?". Maybe you should try and make the last one more explicit: what kind of properties do you need from them? What is your ultimate goal?
Mar
29
comment Open source mathematical software.
@KjetilBHalvorsen I am not sure that Geogebra is open source. Its webpage geogebra.org/cms/en/license claims that: (1) Geogebra is open source (2) Geogebra is free of charge for non-commercial use only (3) the Geogebra source code is available under the GPL. As far as I know, (2) is in contradiction with both (1) (per the OSI open source definition) and (3) (per the GPL terms). Strange indeed.
Mar
27
comment Inserting maple or macaulay script in a paper
The problem is that it's not a question on mathematical writing, but on LaTeX syntax.
Mar
24
comment Is there an example where the error of Gauss-Laguerre quadrature does not vanish?
The Runge function $x\mapsto \frac{1}{1+25x^2}$ is the standard example of hard-to-approximate function for the Gauss-Legendre quadrature in the interval $[-1,1]$ (which is exact for integrating polynomials of degree up to $2n-1$). My first attempt would be trying a change of variable of some form to take that interval into $[0,\infty]$ and Legendre into Laguerre, but I am not an expert in quadrature myself.
Mar
23
comment The height of the Perron-Frobenius eigenvector
ok thanks now I understand better.
Mar
23
comment The height of the Perron-Frobenius eigenvector
probably you misunderstood me; I do know what an eigenvector is, I was asking you to clarify the definition of height, as you did (thanks!). I take it that there should be a similar definition of height or "smallness" for matrices then? What is it if it's not that matrix norm?
Mar
23
comment The height of the Perron-Frobenius eigenvector
I am not familiar with the concept of height and I cannot make sense of your definition on the eigenvector. Could you please formalize it? For the height of the matrix, what you defined is called infinity-norm of a matrix, $\|A\|_{\infty}$, in matrix analysis (see for instance en.wikipedia.org/wiki/Matrix_norm). It satisfies some "typical" inequalities for a norm such as $\|Av\|\leq\|A\|\|v\|$, and one could probably find more results by looking it up with this name.
Mar
22
reviewed Approve suggested edit on symmetric algebra of an ideal and syzygies
Mar
22
comment Stationary Distribution for Markov-like system?
Shameless self-advertising: I have studied a similar problem in sciencedirect.com/science/article/pii/S0024379511004484 and dx.doi.org/10.1137/100796765. Our problem comes from modelling Markovian binary trees with multiple states; I wouldn't be surprised if yours has the same origin. Maybe we can reduce one setting to the other with a change of variable of the form $x=\begin{bmatrix}C\\C\\C\end{bmatrix}-\pi$ for some $C>0$ (a change of variable that we use in the second paper that I have linked).
Mar
22
comment Stationary Distribution for Markov-like system?
@Suvrit $\langle \pi, \pi B\rangle$ is not what you think, from the description it should be the Hadamard product.