If A is a singular matrix nxn, then its value is singular = 0

This is a question about a testexam.

But am I right to assume that a singular matrix has det = 0, which gives it an eigenvalue of 0 and this gives it a singular value of 0?

tags – singular and plural hashtags

It's a bit on the mainly based on the opinion aside, but think about it: a hashtag is a metadata tag that you use to provide a taxonomy to the content. For example, think of a library: you'll have hashtags like Architecture, Drama, Novel, Poetry, and other plural tags, such as Kids, Best Sellers, Classics, etc.

You will notice that IN GENERAL, when the subject has more meaning, it is inscribed in the singular, and when the subject is more diffuse or can be included in a larger taxonomy, it becomes plural. So, you can have Drama -> Top Sellers.

So in your case: if your messages contain only one story, it will probably be better to use the short story hashtag. However, if your posts have mixed content, use the plural.

Is there a theorem showing that De Rham's homology is isomorphic to singular homology?

The only exhibition on Rham's homology that I've found is an appendix to the Uranga and Ibanezs book on Phenomenology of strings. It was brief and gave only a basic overview on how to build this homology.

Now, Rhams' theorem states that there is an isomorphism between the De Rham cohomology of smooth manifolds and that of singular cohomology; and so what appears to be a smooth structure invariant is actually an invariant of topological structure.

Is there a similar theorem showing an isomorphism between Rham's homology and singular homology?

linear algebra – Extending the intuitive meaning of singular matrices

My question is based on "What is the geometric meaning of the singular matrix" published here a few years ago.

To make this a little more intuitive, I would like to add an example.

A vector of three-dimensional force $ F $ applied at a point $ P $ with coordinates $ (x, y, z) $ create a moment $ M $ at the point, for example $ (0, 0, 0) $. This moment $ M $ is still a vector with components $ M_x $, $ M_y $ and $ M_z $ for which, next block:

begin {bmatrix}
0 & F_z & -F_y \
-F_z & 0 & F_x \
F_y & -F_x & 0 \
end {bmatrix}
begin {bmatrix}
X \
y \
z \
end {bmatrix}
begin {bmatrix}
M_x \
My \
M_z \
end {bmatrix}

As can be seen, the matrix above is singular. Math tells me that as a result, it can not be reversed and, therefore, given the forces and moments that can not be solved for the coordinates of $ P $, aka
given the freedom to apply a given force in any place, the moment vectors can not all be produced.

Is this really the case? And if so, what about those who can be created? How do you solve for these?

We can easily build a solution triplet putting $ P $ at $ (1, 1, 1) $ from which, $ M = begin {bmatrix}
F_z – F_y \
F_x – F_z \
F_y – F_x \
end {bmatrix} $

but assuming that instead of $ P $ we knew that $ M $ how could $ P $ to be situated?

Ag.algene – metric (singular) geometry associated with higher cohomology

assume $ X $ is a smooth complex variety and $ L $ is a group of lines with a metric $ h_L $, then a section $ s in H ^ 0 (X, L) $ give another metric $ tilde h_L: = e ^ {- phi} h_L $ or $ phi = log | s | ^ 2_ {h_L} $.

Yes $ u in H ^ q (X, L) $ is a section (or just $ u in H ^ q (X, K_X) $), is there a way to build a metric on $ L $ compared to $ u $?

Aggressive Geometry – singular divisors $ m $ -canonic

Let $ C $ to be a gender curve $ geqslant $ 2.
Let $ K_C $ to be his canonical package.

Let $ m $ to be an integer.
We assume that a generic element in the linear system $ | mK_C | $ is a simple divisor, that is, a divisor without a multiple point.
Let $ S subseteq | mK_C | $ either the set of divisors that are not simple.

is $ S $ always a hypersurface, that is to say without irreducible component of the codimension $ geqslant $ 2 ?
If so, could we calculate the degree of $ S $ in terms of $ g $ and $ m $ (at least for $ m $ wide enough)?

What about higher dimensional varieties?
More precisely, we consider a variety $ X $. Let $ S subseteq | mK_X | $ be the set of singular divisors.

reference request – Filtrations of cellular related spectra and singular homology

I would like to study the spectral filtrations (ie objects of the category homotopy stable "topological" $ SH $; a filtration of a spectrum $ E $ is a sequence of compatible cards $ E _ { le i} to E $) whose levels are "in between" the subsequent levels of cellular filtrations (and the cones of the comparison maps are Moore's shifted spectra). An interesting special case is the filtration that gives the "canonical" filtration on a singular homology (note that this is not the Postnikov $ t $– filtration of the structure, since this one is "much more distant from the cellular"); the levels of this filtration for the finite spectra are also finite (and the "quotients" are the off-axis Moore spectra corresponding to $ H _ * (E) $).

Has anyone ever studied something similar? Can you suggest possible applications for such filtrations?

functional analysis – Singular integral of the composition of the Hilbert transform and the fractional Laplacian

Given $ 0 <s <$ 1, we can define the Fractional Laplacian by

$$ Lambda ^ {- s} f (x): = (- Delta) ^ {- s / 2} (x) = int _ {- infty} ^ {+ infty} | xy | ^ {- 1 + s} f (y) dy $$
or by means of Fourier transform as $$ widehat { Lambda ^ {- s} f} ( xi) = c_s | xi | ^ {- s} widehat {f} ( xi) ; mbox {for all} ; xi neq0. $$

The Hilbert transformation is defined by
$$ Hf (x) = p.v. int _ {- infty} ^ {+ infty} frac {f (y)} {x-y} dy $$
or by means of Fourier transform as
$$ widehat {Hf} ( xi) = – isgn ( xi) widehat {f} ( xi) ; mbox {for all} ; xi neq0. $$

Therefore, we can write
begin {equation} label {eq1} widehat { Lambda ^ {- s} Hf} ( xi) = C_ssgn ( xi) | xi | ^ {- s} widehat {f} ( xi) ; mbox {for all} ; xi neq0.
end {equation}

My question is how can I define $ Lambda ^ {- s} Hf $ by singular integrals.

Sobolev Spaces – Singular Part of Chen-Frid Matching for Divergence Metering Vector Fields

In their fundamental article on vector fields with divergence of measurement, Chen and Frid prove the following theorem (see Thm. 3.2 of the cited document):

Theorem. Let $ F colon mathbb R ^ N to mathbb R ^ N $ to be a bounded and measurable vector field. Assume that its distribution divergence $ text {div} F $ is (distribution represented by a) radon measurement with finite total variation.

Let $ g in text {BV} ( mathbb R ^ N) cap L ^ infty ( mathbb R ^ N) $. Then the distribution
mu: = text {div} (gF) – tilde {g} text {div} F

($ tilde g $ is the precise representative of the function $ g $) is an absolutely continuous measure of radon with respect to $ green Dg green $, whose absolutely continuous part with respect to the measure of Lebesgue coincides with $ F cdot (∇g) _ {ac} $ A.E. .

This interesting theorem is demonstrated in the above-mentioned article. For the research I am currently pursuing, I wonder if anything more is known about $ mu $. In particular, has anyone characterized this measure? $ mu $ (ie its jump part / Cantor part)?

In addition to some results distributed in the literature, I would be interested to guess what is the correct formula (especially for the Cantor part). I have no idea about it … Any ideas? Thank you.

linear algebra – Identity between resolved and singular value density

I read the newspaper

Sengupta, Anirvan M. and Partha P. Mitra. "Distributions of the singular
values ​​for some random matrices. "Physical Review E 60.3 (1999): 3389.

but got stuck in equation (3):

enter the description of the image here

Right here $ lambda_n $ are the singular values ​​of a matrix $ M $ (possibly considered a random matrix), $ rho ( lambda) $ denotes the density of singular values ​​of $ M $, and the resolvent $ mathcal G (z) $ is defined by:

enter the description of the image here

So, how (3) does it come from (2)?