Functional Analysis – Ideal of Strictly Singular Operators

[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. On page 76, after prop. 2.c.3, it is stated that the proof of 2.c.3 shows that an operator $ T: ell_p to ell_p $ is strictly singular if and only if it is compact.

[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $ T: C (K) to X $ is strictly singular, and Theorem 5.2.3 says that a non-weakly compact operator $ T: C (K) to X $ it's not strictly singular.

Note that $ ell_ infty $ is a $ C (K) $ space with K $ Stone-Cech compactification of all positive integers.

st.statistics – Iso integral with weakly singular integrand

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linear algebra – Singular value decomposition of Pauli matrix $ sigma_x $

I'm trying to calculate the singular value decomposition of Pauli's matrix $$ sigma_x = begin {pmatrix} 0 & 1 \ 1 & 0 end {pmatrix}. $$
According to the SVD theorem, this matrix can be decomposed into $ UDV ^ T $ or $ U $ and $ V $ are orthogonal matrices and $ D $ is diagonally. The matrix $ U $ is constructed from the orthonormal eigenvectors of $ sigma_x sigma_x ^ * $ and the matrix $ V $ is constructed from the orthonormal eigenvectors of $ sigma_x ^ * sigma_x $. The problem that I have is that $ sigma_x ^ * sigma_x = sigma_x sigma_x ^ * = I $, the identity matrix and any vector in $ mathbb C ^ 2 $ is a proper vector of the identity matrix. This implies that I can take $ U $ and $ V $ be all $ 2 times $ 2 orthogonal matrix, and I should have a valid singular value decomposition. It does not work, of course, and the problem seems to come from the degeneration of singular values, which in this case are at once $ 1 $, leading to a SVD that is not unique.

My question is: how to calculate the SVD of a matrix of this type, where the right and left eigenvectors are not unique?

List Manipulation – Partition a singular series symmetrically into larger and larger subseries

I've created a symmetric series with powers
$$ {- n, – (n-1), cdots, -1,0,1,2, cdots, n} $$

and wish to study the convergence of the series according to the number of symmetrical terms. So I start with the term 0, then I choose the terms (-1,0,1), then the terms (-2, -1,0,1,2), etc. I'm sure Partition can separate the terms that way, but I do not understand the documentation for coding this and wondering if anyone could help me code it using Partition if that were possible.

thank you,

Aggressive geometry – Can distinct morphisms between the curves induce the same morphism in singular cohomology?

Yes. Since $ Y $ integrates into his jacobian $ B $, just prove the statement for pairs of cards to an abelian variety $ f, g colon X to B $ sending a base point $ x in X $ at $ 0 in B $. Each of these cards is only related to the Albanese variety $ A $ of $ X $, we reduce more to the case of pairs of cards $ f, g colon A to B $ between abelian varieties ( $ 0 at $ 0). Each of these maps is necessarily a group homomorphism, and is uniquely determined by what it does on $ pi_1 = H_1 $Or on $ H ^ 1 (-, mathbf {C}) $.

postgresql – is tsvector the right choice for finding the exact match of two words with variants like the plural and the singular?

I develop a search application that takes a string, then breaks it down into a combination of two words, and then searches the corresponding patterns in a table. An example:

& # 39; My cat ate a rat & # 39;

search patterns: "my cat" "Cat ate" & # 39; "Ate a & # 39; & # 39; rats & # 39;

At the same time, this should correspond to the plural and singular of the combination. So & # 39; my cat & # 39; should match & # 39; my cats & # 39; and also "my cat". The empty words should not be ignored and the two words must be in the same order, without words between them.

My question is this: Is tsvector the right tool for this or can it be done using only the "LIKE " operator? I ask this to decide if I have to spend time diving deeply into tsvector to build my queries.

Thank you!

note: this is my first time here, please let me know if this is not the right place to ask this type of question.

notation – How do you calculate the singular series?

Terence Tao gives on his blog the following formula for something called the singular series:

$$ large mathfrak {S} (h) = 2 Pi_ {2} prod limits_ {p | h; p> 2} frac {p-2} {p-1} $$

I do not understand how to calculate the product:
$$ prod limits_ {p | h; p> 2} frac {p-2} {p-1} $$

Could you possibly calculate numerically some examples and maybe
Explain why $$ p | h; p> 2 $$ means?

habitually $ p | h $ means $ p $ Split $ h $, but since $ h $ can be a small whole number, I do not understand how it works.

singular – Impossible to launch a GARCH BEKK in r

I'm trying to estimate a bivariate GARK BEKK with the mgarchBEKK R pack, but that does not work. The problem is that "H is singular

The error is:

Error in buff.par.transposed[[tmp.count + 1]]% *% as.matrix (eps[count -:
non-compliant arguments
In addition: Warning message:
In BEKK (diferenca.taxas1mo, order = c (1, 1), method = "BFGS"):
negative inverted hessie matrix element

My code is:

bekk.taxas1mo <-BEKK (diferenca.taxas1mo, order = c (1,1), method = "BFGS")

One thing that made me think is that the series I'm working on is about 1/4 zeros. Anyone know if it's a problem?

Is mgarchBEKK a good package to use?

Thank you in advance.

custom publication types – WordPress API – How to display different data in singular or plural responses

I have a "product" type custom publication that returns quite a bit of data in the API response, up to 400 publications with many nodes. Almost all data comes from advanced custom fields (I use the ACF plug-in to API to expose it).

On the 'products' page, I just need to indicate the title and the product image. Is there a way to remove all other fields when requesting all products with https://example.com/wp-json/wp/v2/productbut leave this data in place when you request a specific product with https://example.com/wp-json/wp/v2/product/123 ?

Linear algebra – How to know if a matrix is ​​poorly conditioned or singular using the system function of Eigens (or composition of LUD)?

I use the Eigensystem function and I try to determine if it is singular or poorly conditioned. I use the function as follows:

Electronic system[A]
Composition of LUD[A]

And it returns a list of eigenvalues ​​and eigenvectors, as well as the condition number last. Should the number of conditions be high or low so that we can consider that the corresponding matrix is ​​badly conditioned?

On a matrix, the condition number is $ infty $I'm sure this is badly packaged, but the other numbers are something like 14.555555, and 120.4, etc.