Ciro Santilli @cirosantilli  40

The main interest of this theorem is in classifying the indefinite orthogonal groups, which in turn is fundamental because the Lorentz group is an indefinite orthogonal groups, see: all indefinite orthogonal groups of matrices of equal metric signature are isomorphic.

It also tells us that a change of basis does not the alter the metric signature of a bilinear form, see matrix congruence can be seen as the change of basis of a bilinear form.

The theorem states that the number of 0, 1 and -1 in the metric signature is the same for two symmetric matrices that are congruent matrices.

For example, consider:

The eigenvalues of $A$ are $1$ and $4$, and the associated eigenvectors are:symPy code:and from the eigendecomposition of a real symmetric matrix we know that:

Now, instead of $P$, we could use $PE$, where $E$ is an arbitrary diagonal matrix of type:With this, would reach a new matrix $B$:Therefore, with this congruence, we are able to multiply the eigenvalues of $A$ by any positive number $e_1^2$ and $e_2^2$. Since we are multiplying by two arbitrary positive numbers, we cannot change the signs of the original eigenvalues, and so the metric signature is maintained, but respecting that any value can be reached.

Note that the matrix congruence relation looks a bit like the eigendecomposition of a matrix:but note that $D$ does not have to contain eigenvalues, unlike the eigendecomposition of a matrix. This is because here $S$ is not fixed to having eigenvectors in its columns.

But because the matrix is symmetric however, we could always choose $S$ to actually diagonalize as mentioned at eigendecomposition of a real symmetric matrix. Therefore, the metric signature can be seen directly from eigenvalues.

Also, because $D$ is a diagonal matrix, and thus symmetric, it must be that:

What this does represent, is a general change of basis that maintains the matrix a symmetric matrix.

Related:

Sometimes Ciro Santilli says half jokingly that user interface does not matter.

This software circa 2010-2020 makes that joke not be funny.

How can a UI feel so clunky!

The most aggravating thing is that it is not immediately obvious why it feels so bad.

This is generally good, especially compared to how crappy Evince, the default Ubuntu one, has been around 2014-2020.

This is a good thing. It basically contains an entire website, with HTML and assets inside a single ZIP, and a little bit of metadata.

It is incomprehensible why browsers don't just implement it as they already have all the web part, and also ZIP stuff:

The situation is so sad. Ubuntu 21.04 doesn't come with a reader installed by default:

Dual vectors are the members of a dual space.

In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors $e^i$ are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:

Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector $f$ as:

In the context of quantum mechanics, the bra notation is also used for dual vectors.

DNA detection means determining if a specific DNA sequence is present in a sample.

This can be used to detect if a given species of microorganism is present in a sample, and is therefore a widely used diagnostics technique to see if someone is infected with a virus.

You could of course do full DNA Sequencing to see everything that is there, but since it is as a more generic procedure, sequencing is more expensive and slow.

The alternative is to use a DNA amplification technique.

DNA amplification is one of the key DNA technologies:

Big excitement picture at: molecular biology technologies.

A concrete experiment has been done at Section "Sequencing" on section sequencing.

As of 2018, Ciro Santilli believes that this could be the next big thing in biology technology.

"De novo" means "starting from scratch", that is: you type the desired sequence into a computer, and the synthesize it.

The "de novo" part is important, because it distinguishes this from the already well solved problem of duplicating DNA from an existing DNA template, which is what all our cells do daily, and which can already be done very efficiently in vitro with polymerase chain reaction.

Many companies are attempting to create more efficient de novo synthesis methods:

Notably, the dream of most of those companies is to have a machine that sits on a lab bench, which synthesises whatever you want.

TODO current de novo synthesis costs/time to delivery after ordering a custom sequence.

The initial main applications are likely going to be:but the real pipe dream is building and bootstraping entire artificial chromosomes

News coverage: