Ciro Santilli @cirosantilli 37

Ciro Santilli feels a bit like this guy:

One big divergence: obsession with translating every page into every language.

His Mathematics Genealogy Project entry: www.genealogy.math.ndsu.nodak.edu/id.php?id=56185

Old French website: spoirier.lautre.net/

singlesunion.org/ so cute, he's looking for true love!!! This is something Ciro often thinks about: why it is so difficult to find love without looking people in the eye. The same applies to jobs to some extent. He has an Incel wiki page: incels.wiki/w/Sylvain_Poirier :-)

The strongest are:

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:

Good:

Bad:

Some Obsidian digital gardens that would fit well with OurBigBook:

It good to think about how Euclid's postulates look like in the real projective plane:

Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.

The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.

One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.

Communicating at a distance, from Greek "tele" for distance!

A very cool thing about telecommunication is, besides how incredibly fast it advanced (in this sense it is no cooler than integrated circuit development), how much physics and information theory is involved in it. Applications of telecommunication implementation spill over to other fields, e.g. some proposed quantum computing approaches are remarkably related to telecommunication technology, e.g. microwaves and silicon photonics.

This understanding made Ciro Santilli wish he had opted for telecommunication engineering when he was back in school in Brazil. For some incomprehensible reason, telecommunications was the least competitive specialization in the electric engineering department at the time, behind even power electronics. This goes to show both how completely unrelated to reality university is, and how completely outdated Brazil is/was. Sad stuff.

Not end-to-end encrypted by default, WTF... you have to create "secret chats" for that:

You can't sync secret chats across devices, Signal handles that perfectly by sending E2EE messages across devices:This is a deal breaker because Ciro needs to type with his keyboard.

Desktop does not have secret chats: www.reddit.com/r/Telegram/comments/9beku1/telegram_desktop_secret_chat/ This is likey because it does not store chats locally, it just loads from server every time as of 2019: www.reddit.com/r/Telegram/comments/baqs63/where_are_chats_stored_on_telegram_desktop/ just like the web version. So it cannot have a private key.

Allows you to register a public username and not have to share phone number with contacts: telegram.org/blog/usernames-and-secret-chats-v2.

Has Reproducible builds on Android and iOS: core.telegram.org/reproducible-builds

Self deleting messages added to secret chats in Q1 2021: telegram.org/blog/autodelete-inv2

Can delete messages from the device of the person you sent it to, no matter how old.

This section is about telecommunication systems that are based on top of telephone lines.

Telephone lines were ubiquitous from early on, and many technologies used them to send data, including much after regular phone calls became obsolete with VoIP.

These market forces tended to eventually crush non-telephone-based systems such as telex. Maybe in that case it was just that the name sounded like a thing of the 50's. But still. Dead.

The first working one in 1947 by John Bardeen and walter Brattain in Bell Labs Murray Hill.

People had already patented a lot of stuff before without being able to make them work. Nonsense.

As the name suggests, this is not very sturdy, and was quickly replaced by bipolar junction transistor.

A multilinear form with a domain that looks like:where $V\ast$ is the dual space.

Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:where we remember that the raised indices refer dual vector.

Some examples:

Explain it properly bibliography:

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

Bibliography:

Uses photons!

The key experiment/phenomena that sets the basis for photonic quantum computing is the two photon interference experiment.

The physical representation of the information encoding is very easy to understand:

The perfect Middle Way between command-line interfaces and GUIs. A thing of great beauty.

Basically mean that parallel evolution happened. Some cool ones:

TODO understand this stuff: