Ciro Santilli @cirosantilli 37

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:

As mentioned in True Genius: The Life and Science of John Bardeen page 224, the idea of symmetry breaking was a major motivation in Josephson's study of the Josephson effect.

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.

We don't need to understand a super generalized version of tensor products to know what they mean in basic quantum computing!

Intuitively, taking a tensor product of two qubits simply means putting them together on the same quantum system/computer.

When we write the bra-ket notation: $|00⟩$ that is the same as $|0⟩\otimes |0⟩$.

The tensor product is called a "product" because it distributes over addition.

E.g. consider:

Intuitively, in this operation we just put a Hadamard gate qubit together with a second pure $|0⟩$ qubit.

And the outcome still has the second qubit as always 0, because we haven't made them interact.

The quantum state $\frac{|00⟩+|10⟩}{\sqrt{2}}$ is called a separable state, because it can be written as a single product of two different qubits. We have simply brought two qubits together, without making them interact.

If we then add a CNOT gate to make a Bell state:we can now see that the Bell state is non-separable: we've made the two qubits interact, and there is no way to write this state with a single tensor product. The qubits are fundamentally entangled.

Each term has 8 weeks, and the week number is often used to denote the time at which something happens.

Week 0 is also often used to denote the week before classes officially start. This is especially important in the first term of the year (Michaelmas term) where people are coming back to school and meeting old and new friends.

At the end of the year, after Trinity term, students have exams. These basically account for all of the grades. In certain courses such as the Physics course of the University of Oxford, there is only new material on Michaelmas term and Hilary term, Trinity term being revision-only. So you can imagine that during Trinity term, students are going to be on edge.

Bibliography:

Once upon a time in the 2010's, Ciro Santilli went to an artsy theatre venue in the suburbia of Paris, dragged by his wife then girlfriend of course.

In the venue, there was a politician, who was doing his best to show how much they supported the arts, and there were of course the artists, involved in the play.

The politician would see a political power score on top of every person's head, and would spend an amount of time talking to each person exactly proportional to that score. This meant basically one sentence to us. The words themselves didn't really matter of course, only the time spent, they just have to produce nice sounds.

One of the artists however, and he seemed quite important in the production, for some reason spent a huge amount of time speaking to us. The score the artist saw on our heads was of love, or how interested we were in the art.

Big goals:

Just art:

You have to know the language to appreciate them.

The 60's and 70's were the days, those great proxy wars and CIA dictatorships allowed hippies to make awesome freedom music without too imminent a fear of death.

Songs making fun of things or that are pure Brazil nostalgia are also accepted. No love songs, ever. Except some by Caetano, but that's it!

English:

French:

The BBC 1979-1982 adaptations of John Le Carré's novels are the best miniseries ever made:They are the most realistic depiction of spycraft ever made.

Some honorable mentions:

Bibliography:

Unknown real developer name, claims to be from Canada on YouTube channel about: www.youtube.com/@TheBibitesDigitalLife/about, likely because he's a software developer and wants to keep his employer's claws away from his side project.

Appears to be closed source unfortunately, so not suitable for research.

Video 1. "What will happen after 100h of evolution? by The Bibites (2022)" mentions it was started five years ago, so circa 2017.

Appears to be Unity-based, if you download and extract for Linux you get files named UnityPlayer.so.

Author is named Leo Caussan in game credits at startup: www.linkedin.com/in/l%C3%A9o-caussan-560350136/, a Canadian software engineer.

Was not very Linux compatible: www.reddit.com/r/TheBibites/comments/vqk6ac/program_stalls_at_a_blue_screen/ Trying to run 0.5.0 leads to a blank screen after you click "start simulation".

Ciro Santilli believes that there is a positive correlation between being a software engineer and liking Buddhist-like things.

Maybe it is linked to minimalism and DRY, which software engineers value so greatly.

Even Ciro had to try an unoriginal Buddhist joke intro in one of this Stack Overflow answers.

Ciro also feels that his "minimal reproducible example" scientific language/concept learning method obsession of breaking things into tiny sub-problems has a strong link with Koans.

Some notable Buddhism/programmer examples:

Another thing that points the correlation out is the existence of wattsalan.github.io/ on a github.io about Alan Watts.

Respect, big respect to those people.

Take the group of all Translation in ${\mathbb{R}}^1$.

Let's see how the generator of this group is the derivative operator:

The way to think about this is:

So let's take the exponential map:and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_0$:

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

A slow development test cycle will kill your software.

New developers won't want to learn your project, because they would rather shoot themselves.

This means that build time, and the time to run tests, must be short.

5 seconds to rebuild is the maximum upper limit.

Of course, at some point software gets large enough that things won't fit anymore in 5 seconds. But then you must have either some kind of build caching, or options to do partial builds/tests that will bring things down to that 5 second mark.

You also have to spend some time profiling execution and build from scratch times.

A slow build from scratch will mean that your continuous integration costs a lot, money that could be invested in a new developer!

It also means that people won't bother to reproduce bugs on given commits, or bisect stuff.

One anecdote comes to mind. Ciro Santilli was trying to debug something, and more experience colleague came over.

To reproduce a problem, ciro was running one command, wait 5 seconds, run a second command, wait 5 seconds, run a third command:

The first thing the colleague said: join those three commands into one:And so, Ciro was enlightened.

This is a general principle of software/hardware design that Ciro feels holds wide applicability.

The most extreme case of this is of course the integrated circuit itself, in which it is essentially impossible (?) to observe the specific value of some indidual wire at some point.

Somewhat on the other extreme, we have high level programming languages running on top of an operating system: at this point, you can just GDB step debug your program, print the value of any variable/memory location, and fully understand anything that you want. Provided that you manage to easily reach that point of interest.

And for anything in between we have various intermediate levels of complication. The most notable perhaps being developing the operating system itself. At this level, you can't so easily step debug (although techniques do exist). For early boot or bootloaders for example, you might want to use JTAG for example on real hardware.

In parallel to this, there is also another very important pair of closely linked tradeoffs:

Emulation also has another potential downside: unless you are very careful at implementing things correctly, your model might not be representative of the real thing. Also, there may be important tradeoffs between how much the model looks like the real thing, and how fast it runs. For example, QEMU's use of binary translation allows it to run orders of magnitude faster than gem5. However, you are unable to make any predictions about system performance with QEMU, since you are not modelling key elements like the cache or CPU pipeline.

Instrumentation is another technique that has can be considered to achieve greater observability.

This is the actual main function of university for many people as of the 2020s. And it fulfills it quite well. A breeding ground.

In a closely related sense, university is simply a symbol of personal status. Not a place where you go to learn. And especially in the Anglophone world of fancy colleges, university also doubles down as a form of long term luxury hotel. Even if it ends up meaning debt.

There's nothing wrong with sexual selection. This type of natural eugenics is an important part of humankind. It is however just sad that any type of learning falls so much behind. A close second would be fine. But as it stands, it is just too far off.

Related:

Ciro Santilli has felt that perhaps what is missing in 2020's AGI research is:

Forcing these boundaries to be tested was one of the main design goals of Ciro's 2D reinforcement learning games.

In those games, for example:Therefore, those continuous objects would also have "magic" effects that could not be explained by "simple" "what is touching what" ideas.

Bibliography:

Notable mentions:

Other notable people that are likely also awesome but Ciro has less familiarity with their contributions:

Another notable reference is in Lost Horse LLC, MacKenzie Bezos's charity instrument.

Breakdown:

Related: