Ciro Santilli @cirosantilli 37

Given a matrix $A$ with metric signature containing $m$ positive and $n$ negative entries, the indefinite orthogonal group is the set of all matrices that preserve the associated bilinear form, i.e.:Note that if $A=I$, we just have the standard dot product, and that subcase corresponds to the following definition of the orthogonal group: Section "The orthogonal group is the group of all matrices that preserve the dot product".

As shown at all indefinite orthogonal groups of matrices of equal metric signature are isomorphic, due to the Sylvester's law of inertia, only the metric signature of $A$ matters. E.g., if we take two different matrices with the same metric signature such as:and:both produce isomorphic spaces. So it is customary to just always pick the matrix with only +1 and -1 as entries.

Intuitive definition: real group of rotations + reflections.

Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.

The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.

This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.

Examples:

Demo under: nodejs/sequelize/raw/many_to_many.js.

NO way in the SQL standard apparently, but you'd hope that implementation status would be similar to UPDATE with JOIN, but not even!

The problem of deletionism is that it removes users' confidence that their precious data will be safe. It's almost like having a database that constantly resets itself. Who will be willing to post on a website that deletes the content they created for free half of the time thus wasting people's precious time?

The Klein-Gordon equation directly uses a more naive relativistic energy guess of $p^2+m^2$ squared.

But since this is quantum mechanics, we feel like making $p$ into the "momentum operator", just like in the Schrödinger equation.

But we don't really know how to apply the momentum operator twice, because it is a gradient, so the first application goes from a scalar field to the vector field, and the second one...

So we just cheat and try to use the laplace operator instead because there's some squares on it:

But then, we have to avoid taking the square root to reach a first derivative in time, because we don't know how to take the square root of that operator expression.

So the Klein-Gordon equation just takes the approach of using this squared Hamiltonian instead.

Since it is a Hamiltonian, and comparing it to the Schrödinger equation which looks like:taking the Hamiltonian twice leads to:

We can contrast this with the Dirac equation, which instead attempts to explicitly construct an operator which squared coincides with the relativistic formula: derivation of the Dirac equation.

Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.

TODO also comment on why are complex numbers used in the Schrodinger equation?.

Some approaches:

The derivative of a function gives its slope at a point.

More precisely, it give sthe inclination of a tangent line that passes through that point.

It appears that Maxwell's equations can be derived directly from Coulomb's law + special relativity:This idea is suggested by the charged particle moving at the same speed of electrons thought experiment, which indicates that magnetism is just a consenquence of special relativity.

Name origin: likely because it "determines" if a matrix is invertible or not, as a matrix is invertible iff determinant is not zero.

The cool thing about this notation is that is showed to Ciro Santilli that there is more state to a chess game than just the board itself! Notably:plus some other boring draw rules counters.

Basically what register transfer level compiles to in order to achieve a real chip implementation.

After this is done, the final step is place and route.

They can be designed by third parties besides the semiconductor fabrication plants. E.g. Arm Ltd. markets its Artisan Standard Cell Libraries as mentioned e.g. at: web.archive.org/web/20211007050341/https://developer.arm.com/ip-products/physical-ip/logic This came from a 2004 acquisition: www.eetimes.com/arm-to-acquire-artisan-components-for-913-million/, obviously.

The standard cell library is typically composed of a bunch of versions of somewhat simple gates, e.g.:and so on.

Each of those gates has to be designed by hand as a 3D structure that can be produced in a given fab.

Simulations are then carried out, and the electric properties of those structures are characterized in a standard way as a bunch of tables of numbers that specify things like:Those are then used in power, performance and area estimates.

It is not possible to see stars outside of the Milky Way by naked eye.

With telescopes however, it is possible. www.quora.com/Can-we-distinguish-individual-stars-in-other-galaxies-or-would-it-be-equivalent-to-say-we-know-there-are-other-forests-of-stars-galaxies-but-we-cant-tell-the-individual-trees-stars-What-is-the-farthest-individual/answer/Jerzy-Micha%C5%82-Pawlak contains an amazing answer that mentions two special cases of the furthest ones:

But what we can definitely see are globular clusters of galaxies. E.g. the article en.wikipedia.org/wiki/Messier_87 basically gauges the size of galaxies by the number of globular clusters that they contain.

Can you imagine when those guys started to see moons in other planets? They must have shat bricks. What better evidence can you have that the geocentric model could be wrong?

Ciro Santilli likes this.

He doesn't have the patience to actually watch full episodes with rare exceptions, rather just watching selected scenes from: www.youtube.com/channel/UCdeIGY2DIjpGf0A2m9GSE3g, but still, it is interesting.

What appeals to Ciro in this series is how almost nothing is solved by violence, almost everything is decided in the bridge, at the "cerebral" level of the command structure. This reminds Ciro of a courtroom of law sometimes.

Maybe there's also a bit of 90's nostalgia involved too though, as this is something that would show on some random cable channels a bored young Ciro would have browsed during weekends, never really watching full episodes.

One crime of many episodes is being completely based on some stupid new scientific concept, which any character to back it up.

Another thing that hurt is that due to their obsession with the senior members of the crew, sometimes those senior members are sent in ridiculously risky operations, which is very unrealistic.

Episodes that Ciro watched fully and didn't regret:

Semi worth it:

Not worth it:

Stefan Hell was really excited by this as of 2023.

Instead of shining a light over the entire sample to saturate it, you illuminate just a small bit instead.

He was basically saying that this truly brings the resolution to the actual physical limits, going much much beyond 2014 Nobel prize levels.