By the Open University. "Open" I mean.

Some/all courses expire in 4 weeks: www.futurelearn.com/courses/intro-to-quantum-computing. Ludicrous.

www.reddit.com/r/JordanPeterson/comments/gc23gd/petersons_online_university_still_a_thing/.

Kudos for being a not-for-profit. Also, anyone can create content: e-learning websites must allow students to create learning content. Oh, but TODO is possible for anyone to make content publicly visible? Course join links lik: www.khanacademy.org/join/MJZ6NSV7 require login. webapps.stackexchange.com/questions/165132/how-to-create-a-course-that-is-publicly-visible-without-the-need-to-login-on-kha If that's the case, it is a fatal flaw not shared by OurBigBook.com.

Another cool aspect is that they have the "physical world teacher pull student accounts in" approach built-in quite well at course creation. This is a very good feature.

As of 2021 they were a bit struggling for money it seems: www.youtube.com/watch?v=I8XdUy-wyyM?

These people have good intentions.

The problem is that they don't manage to go critical because there's to way for students to create content, everything is manually curated.

You can't even publicly comment on the textbooks. Or at least Ciro Santilli hasn't found a way to do so. There is just a "submit suggestion" box.

This massive lost opportunity is even shown graphically at: cnx.org/about (archive) where there is a clear separation between:Maybe this wasn't the case in their legacy website, legacy.cnx.org/content?legacy=true, but not sure, and they are retiring that now.

Thus, OurBigBook.com. License: CC BY! So we could re-use their stuff!

By Rice University.

TODO what are the books written in?

It is a shame that they refocused to more applied courses. This also highlights their highly "managed" approach to content creation. Their 2022 pitch on front page says it all:they are focused on the highly paid character of many software engineering jobs.

But one cool point of this website is how they hire tutors to help on the courses. This is a very good thing. It is a fair way of monetizing: e-learning websites must keep content free, only charge for certification.

ALSA can be thought as analogous to physical wires linking up machines.

Except that instead of machines, you have separate programs. One such typical link is:

The advantage of this setup is that separate programs can collaborate to make complex sounds.

The disadvantage of this setup is that it makes it very hard to reproduce results, you basically need a Docker image with the exact same version of everything. And some script to launch and connect all programs correctly.

Some composition systems like LMMS reduce that problem by having synthesizers as plugins, so that you don't have to setup any connections yourself.

aconnect vmpk hello world: askubuntu.com/questions/34391/virtual-midi-piano-keyboard-setup/1298026#1298026

A form is a function from a vector space to elements of the underlying field of the vector space.

Examples:

A Linear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\to \mathbb{R}$.

The set of all linear forms over a vector space forms another vector space called the dual space.

We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.

It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.

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:

Basically a synonym for Lie group which is the way of modelling them.

Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) $GL(n)$, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field $V$ (which can be represented by a matrix infinite dimension), in a way that respects the group operations:

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

Motivation:

Bibliography: