Ciro Santilli @cirosantilli 37

"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.

Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.

Microscope by Antonie van Leeuwenhoek used to first observe microorganisms.

Their websites a bit shitty, clearly a non cohesive amalgamation of several different groups.

E.g. you have to create several separate accounts, and different regions have completely different accounts and websites.

The Europe replacement part website for example is clearly made by a third party called flex.com/ and has Flex written all over it, and the header of the home page has a slightly broken but very obviously broken CSS. And you can't create an account without a VAT number... and they confirmed by email that they don't sell to non-corporate entities without a VAT number. What a bullshit!

This is true. The level of competition in university entry exams in Asia in insane in the early 21st century compared to the West.

This is actually both good and bad. Good because it selects some very good exam passers. And bad because it selects some very good exam passers.

github.com/mdawsonuk/LevelDBDumper/tree/e750a27ff58443ecc410b5c16abbdc539d617387#installation worked on Ubuntu 23.10 Annoying installation, but worked: github.com/mdawsonuk/LevelDBDumper/issues/13

Initial issues off-the-bat:

We can almost reach the Lie algebra of any isometry group in a single go. For every $X$ in the Lie algebra we must have:because $e^{tX}$ has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".

Then:so we reach:With this relation, we can easily determine the Lie algebra of common isometries:

Bibliography:

For this sub-case, we can define the Lie algebra of a Lie group $G$ as the set of all matrices $M\in G$ such that for all $t\in \mathbb{R}$:If we fix a given $M$ and vary $t$, we obtain a subgroup of $G$. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html

And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.

Overview:

TODO there was one which was relly good, can't find it anymore. One day.

The term is not very clear, as it could either mean:

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

Examples:

This is the standard model.