Ciro Santilli @cirosantilli 40

We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:then we derive and evaluate at 0:$L_z$ therefore represents the infinitesimal rotation.

Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator $L_z$:

Repeating the same process for the other directions gives:We have now found 3 linearly independent elements of the Lie algebra, and since $SO(3)$ has dimension 3, we are done.

The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there.

Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.

More precisely: local symmetries of the Lagrangian imply conserved currents.

TODO Ciro Santilli really wants to understand what all the fuss is about:

Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras

The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.

Bibliography:

Recommended from Physics from Symmetry by Jakob Schwichtenberg (2015) page 92:

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

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:

A subset of Spacetime diagram.

The key insights that it gives are:

As of 2025, this company was by far the most well funded in the field.

It was founded by some of the authors of the influential paper Deep learning with coherent nanophotonic circuits.

Presented e.g. at youtu.be/t0yj4hBDUsc?t=456 from Video "Silicon Photonics: The Next Silicon Revolution? by Asianometry (2022)".

Bibliography:

Lance instructor of the 800,000 Imperial Guards (八十万禁军枪棒教头). TODO understand the "枪棒" part: zhidao.baidu.com/question/206659649.html.

The adopted son of Gao Qiu wanted to fuck his wife, and because of this they frame him of planning to take revenge by killing Go Qiu, even though Lin Chong had decided not to take revenge to avoid harming his wife further.

They just keep trying to kill him, until at one point he just gives up and becomes a fugitive.

His story is well told in The Water Margin.

Usually called by others as 林教头 (lin jiaotou, literally Head Instructor Lin, but usually translated as just Instructor Lin).

The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of:and then we just put the unknown $y$ and each derivative into that simple polynomial:except that now the $c_i$ are not just constants, but they can also depend on the argument $x$ (but not on $y$ or its derivatives).

Explicit solutions exist for the very specific cases of:

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

A linear map is a function $f:V_1(F)\to V_2(F)$ where $V_1(F)$ and $V_2(F)$ are two vector spaces over underlying fields $F$ such that:

A common case is $F=\mathbb{R}$, $V_1={\mathbb{R}}_m$ and $V_2={\mathbb{R}}_n$.

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of $F$.

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

Ciro Santilli's LinkedIn profile: www.linkedin.com/in/cirosantilli/ see also: accounts controlled by Ciro Accounts.

LinkedIn fully complies with censorship imposed locally by the Chinese government, and does so in a non-transparent way: cirosantilli.com/china-dictatorship/linkedin.

It is hard to understand what the point of that website is, as it is basically just a more closed version of Facebook, but alas, it has flourished as the only place where people post more useful content compared to Twitter and Facebook. In any case, Ciro just applies the same unfollow policy to all of them: aggressively filter your social media follows.

77K. Low enough for "high temperature superconductors" such as yttrium barium copper oxide, but for "low temperature superconductors", you need to go much lower, typically with liquid helium, which is likely much more expensive. TODO by how much?

www.facebook.com/131402556881886/posts/655763214445815/ gives an origin:The silkqin.com entry: www.silkqin.com/04qart/07sqmp/57ls.htm does not mention this however.