How genes form bodies.
Developmental Genetics 1 by Joseph Ross (2020)
Source. Talks about homeobox genes. The orthogonal group is the group of all matrices that preserve the dot product by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.
More precisely, we know that for any base b, exponentiation satisfies:And we also know that for in particular that we satisfy the exponential function differential equation and so:One interesting fact is that the only thing we use from the exponential function differential equation is the value around , which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.
- .
- .
Now suppose that we want to calculate . The idea is to start from and then then to use the first order of the Taylor series to extend the known value of to .
E.g., if we split into 2 parts, we know that:or in three parts:so we can just use arbitrarily many parts that are arbitrarily close to :and more generally for any we have:
Let's see what happens with the Taylor series. We have near in little-o notation:Therefore, for , which is near for any fixed :and therefore:which is basically the formula tha we wanted. We just have to convince ourselves that at , the disappears, i.e.:
Too restrictive. People should be able to make money from stuff.
The definition of "commercial" could also be taken in extremely broad senses, making serious reuse risky in many applications.
Notably, many university courses use it, notably MIT OpenCourseWare. Ciro wonders if it is because academics are wary of industry, or if they want to make money from it themselves. This reminds Ciro of a documentary he watched about the origins of one an early web browsers in some American university. And then that university wanted to retain copyright to make money from it. But the PhDs made a separate company nonetheless. And someone from the company rightly said something along the lines of:TODO source.
The goal of universities is to help create companies and to give back to society like that. Not to try and make money from inventions.
The GNU project does not like it either www.gnu.org/licenses/license-list.en.html#CC-BY-NC:
en.wikipedia.org/wiki/Creative_Commons_NonCommercial_license#Defining_%22Noncommercial%22 also talks about the obvious confusion this generates: nobody can agree what counts as commercial or not!
In September 2009 Creative Commons published a report titled, "Defining 'Noncommercial'". The report featured survey data, analysis, and expert opinions on what "noncommercial" means, how it applied to contemporary media, and how people who share media interpret the term. The report found that in some aspects there was public agreement on the meaning of "noncommercial", but for other aspects, there is wide variation in expectation of what the term means.
Joe Corneli, of of the contributors, mentions this in a cool-sounding "Peeragogy" context at metameso.org/~joe/:
I earned my doctorate at The Open University in Milton Keynes, with a thesis focused on peer produced support for peer learning in the mathematics domain. The main case study was planetmath.org; the ideas also informed the development of “Peeragogy”.
Collection of coordinate charts.
Shame that the Chinese in the lat 20th early 21st like that bullshit so much. It just weakens everything. Just imaginge those works with more realistic fighting! Would be amazing.
Some courses at least allow you to see material for free, e.g.: www.coursera.org/learn/quantum-optics-single-photon/lecture/UYjLu/1-1-canonical-quantization. Lots of video focus as usual for MOOCs.
Some are paywalled: www.coursera.org/learn/theory-of-angular-momentum?specialization=quantum-mechanics-for-engineers
It is extremely hard to find the course materials without enrolling, even if enrolling for free! By trying to make money, they make their website shit.
The comment section does have a lot of activity: www.coursera.org/learn/statistical-mechanics/discussions/weeks/2! Nice. And works like a proper issue tracker. But it is also very hidden.
November 2023 topics:
- quantum field theory: no
- condensed matter: 1 by Rahul Nandkishore from Colorado Boulder: www.coursera.org/specializations/the-physics-of-emergence-introduction-to-condensed-matter
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.
- "authors", who can create content
- "students", who can consume content
By Rice University.
TODO what are the books written in?
- github.com/openstax/openstax-cms Uses Wagtail CMS. So presumaby they just Wagtail's WYSIWYG.
- github.com/openstax/os-webview
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , 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).
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 (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)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
There are unlisted articles, also show them or only show them.