The exact same problem appears over and over, e.g.:
- transportaion: the last mile of the trip when everyone leaves the train and goes to their different respective offices is the most expensive
- telecommunications: the last mile of wire linking local hubs to actual homes is the most expensive
- electrical grid: same as telecommunications
Ciro Santilli also identified knowledge version of this problem: the missing link between basic and advanced.
Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.
Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.
As the name suggests, this is not very sturdy, and was quickly replaced by bipolar junction transistor.
By William Shockley in 1948 also at Bell Labs Murray Hill.
The function being maximized in a optimization problem.
Does not require entangled particles, unlike E91 which does.
en.wikipedia.org/w/index.php?title=Quantum_key_distribution&oldid=1079513227#BB84_protocol:_Charles_H._Bennett_and_Gilles_Brassard_(1984) explains it well. Basically:
- Alice and Bob randomly select a measurement basis of either 90 degrees and 45 degrees for each photon
- Alice measures each photon. There are two possible results to either measurement basis: parallel or perpendicular, representing values 0 or 1. TODO understand better: weren't the possible results supposed to be pass or non-pass? She writes down the results, and sends the (now collapsed) photons forward to Bob.
- Bob measures the photons and writes down the results
- Alice and Bob communicate to one another their randomly chosen measurement bases over the unencrypted classic channel.This channel must be authenticated to prevent man-in-the-middle. The only way to do this authentication that makes sense is to use a pre-shared key to create message authentication codes. Using public-key cryptography for a digital signature would be pointless, since the only advantage of QKD is to avoid using public-key cryptography in the first place.
- they drop all photons for which they picked different basis. The measurements of those which were in the same basis are the key. Because they are in the same basis, their results must always be the same in an ideal system.
- if there is an eavesdropper on the line, the results of measurements on the same basis can differ.Unfortunately, this can also happen due to imperfections in the system.Alice and Bob must decide what level of error is above the system's imperfections and implies that an attacker is listening.
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:
- youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good SO(3) explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
- www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
- www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
- www.physics.drexel.edu/~bob/LieGroups.html
What is Lie theory? by Mathemaniac 2023
. Source. Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
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- a Wikipedia where each user can have their own version of each article
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
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Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
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