Ciro Santilli @cirosantilli 37

The key is to define only the minimum number of measures: if you define more definitions become over constrained and could be inconsistent.

Learning the modern SI is also a great way to learn some interesting Physics.

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:

Original playlist name: "PHYSICS 68 ADVANCED MECHANICS: LAGRANGIAN MECHANICS"

Author: Michel van Biezen.

High school classical mechanics material, no mention of the key continuous symmetry part.

But does have a few classic pendulum/pulley/spring worked out examples that would be really wise to get under your belt first.

Equivalent to Lagrangian mechanics but formulated in a different way.

Motivation: Lagrangian vs Hamiltonian.

TODO understand original historical motivation, www.youtube.com/watch?v=SZXHoWwBcDc says it is from optics.

Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.

Bibliography:

Like Jimmy Wales, he used to work in finance and then quit. What is it with those successful e-learning people??

Setting: you are sending bits through a communication channel, each bit has a random probability of getting flipped, and so you use some error correction code to achieve some minimal error, at the expense of longer messages.

This theorem sets an upper bound on how efficient you can be in your encoding, for any encoding.

The next big question, which the theorem does not cover is how to construct codes that reach or approach the limit. Important such codes include:

But besides this, there is also the practical consideration of if you can encode/decode fast enough to keep up with the coded bandwidth given your hardware capabilities.

news.mit.edu/2010/gallager-codes-0121 explains how turbo codes were first reached without a very good mathematical proof behind them, but were still revolutionary in experimental performance, e.g. turbo codes were used in 3G/4G.

But this motivated researchers to find other such algorithms that they would be able to prove things about, and so they rediscovered the much earlier low-density parity-check code, which had been published in the 60's but was forgotten, partially because it was computationally expensive.

Besides being useful in engineering, it was very important historically from a "development of mathematics point of view", e.g. it was the initial motivation for the Fourier series.

Some interesting properties:

Sample numerical solutions:

The group of all transformations that preserve some bilinear form, notable examples:

You just map the value (1, 1) $C_m\times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2\times C_3$, the group generated by of (1, 1) is: