Ciro Santilli @cirosantilli 35

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.

These are the key mathematical ideas to understand!!

There are actually a few formulations out there. By far the dominant one as of 2020 has been the Schrödinger picture, which contrasts notably with the Heisenberg picture.

Another well known one is the de Broglie-Bohm theory, which is deterministic, but non-local.

Because a Git commit can have more than 1 parent due to merge commits when you do:

It can even have more than 2, there's no limit. Although that is not so common (with good reason, 2 is already one too many): softwareengineering.stackexchange.com/questions/314215/can-a-git-commit-have-more-than-2-parents/377903#377903

As mentioned by Craig Venter in 100 Greatest Discoveries by the Discovery Channel (2004-2005), the main outcomes of the project were:

Important predecessors:

Let's see:

There are two cases:

Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?

Cryogenic electron microscopy, which was developped in the 70's.

For super-resolution microscopy.

This could have been a Nobel Prize in Physics as well!

Riesz-Fischer theorem is a norm version of it, and Carleson's theorem is stronger pointwise almost everywhere version.

Note that the Riesz-Fischer theorem is weaker because the pointwise limit could not exist just according to it: $L^p$ norm sequence convergence does not imply pointwise convergence.

For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.

Both are harmonic oscillators.

In the LC circuit:

You can kickstart motion in either of those systems in two ways:

TODO how to calculate

Since a matrix $M$ can be seen as a linear map $f_M(\vec{x})$, the product of two matrices $MN$ can be seen as the composition of two linear maps:One cool thing about linear functions is that we can easily pre-calculate this product only once to obtain a new matrix, and so we don't have to do both multiplications separately each time.