Ciro Santilli @cirosantilli 37

There are explicit examples of this. We can have ever thinner disturbances to convergence that keep getting less and less area, but never cease to move around.

If it does converge pointwise to something, then it must match of course.

Once we had the amplified 16S DNA, we were almost ready to start sequencing!

www.velp.com/en/products/lines/3/family/44/vortex_mixers/65/wizard_ir_infrared_vortex_mixer (archive).

Always a bit too much on the superficial side, but sometimes OK, 5-10 minute videos.

Literally: East of the Mountain.

The enemy: github.com/cirosantilli/china-dictatorship.

The cool thing about parallel evolution is that it shows how complex phenotype can evolve from very different initial genetic conditions, highlighting the great power of evolution.

We list some cool ones at: polyphyly.

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.

Subtle is the Lord by Abraham Pais (1982) page 22 mentions that when Einstein saw this in 1915, he was so excited he couldn't work for three days.

This book really tries to recall basic things to ensure that the reader will be able to understand the more advanced ones.

Sometimes it goes a little bit overboard, like defining what a function does several times.

But Ciro Santilli really prefers it when authors error on the side of obvious.