Ciro Santilli @cirosantilli 37

Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.

The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.

The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.

The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument $x$ can now be an operator instead of just a number.

Examples:

This doesn't do a hole lot. Ciro Santilli wouldn't really call it a web framework. It's more like a middleware. Real web frameworks are built on top of it.

Examples under: nodejs/express:

A live example on Heroku can be seen at: github.com/cirosantilli/heroku-node-min

This is quite in-depth, pretty good.

Unrelated to the Khan Academy.

This section groups conjectures that are famous, solved or unsolved.

They are usually conjectures that have a strong intuitive reasoning, but took a very long time to prove, despite great efforts.

The list: en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

In special relativity, it is impossible to travel faster than light.

One argument of why, is that if you could travel faster than light, then you could send a message to a point in Spacetime that is spacelike-separated from the present. But then since the target is spacelike separated, there exists a inertial frame of reference in which that event happens before the present, which would be hard to make sense of.

Even worse, it would be possible to travel back in time:

Uses telephone lines, and therefore were still usable much much after the Internet made them obsolete, which is quite funny.

An alloy of iron and carbon. Because such allys have had such incredible historical importance due to their different properties, different phases of Fe-C have well known names such as steel

The wiki comments: en.wikipedia.org/w/index.php?title=Ferromagnetism&oldid=965600553#Explanation

I think they are a tool to calculate the probability of different types of particle decays and particle collision outcomes. TODO Minimal example of that.

And they can be derived from a more complete quantum electrodynamics formulation via perturbation theory.

Can be used for all of quantum electrodynamics, weak interaction and quantum chromodynamics.

At Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), an intuitive explanation of them in termes of sum of products of propagators is given.

Feynman became a terrible womanizer after his first wife Arline Greenbaum died, involving himself with several married women, and leading to at least two abortions according to Genius: Richard Feynman and Modern Physics by James Gleick (1994).

Ciro Santilli likes to think that he is quite liberal and not a strict follower of Christian morals, but this one shocked him slightly even. Feynman could be a God, but he could also be a dick sometimes.

One particular case that stuck to Ciro Santilli's mind, partly because he is Brazilian, is when Feynman was in Brazil, he had a girlfriend called Clotilde that called him "Ricardinho", which means "Little Richard"; -inho is a diminutive suffix in Portuguese, and also indicates affection. At some point he even promised to take her back to the United States, but didn't in the end, and instead came back and married his second wife in marriage that soon failed.

Richard's third and final wife, Gweneth Howarth, seemed a good match for him though. When they started courting, she made it very clear that Feynman should decide if he wanted her or not soon, because she had other options available and being actively tested. Fight fire with fire.

Simple sines and variants:

2 second 1000 Hz:

A ring where multiplication is commutative and there is always an inverse.

A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.

And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.

Basically the nicest, least restrictive, 2-operation type of algebra.

Examples: