Ciro Santilli @cirosantilli 37

Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in $L^2$, where it always exists and converges pointwise to the function: Carleson's theorem.

This is the future of course, fusion power to generate electricity, and then converting electricity into food somehow.

Hopefully without going through photosynthesis, which feels complicated and wasteful.

Others:

The only cases where formal proof of theorems seem to have had actual mathematical value is for theorems that require checking a very large number of case, so much so that no human can be fully certain that no mistakes were made. Some examples:

A form is a function from a vector space to elements of the underlying field of the vector space.

Examples:

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.

How can you make a translation of the Bible and not put it in the public domain???? You tell me, you tell me.

With Falun Gong it's even more fun: the words of God Himself are will be copyrighted for a while after Li Hongzhi dies!!!

Ciro Santilli's admiration for Dyson goes beyond his "unify all the things approach", which Ciro loves, but also extends to the way he talks and the things he says. Dyson is one of Ciro's favorite physicist.

Besides this, he was also very idealistic compassionate, and supported a peaceful resolution until World War II with United Kingdom was basically inevitable. Note that this was a strategic mistake.

Dyson is "hawk nosed" as mentioned in Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "Dyson". But he wasn't when he was young, see e.g. i2.wp.com/www.brainpickings.org/wp-content/uploads/2016/03/freemandyson_child-1.jpg?resize=768%2C1064&ssl=1 It sems that his nose just never stopped growing after puberty.

He also has some fun stories, like him practicing night climbing while at Cambridge University, and having walked from Cambridge to London (~86km!) in a day with his wheelchair bound friend.

Ciro Santilli feels that the label child prodigy applies even more so to him than to Feynman and Julian Schwinger.

Bibliography:

You need those because it is hard to do the following:

This is hard to do notably because when the update happens, several things might need to change on the webpage at the same time.

Notably, new elements might need to be added to the webpage, which in turn means that new bindings such as button clicks have to be added to those, in a way that keeps the page working.

The only way to do this basically is to have a functional dependency graph that keeps everything in the page in working state as updates come.

Open access with solutions: courses.maths.ox.ac.uk/course/view.php?id=4988

Archive: web.archive.org/web/20240402062301/https://courses.maths.ox.ac.uk/course/view.php?id=4988

Lecturer: Luc Nguyen

This is about functions that take functions as input or output.

Set of ordered pairs. That's it! This is illustrated at: math.stackexchange.com/questions/1480651/is-fx-x-1-x-2-a-function/1481099#1481099