Ciro Santilli @cirosantilli 35

A linear map is a function $f:V_1(F)\to V_2(F)$ where $V_1(F)$ and $V_2(F)$ are two vector spaces over underlying fields $F$ such that:

A common case is $F=\mathbb{R}$, $V_1={\mathbb{R}}_m$ and $V_2={\mathbb{R}}_n$.

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of $F$.

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

Online forums that lock threads after some time are evil. What else needs to be said?

Ciro Santilli dislikes the fact that they take themselves too seriously. Ciro prefers the jokes and tech approach.

"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.

Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.

Grouping by age as done in traditional education as of 2020 is useless.

Rather, we should group students by subject of interest; e.g. natural sciences, social sciences, a sport, etc., just like in any working adult organization!

This way, younger students can actually actively learn from and collaborate with older students about, see notably Jacques Monod's you can learn more from older students than from faculty.

This becomes even more natural when you try to give students must have a flexible choice of what to learn.

This age distinction should be abolished at all stages of the system, not only within K-12, but also across K-12, undergraduate education and postgraduate education.

This idea is part of the ideal that the learning environment should be more like a dojo environment (AKA Peer tutoring, see also dojo learning model), rather than an amorphous checkbox ticking exercise in bureaucracy so that "everyone is educated".

Perhaps, even more importantly, is that we should put much more emphasis on grouping students with other students online, where we can select similar interest amongst the entire population and not just on a per-local-neighbourhood basis.

Text materials are generally superior to video because they:

Only produce video material if:

Never create videos of people just speaking hardcore content for long amounts of time.

If you have to use videos, make them as short as possible, and index them with a textual table of contents.

Also consider using sequences of images or GIFs instead of videos, since those are cheaper.

For example, if you write LaTeX files for you PDFs, give both PDFs and the LaTeX.

This allows other people to:

There is only one thing that can truly motivate you to make good materials: becoming famous.

Strive for that. Make good materials. Publish them. Get good reviews. Loop.

This generates a virtuous loop, which makes you produce better and better material.

What big companies have been created in Europe after World War II, that have not been bought or utterly defeated by American or Japanese companies?Because of all these failures, much fanfare was made as Spotify reached a $50B market capitalization in 2020. An art company, so cute!

As of 2023, the LVMH was the most valuable company in Europe by market capitalization^[ref]. Luxury goods. An area of industry that borders between the useless and the evil.

Europe has basically become an outsourcing hub for the United States. The fact that its starts are all sold if they become large enough just means that R&D is also outsourced.

ASML, and perhaps more maeaningfully its parent/predecessor ASM International from 1964 is perhaps the biggest exception.

The key problem is that there are so many small countries in Europe, that any startup has to deal with too many incompatible legislation and cannot easily sell to the hole of Europe and scale. So then a larger company from a more uniform country comes and eats it up!

Talent mobility is another issue:

So why can't Europe unify its laws?

Because the countries are still essentially walled off by languages. Europe is the perfect example of why having more than one natural language is bad for the world.

There isn't true mobility of people between countries.

You just can't go study or work in any other country (except for the UK, when it was still in the EU) without putting a huge effort into learning its language first.

Without this, there isn't enough mixing to truly make cultures more uniform, and therefore allow the laws to be more uniform.

Europe can't even unify basic things like:

Equally so, it can't force little fiscal paradises who effectively benefit from being in Europe like Ireland, Luxembourg, Monaco, Switzerland ("not European", but should that be allowed?) and Cyprus (the EU can't even maintain its territorial integrity, let alone fiscal) to not offer ridiculously low taxes and incentives which make them entry points for foreign companies to rape Europe.

For this reason, Europe will only continue to go downhill with the years, and the United Kingdom will continue to try and endosymbiose into a state of the United States (although at times it seems that it would rather endosymbiose with China instead).

Historically, this disunion is partly due to the European balance of power, whereby countries would form alliances with old enemies to prevent another country from taking over. Also linked are failed military unification attempts by Napoleon and Hitler, though we are likely better off without the latter succeeding!!! Though those also partly failed due to wider balance of power issues involving the United Kingdom, the Soviet Union and USA, not only due to internal balance. Of course, none of that matters anymore after World War II, where other more unified Europe-sized potencies rose, first the USA and the Soviet Union, and then China, and now European disunion is nothing but a burden.

Evidence such as those makes it clear that the European Union is a failure.

One thing must be said in favour of Europe's mess however: it favours international collaboration in huge projects as a more neutral middle ground. This can be seen more clearly in the ITER and the fiasco that was the Superconducting Super Collider that was cancelled a couple of billion dollars in partly because it failed to attract any foreign investment, compared to the Large Hadron Collider which went on to find the Higgs boson as mentioned at www.scientificamerican.com/article/the-supercollider-that-never-was/.

Bibliography:

Points in the direction in which a wind spinner spins fastest.

Can be denoted either by:Our default symbol is going to be:

Continuous version of the Fourier series.

Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.

Of course, every function defined on a finite line segment (i.e. a compact space).

Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.

As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.