Ciro Santilli @cirosantilli 35

It is a shame that they refocused to more applied courses. This also highlights their highly "managed" approach to content creation. Their 2022 pitch on front page says it all:

for as few as 10 hours a week, you can get the in-demand skills you need to help land a high-paying tech job

they are focused on the highly paid character of many software engineering jobs.

But one cool point of this website is how they hire tutors to help on the courses. This is a very good thing. It is a fair way of monetizing: e-learning websites must keep content free, only charge for certification.

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Linear operator by

Ciro Santilli 35  Updated 2025-01-10  +Created 1970-01-01

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We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

a 2x2 matrix can represent a linear map from $R^{2}$ to $R^{2}$ , so which is a linear operator
the derivative is a linear map from $C^{\infty}$ to $C^{\infty}$ , so which is also a linear operator

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Applications of Lie groups to differential equations by

Ciro Santilli 35  Updated 2025-01-10  +Created 1970-01-01

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Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.

It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.

www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.

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Simple Lie group by

Ciro Santilli 35  Updated 2025-01-10  +Created 1970-01-01

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Molecular biology by

Ciro Santilli 35  Updated 2025-01-10  +Created 1970-01-01

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Ciro Santilli believes that molecular biology technologies will be a large part of the next big things as shown at: Section "Molecular biology technologies".

Bibliography:

www.youtube.com/watch?v=mS563_Teges&list=PLQbPquAyEw4dQ3zOLrdS1eF_KJJbUUyBx Biophysical Techniques Course 2022 by the MRC Laboratory of Molecular Biology. Holy crap that playlist is a tour de force of molecular biology techniques in 2022!

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Lebesgue integral vs Riemann integral by

Ciro Santilli 35  Updated 2025-01-10  +Created 1970-01-01

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Advantages over Riemann:

Lebesgue integral of $L^{p}$ is complete but Riemann isn't.
youtu.be/PGPZ0P1PJfw?t=710 you are able to switch the order of integrals and limits of function sequences on non-uniform convergence. TODO why do we care? This is linked to the Fourier series of course, but concrete example?

Video 1.

Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018)

Source.

youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.

One advantage of that is that the range is always one dimensional.

But the main advantage is that having infinitely many discontinuities does not matter.

Infinitely many discontinuities can make the Riemann partitioning diverge.

But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.

So then the question becomes more of "how to define the measure of a subset of the domain".

Which is why we then fall into measure theory!

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Intellectual property by