Ciro Santilli @cirosantilli 37 

E.g. showing live data from a scientific instrument! TODO:

Real-time polymerase chain reaction Updated 2025-07-16

Also known as: Quantitative PCR (qPCR).

Like PCR, but the amplification machine measures the concentration of DNA at each step.

This describes one possible concentration detection method with fluorescent molecules that only become fluorescent when the DNA is double stranded (SYBR Green)

Polymerase Chain Reaction (PCR) - Quantitative PCR (qPCR) by Applied Biological Materials (2016)

Source.

This allows you to predict the exact initial concentration by extrapolating the exponential curve backwards.

TODO: vs non-real-time PCR. Why can't you just divide by 2 for every heating step to reach back the original concentration? Likely the reaction reach saturation at an unknown step.

TODO: vs non-real-time PCR in medical diagnostics: do you really need to know concentration for diagnostics? Isn't it enough to know if the virus is present or not?

Representation theory Updated 2025-07-16

Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of)

G L (n)

, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field

V

(which can be represented by a matrix infinite dimension), in a way that respects the group operations:

R (g) : G \to G L (V)

(1)

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
3.6 classifies the representations of $S U (2)$ . There is only one possibility per dimension!
3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application

Motivation:

math.stackexchange.com/questions/1628464/what-is-representation-theory

Bibliography:

www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.

Resonance Updated 2025-07-16

Resonance is a really cool thing.

Examples:

mechanical resonance, notably:
- pipe instruments
electronic oscillators, notably:
- LC oscillator, and notably the lossy version RLC circuit

Perhaps a key insight of resonance is that the reonant any lossy system tends to look like the resonance frequency quite quickly even if the initial condition is not the resonant condition itself, because everything that is not the resonant frequency interferes destructively and becomes noise. Some examples of that:

striking a bell or drum can be modelled by applying an impuse to the system
playing a pipe instrument comes down to blowing a piece that vibrates randomly, and then leads the pipe to vibrate mostly in the resonant frequency. Likely the same applies to bowed string instruments, the bow must be creating a random vibration.
playing a plucked string instrument comes down to initializing the system to an triangular wave form and then letting it evolve. TODO find a simulation of that!

Another cool aspect of resonance is that it was kind of the motivation for de Broglie hypothesis, as de Broglie was kind of thinking that electroncs might show discrete jumps on atomic spectra because of constructive interference.

Richard Feynman's drug use Updated 2025-07-16

From Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez!:

The people from the airlines were somewhat bored with their lives, strangely enough, and at night they would often go to bars to drink. I liked them all, and in order to be sociable, I would go with them to the bar to have a few drinks, several nights a week.
One day, about 3:30 in the afternoon, I was walking along the sidewalk opposite the beach at Copacabana past a bar. I suddenly got this treMENdous, strong feeling: "That's just what I want; that'll fit just right. I'd just love to have a drink right now!"
I started to walk into the bar, and I suddenly thought to myself, "Wait a minute! It's the middle of the afternoon. There's nobody here, There's no social reason to drink. Why do you have such a terribly strong feeling that you have to have a drink?" - and I got scared.
I never drank ever again, since then. I suppose I really wasn't in any danger, because I found it very easy to stop. But that strong feeling that I didn't understand frightened me. You see, I get such fun out of thinking that I don't want to destroy this most pleasant machine that makes life such a big kick. It's the same reason that, later on, I was reluctant to try experiments with LSD in spite of my curiosity about hallucinations.

One notable drug early teens Ciro consumed was Magic: The Gathering, see also: Section "Magic: The Gathering is addictive".

Riemann hypothesis Updated 2025-07-16

visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.

One of the Millennium Prize Problems and Hilbert's problems.

What is the Riemann hypothesis REALLY about? by HexagonVideos (2022)

Source.

Ring (mathematics) Updated 2025-07-16

A Ring can be seen as a generalization of a field where:

multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:

we know that 2x2 matrix multiplication is non-commutative in general
some 2x2 matrices have a multiplicative inverse, but others don't

Note that

G L (n)

is not a ring because you can by addition reach the zero matrix.

Saylor Academy Updated 2025-07-16

This is an interesting initiative which has some similarities to Ciro Santilli's OurBigBook project.

The fatal flaw of the initiative in Ciro Santilli's opinion is the lack of user-generated content. We will never get there without UGC and algorithms, never.

Also as of 2021, it mostly useless business courses: learn.saylor.org unfortunately.

But it has several redeeming factors which Ciro Santilli aproves of:

exam as a service-like
they have a GitHub: github.com/saylordotorgo

Licensing appears to be a mixed mess between the dreaded CC BY-NC-SA and the good CC BY, e.g.:

The founder Michael J. Saylor looks a bit crooked, Rich people who create charitable prizes are often crooked comes to mind. But maybe he's just weird.

Michael Saylor interview by Lex Fridman (2022)

Source.

At the timestamp:

When I go, all my assets will flow into a foundation, and the foundation's mission is to make education free for everybody forever.

What statement... maybe he's actually not crooked, maybe it was just an accounting mistake... God, why.

If only Ciro Santilli knew how to contact him and convince him that his current approach is innefective and that Ciro has something better! Michael, please Google into this page some day, Ciro Santilli needs funding for OurBigBook.com. A hopeless Tweet at: twitter.com/cirosantilli/status/1548350114623660035. Also tried to hit his saylor@strategy.com.

Scalable Vector Graphics Updated 2025-07-16

Companies have been really slow to support SVG features in their browsers, and that is very saddening: medium.com/@michaelmangial1/introduction-to-scalable-vector-graphics-6450c03e8d2e

You can't drop SVG support for canvas until there's a way to run untrusted JavaScript on the browser!

SVG does have some compatibility annoyances, notably SVG fonts. But we should as a society work to standardize and implement a fix those, the benefits of SVG are just too great!

Examples:

svg/svg.svg a minimal somewhat sane SVG:
- if the width and height properties were not given, you get the default 300x150, which seems to be set in the SVG standard:
  - stackoverflow.com/questions/40156710/why-does-this-svg-image-have-a-height-of-150px
  - css-tricks.com/scale-svg
how to add na SVG image to a HTML file:
- svg/svg.html: external image. The included file is svg/svg.svg.
- svg/inline.html: inline.
svg/billion-laughs.svg
svg/html.svg
svg/triangle.svg
svg/viewBox.svg: this attribute allows you to control the default SVG svg width= and height= while keeping the coordinates of the drawing untouched. If the viewBox aspect ratio differs from the width/height ratio, you likely want to play with preserveAspectRatio, otherwise you would get white spaces by default on the generated image
CSS with SVG:
- svg/style.svg: inline CSS
- svg/style-external.svg: external CSS with: <?xml-stylesheet type="text/css" href="svg.css" ?>, see also: stackoverflow.com/questions/18434094/how-to-style-svg-with-external-css
  - svg/subdir/style-external.html: is the relative CSS relative to the HTML or to the SVG? Answer: to the SVG... OMG. So how to make it work reliably?
- svg/current-color.html and svg/current-color.svg: illustrates fill="currentColor". Only works for inline SVG however... See also: stackoverflow.com/questions/13000682/how-do-i-have-an-svg-image-inherit-colors-from-the-html-document/13002311
JavaScript with SVG:
- svg/script.svg
- svg/external-js.svg
svg/defs.html hows how defs works
- svg/defs-external.html tries to include external defs from svg/defs.svg, but that fails like everything else related to external SVGs

Scanning electron microscope Updated 2025-07-16

The Scanning Electron Microscope by MaterialsScience2000 (2014)

Source. Shows operation of the microscope really well. Seems too easy, there must have been some extra setup before however. Impressed by how fast the image update, it is basically instantaneous. Produced by Prof. Dr.-Ing. Rainer Schwab from the Karlsruhe University of Applied Sciences.

Video 2.

Mosquito Eye Scanning Electron Microscope Zoom by Mathew Tizard (2005)

Source. Video description mentions is a composite video. Why can't you do it in one shot?

Schrödinger equation for a free one dimensional particle Updated 2025-07-16

Schrödinger equation for a one dimensional particle with

V = 0

. The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle

Then, for each energy

E

, from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:

ψ (x) = \int_{E = - \infty}^{\infty} e^{\frac{2 m E i x}{ℏ}} e^{- i Et /ℏ} = e^{i \frac{2 m E x - Et}{ℏ}}

(1)

Therefore, we see that the solution is made up of infinitely many plane wave functions.

Schrödinger equation for a one dimensional particle Updated 2025-07-16

We select for the general Equation "Schrodinger equation":

$x = x$ , the linear cartesian coordinate in the x direction
$\hat{H} = - \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)$ , which analogous to the sum of kinetic and potential energy in classical mechanics

giving the full explicit partial differential equation:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = [- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)] ψ (x, t)

(1)

Equation 1.

Schrödinger equation for a one dimensional particle

The corresponding time-independent Schrödinger equation for this equation is:

[- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x)] ψ (x) = E ψ (x)

(2)

Equation 2.

time-independent Schrödinger equation for a one dimensional particle

Schrödinger equation solution for the helium atom Updated 2025-07-16

No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.

Bibliography:

Quantum Chemistry 9.2 - Helium Atom Energy Approximations by TMP Chem (2016)

Source. Video gives the actual numerical value of various methods, second order perturbation theory being very close. But it the says that in the following videos will only do Hartree-Fock method.

Schrödinger picture Updated 2025-07-16

To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: Schrödinger picture example: quantum harmonic oscillator.

The state of a quantum system is a unit vector in a Hilbert space.

"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:

the state collapses to an eigenvector of the self adjoint operator
the result of the measurement is the eigenvalue of the self adjoint operator
the probability of a given result happening when the spectrum is discrete is proportional to the modulus of the projection on that eigenvector.
For continuous spectra such as that of the position operator in most systems, e.g. Schrödinger equation for a free one dimensional particle, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.
In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.
The continuous position operator case is well illustrated at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"

Those last two rules are also known as the Born rule.

Self adjoint operators are chosen because they have the following key properties:

their eigenvalues form an orthonormal basis
they are diagonalizable

Perhaps the easiest case to understand this for is that of spin, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a relativistic quantum theory such as the Dirac equation.

The next steps are to look at simple 1D bound states such as particle in a box and quantum harmonic oscillator.

This naturally generalizes to Schrödinger equation solution for the hydrogen atom.

The solution to the Schrödinger equation for a free one dimensional particle is a bit harder since the possible energies do not make up a countable set.

This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.

Quantum Field Theory lecture notes by David Tong (2007) mentions that:

if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field $ϕ$ .

Science fiction film Updated 2025-07-16

Science is the reverse engineering of nature Updated 2025-07-16

How software engineers view science:

Science is the reverse engineering of nature.

Ciro Santilli had once assigned this as one of Ciro Santilli's best random thoughts, but he later found that Wikipedia actually says exactly that: en.wikipedia.org/wiki/Reverse_engineering ("similar to scientific research, the only difference being that scientific research is about a natural phenomenon") so maybe that is where Ciro picked it up unconsciously in the first place.

Science makes progress funeral by funeral Updated 2025-07-16