Ciro Santilli @cirosantilli 37

Ciro Santilli 37 Updated 2025-07-16

Starting tx a87d406fae047258a12923b3c11a797a5765bd8f868df5c7e9b1cead0e92c9c1: the message:

503: Bitcoin over capacity!

appears about 13 thousand times. WTF happened?

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Conservation of the square amplitude in the Schrodinger equation by

Ciro Santilli 37 Updated 2025-07-16

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Proof that the probability 1 is conserved by the time evolution:

It can be derived directly from the Schrödinger equation.

Bibliography:

quantummechanics.ucsd.edu/ph130a/130_notes/node127.html from quantum physics by Jim Branson (2003).
That proof also mentions that if the potential V is not real, then there is no conservation of probability! Therefore the potential must be real valued!

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Parasites tend to have smaller DNAs by

Ciro Santilli 37 Updated 2025-07-16

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If you live in the relatively food abundant environment of another cell, then you don't have to be able to digest every single food source in existence, of defend against a wide range of predators.

So because DNA replication is a key limiting factor of bacterial replication time, you just reduce your genome to a minimum.

And likely you also want to be as small as possible to evade the host's immune system.

Power, Sex, Suicide by Nick Lane (2006) section "Gene loss as an evolutionary trajectory" puts it well:

One of the most extreme examples of gene loss is Rickettsia prowazekii, the cause of typhus. [...] Over evolutionary time Rickettsia has lost most of its genes, and now has a mere  protein-coding genes left. [...] Rickettsia is a tiny bacterium, almost as small as a virus, which lives as a parasite inside other cells. It is so well adapted to this lifestyle that it can no longer survive outside its host cells. [...] It was able to lose most of its genes in this way simply because they were not needed: life inside other cells, if you can survive there at all, is a spoonfed existence.

and also section "How to lose the cell wall without dying" page 184 has some related mentions:

While many types of bacteria do lose their cell wall during parts of their life cycle only two groups of prokaryotes have succeeded in losing their cell walls permanently, yet lived to tell the tale. It's interesting to consider the extenuating circumstances that permitted them to do so.
[...]
One group, the Mycoplasma, comprises mostly parasites, many of which live inside other cells. Mycoplasma cells are tiny, with very small genomes. M. genitalium, discovered in 1981, has the smallest known genome of any bacterial cell, encoding fewer than 500 genes. M. genitalium, discovered in 1981, has the smallest known genome of any bacterial cell, encoding fewer than 500 genes. [...] Like Rickettsia, Mycoplasma have lost virtually all the genes required for making nucleotides, amino acids, and so forth.

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Arecibo message by

Ciro Santilli 37 Updated 2025-07-16

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Google Cloud Platform by

Ciro Santilli 37 Updated 2025-07-16

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Clifford plus T by

Ciro Santilli 37 Updated 2025-07-16

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Set of quantum logic gate composed of the Clifford gates plus the Toffoli gate. It forms a set of universal quantum gates.

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Ainan Celeste Cawley by

Ciro Santilli 37 Updated 2025-07-16

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His father fought a lot with the stupid educational system to try and move his son to his full potential and move to more advanced subjects early.

A crime of society to try and prevent it. They actually moved the family from Singapore to Malaysia for a learning opportunity for the son. Amazing.

This is the perfect illustration of one of Ciro Santilli's most important complaints about the 2020 educational system:

and why Ciro created OurBigBook.com to try and help fix the issue.

Possible social media

www.instagram.com/ainancawley

Bibliography:

Video 1.

The Most Talented Children And Adults On The Planet by Our Life (2008)

Source. Has some good mentions of Ainan and others.

Video 2.

Ainan Cawley: Child prodigy (2013)

Source.

youtu.be/Ugi74cwPMSg?t=137 finding the right school

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CC BY-SA 4.0 by

Ciro Santilli 37 Updated 2025-07-16

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Euler-Lagrange equation by

Ciro Santilli 37 Updated 2025-07-16

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Let's start with the one dimensional case. Let the

q : R \to R

and a Functional

I

defined by a function of three variables

F : R^{3} \to R

I (q) = \int_{t_{0}}^{t} F (t, q (t), \overset{q}{˙} (t)) d t

(1)

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given

F

, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that

F

such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙} = 0

(2)

\frac{\partial F}{\partial q}

and

\frac{\partial F}{\partial q ˙}

we simply mean "the partial derivative of

F

with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where

q

is the unknown, since:

the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q}$ is a function of $t, q (t), \overset{q}{˙} (t)$
the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙}$ is a function of $t, q (t), \overset{q}{˙} (t)$ . And so it's derivative with respect to time will contain only up to $\overset{q}{¨}$

Now let's think about the multi-dimensional case. Instead of having

q : R \to R

, we now have

q : R \to R^{n}

. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case,

F

is going to be a function of 5 variables

F : R^{5} \to R

rather than 3 as in the one dimensional case, and the functional looks like:

I (q) = \int_{t_{0}}^{t} F (t, q_{1} (t), q_{2} (t), \overset{q_{1}}{˙} (t), \overset{q_{2}}{˙}) (t) d t

(3)

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions

q_{1}

and

q_{2}

of order up to 2 in both variables:

\frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{2} ˙} = 0

(4)

At this point, notation is getting a bit clunky, so people will often condense the

q_{i}

vector

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2} ˙} = 0

(5)

or just omit the arguments of

F

entirely:

\frac{\partial F}{\partial q _{i}} - \frac{d}{d t} \frac{\partial F}{\partial q _{i} ˙} = 0

(6)

Video 1.

Calculus of Variations ft. Flammable Maths by vcubingx (2020)

Source.

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Ultraviolet catastrophe by

Ciro Santilli 37 Updated 2025-07-16

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Video 1.

What is the Ultraviolet Catastrophe? by Physics Explained (2020)

Source.

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Big O notation by

Ciro Santilli 37 Updated 2025-07-16

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Module bound above, possibly multiplied by a constant:

f (x) = O (g (x))

(1)

is defined as:

\exists M > 0\exists x_{0} \forall x > x_{0} : ∣ f (x) ∣ \leq M g (x)

(2)

E.g.:

$\forall c \in R x + c = O (x)$ . For $c < 0$ , $M = 1$ is enough. Otherwise, any $M > 1$ will do, the bottom line will always catch up to the top one eventually.