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Ciro Santilli 40 Updated 2025-07-16

The differential equation that is solved by the exponential function:

y^{'} (x) = y (x)

(1)

with initial condition:

y (0) = 1

(2)

TODO find better name for it, "linear homogenous differential equation of degree one" almost fully constrainst it except for the exponent constant and initial value.

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List of electric guitar players by

Ciro Santilli 40 Updated 2025-07-16

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Taylor expansion definition of the exponential function by

Ciro Santilli 40 Updated 2025-07-16

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The Taylor series expansion is the most direct definition of the expontial as it obviously satisfies the exponential function differential equation:

the first constant term dies
each other term gets converted to the one before
because we have infinite many terms, we get what we started with!

e^{x} = \sum_{n = 0}^{\infty} \frac{x ^{n}}{n !} = 1 + \frac{x}{1} + \frac{x ^{2}}{2} + \frac{x ^{3}}{2 \times 3} + \frac{x ^{4}}{2 \times 3 \times 4} + \dots

(1)

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Product definition of the exponential function by

Ciro Santilli 40 Updated 2025-07-16

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e^{x} = lim_{n \to \infty} (1 + \frac{x}{n})^{n}

(1)

The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.

More precisely, we know that for any base b, exponentiation satisfies:

$b^{x + y} = b^{x} b^{y}$ .
$b^{0} = 1$ .

And we also know that for

b = e

in particular that we satisfy the exponential function differential equation and so:

\frac{d e ^{x}}{d x} (0) = 1

(2)

One interesting fact is that the only thing we use from the exponential function differential equation is the value around

x = 0

, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate

e^{1}

. The idea is to start from

e^{0}

and then then to use the first order of the Taylor series to extend the known value of

e^{0}

e^{1}

E.g., if we split into 2 parts, we know that:

e^{1} = e^{1/2} e^{1/2}

(3)

or in three parts:

e^{1} = e^{1/3} e^{1/3} e^{1/3}

(4)

so we can just use arbitrarily many parts

e^{1/ n}

that are arbitrarily close to

x = 0

e^{1} = (e^{1/ n})^{n}

(5)

and more generally for any

x

we have:

e^{x} = (e^{x / n})^{n}

(6)

Let's see what happens with the Taylor series. We have near

y = 0

in little-o notation:

e^{y} = 1 + y + o (y)

(7)

Therefore, for

y = x / n

, which is near

y = 0

for any fixed

x

e^{x / n} = 1 + x / n + o (1/ n)

(8)

and therefore:

e^{x} = (e^{x / n})^{n} = (1 + x / n + o (1/ n))^{n}

(9)

which is basically the formula tha we wanted. We just have to convince ourselves that at

lim_{n \to \infty}

, the

o (1/ n)

disappears, i.e.:

(1 + x / n + o (1/ n))^{n} = (1 + x / n)^{n}

(10)

To do that, let's multiply

e^{y}

by itself once:

e^{y} e^{y} = (1 + y + o (y)) (1 + y + o (y)) = 1 + 2 y + o (y)

(11)

and multiplying a third time:

e^{y} e^{y} e^{y} = (1 + 2 y + o (y)) (1 + y + o (y)) = 1 + 3 y + o (y)

(12)

TODO conclude.

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Expectation value by

Ciro Santilli 40 Updated 2025-07-16

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Logarithm by

Ciro Santilli 40 Updated 2025-07-16

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Debugging by

Ciro Santilli 40 Updated 2025-07-16

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Debugging sucks. But there's also nothing quite that "oh fuck, that's why it doesn't work" moment, which happens after you have examined and placed everything that is relevant to the problem into your brain. You just can't see it coming. It just happens. You just learn what you generally have to look at so it happens faster.

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Ciro's call hierarchy notation by

Ciro Santilli 40 Updated 2025-07-16

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This is a simple hierarchical plaintext notation Ciro Santilli created to explain programs to himself.

It is usuall created by doing searches in an IDE, and then manually selecting the information of interest.

It attempts to capture intuitive information not only of the call graph itself, including callbacks, but of when things get called or not, by the addition of some context code.

For example, consider the following pseudocode:

f1() {
}

f2(i) {
  if (i > 5) {
    f1()
  }
}

f3() {
  f1()
  f2_2()
}

f2_2() {
  for (i = 0; i < 10; i++) {

    f2(i)
  }
}

main() {
  f2_2()
  f3()
}

Supose that we are interested in determining what calls f1.

Then a reasonable call hierarchy for f1 would be:

f2(i)
  if (i > 5) {
    f1()

  f2_2()
    for (i = 0; i < 10; i++) {
      f2(i)

    main
    f3
f3()
  main()

Some general principles:

start with a regular call tree
to include context:
- remove any blank lines from the snippet of interest
- add it indented below the function
- and then follow it up with a blank line
- and then finally add any callers at the same indentation level

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AlphaFold by

Ciro Santilli 40 Updated 2025-07-16

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Bisection (software engineering) by

Ciro Santilli 40 Updated 2025-07-16

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One of the Holiest age old debugging techniques!

Git has some helpers to help you achieve bisection Nirvana: stackoverflow.com/questions/4713088/how-to-use-git-bisect/22592593#22592593

Obviously not restricted to software engineering alone, and used in all areas of engineering, e.g. Video "Air-tight vs. Vacuum-tight by AlphaPhoenix (2020)" uses it in vacuum engineering.

The cool thing about bisection is that it is a brainless process: unlike when using a debugger, you don't have to understand anything about the system, and it incredibly narrows down the problem cause for you. Not having to think is great!

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Matrix exponential by

Ciro Santilli 40 Updated 2025-07-16

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Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.

Video 1.

How (and why) to raise e to the power of a matrix by 3Blue1Brown (2021)

Source.

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Polynomial by

Ciro Santilli 40 Updated 2025-07-16

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Robert Noyce: The Man Behind the Microchip by Leslie Berlin (2006) by

Ciro Santilli 40 Updated 2025-07-16

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Borrow from the Internet Archive for free: archive.org/details/manbehindmicroc000berl/page/n445/mode/2up

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Debugger by

Ciro Santilli 40 Updated 2025-07-16

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Human body by

Ciro Santilli 40 Updated 2025-07-16

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Cubic equation by

Ciro Santilli 40 Updated 2025-07-16

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GNU Debugger by

Ciro Santilli 40 Updated 2025-07-16

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Just add GDB Dashboard, and you're good to go.

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Pernosco by

Ciro Santilli 40 Updated 2025-07-16

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pernos.co/

Proprietary extension to Mozilla rr by rr lead coder Robert O'Callahan et. al, started in 2016 after he quit Mozilla.

TODO what does it add to rr?

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2019 redefinition of the SI base units by

Ciro Santilli 40 Updated 2025-07-16

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web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf gives it in raw:

the unperturbed ground state hyperfine transition frequency of the caesium-133 atom $Δ v_{C s}$ is 9 192 631 770 Hz
the speed of light in vacuum c is 299 792 458 m/s
the Planck constant h is 6.626 070 15 × $1 0^{- 34}$ J s
the elementary charge e is 1.602 176 634 × $1 0^{- 19}$ C
the Boltzmann constant k is 1.380 649 × $1 0^{- 23}$ J/K
the Avogadro constant NA is 6.022 140 76 × $1 0^{23}$ mol
the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W,

The breakdown is:

actually use some physical constant:
- the unperturbed ground state hyperfine transition frequency of the caesium-133 atom $Δ v_{C s}$ is 9 192 631 770 Hz
  Defines the second in terms of caesium-133 experiments. The beauty of this definition is that we only have to count an integer number of discrete events, which is what allows us to make things precise.
- the speed of light in vacuum c is 299 792 458 m/s
  Defines the meter in terms of speed of light experiments. We already had the second from the previous definition.
- the Planck constant h is 6.626 070 15 × $1 0^{- 34}$ J s
  Defines the kilogram in terms of the Planck constant.
- the elementary charge e is 1.602 176 634 × $1 0^{- 19}$ C
  Defines the Coulomb in terms of the electron charge.
arbitrary definitions based on the above just to match historical values as well as possible:
- the Boltzmann constant k is 1.380 649 × $1 0^{- 23}$ J/K
  Arbitrarily defines temperature from previously defined energy (J) to match historical values.
- the Avogadro constant NA is 6.022 140 76 × $1 0^{23}$ mol
  Arbitrarily defines the mol to match historical values. In particular, the kilogram is not an exact multiple of the weight of an atom of hydrogen.
- the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W
  Arbitrarily defines the Candela in terms of previous values to match historical records. The most useless unit comes last as you'd expect.

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GDB Dashboard by

Ciro Santilli 40 Updated 2025-07-16

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github.com/cyrus-and/gdb-dashboard

GDB Nirvana?

stackoverflow.com/questions/10115540/gdb-split-view-with-code/51301717#51301717

Figure 1.
Screenshot of terminal running GDB Dashboard
. Source.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade