Ciro Santilli @cirosantilli  40

Ciro Santilli  40 Updated 2025-07-16

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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Positive definite matrix by

Ciro Santilli  40 Updated 2025-07-16

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The dot product is a positive definite matrix, and so we see that those will have an important link to familiar geometry.

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Halting problem by

Ciro Santilli  40 Updated 2025-07-16

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The canonical undecidable problem.

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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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Debugger by

Ciro Santilli  40 Updated 2025-07-16

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De novo DNA synthesis company by

Ciro Santilli  40 Updated 2025-07-16

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Artificial gene synthesis by

Ciro Santilli  40 Updated 2025-07-16

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Using de novo DNA synthesis to synthesize a genes to later insert somewhere.

Note that this is a specific application of de novo DNA synthesis, e.g. polymerase chain reaction primers is another major application that does not imply creating genes.

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Species bootstrapping from DNA by

Ciro Santilli  40 Updated 2025-07-16

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Synthesizing the DNA itself is not the only problem however.

You then have to get that DNA into a working living form state so that normal cell processes can continue:

for viri see: synthetic virus
for bacteria, you have to inject it into a cell
for placental animals, you also have to somehow simulate a compatible placenta. It is likely easier for eggs.

Multicellular questions:

biology.stackexchange.com/questions/8590/can-extinct-animals-be-cloned

Apparently achieved for the first time in 2021: www.jcvi.org/research/first-self-replicating-synthetic-bacterial-cell by the J. Craig Venter Institute.

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WikiLeaks by

Ciro Santilli  40 Updated 2025-07-16

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List of data breaches by

Ciro Santilli  40 Updated 2025-07-16

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Eat Just by

Ciro Santilli  40 Updated 2025-07-16

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Tearjerker by

Ciro Santilli  40 Updated 2025-07-16

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CC BY by

Ciro Santilli  40 Updated 2025-07-16

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CC BY-SA by

Ciro Santilli  40 Updated 2025-07-16

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This one strikes the right balance between restriction and permissions. NC and ND are simply too restrictive.

TODO where does the SA boundary end? E.g.:

Does source code need to be redistributed: opensource.stackexchange.com/questions/1849/does-the-cc-by-sa-license-require-that-source-code-of-derivative-works-be-shared

Case law list on the CC wiki: wiki.creativecommons.org/wiki/Category:Case_Law

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CC BY-NC-SA by

Ciro Santilli  40 Updated 2025-07-16

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Too restrictive. People should be able to make money from stuff.

The definition of "commercial" could also be taken in extremely broad senses, making serious reuse risky in many applications.

Notably, many university courses use it, notably MIT OpenCourseWare. Ciro wonders if it is because academics are wary of industry, or if they want to make money from it themselves. This reminds Ciro of a documentary he watched about the origins of one an early web browsers in some American university. And then that university wanted to retain copyright to make money from it. But the PhDs made a separate company nonetheless. And someone from the company rightly said something along the lines of:

The goal of universities is to help create companies and to give back to society like that. Not to try and make money from inventions.

TODO source.

The GNU project does not like it either www.gnu.org/licenses/license-list.en.html#CC-BY-NC:

This license does not qualify as free, because there are restrictions on charging money for copies. Thus, we recommend you do not use this license for documentation.
In addition, it has a drawback for any sort of work: when a modified version has many authors, in practice getting permission for commercial use from all of them would become infeasible.

en.wikipedia.org/wiki/Creative_Commons_NonCommercial_license#Defining_%22Noncommercial%22 also talks about the obvious confusion this generates: nobody can agree what counts as commercial or not!

In September 2009 Creative Commons published a report titled, "Defining 'Noncommercial'". The report featured survey data, analysis, and expert opinions on what "noncommercial" means, how it applied to contemporary media, and how people who share media interpret the term. The report found that in some aspects there was public agreement on the meaning of "noncommercial", but for other aspects, there is wide variation in expectation of what the term means.

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Counter-Strike: Global Offensive by

Ciro Santilli  40 Updated 2025-07-16

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Coronavirus Replication Cycle by

Ciro Santilli  40 Updated 2025-07-16

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www.youtube.com/watch?v=zvuYJTL90J8&t=166s The Coronavirus Replication Cycle by Kevin Tokoph (2020)

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COVID-19 by

Ciro Santilli  40 Updated 2025-07-16

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COVID happens in two stages:

viral infection
inflammatory phase, where the body takes over, and sometimes harms itself. It seems that people are not generally contagious at this point?

This distinction is one of the reasons why separating the virus name (SARS-CoV-2) from the disease makes sense: the disease is much broader than the viral infection.

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Dan Kaminsky by

Ciro Santilli  40 Updated 2025-07-16

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A superstar security researcher with some major exploits from in the 2000's.

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