Mentinoned at en.wikipedia.org/wiki/Jim_Baggott quoting popsciencebooks.blogspot.com/2012/09/jim-baggott-four-way-interview.html

Ciro Santilli and Jim would get along mighty well: there is value in tutorials written by beginners.

This is actually pretty good! Makes a small first step into The missing link between basic and advanced.

By the Simons Foundation.

Unfortunately does not use a free license for content.

One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for K-12, lower undergrad levels), but extremely hard to prove.

This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.

Bibliography:

Randomly reproduced at: web.archive.org/web/20080105074243/http://personal.stevens.edu/~nkahl/Top100Theorems.html

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:And we also know that for $b=e$ in particular that we satisfy the exponential function differential equation and so: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 $e^0$ to $e^1$.

E.g., if we split into 2 parts, we know that:or in three parts:so we can just use arbitrarily many parts $e^{1/n}$ that are arbitrarily close to $x=0$:and more generally for any $x$ we have:

Let's see what happens with the Taylor series. We have near $y=0$ in little-o notation:Therefore, for $y=x/n$, which is near $y=0$ for any fixed $x$:and therefore: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.:

To do that, let's multiply $e^y$ by itself once:and multiplying a third time:TODO conclude.

Direct consequence of Euclid's formula.

Multivariate polynomial where each term has degree 2, e.g.:is a quadratic form because each term has degree 2:but e.g.:is not because the term $x^3$ has degree 3.

More generally for any number of variables it can be written as:

There is a 1-to-1 relationship between quadratic forms and symmetric bilinear forms. In matrix representation, this can be written as:where $\vec{x}$ contains each of the variabes of the form, e.g. for 2 variables:

Strictly speaking, the associated bilinear form would not need to be a symmetric bilinear form, at least for the real numbers or complex numbers which are commutative. E.g.:But that same matrix could also be written in symmetric form as:so why not I guess, its simpler/more restricted.

The book is very dry, extremelly boring unfortunately. Definition and theorem only for the most part.

This is what society gets for not using open knowledge: some of its best minds will be bound to waste endless hours reversing some useless technology.

With that said, even when you do have the source code, reading run logs and using debuggers are a sort of reverse engineering at heart.

One of the most jaw dropping reverse engineering projects Ciro has ever seen is the Super Mario 64 reverse engineering project.

A cool thought about cancer expressed at Power, Sex, Suicide by Nick Lane (2006) is that cancer it is the direct product of natural selection gone wrong!

Cancer cells are obviously selected against anti-cancer mechanism, which when they manage to evade, they reproduce uncontrollably, gaining more and more momentum.