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.

en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence mentions it always exists for all invertible complex matrices. But the real condition is more complicated. Notable counter example: -1 cannot be reached by any real $e^{tk}$.

The Lie algebra exponential covering problem can be seen as a generalized version of this problem, because

Physics (like all well done science) is the art of predicting the future by modelling the world with mathematics.

And predicting the future is the first step towards controlling it, i.e.: engineering.

Ciro Santilli doesn't know physics. He writes about it partly to start playing with some scientific content for: OurBigBook.com, partly because this stuff is just amazingly beautiful.

Ciro's main intellectual physics fetishes are to learn quantum electrodynamics (understanding the point of Lie groups being a subpart of that) and condensed matter physics.

Every science is Physics in disguise, but the number of objects in the real world is so large that we can't solve the real equations in practice.

Luckily, due to emergence, we can use uglier higher level approximations of the world to solve many problems, with the complex limits of applicability of those approximations.

Therefore, such higher level approximations are highly specialized, and given different names such as:

As of 2019, all known physics can be described by two theories:

Unifying those two into the theory of everything one of the major goals of modern physics.

RIP: www.quora.com/Is-SourceForge-still-relevant-to-open-source-projects/answer/Ciro-Santilli

The set of all algebraic numbers forms a field.

This field contains all of the rational numbers, but it is a quadratically closed field.

Like the rationals, this field also has the same cardinality as the natural numbers, because we can specify and enumerate each of its members by a fixed number of integers from the polynomial equation that defines them. So it is a bit like the rationals, but we use potentially arbitrary numbers of integers to specify each number (polynomial coefficients + index of which root we are talking about) instead of just always two as for the rationals.

Each algebraic number also has a degree associated to it, i.e. the degree of the polynomial used to define it.

TODO understand.

Because a Git commit can have more than 1 parent due to merge commits when you do:

It can even have more than 2, there's no limit. Although that is not so common (with good reason, 2 is already one too many): softwareengineering.stackexchange.com/questions/314215/can-a-git-commit-have-more-than-2-parents/377903#377903

Sometimes mathematicians go a little overboard with their naming.

Bibliography:

There's a billion simple looking expressions which are not known to be transcendental numbers or not. It's cute simple to state but hard to prove at its best.

Open as of 2020:

Bibliography: