Ciro Santilli @cirosantilli 35

This is the classic result of formal language theory, but there is too much slack between context free and context sensitive, which is PSPACE (larger than NP!).

By Noam Chomsky.

A good summary table that opens up each category much more can be seen e.g. at the bottom of en.wikipedia.org/wiki/Automata_theory under the summary thingy at the bottom entitled "Automata theory: formal languages and formal grammars".

The opposite of from first principles.

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as $GF(n)$.

Every element of a finite field satisfies $x^{order}=x$.

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

$4=2^2$, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

Multiplication is done modulo $X^2+X+1$, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

Without modulo, that would not be one of the elements of the field anymore due to the $1X^2$!

So we take the modulo, we note that:and by the definition of modulo:which is the final result of the multiplication.

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

GitHub account: github.com/hplgit

It should be mentioned that when you start Googling for PDE stuff, you will reach Han's writings a lot under his GitHub Pages: hplgit.github.io/, and he is one of the main authors of the FEniCS Project.

Unfortunately he died of cancer in 2016, shame, he seemed like a good educator.

He also published to GitHub pages with his own crazy markdown-like multi-output markup language: github.com/hplgit/doconce.

Rest in peace, Hans.

Subtle is the Lord by Abraham Pais (1982) chapter 4 "Entropy and Probability" mentions well how Boltzmann first thought that the second law was an actual base physical law of the universe while he was calculating numerical stuff for it, including as late as 1872.

But then he saw an argument by Johann Joseph Loschmidt that given the time reversibility of classical mechanics, and because they were thinking of atoms as classical balls as in the kinetic theory of gases, then there always exist a valid physical state where entropy decreases, by just reversing the direction of time and all particle speeds.

So from this he understood that the second law can only be probabilistic, and not a fundamental law of physics, which he published clearly in 1877.

The general result from eigendecomposition of a matrix:becomes:where $O$ is an orthogonal matrix, and therefore has $O^{-1}=O^T$.

This is a good thing. It basically contains an entire website, with HTML and assets inside a single ZIP, and a little bit of metadata.

It is incomprehensible why browsers don't just implement it as they already have all the web part, and also ZIP stuff:

The situation is so sad. Ubuntu 21.04 doesn't come with a reader installed by default:

DNA amplification is one of the key DNA technologies:

Big excitement picture at: molecular biology technologies.

A concrete experiment has been done at Section "Sequencing" on section sequencing.

Given a bunch of points in $n$ dimensions, PCA maps those points to a new $p$ dimensional space with $p\le n$.

$p$ is a hyperparameter, $p=1$ and $p=2$ are common choices when doing dataset exploration, as they can be easily visualized on a planar plot.

The mapping is done by projecting all points to a $p$ dimensional hyperplane. PCA is an algorithm for choosing this hyperplane and the coordinate system within this hyperplane.

The hyperplane choice is done as follows:

www.sartorius.com/en/knowledge/science-snippets/what-is-principal-component-analysis-pca-and-how-it-is-used-507186 provides an OK-ish example with a concrete context. In there, each point is a country, and the input data is the consumption of different kinds of foods per year, e.g.:so in this example, we would have input points in 4D.

The question is then: we want to be able to identify the country by what they eat.

Suppose that every country consumes the same amount of flour every year. Then, that number doesn't tell us much about which country each point represents (has the least variance), and the first PCA axes would basically never point anywhere near that direction.

Another cool thing is that PCA seems to automatically account for linear dependencies in the data, so it skips selecting highly correlated axes multiple times. For example, suppose that dry codfish and olive oil consumption are very high in Portugal and Spain, but very low in Germany and Poland. Therefore, the variation is very high in those two parameters, and contains a lot of information.

However, suppose that dry codfish consumption is also directly proportional to olive oil consumption. Because of this, it would be kind of wasteful if we selected:since the information about codfish already tells us the olive oil. PCA apparently recognizes this, and instead picks the first axis at a 45 degree angle to both dry codfish and olive oil, and then moves on to something else for the second axis.

We can see that much like the rest of machine learning, PCA can be seen as a form of compression.

Forms a normal subgroup of the general linear group.