Ciro Santilli @cirosantilli 37

Kind of works! Notably, has the amazing cycling database offline for you, if you fall within the 6 area downloads. It is worth supporting these people beyond the 6 free downloads however.

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.

This game is quite detailed: www.youtube.com/watch?v=w4Jmqp8a_bU

When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.

This is perhaps the best "default definition".

Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Haven't found the one yet:

Optional but really ideal:

The state of messaging is ridiculous as of 2020.

Many/most companies are unable to give any beauty to its employees.

Hiring is simply a process of "let's get this money making project working ASAP", bring people in, without considering Brooks's law.

And then when that happens, companies put people in extremely narrow knowledge areas, making them unable to see or participate in the bigger picture of things, unless they spend 10 years there and reach architect status.

This is perhaps particularly painful for high flying birds like Ciro Santilli.

Companies need a higher top to down force that attempts to actually teach the business and tech to every employee to counter the low level manager get things done now pressure.

Companies that are able to do that, will have many more employees with a sense of purpose, and with the ability to innovate. Those companies will win.

Applications:

It is harder to measure the impact of nonprofits than of for-profits, since you can't just look at their bank balances.

This is one fundamental difficulty of nonprofit work, how to prove that you deserve the investments and not someone else.

Initially there were mathematical reasons why people suspected that all boson needed to have 0 mass as is the case for photons a gluons, see Goldstone's theorem.

However, experiments showed that the W boson and the Z boson both has large non-zero masses.

So people started theorizing some hack that would fix up the equations, and they came up with the higgs mechanism.

Movies that are very expensive to make tend to be bad, because they have to make returns and thus appeal to a large amorphous population without any specialization, i.e. the lowest common denominator but in TV Tropes terminology rather than mathematics: tvtropes.org/pmwiki/pmwiki.php/Main/LowestCommonDenominator.

See e.g.:

Looking down the largest flops of all time list didn't help much, only Heaven's gate appears reasonable from the top 20.

On TV Tropes: it's Popular Now It Sucks.

As of 2020, basically means "liquid nitrogen temperature", which is much cheaper than liquid helium.

The dream of course being room temperature and pressure superconductor.

MRI is using NMR to image inside peoples bodies!

Microbit simulator using some Microsoft framework.

TODO the Python code from there does not seem to run on the microbit via uflash, because it is not MicroPython.

support.microbit.org/support/solutions/articles/19000111744-makecode-python-and-micropython explains.

forum.makecode.com/t/help-understanding-local-build-options/6130 asks how to compile locally and suggests it is possible. Seems to require Yotta, so presumably compiles?

Presumably this is because Microsoft ported their MakeCode thing to the MicroBit, and the Micro Bit foundation accepted them.

E.g. there toggling a LED:but the code that works locally is a completely differently named API set_pixel:Microsoft going all in on adopt extend extinguish from an early age!

Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!

Each $M_n$ is a permutation group on $n$ elements. There isn't an obvious algorithmic relationship between $n$ and the actual group.

TODO initial motivation? Why did Mathieu care about k-transitive groups?

Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of $M_{23}$ and be done with it.