Ciro Santilli @cirosantilli 37

Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.

The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.

The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.

The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument $x$ can now be an operator instead of just a number.

Examples:

The main interest of this theorem is in classifying the indefinite orthogonal groups, which in turn is fundamental because the Lorentz group is an indefinite orthogonal groups, see: all indefinite orthogonal groups of matrices of equal metric signature are isomorphic.

It also tells us that a change of basis does not the alter the metric signature of a bilinear form, see matrix congruence can be seen as the change of basis of a bilinear form.

The theorem states that the number of 0, 1 and -1 in the metric signature is the same for two symmetric matrices that are congruent matrices.

For example, consider:

The eigenvalues of $A$ are $1$ and $4$, and the associated eigenvectors are:symPy code:and from the eigendecomposition of a real symmetric matrix we know that:

Now, instead of $P$, we could use $PE$, where $E$ is an arbitrary diagonal matrix of type:With this, would reach a new matrix $B$:Therefore, with this congruence, we are able to multiply the eigenvalues of $A$ by any positive number $e_1^2$ and $e_2^2$. Since we are multiplying by two arbitrary positive numbers, we cannot change the signs of the original eigenvalues, and so the metric signature is maintained, but respecting that any value can be reached.

Note that the matrix congruence relation looks a bit like the eigendecomposition of a matrix:but note that $D$ does not have to contain eigenvalues, unlike the eigendecomposition of a matrix. This is because here $S$ is not fixed to having eigenvectors in its columns.

But because the matrix is symmetric however, we could always choose $S$ to actually diagonalize as mentioned at eigendecomposition of a real symmetric matrix. Therefore, the metric signature can be seen directly from eigenvalues.

Also, because $D$ is a diagonal matrix, and thus symmetric, it must be that:

What this does represent, is a general change of basis that maintains the matrix a symmetric matrix.

Related:

As of 2018, Ciro Santilli believes that this could be the next big thing in biology technology.

"De novo" means "starting from scratch", that is: you type the desired sequence into a computer, and the synthesize it.

The "de novo" part is important, because it distinguishes this from the already well solved problem of duplicating DNA from an existing DNA template, which is what all our cells do daily, and which can already be done very efficiently in vitro with polymerase chain reaction.

Many companies are attempting to create more efficient de novo synthesis methods:

Notably, the dream of most of those companies is to have a machine that sits on a lab bench, which synthesises whatever you want.

TODO current de novo synthesis costs/time to delivery after ordering a custom sequence.

The initial main applications are likely going to be:but the real pipe dream is building and bootstraping entire artificial chromosomes

News coverage:

A software that implements some database system, e.g. PostgreSQL or MySQL are two (widely extended) SQL implementations.

Marc Verdiell is a French electrical engineer born in 1963 or 1964^[ref] and best known for being the creator and host of the CuriousMarc YouTube channel where he does mind blowing repairs and reverse engineering of vintage computers and other electronic equipment.

Marc sold his company LightLogic, an optoelectronics company he founded, to Intel in April 2001. This was just after the dot-com crash, but Intel apparently still correctly believed that the networking and the Internet would continue to grow and was investing in the area. His associate Frank Shum sued claiming he should be credited for some of the inventions sold but lost and Marc got it all.^{[ref][ref][ref]}. Marc was then almost immediately appointed an Intel fellow at the extremelly early age of 37, and then stayed for a few years at Intel until 2006 according to his LinkedIn.^[ref][ref]

Marc's LinkedIn profile: www.linkedin.com/in/marc-verdiell-9742795/

Marc's full name is actualy Jean-Marc Verdiell, but Ciro Santilli remembers there was one YouTube video where he mentions he gave up on "Jean" partly because anglophones would murder its pronounciation all the time.

Marc's PhD thesis is listed at: theses.fr/1990PA112048 and it is entitled:which is translated into English as:but the full text is not available online.

2.5.0 manual prebuilt download install on Ubuntu 20.04 just worked. Launch directly from unzip without install. Play with examples under install/Examples

Their docs are a reasonable way to learn Csound: cabbageaudio.com/docs/introduction/