The most powerful GUI file manager ever?? Infinite configurability??
Ciro Santilli wasted some time on it before he gave up on file managers altogether and started using only the CLI with a few aliases.
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 can now be an operator instead of just a number.
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 are and , and the associated eigenvectors are:symPy code:and from the eigendecomposition of a real symmetric matrix we know that:
A = Matrix([[2, sqrt(2)], [sqrt(2), 3]])
A.eigenvects()
Now, instead of , we could use , where is an arbitrary diagonal matrix of type:With this, would reach a new matrix :Therefore, with this congruence, we are able to multiply the eigenvalues of by any positive number and . 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 does not have to contain eigenvalues, unlike the eigendecomposition of a matrix. This is because here is not fixed to having eigenvectors in its columns.
But because the matrix is symmetric however, we could always choose to actually diagonalize as mentioned at eigendecomposition of a real symmetric matrix. Therefore, the metric signature can be seen directly from eigenvalues.
What this does represent, is a general change of basis that maintains the matrix a symmetric matrix.
"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.
Notably, the dream of most of those companies is to have a machine that sits on a lab bench, which synthesises whatever you want.
The initial main applications are likely going to be:but the real pipe dream is building and bootstraping entire artificial chromosomes
- polymerase chain reaction primers (determine which region will be amplified
- creating a custom sequence to be inserted in a plasmid, i.e. artificial gene synthesis
News coverage:
- 2023-03 twitter.com/sethbannon/status/1633848116154880001
- 2020-10-05 www.nature.com/articles/s41587-020-0695-9 "Enzymatic DNA synthesis enters new phase"
Nuclera eDNA enzymatic de novo DNA synthesis explanatory animation (2021)
Source. The video shows nicely how Nuclera's enzymatic DNA synthesis works:- they provide blocked nucleotides of a single type
- add them with the enzyme. They use a werid DNA polymerase called terminal deoxynucleotidyl transferase that adds a base at a time to a single stranded DNA strand rather than copying from a template
- wash everything
- do deblocking reaction
- and then repeat until done
There are unlisted articles, also show them or only show them.