Ciro Santilli @cirosantilli 37

The change of basis matrix $C$ is the matrix that allows us to express the new basis in an old basis:

Mnemonic is as follows: consider we have an initial basis $(x_{old},y_{old})$. Now, we define the new basis in terms of the old basis, e.g.:which can be written in matrix form as:and so if we set:we have:

The usual question then is: given a vector in the new basis, how do we represent it in the old basis?

The answer is that we simply have to calculate the matrix inverse of $M$:

That $M^{-1}$ is the matrix inverse.

Power, Sex, Suicide by Nick Lane (2006) part 5 "Murder or suicide" mentions that apoptosis has two main functions:

In simple terms, if you believe in the Schrödinger equation and its modern probabilistic interpretation as described in the Schrödinger picture, then at first it seem that there is no strict causality to the outcome of experiments.

People have then tried to recover that by assuming that there is some inner sate beyond the Schrödinger equation, but these ideas are refuted by Bell test experiments, unless we give up the principle of locality, which feels more important, especially in special relativity, where faster-than-light implies time travel, which breaks causality even more dramatically.

The de Broglie-Bohm theory is a deterministic but non-local formulation of quantum mechanics.

This feels so right. Doesn't have to be taken so literally for non-Monks, but all have clear less-extreme applications to non monks.

Ah, the Holy Grail of both biology and computer science!

Ciro Santilli feels it is not for his generation though, and that is one of the philosophical things that saddens him the most in this world.

On the other hand, Ciro's playing with the Linux kernel and other complex software which no single human can every fully understand cheer him up a bit. But still, the high level view, that we can have...

For now, Ciro's 2D reinforcement learning games.

Database of promoter.

E.g. for E. Coli K-12 MG1655: biocyc.org/group?id=:ALL-PROMOTERS&orgid=ECOLI For some context see e. Coli K-12 MG1655 gene thrL + e. Coli K-12 MG1655 gene thrA + thrB + thrC all of which are in the same transcription unit.

Stack overflow allows deleting content/making it visible only to 10k rep users.

Ciro Santilli is strictly against this, and this is an intended core policy of OurBigBook.com.

If you delete people's content randomly, they will be much less likely to write anything.

Getting downvoted to oblivion is one thing, but data loss? Unacceptable.

Only illegal content must ever be deleted. Or extremely obvious spam. But anything in a gray area should never be removed.

Deletion can be done by either:

Examples:

The voltage changes perpendicular to the current when magnetic field is applied.

An intuitive video is:

The key formula for it is:where:

Applications:

Other more precise non-classical versions:

Bibliography:

For this sub-case, we can define the Lie algebra of a Lie group $G$ as the set of all matrices $M\in G$ such that for all $t\in \mathbb{R}$:If we fix a given $M$ and vary $t$, we obtain a subgroup of $G$. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

Notation used in quantum mechanics.

Ket is just a vector. Though generally in the context of quantum mechanics, this is an infinite dimensional vector in a Hilbert space like $L^2$.

Bra is just the dual vector corresponding to a ket, or in other words projection linear operator, i.e. a linear function which can act on a given vector and returns a single complex number. Also known as... dot product.

For example:is basically a fancy way of saying:that is: we are taking the projection of $y$ along the $x$ direction. Note that in the ordinary dot product notation however, we don't differentiate as clearly what is a vector and what is an operator, while the bra-ket notation makes it clear.

The projection operator is completely specified by the vector that we are projecting it on. This is why the bracket notation makes sense.

It also has the merit of clearly differentiating vectors from operators. E.g. it is not very clear in $x\cdot y$ that $x$ is an operator and $y$ is a vector, except due to the relative position to the dot. This is especially bad when we start manipulating operators by themselves without vectors.

This notation is widely used in quantum mechanics because calculating the probability of getting a certain outcome for an experiment is calculated by taking the projection of a state on one an eigenvalue basis vector as explained at: Section "Mathematical formulation of quantum mechanics".

Making the projection operator "look like a thing" (the bra) is nice because we can add and multiply them much like we can for vectors (they also form a vector space), e.g.:just means taking the projection along the $x+y$ direction.

Ciro Santilli thinks that this notation is a bit over-engineered. Notably the bra's are just vectors, which we should just write as usual with $\overrightarrow{v}$... the bra thing makes it look scarier than it needs to be. And then we should just find a different notation for the projection part.

Maybe Dirac chose it because of the appeal of the women's piece of clothing: bra, in an irresistible call from British humour.

But in any case, alas, we are now stuck with it.