Linear map of two variables.
More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from:that is linear on the first two arguments from X and Y, i.e.:Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.
The most important example by far is the dot product from , which is more specifically also a symmetric bilinear form.
If you are going to make a television series, do make it a limited one. Plan one story, and execute it amazingly. Don't let things drag on and on.
For the typical case of a linear form over , the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.
There seems to be nothing of particular artistic value as far as we've seen so far, the only interest in such tokens seems to be that:
- there are some examples that came earlier than those in the Bitcoin blockchain, notably a bit earlier than Section "BitLen"
- Namecoin is a NFT system unlike Bitcoin which is fungible, so those assets are naturally tradable
The first really good quantum mechanics theory made compatible with special relativity was the Dirac equation.
And then came quantum electrodynamics to improve it: Dirac equation vs quantum electrodynamics.
TODO: does it use full blown QED, or just something intermediate?
www.youtube.com/watch?v=NtnsHtYYKf0 "Mercury and Relativity - Periodic Table of Videos" by Periodic Videos (2013). Doesn't give the key juicy details/intuition. Also mentioned on Wikipedia: en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Mercury
Analects translation by Robert Eno (2015) 16.7:
Instead of trying to dominate the sequencing market and gain trillions of dollars from it, they local British early stage investors were more than happy to get a 20x return on their small initial investments, and sold out to the Americans who will then make the real profit.
And now Solexa doesn't even have its own Wikipedia page, while Illumina is set out to be the next Microsoft. What a disgrace.
Cambridge visitors can still visit the Panton Arms pub, which was the location of the legendary "hey we should talk" founders meeting, chosen due to its proximity to the chemistry department of the University of Cambridge.
In 2021 the founders were awarded the Breakthrough Prize. The third person awarded was Pascal Mayer. He was apparently at Serono Pharmaceutical Research Institute at the time of development. They do have a wiki page unlike Solexa: en.wikipedia.org/wiki/Serono. They paid a 700 million fine in 2005 in the United States, and sold out in 2006 to Merck for 10 billion USD.
Allan Holdsworth performing in Central Park in (1978)
Source. Secrets Allan Holdsworth album
. Source. More important than the monk itself in Chinese culture, it is used as a Budhist Amen. TODO find some usage in some Chinese television series.
Television series that Ciro Santilli watched all episodes of Updated 2025-04-24 +Created 1970-01-01
The Fall of The Simpsons: How it Happened by Super Eyepatch Wolf (2017)
Source. This interesting video essay makes the main point that the Simpsons used to be good because they mocked mainstream. But then they became mainstream, which basically defeated their purpose. Think different from Apple comes to mind.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 .
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 along the 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 that is an operator and 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 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 ... 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.
One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.
His combination of politically incorrect dirt talk with amazing quirky decks captures Ciro's imagination.
Anonymous no face-reveal.
The videos are heavily edited with all pauses cut out, which makes them very quick to watch and saves viewer time.
Modern focused, with some occasional newer formats mixed in.
When Wizards publishes several useless sets in a row without a single modern playable card, he's just forced into Standard.
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