Dual vectors are the members of a dual space.
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:
Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector as:
In the context of quantum mechanics, the bra notation is also used for dual vectors.
See also: covariance.
Experiments explained by QED but not by the Dirac equation:
- Lamb shift: by far the most famous one
- hyperfine structure TODO confirm
- anomalous magnetic dipole moment of the electron
Given experiments such as the Fizeau experiment and the Michelson-Morley experiment that couldn't really detect the Earth's movement across aether, people started to wonder if the Earth wasn't dragging the luminiferous aether.
Sequence alignment is trying to match a DNA or amino acid sequence, even though the sequences might not be exactly the same, otherwise it would be a straight up string-search algorithm.
This is fundamental in bioinformatics for two reasons:
- when you sequence the DNA of a new species, you can guess what each protein does by comparing it with similar proteins in other species that you have already studied
- when doing DNA sequencing, and specially short-read DNA sequencing, you generally need to align the reads to reference genomes to know where you are inside the entire genome, and then be able to spot mutations, notably single-nucleotide polymorphisms
Lecture videos: podcast.ucsd.edu/ Mostly paywalled, but a few not.
Superconducting qubits are bad because of fabrication variation Updated 2025-01-06 +Created 1970-01-01
Atom-based qubits like trapped ion quantum computers have parameters fixed by the laws of physics.
However superconducting qubits have a limit on how precise their parameters can be set based on how well we can fabricate devices. This may require per-device characterisation.
The problem of deletionism is that it removes users' confidence that their precious data will be safe. It's almost like having a database that constantly resets itself. Who will be willing to post on a website that deletes the content they created for free half of the time thus wasting people's precious time?
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.
Looking down the largest flops of all time list didn't help much, only Heaven's gate appears reasonable from the top 20.
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.
So people started theorizing some hack that would fix up the equations, and they came up with the higgs mechanism.
List databases:
echo 'show dbs' | mongoDelete database:or:
use mydb
db.dropDatabase()echo 'db.dropDatabase()' | mongo mydbView collections within a database:
echo 'db.getCollectionNames()' | mongo mydbShow all data from one of the collections: stackoverflow.com/questions/24985684/mongodb-show-all-contents-from-all-collections
echo 'db.collectionName.find()' | mongo mydb Unlisted articles are being shown, click here to show only listed articles.