The definition of the "dot product" of a general space varies quite a lot with different contexts.
Most definitions tend to be bilinear forms.
We use the unqualified generally refers to the dot product of Real coordinate spaces, which is a positive definite symmetric bilinear form. Other important examples include:The rest of this section is about the case.
- the complex dot product, which is not strictly symmetric nor linear, but it is positive definite
- Minkowski inner product, sometimes called" "Minkowski dot product is not positive definite
The positive definite part of the definition likely comes in because we are so familiar with metric spaces, which requires a positive norm in the norm induced by an inner product.
The default Euclidean space definition, we use the matrix representation of a symmetric bilinear form as the identity matrix, e.g. in :so that:
The dual space of a vector space , sometimes denoted , is the vector space of all linear forms over with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
- stackoverflow.com/questions/600079/how-do-i-clone-a-subdirectory-only-of-a-git-repository/52269934#52269934
- summaries:
- dupes:
- file or directory
- file
- only small files:
Bibliography review:
- Quantum Field Theory lecture notes by David Tong (2007) is the course basis
- quantum field theory in a nutshell by Anthony Zee (2010) is a good quick and dirty book to start
Course outline given:
- classical field theory
- quantum scalar field. Covers bosons, and is simpler to get intuition about.
- quantum Dirac field. Covers fermions
- interacting fields
- perturbation theory
- renormalization
Non-relativistic QFT is a limit of relativistic QFT, and can be used to describe for example condensed matter physics systems at very low temperature. But it is still very hard to make accurate measurements even in those experiments.
Defines "relativistic" as: "the Lagrangian is symmetric under the Poincaré group".
Mentions that "QFT is hard" because (a finite list follows???):But I guess that if you fully understand what that means precisely, QTF won't be too hard for you!
There are no nontrivial finite-dimensional unitary representations of the Poincaré group.
Notably, this is stark contrast with rotation symmetry groups (SO(3)) which appears in space rotations present in non-relativistic quantum mechanics.
www.youtube.com/watch?v=T58H6ofIOpE&t=5097 describes the relativistic particle in a box thought experiment with shrinking walls
- symmetry in classical field theory
- from Lagrangian density we can algorithmically get equations of motion, but the Lagrangian density is a more compact way of representing the equations of motion
- definition of symmetry in context: keeps Lagrangian unchanged up to a total derivative
- Noether's theorem
- youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3062 Lagrangian and conservation example under translations
- youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3394 same but for Poincaré transformations But now things are harder, because it is harder to describe general infinitesimal Poincare transforms than it was to describe the translations. Using constraints/definition of Lorentz transforms, also constricts the allowed infinitesimal symmetries to 6 independent parameters
- youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=4525 brings out Poisson brackets, and concludes that each conserved current maps to a generator of the Lie algebraThis allows you to build the symmetry back from the conserved charges, just as you can determine conserved charges starting from the symmetry.
In the Galilean transformation, there are two separate invariants that two inertial frame of reference always agree on between two separate events:
- time
- length, given by the Pythagorean theorem
However, in special relativity, neither of those are invariant separately, since space and time are mixed up together.
Instead, there is a new unified invariant: the spacetime-interval, given by:
Note that this distance can be zero for two events separated.
The projects you do must always aim to achieving some novel result.
You don't have to necessarily reach it. But you must aim for it.
Novel result can be taken broadly.
E.g., a new tutorial that explains something in a way never done before is novel.
But there must be something to your project that has never been done before.
You can start by reproducing other's work.
One of the most powerful chess engine as of 2023: computer chess competition.
CLI program implementing Universal Chess Interface: www.reddit.com/r/ComputerChess/comments/b6rdez/commandline_options_for_stockfish/
How to actually play against it: chess.stackexchange.com/questions/4353/how-to-install-stockfish-on-ubuntu So hard!
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.
- the advantage of using Lagrangian mechanics instead of directly trying to work out the equations of motion is that it is easier to guess the Lagrangian correctly, while still imposing some fundamental constraints
- youtu.be/bTcFOE5vpOA?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3375
- Lagrangian mechanics is better for path integral formulation. But the mathematics of that is fuzzy, so not going in that path.
- Hamiltonian mechanics is better for non-path integral formulation
- youtu.be/bTcFOE5vpOA?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3449 Hamiltonian formalism requires finding conjugate pairs, and doing a
TODO: the interrupt is firing only once:
Adapted from: danielmangum.com/posts/risc-v-bytes-timer-interrupts/
Tested on Ubuntu 23.10:Then on shell 1:and on shell 2:GDB should break infinitel many times on
sudo apt install binutils-riscv64-unknown-elf qemu-system-misc gdb-multiarch
cd riscv
makeqemu-system-riscv64 -machine virt -cpu rv64 -smp 1 -s -S -nographic -bios none -kernel timer.elfgdb-multiarch timer.elf -nh -ex "target remote :1234" -ex 'display /i $pc' -ex 'break *mtrap' -ex 'display *0x2004000' -ex 'display *0x200BFF8'mtrap as interrupts happen.Official Bitcoin domain registered by Satoshi Nakamoto.
Registration: 2008-08-18 by www.namecheap.com, an American company. But using a privacy oriented registrar: bitcoin.stackexchange.com/questions/89532/how-did-nakamoto-untraceably-pay-for-registering-bitcoin-org It is unknown how he could have paid anonymously, so it seems likely that the true identity could be obtained by law enforcement if needed.
First archive 2009-01-31: web.archive.org/web/20090131115053/http://bitcoin.org/ Also from the archive history web.archive.org/web/20100701000000*/bitcoin.org, things really started picking up on July 2010. This is almost certainly due to the opening of
Ciro Santilli often sees all those genius who are much smarter than him making shitty forum/mailing list posts, they need to learn this:
- The apparently most important one liner error message must appear in the title, and fuller apparently relevant logs must appear on the body
- You must always give the version of the software that you are using as either a tag or git SHAThese are an important part of the minimal working example.
- For build errors, you must give your OS and compiler version and version of any relevant external library
By writing in English you reach more people.
Writing in any other language is a waste of time.
The reason is simple: English speakers control a huge proportion of the world's GDP.
English is the de-facto Lingua Franca of the second half of the 20th Century, it is the new lingua franca, the new Latin, and there is no escaping it.
Students who don't know English will never do anything truly useful in science and technology. So it is pointless to teach them anything (besides English itself).
For example, if you write LaTeX files for you PDFs, give both PDFs and the LaTeX.
This allows other people to:
- modify and reuse your material
- make improvement suggestions that you can accept by clicking a buttonThe perfect way to do this is to use GitHub pull requests
NCBI taxonomy entry: www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cgi?id=511145 This links to:
- genome: www.ncbi.nlm.nih.gov/genome/?term=txid511145 From there there are links to either:
- Download the FASTA: "Download sequences in FASTA format for genome, protein"For the genome, you get a compressed FASTA file with extension
.fnacalledGCF_000005845.2_ASM584v2_genomic.fnathat starts with:>NC_000913.3 Escherichia coli str. K-12 substr. MG1655, complete genome AGCTTTTCATTCTGACTGCAACGGGCAATATGTCTCTGTGTGGATTAAAAAAAGAGTGTCTGATAGCAGCTTCTGAACTG
- Interactively browse the sequence on the browser viewer: "Reference genome: Escherichia coli str. K-12 substr. MG1655" which eventually leads to: www.ncbi.nlm.nih.gov/nuccore/556503834?report=graphIf we zoom into the start, we hover over the very first gene/protein: the famous (just kidding) e. Coli K-12 MG1655 gene thrL, at position 190-255.The second one is the much more interesting e. Coli K-12 MG1655 gene thrA.
- Gene list, with a total of 4,629 as of 2021: www.ncbi.nlm.nih.gov/gene/?term=txid511145
She's truly passionate about health research and keeping healthy, almost obsessed by it. Also she's strong willed, and energetic. Good traits for founding 23andMe.
- www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair Fantastic painting of the people.
As www.nytimes.com/2017/11/18/style/anne-wojcicki-23andme-genetics.html puts it well:
The Wojcickis grew into Silicon Valley royalty. It’s the sort of family, Anne jokes, where “you’re only a viable fetus once you have your Ph.D.
Anne Wojcicki interview by Talks at Google (2018)
Source. She's athletic! As mentioned at: www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair. And despite the name, and unlike Sergey, she's completely american as seen from her perfect accent!- youtu.be/pDoALM0q1LA?t=173 coding on garage while they do dishes and burritos
- youtu.be/pDoALM0q1LA?t=331 why she's obsessed with healthcare. Also mentioned at: www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair how she was trying to save Sergei from some of his genetic predispositions
- youtu.be/pDoALM0q1LA?t=571 she really cared about 23andMe, but the public didn't as much as her. She's truly passionate about mining genetic data. Maybe she came a bit early.
- youtu.be/pDoALM0q1LA?t=1038 doc in a box workaround
Project trying to compute BB(5) once and for all. Notably it has better presentation and organization than any other previous effort, and appears to have grouped everyone who cares about the topic as of the early 2020s.
Very cool initiative!
By 2023, they had basically decided every machine: discuss.bbchallenge.org/t/the-30-to-34-ctl-holdouts-from-bb-5/141
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