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We define this as the functional equation:It is a bit like cauchy's functional equation but with multiplication instead of addition.
This section groups conjectures that are famous, solved or unsolved.
They are usually conjectures that have a strong intuitive reasoning, but took a very long time to prove, despite great efforts.
Alain Aspect in 2016
. Source. Alain Aspect in Quantum entanglement Documentary (1985)
Source. The moustache and broken English were already his trademarks back then!!! Also cool to get a glimpse of his lab, and good schematics of the experiment. TODO exact lab location? Documentary says in Paris, but where?An efficient algorithm to calculate the discrete Fourier transform.
NCBI entry: www.ncbi.nlm.nih.gov/gene/945803.
Part of a reaction that produces threonine.
This protein is an enzyme. The UniProt entry clearly shows the chemical reactions that it catalyses. In this case, there are actually two! It can either transforming the metabolite:Also interestingly, we see that both of those reaction require some extra energy to catalyse, one needing adenosine triphosphate and the other nADP+.
- "L-homoserine" into "L-aspartate 4-semialdehyde"
- "L-aspartate" into "4-phospho-L-aspartate"
TODO: any mention of how much faster it makes the reaction, numerically?
Since this is an enzyme, it would also be interesting to have a quick search for it in the KEGG entry starting from the organism: www.genome.jp/pathway/eco01100+M00022 We type in the search bar "thrA", it gives a long list, but the last entry is our "thrA". Selecting it highlights two pathways in the large graph, so we understand that it catalyzes two different reactions, as suggested by the protein name itself (fused blah blah). We can now hover over:Note that common cofactor are omitted, since we've learnt from the UniProt entry that this reaction uses ATP.
- the edge: it shows all the enzymes that catalyze the given reaction. Both edges actually have multiple enzymes, e.g. the L-Homoserine path is also catalyzed by another enzyme called metL.
- the node: they are the metabolites, e.g. one of the paths contains "L-homoserine" on one node and "L-aspartate 4-semialdehyde"
If we can now click on the L-Homoserine edge, it takes us to: www.genome.jp/entry/eco:b0002+eco:b3940. Under "Pathway" we see an interesting looking pathway "Glycine, serine and threonine metabolism": www.genome.jp/pathway/eco00260+b0002 which contains a small manually selected and extremely clearly named subset of the larger graph!
But looking at the bottom of this subgraph (the UI is not great, can't Ctrl+F and enzyme names not shown, but the selected enzyme is slightly highlighted in red because it is in the URL www.genome.jp/pathway/eco00260+b0002 vs www.genome.jp/pathway/eco00260) we clearly see that thrA, thrB and thrC for a sequence that directly transforms "L-aspartate 4-semialdehyde" into "Homoserine" to "O-Phospho-L-homoserine" and finally tothreonine. This makes it crystal clear that they are not just located adjacently in the genome by chance: they are actually functionally related, and likely controlled by the same transcription factor: when you want one of them, you basically always want the three, because you must be are lacking threonine. TODO find transcription factor!
The UniProt entry also shows an interactive browser of the tertiary structure of the protein. We note that there are currently two sources available: X-ray crystallography and AlphaFold. To be honest, the AlphaFold one looks quite off!!!
By inspecting the FASTA for the entire genome, or by using the NCBI open reading frame tool, we see that this gene lies entirely in its own open reading frame, so it is quite boring
From the FASTA we see that the very first three Codons at position 337 arewhere
ATG CGA GTGATG is the start codon, and CGA GTG should be the first two that actually go into the protein:ecocyc.org/gene?orgid=ECOLI&id=ASPKINIHOMOSERDEHYDROGI-MONOMER mentions that the enzime is most active as protein complex with four copies of the same protein:TODO image?
Aspartate kinase I / homoserine dehydrogenase I comprises a dimer of ThrA dimers. Although the dimeric form is catalytically active, the binding equilibrium dramatically favors the tetrameric form. The aspartate kinase and homoserine dehydrogenase activities of each ThrA monomer are catalyzed by independent domains connected by a linker region.
As per stackoverflow.com/a/52351480/895245 our standard test setup is:
while true; do
resp=$"$(date): hello\n"
len="$(printf '%s' "$resp" | wc -c)"
printf "HTTP/1.1 200 OK\r\nContent-Length: $len\r\n\r\n${resp}\n" | nc -Nl 8000
doneUnsurprisingly the term "computer" became a synonym for this from the 1960s onwards!
As of 2022, this tends to be the more "default" when you talk about a quantum computer.
But there are some serious analog quantum computer contestants in the field as well.
x.com/Birdyword/status/1777591446612193667Singaporean taxi drivers are the cutest.
A while ago a Singaporean taxi driver asked me what a UK hawker centre lunch might cost. I explained that there aren't any. He asked what office workers ate, and I explained the concept of Meal deals. And he looked at me with such a powerful combination of pity and revulsion.
Radio Wave Properties: Electric and Magnetic Dipole Antennae by Harvard Natural Sciences Lecture Demonstrations (2020)
Source. The dude lights bulbs on an antenna made of a single piece of copper, powered with EM radiation. Amazing.
Dipole antenna receiver animation
. Source. 
Dipole antenna transmitter animation
. Source. 
A Cartesian product that carries over some extra structure of the input groups.
E.g. the direct product of groups carries over group structure on both sides.
Discrete Fourier transform of a real signal by 
Ciro Santilli 35 Updated 2024-12-23 +Created 1970-01-01
Ciro Santilli 35 Updated 2024-12-23 +Created 1970-01-01See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal
The transform still has complex numbers.
Summary:Therefore, we only need about half of to represent the signal, as the other half can be derived by conjugation.
- is real
"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of .
NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts
DFT of with 25 points
. Source at: numpy/fft_plot.py. This plot illustrates how the DFT of a real signal is symmetric around the middle point, and so only half of the transform points are needed to reconstruct the original signal. We also see how the phase of the sinusoids determines if their DFT components are real or imaginary. Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
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