Continuous problems are simpler than discrete ones Updated 2025-07-16
Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.
Some good examples of that:
- complex discrete problems:
- simple continuous problems:
- characterization of Lie groups
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Discrete Fourier transform Updated 2025-07-16
Output: another sequence of complex numbers such that:Intuitively, this means that we are braking up the complex signal into sinusoidal frequencies:and is the amplitude of each sine.
- : is kind of magic and ends up being a constant added to the signal because
- : sinusoidal that completes one cycle over the signal. The larger the , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
- : sinusoidal that completes two cycles over the signal
- ...
- : sinusoidal that completes cycles over the signal
Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.
- compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
- noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise
Sample software implementations:
- numpy.fft, notably see the example: numpy/fft.py
DFT of with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point. Flatpak Updated 2025-07-16
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React setups:
DNA sequencing milestone Updated 2025-07-16
Most of these are going to be Whole-genome sequencing of some model organism:en.wikipedia.org/wiki/Whole_genome_sequencing#History lists them all. Basically th big "firsts" all happened in the 1990s and early 2000s.
- 1975 by Sanger et al.: 5 kbp of the single-stranded bacteriophage ΦX174 using Sanger's radiolabelling method
- 1981 by Sanger et al.: 17 kbp of human mitochondrial DNA via Sanger method, known as the Cambridge Reference Sequence
- 2003: Human Genome Project (3 Gbp)
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This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.
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Power, Sex, Suicide by Nick Lane (2006) Updated 2025-07-16
All pages below are from the second edition from 2018. It seems that there weren't any changes in the text, the updated preface mentions
As it happens, nearly 15 years have passed since the 1st edition of Power, Sex, Suicide was published, and I am resisting the temptation to make any lame revisions. Some say that even Darwin lessened the power of his arguments in the Origin of Species through his multiple revisions, in which he dealt with criticisms and sometimes shifted his views in the wrong direction. I prefer my original to speak for itself, even if it turns out to be wrong.
This is partly addressed in the preface of the second edition from 2018.
Central thesis:
- there are two sexes because of mitochondria
- the acquisition of mitochondria was one of the most important steps in the evolution of eukaryotes.There are no known eukaryotes which never had mitochondria. Having mitochondria appears to be a requisite for being an eukaryote.Contrast this for example with multicellularity, which is highly polyphyletic.
- Apoptosis is largely regulated by mitochondria
- there are two main theories for how the mitochondria endosymbiosis started:
- parsitic hypothesis of mitochondrial endosymbiosis: a parasitic option rather than cooperative
- hydrogen hypothesis: a cooperative option rather than parasitic
Smaller points:
- 10% of our body weight (dry presumably?) is mitochondria. Also quoted at: www.nature.com/scitable/blog/student-voices/mighty_mitochondria. TODO confirm.
- eukaryotes can do phatocytosis due to their cytoskeleton
- paints a colorful picture of Peter Mitchell. Some Wikipedia edits are warranted!
- it is hard for complex organisms to evolve because longer DNA means longer replication time
- cancer is natural selection gone wrong
- multicellular organisms are not utopias where every cell lives happily. Rather, they are dictatorships, where any dissident is forced to commit seppuku. Lu Xun's petition quote comes to mind.
Nitpicks:
- the book calls ATP synthase "ATPase" in several points, which is confusing because -ase means "something that breaks", and in 2020 parlance, there are ATPases which actually break ATP: en.wikipedia.org/wiki/ATPase. The book itself acknowledges that on page 135:
The ATPase is freely reversible. Under some circumstances it can go into reverse, whereupon it splits ATP, and uses the energy released to pump protons up the drive shaft, back across the membrane against the pressure of the reservoir. In fact the very name ATPase (rather than ATP synthase) signifies this action, which was discovered first. This bizarre trait hides a deep secret of life, and we’ll return to it in a moment.
Some criticisms:
- some of the later chapters are a bit more boring, like the stuff about warm-blooded animals. Perhaps is it that Ciro Santilli is more interested in the molecular aspects than macro
- the author talks about some very recent research at the time. While this does highlight his expertise, some of the points mentioned might still be in a state of flow. This is acknowledged by the author himself on the 2018 updated preface however.
SoundFont Updated 2025-07-16
Contains instructions on how to synthesize MIDI.
Extension: sf2.
Can be imported for example by:
- vmpk
- Csound with
sfload: www.csounds.com/manual/html/sfload.html
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