The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
- Stern-Gerlach experiment
- fine structure split in energy levels
- anomalous Zeeman effect
- of a more statistical nature, but therefore also macroscopic and more dramatically observable:
- ferromagnetism
- Bose-Einstein statistics vs Fermi-Dirac statistics. A notable example is the difference in superfluid transition temperature between superfluid helium-3 and superfluid helium-4.
Haven't found the one yet:
- open source software, doh
- end-to-end encryption...
- has browser frontend and Android app
- public URL without sharing your mobile phone: messaging software that force you to have a mobile phone
- self-destroying messages (turned on by default please)
- user base large enough to give some confidence that it was reviewed for security issues
- easy/built-in setup over Tor
Optional but really ideal:
- can delete messages from the device of the person you sent it to, no matter how old
- decentralized, your username is a public key
The state of messaging is ridiculous as of 2020.
If you can reduce a mathematical problem to the Halting problem of a specific turing machine, as in the case of a few machines of the Busy beaver scale, then using Turing machine deciders could serve as a method of automated theorem proving.
That feels like it could be an elegant proof method, as you reduce your problem to one of the most well studied representations that exists: a Turing machine.
However it also appears that certain problems cannot be reduced to a halting problem... OMG life sucks (or is awesome?): Section "Turing machine that halts if and only if Collatz conjecture is false".
Node.js does have Node.js
worker_threads however.async is all present in JavaScript for two reasons:- you make network requests all the time
- JavaScript is single threaded, so if you are waiting for a network request, the UI freezes, see remarks on the deprecation of synchronous HTTP request at: developer.mozilla.org/en-US/docs/Web/API/XMLHttpRequest/Synchronous_and_Asynchronous_Requests
However, it is also Hell: how to convert
async to sync in JavaScript.Mathematically, we can decide if two events are timelike-separated or spacelike-separated by just looking at the sign of the spacetime interval between them.
On the light cone, these are events on the left/right part of the cone.
Different observers might not agree on the order of two spacelike-separated events.
Further discussion at Section "Light cone".
The opposite of those events are timelike-separated events.
Off-the-shelf techniques to become a teaching superhero.
Customized website idea at: OurBigBook.com.
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
We use Zero-based numbering in our definitions because it just makes every formula simpler.
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.There are apparently two methods:
- in the script, e.g. as in the Genesis block message
- in output addresses
Specific implementations:
- eternitywall.it/ Eternity WallLaunched 2015 www.newsbtc.com/news/bitcoin/eternity-wall-records-1150-documents-blockchain-first-year/TODO find sample transactions. Did it support images?Shutdown sometime after 2019, working archive: web.archive.org/web/20190417074034/https://eternitywall.it/ says "Sorry, the service is not properly working at the moment..." and last working message timestamped "April 16, 2019 8:02 PM GMT".
scholarworks.sjsu.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=5051&context=etd_theses proves that the Mathieu group is simple in just 200 pages. Nice.
Our examples are located under nodejs/next:
- nodejs/next/hello-world: a hello world. There's an in-tree one at: github.com/vercel/next.js/tree/e75361fd03872b097e817634c049b3185f24cf56/examples/hello-world, but ours is truly minimal
- nodejs/next/hoc: shows how to use a higher order component (HOC) to factor out
getStaticPropsacross two pages: nodejs/next/hoc/pages/index.js and nodejs/next/hoc/pages/notindex.js - nodejs/next/typescript: simple TypeScript example, minimized from: github.com/vercel/next.js/tree/d61b0761efae09bd9cb1201ff134ed8950d9deca/examples/with-typescriptNotably, that shows how
requireerrors are avoided in that case as mentioned at: stackoverflow.com/questions/64926174/module-not-found-cant-resolve-fs-in-next-js-application/70363153#70363153 - nodejs/next/localStorage: a counter that is persistent across page reloads by using
localStorage. Used in: stackoverflow.com/questions/54819721/next-js-access-localstorage-before-rendering-page/68136224#68136224
Solved ones:
- solved by preview mode in Next.js 12:
- ISR was basically unusable for CRUD websites because you can't force a one-off immediate page update:
Interesting because of the Cook-Levin theorem: if only a single NP-complete problem were in P, then all NP-complete problems would also be P!
We all know the answer for this: either false or independent.
Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,
It therefore requires you to redefine and reprove all of calculus.
For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.
One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.
Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.
There is only one thing that can truly motivate you to make good materials: becoming famous.
Strive for that. Make good materials. Publish them. Get good reviews. Loop.
This generates a virtuous loop, which makes you produce better and better material.
And if you really can't make money from a subject, there is only one other thing people crave: beauty.
You have to give the beauty motivations upfront, before boring people to death with endless prerequisites, otherwise no one will ever want to learn it.
Free borrow on the Internet Archive: archive.org/details/inwardboundofmat0000pais/page/88/mode/2up
The book unfortunately does not cover the history of quantum mechanics very, the author specifically says that this will not be covered, the focus is more on particles/forces. But there are still some mentions.
Check which you you have:Tested on Ubuntu 23.10 I see:which means I have GNOME Display Manager.
systemctl status display-manager.service● gdm.service - GNOME Display Manager
Loaded: loaded (/lib/systemd/system/gdm.service; static)
Active: active (running) since Sun 2023-12-24 10:34:50 GMT; 23min ago
Process: 1827 ExecStartPre=/usr/share/gdm/generate-config (code=exited, status=0/SUCCESS)
Main PID: 1850 (gdm3)
Tasks: 4 (limit: 71817)
Memory: 6.8M
CPU: 119ms
CGroup: /system.slice/gdm.service
└─1850 /usr/sbin/gdm3They appear to be included, with rationale that you can already include syntactically valid crap in an unprovable way: github.com/bitcoin/bitcoin/issues/320 Better then have syntactically invalid crap that is provable.
The outputs of this transaction seem to be the first syntactically incorrect scripts of the blockchain: blockchain.info/tx/ebc9fa1196a59e192352d76c0f6e73167046b9d37b8302b6bb6968dfd279b767?format=json, found by parsing everything locally. The transaction was made in 2013 for 0.1 BTC, which then became unspendable.
The first invalid script is just e.g. "script":"01", which says will push one byte into the stack, but then ends prematurely.
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