Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.
Group of the unitary matrices.
Complex analogue of the orthogonal group.
One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. is the unit circle.
The opposite of open source software.
Knock knock.
Ciro Santilli once visited the chemistry department of a world leading university, and the chemists there were obsessed with NMR. They had small benchtop NMR machines. They had larger machines. They had a room full of huge machines. They had them in corridors and on desk tops. Chemists really love that stuff. More precisely, these are used for NMR spectroscopy, which helps identify what a sample is made of.
Introduction to NMR by Allery Chemistry
. Source. - only works with an odd number of nucleons
- apply strong magnetic field, this separates the energy of up and down spins. Most spins align with field.
- send radio waves into sample to make nucleons go to upper energy level. We can see that the energy difference is small since we are talking about radio waves, low frequency.
- when nucleon goes back down, it re-emits radio waves, and we detect that. TODO: how do we not get that confused with the input wave, which is presumably at the same frequency? It appears to send pulses, and then wait for the response.
How to Prepare and Run a NMR Sample by University of Bath (2017)
Source. This is a more direct howto, cool to see. Uses a Bruker Corporation 300. They have a robotic arm add-on. Shows spectrum on computer screen at the end. Shame no molecule identification after that!This video has the merit of showing real equipment usage, including sample preparation.
Says clearly that NMR is the most important way to identify organic compounds.
- youtu.be/uNM801B9Y84?t=41 lists some of the most common targets, including hydrogen and carbon-13
- youtu.be/uNM801B9Y84?t=124 ethanol example
- youtu.be/uNM801B9Y84?t=251 they use solvents where all protium is replaced by deuterium to not affect results. Genius.
- youtu.be/uNM801B9Y84?t=354 usually they do 16 radio wave pulses
Introductory NMR & MRI: Video 01 by Magritek (2009)
Source. Precession and Resonance. Precession has a natural frequency for any angle of the wheel.Introductory NMR & MRI: Video 02 by Magritek (2009)
Source. The influence of temperature on spin statistics. At 300K, the number of up and down spins are very similar. As you reduce temperature, we get more and more on lower energy state.Introductory NMR & MRI: Video 03 by Magritek (2009)
Source. The influence of temperature on spin statistics. At 300K, the number of up and down spins are very similar. As you reduce temperature, we get more and more on lower energy state.NMR spectroscopy visualized by ScienceSketch
. Source. 2020. Decent explanation with animation. Could go into a bit more numbers, but OK. Equivalent alternatives to the Schrodinger equation by 
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01 Time invariance implies energy conservation by Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01One of the main children cartoons Ciro Santilli liked to watch. Part of the Pokemon Mania of the 90s of course.
Ciro could not understand why Nintendo won't make a proper 3D MMORPG Pokemon with actually 3D Pokemon roaming the land, which is obviously what everyone wants. There are even fan games getting there!until this explaiend it beautifully Video 1. "The Downfall Of Mainline Pokemon Games by GONZ media (2020)":
- pokemon-mmo-3d.com Interesting concept here, where when you deploy a Pokemon, you start controlling the Pokemon rather than the main character
- www.quora.com/Why-won-t-Nintendo-make-an-open-world-3D-Pokémon-RPG/answer/Ciro-Santilli
It is hard to distinguish between colonies of unicellular organism and multicellular organism as there is a continuum between both depending on how well integrated they cells are.
Molecular biology laboratory equipment by Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01The first proper galaxy near the Milky Way is the Andromeda Galaxy. Everything else in the middle is a satellite of either of of those.
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