TODO clear example of the computational problem that it solves.
A unique projective space can be defined for any vector space.
The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e.
The most important initial example to study is the real projective plane.
www.quora.com/Why-do-successful-geeky-white-men-have-Asian-wives-This-seems-to-be-the-norm-in-Silicon-Valley suggests it is just an statistical inevitability.
This is a neologism by Ciro Santilli, it refers to the fact that Zatoichi was not fully blind, but extremely hard of sight, which makes him:and metaphorically refers to similar situations where a person or group of people are in the middle of two groups and not part of either of them.
- too capable for the blind people, who did not trust him
- too incapable for non-blind people, who despised him
A related thing that comes to mind is Aum Shinrikyo's Prophet Shoko Asahara, who was semi blind, and would bully the fully blind people of his school for blind people.
- quantumtech.blog/2023/01/17/quantum-computing-with-neutral-atoms/ OK this one hits it:So we understand that it is truly like the classical computer analog vs digital case.
As Alex Keesling, CEO of QuEra told me, "... whereas in gate-based [digital] quantum computing the focus is on the sequence of the gates, in analog quantum processing it's more about the position of the atoms and where you place them so they can mirror real life problems. We arrange the atoms and define the forces that drive them and then measure the result... so it’s a geometric encoding of the problem itself."
- thequantuminsider.com/2022/06/28/why-analog-neutral-atoms-quantum-computing-is-a-promising-direction-for-early-quantum-advantage on The Quantum Insider useless article mostly by Pasqal
TensorFlow quantum by Masoud Mohseni (2020)
Source. At the timestamp, Masoud gives a thought experiment example of the perhaps simplest to understand analog quantum computer: chained double-slit experiments with carefully calculated distances between slits. Calulating the final propability distribution of that grows exponentially.This is actually pretty good! Makes a small first step into The missing link between basic and advanced.
By the Simons Foundation.
Unfortunatly does not use a free license for content.
This is one of the first examples in most quantum field theory.
It usually does not involve any forces, just the interpretation of what the quantum field is.
www.youtube.com/watch?v=zv94slY6WqY&list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou&index=2 Quantization Of A Free Real Scalar Field by Dietterich Labs (2019)
There is no fundamental difference between them, a quantum algorithm is a quantum circuit, which can be seen as a super complicated quantum gate.
Perhaps the greats practical difference is that algorithms tend to be defined for an arbitrary number of N qubits, i.e. as a function for that each N produces a specific quantum circuit with N qubits solving the problem. Most named gates on the other hand have fixed small sizes.
A subset of Spacetime diagram.
The key insights that it gives are:
- future and past are well defined: every reference frame sees your future in your future cone, and your past in your past coneOtherwise causality could be violated, and then things would go really bad, you could tell your past self to tell your past self to tell your past self to do something.You can only affect the outcome of events in your future cone, and you can only be affected by events in your past cone. You can't travel fast enough to affect.Two spacetime events with such fixed causality are called timelike-separated events.
- every other event (to right and left, known as spacelike-separated events) can be measured to happen before or after your current spacetime event by different observers.But that does not violate causality, because you just can't reach those spacetime points anyways to affect them.
Animation showing how space-separated events can be observed to happen in different orders by observers in different frames of reference
. Source. 
Split in the spectral line when a magnetic field is applied.
Non-anomalous: number of splits matches predictions of the Schrödinger equation about the number of possible states with a given angular momentum. TODO does it make numerical predictions?
www.pas.rochester.edu/~blackman/ast104/zeeman-split.html contains the hello world that everyone should know: 2p splits into 3 energy levels, so you see 3 spectral lines from 1s to 2p rather than just one.
p splits into 3, d into 5, f into 7 and so on, i.e. one for each possible azimuthal quantum number.
It also mentions that polarization effects become visible from this: each line is polarized in a different way. TODO more details as in an experiment to observe this.
Well explained at: Video "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)".
Experimental physics - IV: 22 - Zeeman effect by Lehrportal Uni Gottingen (2020)
Source. This one is decent. Uses a cadmium lamp and an etalon on an optical table. They see a more or less clear 3-split in a circular interference pattern,
They filter out all but the transition of interest.
- youtu.be/ZmObNFAqkBE?t=165 passes the lines through a polarizer, which shows how orbital angular momentum is carried by photon polarization
- youtu.be/ZmObNFAqkBE?t=370 says they are looking at 1D2 to 1P1 changes.
Zeeman Effect - Control light with magnetic fields by Applied Science (2018)
Source. Does not appear to achieve a crystal clear split unfortunately.If you shine microwave radiation on a Josephson junction, it produces a fixed average voltage that depends only on the frequency of the microwave. TODO how is that done more precisely? How to you produce and inject microwaves into the thing?
It acts therefore as a perfect frequency to voltage converter.
The Wiki page gives the formula: en.wikipedia.org/wiki/Josephson_effect#The_inverse_AC_Josephson_effect You get several sinusoidal harmonics, so the output is not a perfect sine. But the infinite sum of the harmonics has a fixed average voltage value.
And en.wikipedia.org/wiki/Josephson_voltage_standard#Josephson_effect mentions that the effect is independent of the junction material, physical dimension or temperature.
All of the above, compounded with the fact that we are able to generate microwaves with extremely precise frequency with an atomic clock, makes this phenomenon perfect as a Volt standard, the Josephson voltage standard.
TODO understand how/why it works better.
Highlighted at the Origins of Precision by Machine Thinking (2017).
The 3D structure of GFP is so cool. It is so clearly a bottle with a fluorescent bit well isolated right in the middle. Like a little lamp.
Experiments explained by QED but not by the Dirac equation:
- Lamb shift: by far the most famous one
- hyperfine structure TODO confirm
- anomalous magnetic dipole moment of the electron
This point is beautifully argued in lots of different sources, and is clearly a pillar of AGI.
Perhaps one may argue that our deep learning layers do form some kind of hierarchy, e.g. this is very clear in certain models such as convolutional neural network. But many of those models cannot have arbitrarily deep hierarchies, which appears to be a fundamental aspect of intelligence.
How to Create a Mind:
The lists of steps in my mind are organized in hierarchies. I follow a routine procedure before going to sleep. The first step is to brush my teeth. But this action is in turn broken into a smaller series of steps, the first of which is to put toothpaste on the toothbrush. That step in turn is made up of yet smaller steps, such as finding the toothpaste, removing the cap, and so on. The step of finding the toothpaste also has steps, the first of which is to open the bathroom cabinet. That step in turn requires steps, the first of which is to grab the outside of the cabinet door. This nesting actually continues down to a very fine grain of movements, so that there are literally thousands of little actions constituting my nighttime routine. Although I may have difficulty remembering details of a walk I took just a few hours ago, I have no difficulty recalling all of these many steps in preparing for bed - so much so that I am able to think about other things while I go through these procedures. It is important to point out that this list is not stored as one long list of thousands of steps - rather, each of our routine procedures is remembered as an elaborate hierarchy of nested activities.
Human Compatible: TODO get exact quote. It was something along: life goal: save world from hunger. Subgoal: apply for some grant. Sub-sub-goal: eat, sleep, take shower. Sub-sub-sub-goal: move muscles to get me to table and open a can.
The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by:
Intuitively, is a direction, and is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
There are unlisted articles, also show them or only show them.