Point-contact transistor Updated 2025-07-16
As the name suggests, this is not very sturdy, and was quickly replaced by bipolar junction transistor.
Tensor Updated 2025-07-16
Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:where we remember that the raised indices refer dual vector.
Explain it properly bibliography:
- www.reddit.com/r/Physics/comments/7lfleo/intuitive_understanding_of_tensors/
- www.reddit.com/r/askscience/comments/sis3j2/what_exactly_are_tensors/
- math.stackexchange.com/questions/10282/an-introduction-to-tensors?noredirect=1&lq=1
- math.stackexchange.com/questions/2398177/question-about-the-physical-intuition-behind-tensors
- math.stackexchange.com/questions/657494/what-exactly-is-a-tensor
- physics.stackexchange.com/questions/715634/what-is-a-tensor-intuitively
Lie algebra Updated 2025-07-16
Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".
Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.
Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.
To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.
- the dimension
- the Lie bracket
As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
Photonic quantum computer Updated 2025-07-16
The key experiment/phenomena that sets the basis for photonic quantum computing is the two photon interference experiment.
The physical representation of the information encoding is very easy to understand:
- input: we choose to put or not photons into certain wires or no
- interaction: two wires pass very nearby at some point, and photons travelling on either of them can jump to the other one and interact with the other photons
- output: the probabilities that photos photons will go out through one wire or another
Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)
Source. This is a good introduction to a photonic quantum computer. Highly recommended.- youtube.com/watch?v=7wCBkAQYBZA&t=1285 shows an experimental curve for a two photon interference experiment by Hong, Ou, Mandel (1987)
- youtube.com/watch?v=7wCBkAQYBZA&t=1440 shows a KLM CNOT gate
- youtube.com/watch?v=7wCBkAQYBZA&t=2831 discusses the quantum error correction scheme for photonic QC based on the idea of the "Raussendorf unit cell"
Text-based user interface Updated 2025-07-16
Political parties in the United States Updated 2025-07-16
Polyphyly Updated 2025-07-16
Basically mean that parallel evolution happened. Some cool ones:
- homeothermy: mammals and birds
- animal flight: bats, birds and insects
- multicellularity: evolved a bunch of times
Positrons are electrons travelling back in time Updated 2025-07-16
Primality test Updated 2025-07-16
Primate Updated 2025-07-16
Probably quantum secure encryption algorithm Updated 2025-07-16
Program optimization Updated 2025-07-16
Mathematical formulation of quantum field theory Updated 2025-07-16
The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.
But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:
The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:
The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.
Projective elliptic geometry Updated 2025-07-16
Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.
Projective space Updated 2025-07-16
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.
Mathematical Sciences masters course of the University of Oxford Updated 2025-07-16
Endocytosis Updated 2025-07-16
Mathematician Updated 2025-07-16
Poet, scientists and warriors all in one? Conquerors of the useless.
A wise teacher from University of São Paulo once told the class Ciro Santilli attended an anecdote about his life:It turned out that, about 10 years later, Ciro ended up following this advice, unwittingly.
I used to want to learn Mathematics.But it was very hard.So in the end, I became an engineer, and found an engineering solution to the problem, and married a Mathematician instead.
Mathematics and Computer science course of the University of Oxford Updated 2025-07-16
Public landing page: www.ox.ac.uk/admissions/undergraduate/courses/course-listing/mathematics-and-computer-science
A mixed cross department course with the Mathematical Institute of the University of Oxford.. Its corresponding masters is known as Oxford MMathCompSci. The handbook is together with the computer science one: Section "Computer science course of the University of Oxford".
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