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
Sample software implementations:
www.aip.org/history-programs/niels-bohr-library/ex-libris-universum/harvard-project-physics-role-history-science mentions that they have a special focus to the history of physics, as can be seen e.g. on The World Of Enrico Fermi by Harvard Project Physics (1970).
Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!
Each is a permutation group on elements. There isn't an obvious algorithmic relationship between and the actual group.
TODO initial motivation? Why did Mathieu care about k-transitive groups?
Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:
Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of and be done with it.
Mathieu group section of Why Do Sporadic Groups Exist? by Another Roof (2023)
. Source. Only discusses Mathieu group but is very good at that.Examples:
- mechanical resonance, notably:
- pipe instruments
- electronic oscillators, notably:
- LC oscillator, and notably the lossy version RLC circuit
Perhaps a key insight of resonance is that the reonant any lossy system tends to look like the resonance frequency quite quickly even if the initial condition is not the resonant condition itself, because everything that is not the resonant frequency interferes destructively and becomes noise. Some examples of that:
- striking a bell or drum can be modelled by applying an impuse to the system
- playing a pipe instrument comes down to blowing a piece that vibrates randomly, and then leads the pipe to vibrate mostly in the resonant frequency. Likely the same applies to bowed string instruments, the bow must be creating a random vibration.
- playing a plucked string instrument comes down to initializing the system to an triangular wave form and then letting it evolve. TODO find a simulation of that!
Another cool aspect of resonance is that it was kind of the motivation for de Broglie hypothesis, as de Broglie was kind of thinking that electroncs might show discrete jumps on atomic spectra because of constructive interference.
For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.
To see why this is called a plane, move he center of the sphere to , and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator . Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.
Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!
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.
Mathematics and Computer science course of the University of Oxford by
Ciro Santilli 40 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".
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.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.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.:has determinant 1, but:so is not orthogonal.
The normal navigation to them was paywalled, but the static files are served without login checks if you know their URL. One way to go about it is to search by prefix on the Wayback Machine: web.archive.org/web/*/https://www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/*
The last handbooks we can find are 2020/2021, they might have move to a new more properly paywalled location after that year.
- 2020/2021:
- Year 1: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y1-ug-handbook-2020-2021-final-47501.pdf
- Year 2: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y2-ug-handbook-2020-2021-final-47495.pdf
- Year 3: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y3-ug-handbook-2020-2021-final-47496.pdf
- Year 4: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y4-ug-handbook-2020-2021-final-47497.pdf
- Physics and Philosophy: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/pphandbook-47524.pdf
- 2019/2020. They seem to have split the handbook up per year after some point.
- Year 1: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y1-ug-handbook-2019-2020-final-8october2019-45541.pdf
- Year 2: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y2-ug-handbook-2019-2020-final-8-october2019-45542.pdf
- Year 3: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y3-ug-handbook-2019-2020-updated-21november2019-45955.pdf
- Year 4: www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/y4-ug-handbook-2019-2020-final-8october2019-45544.pdf
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
users.physics.ox.ac.uk/~lvovsky/B3/ contain assorted PDFs from between 2015 and 2019
Syllabus reads:
- Multi-electron atoms: central field approximation, electron configurations, shell structure, residual electrostatic interaction, spin orbit coupling (fine structure).
- Spectra and energy levels: Term symbols, selection rules, X-ray notation, Auger transitions.
- Hyperfine structure; effects of magnetic fields on fine and hyperfine structure. Presumably Zeeman effect.
- Two level system in a classical light field: Rabi oscillations and Ramsey fringes, decaying states; Einstein
- A and B coefficients; homogeneous and inhomogeneous broadening of spectral lines; rate equations.
- Optical absorption and gain: population inversion in 3- and 4-level systems; optical gain cross section; saturated absorption and gain.
Professor in 2000s seems to beBut as of 2023 marked emeritus, so who took over?
- en.wikipedia.org/wiki/Paul_Ewart. He actually fought not to be dismissed by age and won!
- www.physics.ox.ac.uk/our-people/ewart
Ewart is actually religious:This dude is pure trouble for Oxford!
- www.youtube.com/watch?v=aulL-Qa65i0 Paul Ewart, Chance, Science and Spirituality by Faraday Institute for Science and Religion. Oh, he is/was actually chairman of that crap
- www.youtube.com/watch?v=PVX2F4XvGYo Chaos and the Character of God by Prof. Paul Ewart
Undated materials Ewart:
- users.physics.ox.ac.uk/~ewart/index.htm
- users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20lecture%20notes%20C%20port.pdf
- slides: users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20Lecture%20PPT%20slides%201_8.pdf. Also under: www2.physics.ox.ac.uk/sites/default/files/2011-10-19/atomic_physics_lectures_1_8_09_pdf_pdf_18283.pdf. The course was previously B1, they just change the IDs randomly from time to time to fit the B1-7 numbering.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.Figure 4. Visual Studio Code extension tree navigation.Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact





