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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

\pm 1

looks like in

SO (3)

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:

O (n) ≅ SO (n) \times C_{2}

(1)

A low dimensional example:

O (1) ≅ SO (2) \times C_{2}

(2)

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.:

M = [2132]

(3)

has determinant 1, but:

M^{T} M = [58811]

(4)

M

is not orthogonal.

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Discrete Fourier transform by

Ciro Santilli 40 Updated 2025-07-16

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Input: a sequence of

N

complex numbers

x_{k}

Output: another sequence of

N

complex numbers

X_{k}

such that:

x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}

(1)

Intuitively, this means that we are braking up the complex signal into

N

sinusoidal frequencies:

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{kn}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

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

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.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ 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.

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Quantum Processes and Computation course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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2022 page: www.cs.ox.ac.uk/teaching/courses/2022-2023/quantum/ (archive). Assignments are available:

2022 lecturer: Aleks Kissinger

The course would be better named ZX-calculus as it appears to be the only subject covered.

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Year 4 of the computer science course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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Quantum Information course of the University of Oxford Hilary 2023 by

Ciro Santilli 40 Updated 2025-07-16

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This section is about the version of the course offered on Hilary term 2023 (January).

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Quantum Information course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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www.cs.ox.ac.uk/teaching/courses/qi/

2023: Jonathan Barrett

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Models of computation course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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www.cs.ox.ac.uk/teaching/courses/modelsofcomputation/

Materials paywalled e.g. www.cs.ox.ac.uk/teaching/courses/2022-2023/modelsofcomputation/

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Concurrent programming course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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Compilers course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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www.cs.ox.ac.uk/teaching/courses/com

Materials paywalled: www.cs.ox.ac.uk/teaching/courses/2022-2023/com

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Algorithms and Data Structures course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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www.cs.ox.ac.uk/teaching/courses/algorithms/

Materials paywalled: www.cs.ox.ac.uk/teaching/courses/2022-2023/algorithms/

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Year 2 or 3 course of the computer science course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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Computer science and philosophy masters course of the University of Oxford by

Ciro Santilli 40 Updated 2025-07-16

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Public landing page: www.ox.ac.uk/admissions/undergraduate/courses/course-listing/computer-science-and-philosophy

Corresponding undergrad: Computer Science and Philosophy course of the University of Oxford.

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 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!

Video 1.

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-calculus
Articles 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/derivative
Video 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:
- 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
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.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
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.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade