New articles - OurBigBook.com

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:

T_{i_{1} \dots i_{m}}^{j_{1} \dots j_{n}} = T (e_{i_{1}}, \dots, e_{i_{m}}, e^{j_{1}}, \dots, e^{j_{m}})

(2)

where we remember that the raised indices refer dual vector.

Some examples:

Explain it properly bibliography:

 Read the full article

Mathieu group by

Ciro Santilli 37 Updated 2025-07-16

 View more

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

M_{n}

is a permutation group on

n

elements. There isn't an obvious algorithmic relationship between

n

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:

22 is 3-transitive but not 4-transitive.
four of them (11, 12, 23 and 24) are the only sporadic 4-transitive groups as per the classification of 4-transitive groups (no known simpler proof as of 2021), which sounds like a reasonable characterization. Note that 12 and 25 are also 5 transitive.

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

M_{23}

and be done with it.

Video 1.

Mathieu group section of Why Do Sporadic Groups Exist? by Another Roof (2023)

. Source. Only discusses Mathieu group but is very good at that.

 Read the full article

Discrete Fourier transform by

Ciro Santilli 37 Updated 2025-07-16

 View more

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.

 Read the full article

Series LC circuit by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Robert O'Callahan by

Ciro Santilli 37 Updated 2025-07-16

 View more

Creator of Mozilla rr, of which Ciro Santilli is a huge fan of!

He quit Mozilla in 2016 to try and commercialize an rr extension called Pernosco.

But as of 2022, he advertised himself as part of "Google Research", so maybe that went under, sample source: archive.ph/o9622. TODO when did he start? There's apparently an unrelated homonym: www.linkedin.com/in/rob-ocallahan/

He's apparently very religious, and very New Zelandish, twitter.com/rocallahan auto-describes:

Christian. Repatriate Kiwi.

Terry A. Davis and D. Richard Hipp come to mind. One is tempted to speculate a correlation even, the proportion amongst systems programmers feels so much higher than in other areas of programming! Maybe it is because you have to be a God to do it in the first place.

Video 1.

Robert O'Callahan interview by Toby Ho (2022)

Source.

 Read the full article

 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

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

 Site settings

 Pinned article: Introduction to the OurBigBook Project  Hide

 Pinned article: Introduction to the OurBigBook Project