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Logic by

Ciro Santilli 37 Updated 2025-07-16

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Japan by

Ciro Santilli 37 Updated 2025-07-16

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Linear function by

Ciro Santilli 37 Updated 2025-07-16

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The term is not very clear, as it could either mean:

a real number function whose graph is a line, i.e.:
$f (x) = a x + b$
(1)
or for higher dimensions, a hyperplane:
$f (x_{1}, x_{2}, \dots, x_{n}) = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n} + b$
(2)
a linear map. Note that the above linear functions are not linear maps unless $b = 0$ (known as the homogeneous case), because e.g.:
$f (x + y) = a x + a y + b$
(3)
but
$f (x) + f (y) = a x + b + a y + b$
(4)
For this reason, it is better never to refer to linear maps as linear functions.

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Symmetric multilinear map by

Ciro Santilli 37 Updated 2025-07-16

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Same value if you swap any input arguments.

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Projection (mathematics) by

Ciro Santilli 37 Updated 2025-07-16

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Matrix by

Ciro Santilli 37 Updated 2025-07-16

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Dense matrix by

Ciro Santilli 37 Updated 2025-07-16

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Non-commutative by

Ciro Santilli 37 Updated 2025-07-16

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Classification of 5-transitive groups by

Ciro Santilli 37 Updated 2025-07-16

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Apparently only Mathieu group

M_{12}

and Mathieu group

M_{24}

www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:

The automorphism group of the extended Golay code is the 54-transitive Mathieu group $M_{24}$ . This is one of only two finite 5-transitive groups other than symmetric and alternating groups

Hmm, is that 54, or more likely 5 and 4?

scite.ai/reports/4-homogeneous-groups-EAKY21 quotes link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and en.wikipedia.org/wiki/Mathieu_group_M12 mentions

M_{12}

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GF(2) by

Ciro Santilli 37 Updated 2025-07-16

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Double cover by

Ciro Santilli 37 Updated 2025-07-16

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Coordinate chart by

Ciro Santilli 37 Updated 2025-07-16

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Torus by

Ciro Santilli 37 Updated 2025-07-16

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Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) by

Ciro Santilli 37 Updated 2025-07-16

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Good ultra quick visual non-mathematical introduction to the Riemann hypothesis and analytic continuation.

Video 1. Source.

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Polarization by

Ciro Santilli 37 Updated 2025-07-16

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Start with: Section "String polarization".

Then go to: Section "Polarization of light".

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Boundary condition by

Ciro Santilli 37 Updated 2025-07-16

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Chart type by

Ciro Santilli 37 Updated 2025-07-16

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Convex polytope by

Ciro Santilli 37 Updated 2025-07-16

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4-polytope by

Ciro Santilli 37 Updated 2025-07-16

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Lie algebra of a isometry group by

Ciro Santilli 37 Updated 2025-07-16

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We can almost reach the Lie algebra of any isometry group in a single go. For every

X

in the Lie algebra we must have:

\forall v, w \in V, t \in R (e^{tX} v ∣ e^{tX} w) = (v ∣ w)

(1)

because

e^{tX}

has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".

Then:

\frac{d ( e ^{tX} v ∣ e ^{tX} w )}{d t}_{0} = 0 ⟹ (X e^{tX} v ∣ e^{tX} w) + (e^{tX} v ∣ X e^{tX} w)_{0} = 0 ⟹ (X v ∣ w) + (v ∣ Xw) = 0

(2)

so we reach:

\forall v, w \in V (X v ∣ w) = - (v ∣ Xw)

(3)

With this relation, we can easily determine the Lie algebra of common isometries:

Lie algebra of $O (n)$

Bibliography:

An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) page 151

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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