New articles - OurBigBook.com

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Fraction by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Orthonormality by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Tensor by

Ciro Santilli 37 Updated 2025-07-16

 View more

A multilinear form with a domain that looks like:

V^{m} \times V *^{n} \to R

(1)

where

V *

is the dual space.

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

torchvision ResNet by

Ciro Santilli 37 Updated 2025-07-16

 View more

pytorch.org/vision/0.13/models.html has a minimal runnable example adapted to python/pytorch/resnet_demo.py.

That example uses a ResNet pre-trained on the COCO dataset to do some inference, tested on Ubuntu 22.10:

cd python/pytorch
wget -O resnet_demo_in.jpg https://upload.wikimedia.org/wikipedia/commons/thumb/6/60/Rooster_portrait2.jpg/400px-Rooster_portrait2.jpg
./resnet_demo.py resnet_demo_in.jpg resnet_demo_out.jpg

This first downloads the model, which is currently 167 MB.

We know it is COCO because of the docs: pytorch.org/vision/0.13/models/generated/torchvision.models.detection.fasterrcnn_resnet50_fpn_v2.html which explains that

FasterRCNN_ResNet50_FPN_V2_Weights.DEFAULT

is an alias for:

FasterRCNN_ResNet50_FPN_V2_Weights.COCO_V1

The runtime is relatively slow on P51, about 4.7s.

After it finishes, the program prints the recognized classes:

['bird', 'banana']

so we get the expected bird, but also the more intriguing banana.

By looking at the output image with bounding boxes, we understand where the banana came from!

Figure 2.
python/pytorch/resnet_demo_out.jpg
. The beak was of course a banana, not a beak!

 Read the full article

Apache Jena CLI tools setup by

Ciro Santilli 37 Updated 2025-07-16

 View more

The CLI tools don't appear to be packaged for Ubuntu 23.10? Annoying... There is a package libapache-jena-java but it doesn't contain any binaries, only Java library files.

To run the CLI tools easily we can download the prebuilt:

sudo apt install openjdk-22-jre
wget https://dlcdn.apache.org/jena/binaries/apache-jena-4.10.0.zip
unzip apache-jena-4.10.0.zip
cd apache-jena-4.10.0
export JENA_HOME="$(pwd)"
export PATH="$PATH:$(pwd)/bin"

and we can confirm it works with:

sparql -version

which outputs:

Apache Jena version 4.10.0

If your Java is too old then then running sparql with the prebuilts fails with:

Error: A JNI error has occurred, please check your installation and try again
Exception in thread "main" java.lang.UnsupportedClassVersionError: arq/sparql has been compiled by a more recent version of the Java Runtime (class file version 55.0), this version of the Java Runtime only recognizes class file versions up to 52.0
        at java.lang.ClassLoader.defineClass1(Native Method)
        at java.lang.ClassLoader.defineClass(ClassLoader.java:756)
        at java.security.SecureClassLoader.defineClass(SecureClassLoader.java:142)
        at java.net.URLClassLoader.defineClass(URLClassLoader.java:473)
        at java.net.URLClassLoader.access$100(URLClassLoader.java:74)
        at java.net.URLClassLoader$1.run(URLClassLoader.java:369)
        at java.net.URLClassLoader$1.run(URLClassLoader.java:363)
        at java.security.AccessController.doPrivileged(Native Method)
        at java.net.URLClassLoader.findClass(URLClassLoader.java:362)
        at java.lang.ClassLoader.loadClass(ClassLoader.java:418)
        at sun.misc.Launcher$AppClassLoader.loadClass(Launcher.java:352)
        at java.lang.ClassLoader.loadClass(ClassLoader.java:351)
        at sun.launcher.LauncherHelper.checkAndLoadMain(LauncherHelper.java:621)

Build from source is likely something like:

sudo apt install maven openjdk-22-jdk
git clone https://github.com/apache/jena --branch jena-4.10.0 --depth 1
cd jena
mvn clean install

TODO test it.

If you make the mistake of trying to run the source tree without build:

git clone https://github.com/apache/jena --branch jena-4.10.0 --depth 1
cd jena
export JENA_HOME="$(pwd)"
export PATH="$PATH:$(pwd)/apache-jena/bin"

it fails with:

Error: Could not find or load main class arq.sparql

as per: users.jena.apache.narkive.com/T5TaEszT/sparql-tutorial-querying-datasets-error-unrecognized-option-graph

 Read the full article

Gvim by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Messaging software by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Discord (software) by

Ciro Santilli 37 Updated 2025-07-16

 View more

Ciro Santilli's discord ID: cirosantilli#8921. See also: how to contact Ciro Santilli.

You gotta be born after the year 2000 to understand it.

This is becoming more and more popular as a group chat with channels and threads possibility as of 2020.

Very similar to Slack.

No user URLs? support.discord.com/hc/en-us/community/posts/360041519131-UserProfilesLinks

They force your username to have 4 random digits? www.reddit.com/r/discordapp/comments/43kjdl/whats_the_number_next_to_the_username/

Not possible to anonymously join just one server without creating a new account? What's the point of servers then! www.reddit.com/r/discordapp/comments/6gmjl7/changing_nick_before_joining_a_new_server/ Oh, also nicks don't hide your username from the server in any way, you can get the original username by just clicking on the person's username.

No proper threaded discussion without creating new channels? As of 2022 there is kind of a way, but it was a bit obtuse.

As of 2022 they also have a school hub: support.discord.com/hc/en-us/articles/4406046651927-Discord-Student-Hubs-FAQ which auto creates groups by university email access. Good idea, and shows popularity amongst that user group.

Servers don't have an ID to join them? www.reddit.com/r/discordapp/comments/b9zdt6/join_discord_server_from_id/

Can only make public servers if you have 1000 members?? support.discord.com/hc/en-us/articles/360023968311 Why so much bullshit?? www.reddit.com/r/discordapp/comments/6jouf8/how_do_i_make_my_server_public/

Also asked at: webapps.stackexchange.com/questions/163441/how-to-create-a-public-discord-server-that-anyone-can-join-without-an-invite

 Read the full article

FFmpeg filter graph by

Ciro Santilli 37 Updated 2025-07-16

 View more

Filter graphs are a thing of great beauty. What an amazingly obscure domain-specific language, but which can produce striking results with very little!!!

A quick example from stackoverflow.com/questions/59551013/how-to-generate-stereo-sine-wave-using-ffmpeg-with-different-frequencies-for-eac/77730492#77730492 illustrates some of the fundamentals:

ffplay -autoexit -nodisp -f lavfi -i '
sine=frequency=500[a];
sine=frequency=1000[b];
[a][b]amerge, atrim=end=2
'

which creates a graph:

                              +--------+
[sine=frequency=500]--->[a]-->|        |
                              | amerge |-->[atrim]-->[output]
[sine=frequency=1000]-->[b]-->|        |
                              +--------+

and plays 500 Hz on the left channel and 1000 Hz on the right channel for 2 seconds.

So we see the following syntax patterns:

sine, amerge and atrim are filters
sine=frequency=500: the first = says "araguments follow"
- frequency=500 sets the frequency argument of the sine filter
- for multiple arguments the syntax is to separate arguments with colons e.g. sine=frequency=500:duration=2
;: separates statements
[a], [b]: sets the name of an edge
,: creates unnamed edge between filters that have one input and one output

A list of all filters can be obtained ith:

ffmpeg -filters

and parameters for a single filter can be obtained with:

ffmpeg --help filter=sine

TODO dump graph to ASCII art? trac.ffmpeg.org/wiki/FilteringGuide#Visualizingfilters mentions a -dumpgraph option, but haven't managed to use it yet.

Bibliography:

ffmpeg.org/ffmpeg-filters.html official documentation
trac.ffmpeg.org/wiki/FilteringGuide some handy tips from the FFMpeg Wiki

 Read the full article

Rust library by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Web crawling by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Scientific software by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

SymPy by

Ciro Santilli 37 Updated 2025-07-16

 View more

This is the dream cheating software every student should know about.

It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!

The code in this section was tested on sympy==1.8 and Python 3.9.5.

Let's start with some basics. fractions:

from sympy import *
sympify(2)/3 + sympify(1)/2

outputs:

7/6

Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!

We can also do everything with symbols:

from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)

outputs:

x/3 + y/2

We can now evaluate that expression object at any time:

expr.subs({x: 1, y: 2})

outputs:

4/3

How about a square root?

x = sqrt(2)
print(x)

outputs:

sqrt(2)

so we understand that the value was kept without simplification. And of course:

sqrt(2)**2

outputs 2. Also:

sqrt(-1)

outputs:

I is the imaginary unit. We can use that symbol directly as well, e.g.:

I*I

gives:

-1

Let's do some trigonometry:

cos(pi)

gives:

-1

and:

cos(pi/4)

gives:

sqrt(2)/2

The exponential also works:

exp(I*pi)

gives;

-1

Now for some calculus. To find the derivative of the natural logarithm:

from sympy import *
x = symbols('x')
print(diff(ln(x), x))

outputs:

1/x

Just read that. One over x. Beauty. And now for some integration:

print(integrate(1/x, x))

outputs:

log(x)

OK.

Let's do some more. Let's solve a simple differential equation:

y''(t) - 2y'(t) + y(t) = sin(t)

Doing:

from sympy import *
x = symbols('x')
f, g = symbols('f g', cls=Function)
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)**4)
print(dsolve(diffeq, f(x)))

outputs:

Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)

which means:

f (x) = C_{1} + C_{2} x e^{x} + cos (x) /2

(1)

To be fair though, it can't do anything crazy, it likely just goes over known patterns that it has solvers for, e.g. if we change it to:

diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)

it just blows up:

NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2

Sad.

Let's try some polynomial equations:

from sympy import *
x, a, b, c = symbols('x a b c d e f')
eq = Eq(a*x**2 + b*x + c, 0)
sol = solveset(eq, x)
print(sol)

which outputs:

FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))

which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:

sol.subs({a: 1, b: 2, c: 3})

which outputs

FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)

Let's see if it handles the quartic equation:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

and this time it spits out something more magic:

ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)

Oh well.

Let's try some linear algebra.

m = Matrix([[1, 2], [3, 4]])

Let's invert it:

m**-1

outputs:

Matrix([
[ -2,    1],
[3/2, -1/2]])

 Read the full article

Zero-based numbering by

Ciro Santilli 37 Updated 2025-07-16

 Read the full article

Android (operating system) by

Ciro Santilli 37 Updated 2025-07-16

 View more

the best full featured free OS we have today, since POSIX gave up short of any UI specification, and Chrome OS is not there yet
usable and likely efficient Java API for apps if Oracle doesn't manage to destroy it with its lawsuit

However, many, many, many terrible horrors come with it:

it hasn't made the move to desktop for too many years. It could destroy Microsoft Windows and replace it with open source, but they just won't budge towards an unified mobile/desktop setup.
vendors litter it with uninstallable bloatware that should be illegal. European Union to the rescue!!! www.cnbc.com/2020/12/15/digital-markets-act-eus-new-rules-on-big-tech.html
vendors lock down devices so it is very hard to get sudo, let alone to modify their images!
there isn't enough hardware standardization for open source distros to thrive like on desktop
code drops mean that "master" is useless and trying to contribute from outside vendors' closed walls is a waste of time: stackoverflow.com/questions/1809774/how-to-compile-the-android-aosp-kernel-and-test-it-with-the-android-emulator/48310014#48310014
if you ever go below the Java API, e.g. to C++ or AOSP build, everything is horrendous and undocumented
Google doesn't care about the CLI, even the hello world requires creating infinite out-of-control boilerplate from a GUI: stackoverflow.com/questions/20801042/how-to-create-android-project-with-gradle-from-command-line/46994747#46994747
the boot is uber bloated and takes forever in cycle simulators

 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