Set function by Ciro Santilli 37 Updated 2025-07-16
This section is about functions that operates on arbitrary sets.
SQL example by Ciro Santilli 37 Updated 2025-07-16
We have some runnable SQL examples with assertion under the sequelize/raw directory.
These examples are written in the Sequelize library using raw queries.
Sequelize is used minimally, just to feed raw queries in transparently to any underlying database, and get minimally parsed results out for us, which we then assert with standard JavaScript. The queries themselves are all written by hand.
By default the examples run on SQLite. Just like the examples from sequelize example, you can set the database at runtime as:
Here we list only examples which we believe are standard SQL, and should therefore work across different SQL implementations:
MariaDB by Ciro Santilli 37 Updated 2025-07-16
Dude's a legend. Sells company for a few million. Then forks the open source project next year. Love it.
On Ubuntu 20.10 PostgreSQL 12.6, login with psql on my default username without sudo fails with: stackoverflow.com/questions/11919391/postgresql-error-fatal-role-username-does-not-exist
This is the one that worked on Ubuntu 21.04: stackoverflow.com/questions/11919391/postgresql-error-fatal-role-username-does-not-exist/38444152#38444152
sudo -u postgres createuser -s $(whoami)
createdb $(whoami)
Explanation:
  • sudo -u postgres uses the postgres user via peer authentication
  • -s in createuser -s: make it a superuser
  • createdb: TODO why do we have to create a table with the same name as the user? Otherwise login fails.
You can now run psql without any password. This works without password due to peer authentication:
sudo cat /etc/postgresql/12/main/pg_hba.conf
shows that peer authentication is available to all users apparently:
local   all             postgres                                peer

# TYPE  DATABASE        USER            ADDRESS                 METHOD

# "local" is for Unix domain socket connections only
local   all             all                                     peer
List users:
psql -c '\du'
output:
                                    List of roles
  Role name  |                         Attributes                         | Member of
-------------+------------------------------------------------------------+-----------
 ciro        | Superuser, Create role, Create DB                          | {}
 owning_user |                                                            | {}
 postgres    | Superuser, Create role, Create DB, Replication, Bypass RLS | {}
Delete user later on:
psql -c 'DROP USER username;'
Create a database:
createdb testdb0
Help toplevel:
help
Get help for Postgres commands such as \h and so on:
\?
List supported SQL commands:
\h
Show syntax for one type of command:
\h SELECT
List all databases:
psql -c '\l'
which shows:
    Name     |  Owner   | Encoding |   Collate   |    Ctype    |   Access privileges
-------------+----------+----------+-------------+-------------+-----------------------
 ciro        | postgres | UTF8     | en_GB.UTF-8 | en_GB.UTF-8 |
 postgres    | postgres | UTF8     | en_GB.UTF-8 | en_GB.UTF-8 |
 template0   | postgres | UTF8     | en_GB.UTF-8 | en_GB.UTF-8 | =c/postgres          +
             |          |          |             |             | postgres=CTc/postgres
 template1   | postgres | UTF8     | en_GB.UTF-8 | en_GB.UTF-8 | =c/postgres          +
             |          |          |             |             | postgres=CTc/postgres
 testdb0     | postgres | UTF8     | en_GB.UTF-8 | en_GB.UTF-8 |
(6 rows)
Delete a database:
psql -c 'DROP DATABASE "testdb0";'
If you didn't give a database from the command line e.g.:
psql
you can do that afterwards with:
\c testdb0
Let's create a table and test that it is working:
psql testdb0 -c 'CREATE TABLE table0 (int0 INT, char0 CHAR(16));'
List tables, no special tables:
psql testdb0 -c '\dt'
gives:
        List of relations
 Schema |  Name  | Type  | Owner
--------+--------+-------+-------
 public | table0 | table | ciro
(1 row)
View table schema: stackoverflow.com/questions/109325/postgresql-describe-table
psql testdb0 -c '\d+ table0'
output:
                                      Table "public.table0"
 Column |     Type      | Collation | Nullable | Default | Storage  | Stats target | Description
--------+---------------+-----------+----------+---------+----------+--------------+-------------
 int0   | integer       |           |          |         | plain    |              |
 char0  | character(16) |           |          |         | extended |              |
Insert some data into it and get the data out:
psql testdb0 -c "INSERT INTO table0 (int0, char0) VALUES (2, 'two'), (3, 'three'), (5, 'five'), (7, 'seven');"
psql testdb0 -c 'SELECT * FROM table0;'
output:
 int0 |      char0
------+------------------
    2 | two
    3 | three
    5 | five
    7 | seven
(4 rows)
Delete the table:
psql testdb0 -c 'DROP TABLE table0;'
Hamilton's equations by Ciro Santilli 37 Updated 2025-07-16
Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:
So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.
Mean Absolute Error (MAE) is a common metric used to evaluate the performance of regression models. It measures the average magnitude of the errors in a set of predictions, without considering their direction (i.e., it takes the absolute values of the errors).
The term "spin representation" is commonly used in the context of quantum mechanics and refers to a mathematical framework for describing the intrinsic angular momentum (spin) of quantum particles. Spin is a fundamental property of quantum particles like electrons, protons, neutrons, and other elementary and composite particles. ### Key Elements of Spin Representation: 1. **Quantum States**: Spin states are represented as vectors in a Hilbert space.
Euclidean symmetries refer to the transformations that preserve the structure of Euclidean space, which is the familiar geometry of flat spaces (typically two-dimensional and three-dimensional spaces). These symmetries encompass various operations that can be applied to geometric figures without altering their fundamental properties, such as distances and angles. The main types of Euclidean symmetries include: 1. **Translations**: Shifting a figure from one location to another without rotation or reflection.
The affine symmetric group, often denoted as \( \text{Aff}(\mathbb{Z}/n\mathbb{Z}) \) or \( \text{Aff}(n) \), is an extension of the symmetric group that includes not only permutations of a finite set but also affine transformations. Specifically, it refers to a group of transformations that act on a finite cyclic group, typically represented as \( \mathbb{Z}/n\mathbb{Z} \).
In geometry, axiality refers to a property or characteristic related to axes, particularly concerning symmetry and orientation. While the term isn't frequently used in mainstream geometry literature, it often relates to how certain objects or shapes are organized around an axis. In the context of geometry, axiality can describe: 1. **Symmetry**: An object is said to have axiality if it exhibits symmetry about an axis.
Circular symmetry, often referred to as radial symmetry, is a type of symmetry where an object or shape appears the same when rotated around a central point. In other words, if you were to rotate the object through any angle about that central point, it would look unchanged. In the context of two-dimensional shapes, examples of circular symmetry include circles, wheels, and starfish. In three dimensions, objects like spheres and some types of flower arrangements exhibit circular symmetry.
Facial symmetry refers to the degree to which one side of a person's face is a mirror image of the other side. In a perfectly symmetrical face, corresponding features (such as eyes, eyebrows, lips, and jawline) match in size, shape, and position on both sides. However, most human faces are not perfectly symmetrical; slight asymmetries are common and can even contribute to an individual's uniqueness and attractiveness.

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!
We have two killer features:
  1. 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
  2. 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.
    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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    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