OpenLLM-France/wikimedia huggingface dataset:
# Algebra/Chapter 9/Graphing

A Quadratic function is a polynomial where the highest power is two. The basic form of this function is: F(X) = ax² + bx + c. Where, ax² is the quadratic term, bx is the linear term and c is the independent term or "constant", which does not depend on the variable, x. The letters a and b are called "coefficients", a being the "leading coefficient". The standard form is F(X) = ax² + bx + c. The x-intercepts of the function are:
$x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}$ OR $x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}$
The independent term is also the y-coordinate of the point of intersection with the y-axis (when X=0, F(X)=C).
A quadratic function has a "vertex" or "turning point", which is the point where the function has either a maximum or minimum value. If a is greater than zero, then there will be a minimum and the curve will be concave. If a is less than zero, then there will be a maximum and the curve will be convex. If a = 0, then we have a linear function rather than a quadratic function.The x-coordinate of the vertex is $x=-{\frac {b}{2a}}$ The y-coordinate of the vertex is $F(-{\frac {b}{2a}})$
The general form of a quadratic equation is actually F(X) = ax² + bxy + cy² + dx + ey + f = 0, which can take many shapes including circles, ellipses and parabolas, but in most Western high schools, quadratic equation refers only to those of the form F(X) = ax² + bx + c, which forms a parabola.
# Using Ubuntu Linux/Virtualization

Virtualization is a system where an operating system can be installed into another host operating system. The technology involves using some space on the hard disk where the virtual system would be installed and operated. There are many free and proprietary software for virtualization. VMware and VirtualBox are 2 such examples.
When an operating system is installed in a virtual box it behaves exactly the same and the end user would not even know that a system is actually running in another operating system. This way one need not partition the hard disk for another operating system.
This becomes particularly useful for testing an operating system before deciding to actually use it as a complete replacement, or using virtual machines for software development. When we think of this option as an intermediate phase in the migration process, one can think of installing Ubuntu inside Windows and using it for some days with all its features and capabilities. When users get familiar with the partitioning and other complex tasks they can confidently install Ubuntu directly.

### Running Ubuntu in a VirtualBox

* Download the Ubuntu ISO from Canonical's website.
* Download the Oracle VM VirtualBox.
* Create a new virtual machine. You should give it the name "Ubuntu" and, if needed, select Ubuntu in the OS dropdown list.
* Select how much RAM you will allocate to your virtual machine. Too much RAM and your host operating system will not run while the guest is running, which causes it to fail sometimes.
* Select "Create a virtual hard drive now".
* Select a dynamically allocated or a fixed size disk. A dynamically allocated virtual hard drive will keep growing and growing until your computer hits its hard drive limit. A fixed size virtual hard drive will keep growing until it hits the fixed size.
* Select the limit for the hard drive. Let it create the virtual hard drive.
USE THE ISO:
* Click on "Settings" > "Storage".
* Click the CD/DVD icon with a + on it to select the ISO.
* Choose the Ubuntu ISO.
* Click on "System" and set the boot priority to CD/DVD first, so the virtual machine will boot into Ubuntu.
INSTALL UBUNTU
* Open the Ubuntu virtual machine.
* Select the language.
* Double-click Install Ubuntu.
* Install Ubuntu while answering the questions, and voila!
For more details, see
* VirtualBox/Setting up a Virtual Machine/Ubuntu

### Installing VirtualBox on Ubuntu

* VirtualBox/Installation/Linux#Installation for Ubuntu

## Further reading

* "Welcome : Links to get started with Virtualization" has a section on how to run Ubuntu in a virtual machine on Windows.
* Windows XP under QEMU: how to run Windows in a virtual machine on top of Ubuntu
* Overview: virtualization on Ubuntu servers
* Guide: virtualization on Ubuntu servers
* SeamlessVirtualization: rather than 2 separate "desktops", it's possible have application windows on the same "desktop", applications running in 2 different OSes. (Running Windows applications on an Ubuntu host system).
# Algebra/Chapter 9/Quadratic Equation
## Derivation

The solutions to the general-form quadratic function $ax^{2}+bx+c=0$ can be given by a simple equation called the quadratic equation. To solve this equation, recall the completed square form of the quadratic equation derived in the previous section:
    $y=a\left(x+{\frac {b}{2a}}\right)^{2}+c-{\frac {b^{2}}{4a}}$
In this case, $y=0$ since we're looking for the root of this function. To solve, first subtract c and divide by a:
    $\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}$
Take the (plus and minus) square root of both sides to obtain:
    $x+{\frac {b}{2a}}=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}$
Subtracting ${\frac {b}{2a}}$ from both sides:
    $x=-{\frac {b}{2a}}\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}$
This is the solution but it's in an inconvenient form. Let's rationalize the denominator of the square root:
    ${\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}={\sqrt {\frac {b^{2}-4ac}{4a^{2}}}}={\frac {\sqrt {b^{2}-4ac}}{2|a|}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}$
Now, adding the fractions, the final version of the quadratic formula is:
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$
This formula is very useful, and it is suggested that the students memorize it as soon as they can.

## Discriminant

The part under the radical sign, ${b^{2}-4ac}$ , is called the discriminant, $\Delta$ . The value of the discriminant tells us some useful information about the roots.
* If $\Delta >0$ , there are two unique real solutions.
* If $\Delta =0$ , there is one unique real solution.
* If $\Delta <0$ , there are two unique, conjugate imaginary solutions.
* If $\Delta$ is a perfect square then the two solutions are rational, otherwise they are irrational conjugates.

## Word Problems

Need to pull word problems from http://teachers.yale.edu/curriculum/search/viewer.php?id=initiative_07.06.12_u&skin=h
# Algebra/Chapter 15/Arithmetic Progression (AP)

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Example 1:
A frog jumps every 2 seconds. If we start from the very first second the frog jump we take it as $t=0$. The next jump will be on $t=2$. As the frog jumps every 2 seconds so; $t=0+2\rightarrow t=2$
So can you tell when the next jump will be? If you said $t=4$ then yes, you are correct. Do you remember how we got $t=2$? Now how did you get $t=4$? The same way right? Well what you did was you added the difference $d=2$ to previous term (which was 2). So to get next term of AP you always add the common difference to current term, and to get previous term you always subtract the common difference from current term. A more general formula to get the ‘n’th term of an AP is:
$a_{n}=a_{1}+(n-1)d$
Where $a_{1}$ is the first term and d is the common difference of the AP. So the general terms of AP are:
$a_{1},\;a_{1}+d,\;a_{1}+2d,\;a_{1}+3d,\;\dots \;a_{1}+(n-1)d$

### Finding the sum of ‘n’ terms of an arithmetic sequence

When needed to find the sum of ‘n’ terms of an AP, you use:
$s_{n}={\frac {n}{2}}(2a_{1}+(n-1)d)$
!If you have the first and last term you can use a short-cut formula to obtain the sum.
$s_{n}={\frac {n(a_{1}+a_{n})}{2}}$
Where $a_{n}$ is the last term.

## Notes

* The sum of infinite arithmetic sequence is essentially infinite, but if $a=b=0$, then the sum is 0.
# A Link Rot Bestiary/Chapter 5 : Archived Redirect

Archived redirect or Ghost redirect is redirect link rot. A redirect once existed on the live web and was later deleted. Fortunately while the redirect was live, it was saved into the Wayback Machine, where it's ghostly existence is preserved.
Example: http://www.ew.com/ew/article/0,,286000,00.html is 404 on the live web .. but there is an old redirect saved in the Wayback Machine at https://web.archive.org/web/20150619213058/https://www.ew.com/ew/article/0,,286000,00.html which redirects to http://www.ew.com/article/1998/12/04/twelve-songs-christmas which is the new live URL.
By searching for and discovering archived redirects it is possible to convert dead URLs to live URLs.
A command line tool to automate the above steps is Ghostredir
# A Link Rot Bestiary/Chapter 6 : Usurpation

Usurpation is when a domain has been hijacked by a spammer. They re-register an expired domain and redirect traffic to their servers which contain unwanted content. It is a variety of soft 404.
As the Internet ages and more domains go offline, spammers look for dead links on sites like Wikipedia and elsewhere to determine which domains have a large number of links. They then re-register the domain which now contains spam content. The classic example is Asian gambling sites, who control thousands of domains scoured from Wikipedia. In this way they broadly spam Wikipedia without a single edit.
A system has been established on English Wikipedia to report the domains so they can be converted to usurped status.
Nevertheless, the problem exists Internet-wide. For example there is a domain on GitHub with over 700 impressions in repo README files that has been usurped - fixing these would require submitting 700 push requests to each repo.
As can be seen, the problem is large, intractable and will only increase as bad actors catch on. The long term solution will be cooperation from domain registars. Until then, this special class of soft 404 is notable.
# GIMP/Create a Metal Effect

This article will describe you how to create a brushed metal effect.

### Step 1

First, we fill the canvas using the hurl filter (Filters > Noise > Hurl). Set Randomization (%) to 100.
This will do the same thing as the Scatter RGB filter (Filters > Noise > Scatter RGB) with all channels (Red, Green, and Blue) set to 1.0, However, Scatter RGB has slightly more options, and if you turn off Independent RGB in the settings dialogue, may allow you to skip the desaturation step.

### Step 2

Then, apply a motion blur to the image (Filters > Blur > Motion Blur...) with an angle of 0 and a length of about 50. Change the angle to 180 and repeat the effect. Why? Notice how the right edge of the final image isn't nice and brushed looking? Doing it in both directions will smooth this out. You may have to adjust the length setting so as not to "overblur".
Alternatively, you can use the Gaussian Blur filter (Filters > Blur > Gaussian Blur). Just click the chain icon and turn the vertical value down and adjust the horizontal value until you are satisfied. This effect will not leave a side of the image left unblurred and may have an even more satisfying final result with proper adjustment.

### Step 3

This is the shortest step. Now we only need to desaturate the image. Go to Colours > Desaturate and press Lightness, although Luminosity may sometimes look better.
Apply cropping / tiling / brightness / lighting filters to suit. Cropping may not be necessary if you've used the Gaussian Blur approach as opposed to the Motion Blur approach.
← Body Shifting · Create a Balloon →
# LaTeX

This is a guide to the LaTeX typesetting system. It is intended as a useful resource for everybody, from new users who wish to learn, to old hands who need a quick reference.
TeX and LaTeX
TeX is a typesetting computer program created by Donald Knuth, originally for his magnum opus, The Art of Computer Programming. It takes a "plain" text file and converts it into a high-quality document for printing or on-screen viewing. LaTeX is a macro system built on top of TeX that aims to simplify its use and automate many common formatting tasks. It is the de-facto standard for academic journals and books in several fields, such as mathematics and physics, and provides some of the best typography free software has to offer.
This book is organized into different parts:

## Contents

Edit the TOC or the stages
* Getting Started
** Introduction
** Installation
** Installing Extra Packages
** Basics
** How to get help
* Common Elements
** Document Structure
** Text Formatting
** Paragraph Formatting
** Colors
** Fonts
** List Structures
** Special Characters
** Internationalization
** Rotations
** Tables
** Title creation
** Page Layout
** Customizing Page Headers and Footers‎
** Importing Graphics
** Floats, Figures and Captions
** Footnotes and Margin Notes
** Hyperlinks
** Labels and Cross-referencing
** Initials
* Mechanics
** Errors and Warnings
** Lengths
** Counters
** Boxes
** Rules and Struts
* Technical Text
** Mathematics
** Advanced Mathematics
** Theorems
** Chemical Graphics
** Algorithms
** Source Code Listings
** Linguistics
* Special Pages
** Indexing
** Glossary
** Bibliography Management
** More Bibliographies
* Special Documents
** Scientific Reports (Bachelor Report, Master Thesis, Dissertation)
** Letters
** Presentations
** Teacher's Corner
** Curriculum Vitae
** Academic Journals (MLA, APA, etc.)
* Creating Graphics
** Introducing Procedural Graphics
** MetaPost
** Picture
** PGF/TikZ
** PSTricks
** Xy-pic
** Creating 3D graphics
* Programming
** Macros
** Plain TeX
** Creating Packages
** Creating Package Documentation
** Themes
* Miscellaneous
** Modular Documents
** Collaborative Writing of LaTeX Documents
** Export To Other Formats
* Help and Recommendations
** FAQ
** Tips and Tricks
* Appendices
** Authors
** Links
** Package Reference
** Sample LaTeX documents
** Index
** Command Glossary

## Other Wikibooks

* TeX
* LaTeX in Haskell
* ConTeXt
# Associative Composition Algebra/Bibinarions

A construction of a doubled algebra was initiated by L. E. Dickson and recounted by A. A. Albert. The method presumes an algebra with conjugate * and produces one of double the dimension and a new conjugation: (u, v)* = (u*, −v) where u* denotes the original conjugation. The new algebra has products given by
    $(u,v)\times (w,x)=(uw-vx^{*},ux+vw^{*}).$
Starting with a field and its conjugation, a sequence of algebras can be so constructed. The division binarions arises when the starting field is the real numbers $\mathbb {R} .$ Since the reals have no conjugation, the identity is substituted, and the conjugation arises as above. Continuing with the construction, using the binarion conjugate, quaternions are obtained as will be seen. However, the binarion conjugation may be forgotten (identity substituted), and bibinarions produced according to the Dickson/Albert method given above.
A bibinarion is then a pair of division binarions (u, v) with conjugation (u, v)* = (u, −v). The norm of a bibinarion is then
    $(u,v)(u,v)^{*}=(u,v)(u,-v)=(u^{2}-v(-v),\ u(-v)+vu)=(u^{2}+v^{2},\ 0).$
Notice that this norm is a division binarion, and is not the kind of norm that produces a metric.
Furthermore, with i$^2$ = −1 in C, a bibinarion (u, i u) has zero norm. Such an element is called a null vector. The bibinarions form a split algebra since some elements are null vectors.
The product of two bibinarions is commutative since the generating conjugation is the identity. Most remarkably, there is bibinarion j = (0, i) with j$^2$ = (0, i)$^2$ = (−i$^2$, 0) = +1. The two-dimensional subalgebra of bibinarions on basis {1, j } is called split binarions.
Let u = i sin θ and v = i cos θ. Then for any θ the norm of (u,v) is
    $-\sin ^{2}\theta -\cos ^{2}\theta =-1.$ The "anti-circle" can be formed by {(u,v) : 0 ≤ θ < 2 π }, a circle with imaginary radius in the space of bibinarions.

## History

The idea of an algebra with two imaginary units that commute was considered in mid-19th century Britain. Hamilton used a commuting h with his biquaternions. James Cockle saw that the square of the product hi of imaginary units was plus one, thus creating "a new imaginary in algebra" as he wrote in Philosophical Magazine in 1848. His use of the letter j, j$^2$ = +1, has been widely adopted. Although Hamilton provided a vocabulary of vector operations (including the del operator), these explorations preceded set theory, group theory, and the unfolding of mathematical notation. With 1 on the real axis, the two imaginary units h and i, and their product hi, Cockle's commutative algebra T (tessarines) has a real basis of four elements. By the end of the 19th century tessarines and quaternions were referred to as hypercomplex numbers.
In 1892 Corrado Segre introduced bicomplex numbers in Mathematische Annalen (v 40: 455 to 67).
The division binarion basis of this algebra is used in the Dickson-style construction of biquaternions.
← Binarions · Split-binarions →
# Android
## Contents

* Introduction
* Android Applications
* Android SDK
* API Overview
* PhoneGap (JavaScript)
* Kivy (Python)
* Developing on Android
* Testing
* Low Level Graphics
# A Link Rot Bestiary/Preface

The Internet Archive team began an effort to repair link on Wikipedia in late 2015. Two notable tools were developed, Internet Archive Bot, and Wayback Medic. They evolved through practical experience and iteration. This book describes a system of first principles, and some practical solutions.
The concepts described here are applicable to link rot, wherever it might exist. Wikipedia is the fertile testing ground for link rot discovery and repair.
# Algebra/Chapter 21

Chapter 21: Approximation Theory
← Chapter 20 · Chapter 22 →
Introduction

## Sections

* Iteration
* Polynomial Interpolation
* Optimal Polynomials
* Symmetric Equations
* Chebyshev Polynomials

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Algebra/Chapter 18

Chapter 18: Parametric Equations and Polar Equations
← Chapter 17 · Chapter 19 →

## Sections

* Parametric Equations
# Algebra/Chapter 13

Chapter 13: Rational and Radical Functions
← Chapter 12 · Chapter 14 →
Introduction

## Sections

* Dividing Polynomials
# Geometry/Chapter 1

Chapter 1: An Introduction to Geometry
Introduction

## Sections

* Points, Lines, Line Segments and Rays
* Proofs
* Logical Statements and Theorems
* Angles
* Sets of Lines

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
← Introduction · Chapter 2 →
# Algebra/Cover

Chapter 0 →

# Welcome To Algebra!

This Wikibook helps you to learn algebra. If you wish to read this book, click the link above.
# Geometry/Chapter 0

Chapter 0: The World Around Us
← Cover · Chapter 1 →
Introduction

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# CAT-Tools

CAT means "Computer Aided Translation", referring not to machine translation, but to translation done by human translators with specialized software providing functions to increase workflow while providing quality assurance, glossaries and other assistance.
Wikibooks was chosen for these manuals to be written by translators for translators to make sure the content is open to the public and subject to a continuous quality improvement. Wikibooks also allows FAQs and other useful informational tools.
Expanding this to other languages would of course be welcome. If you see something that should be improved, please jump in. Every bit that improves this content is helpful to everyone else who reads it. It is hoped comparative pages will be added to provide guidance among different CAT programs.

## Contents
### Tools

* Across
* AnyMem
* AppleTrans
* Cafetran
* DéjàVu 3
* DéjàVu X
* Felix
* Fluency
* Heartsome CAT Bundle
* Heartsome Translation Studio
* MemoQ
* Memsource
* Metatexis
* Mneme
* MultiTrans
* Okapi Framework
* OmegaT
* Open Language Tools
* SmartCAT
* SDL Trados
* STAR Transit
* Similis
* Termbases.eu
* TM-database
* Transolution
* Virtaal
* Visual-Localize
* Wordfast
* Wordfast free version (dated 21 September 2002)
* Wordfisher
* Crowdin
* Redokun

## External links

* Cat Tools Discussions - Cat Tools Community
* Open-Source CAT-Tools - Links to open source resources
* ProZ.com forum on CAT Tools - ProZ.com Translation Workplace forum dedicated to CAT Tools
* TranslatorsTraining - Video-based comparisons and introductions and other information about 15 different CAT tools

### Similar projects

* TranslateWiki - specialized in software localization
# PSP/cIPL Motherboards

cIPL supports:
* PSP 1000 Fat: (All motherboards) TA-079v1, TA-079v2, TA-079v3, TA-081, TA-082, TA-086
* PSP 2000 Slim: (Everything except TA-088v3): TA-085v1, TA-085v2, TA-088v1, TA-088v2, TA-090v1
All of the cIPL compatible PSPs are also capable of using LCFW and 6.20 Permapatch (though cIPL is preferred).
# Linear Algebra/OLD/Eigenvalues and Eigenvectors

Let $T:V\rightarrow V$ be a Linear Map where $V$ is a vector space over a field $K$. Then if some non-zero vector $v\in V$ satisfies the equation $T(v)=\lambda v$ with $\lambda \in K$ then $v$ is an eigen vector of $T$ and $\lambda$ is its corresponding eigen value.
Equivalently in Matrices we have $Av=\lambda v$.
For example, the matrix ${\begin{pmatrix}2&0\\0&3\end{pmatrix}}$ will double the vector ${\begin{pmatrix}1\\0\end{pmatrix}}$ and triple the vector ${\begin{pmatrix}0\\1\end{pmatrix}}$.

## How to find eigenvectors and eigenvalues

Finding the eigenvalues and eigenvectors is fairly simple. First we will find the eigenvalues, and then we will see how to find the eigenvectors that correspond to a specific eigenvalue.
If V is an eigenvector of matrix A with eigenvalue $\lambda$ then by definition AV=$\lambda$V. after subtracting $\lambda$V we will get AV-$\lambda$V=0. since $\lambda$V equals $\lambda$IV where I is the identity matrix we finally get (A-$\lambda$I)V=0. so $\lambda$ is an eigenvalue if and only if the matrix (A-$\lambda$I) sends some vector to 0. This is luckily easy to determine, because a matrix sends some vectors to zero if and only if its determinant is equal to zero.
Let's do an example and find the eigenvalues of ${\begin{pmatrix}-1&-3&-6\\0&2&0\\3&3&8\end{pmatrix}}$ we will subtract tI from the matrix (t is a variable) and find out when the determinant |A-tI| equals 0 (this is called the characteristic polynom).
A-tI = ${\begin{pmatrix}-1-t&-3&-6\\0&2-t&0\\3&3&8-t\end{pmatrix}}$
|A-tI| is (calculating with the first row): (-1-t)*(2-t)*(8-t)+3*6*(2-t) simplifying a bit we get (2-t)$^2$*(5-t) since |A-tI|=0 when t=2,5, the eigenvalues are 2 and 5.
let's find the eigenvectors space of the eigenvalue 2: by subtracting 2 from the diagonal, we get A-2I: ${\begin{pmatrix}-3&-3&-6\\0&0&0\\3&3&6\end{pmatrix}}$
the vectors ${\begin{pmatrix}2\\0\\-1\end{pmatrix}}$ and ${\begin{pmatrix}0\\2\\-1\end{pmatrix}}$ form a kernel space of A-2I, and therefore any linear combination of them is an eigenvector with an eigenvalue of 2. for example 10*${\begin{pmatrix}2\\0\\-1\end{pmatrix}}$ + 3*${\begin{pmatrix}0\\2\\-1\end{pmatrix}}$=${\begin{pmatrix}20\\6\\-13\end{pmatrix}}$ and ${\begin{pmatrix}-1&-3&-6\\0&2&0\\3&3&8\end{pmatrix}}$*${\begin{pmatrix}20\\6\\-13\end{pmatrix}}$=${\begin{pmatrix}40\\12\\-26\end{pmatrix}}$=2*${\begin{pmatrix}20\\-6\\-13\end{pmatrix}}$
you can similliary find the eigenvectors space of the eigenvalue 5 by computing A-5I (subtracting 5 from the diagonal) and finding it's kernel space.
* You can also use a mathematical software like octave to find the eigenvalues: