OpenLLM-France/wikimedia huggingface dataset:
# Topology/Cantor Space

The Cantor space is a generalisation of the Cantor set, which can be seen as any element as the homeomorphism class of the Cantor set defined below.

## Definition of the Cantor set

The Cantor set is the limit of an infinite process of removing subsets of the unit interval depicted here.
So at any given step we remove the middle thirds of the unit interval, let $C_{0}=[0,1]$ so at the n-th step we have:
    $C_{n}={\frac {C_{n-1}}{3}}\cup {\big{\frac {2}{3}}+{\frac {C_{n-1}}{3}}{\big )}$
So the effect is scaling the previous step down by a third and putting two copies next to each other. Then the Cantor set is
    $C=\bigcap _{i=0}^{\infty }C_{i}$

## Definition of the Cantor space

A Cantor space is any space homeomorphically equivalent to $C$ defined above. The value of this definition is elucidated by the following theorem.

## Brouwer's Theorem

An equivalent form of Brouwer's original theorem is: "A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected, and metrizable."
To understand this let's fill in the definition not yet covered.
1. Clearly, non-empty means it is not the empty set.
2. Perfect means the space has no isolated points (no points whose singleton is an open set in the ambient space).
3. Compactness has been covered.
4. Totally disconnected means the connected components (the maximal connected subsets) of the space are all points.
5. Metrisable means there exists a metric for this space.
Thus the theorem states that the combination of these conditions leaves a space homeomorphic to the Cantor set.

## Example

Consider the map from the set of infinite binary codes $\{0,1\}^{\mathbb {N} }$ often written $2^{\mathbb {N} }$ to the Cantor set defined above: $f:2^{\mathbb {N} }\to C$ defined as for a sequence $a=(a_{i})_{i=0}^{\infty }$
    $f(a)=\sum _{i=0}^{\infty }{\frac {2a_{i}}{3^{i+1}}}$
Then f is a homeomorphism of metric spaces, meaning that all the properties above hold for the space $2^{\mathbb {N} }$.
# Topology/Normed Vector Spaces

A normed vector space $(V,|\cdot |)$ is a vector space V with a function $|\cdot |:V\times V\to \mathbb {R}$ that represents the length of a vector, called a norm.

## Definition

We know the vector space definition, so we need to define the norm function. $|\cdot |$ is a norm if these three conditions hold.
1. Only the zero vector has zero length, with all others being positive. $|v|\geq 0,|v|=0\iff v=0$ for all $v\in V$.
2. For $a\in \mathbb {R}$ and $v\in V$ we have $|av|=|a||v|$.
3. The triangle inequality holds: $|v+w|\leq |v|+|w|$ for all $v,w\in V$.

## Example

For a given $n\in \mathbb {N}$ we know that $\mathbb {R} ^{n}$ is a vector space and its norm can be defined to be $|v-w|=d(v,w)$ ie. $|v|=d(v,0)$. This is not unusual, in fact we say that a norm induces a metric with the first equation. So normed vector spaces are always metric spaces. Let's prove this.

## Theorem

Normed vector spaces are metric spaces.
Proof
It suffices to show that $d(v,w)=|v-w|$ satisfies the metric axioms. Let $u,v,w\in V$
1. $d(v,w)=|v-w|\geq 0$ holds by definition and $d(v,w)=|v-w|=0\iff v-w=0\iff v=w$ as required.
2. $d(v,w)=|v-w|=|-1||w-v|=|w-v|=d(w,v)$
3. $d(u,v)+d(v,w)=|u-v|+|v-w|\geq |u-v+v-w|=d(u,w)$ so the triangle inequality translates correctly.
Since the axioms hold, we conclude that V is a metric space.

## Exercises

(under construction)
# Lua in SpringRTS/Units
## Units

Each unit in the game has a corresponding so called unit definition file. All unit definition files can be found in the "units" folder of a mod. Each unit definition file is practically just a text file that holds information about a specific unit. For example in the BA mod there is a file called "armcom.fbi" in the units folder that hols all the information for the arm commander. If this file is deleted then the arm commander won't be able to appear in the game. This file is necessary for the unit because this file is what the engine uses to define a unit. With defining a unit we just mean some way so the engine knows that there should be this specific unit in the game and an explanation of how it should look and its general behaviour.
So what the unit definition file does is that it keeps all information regarding a unit. The most necessary information kept in the file is the line that links the unit with the 3d model. A unit without a 3d model cannot exist.
That line looks like this:
This informs the engine that the 3d model has the name "ARMCOM". Keep in mind that names are not case sensitive and don't need the format extension of the file. In the above example the file armcom.3do is linked to the unit. That file as all 3d models are found in the folder Objects3d of the mod.
# XRX/What is XRX

Although the term XRX was originally intended to connote the three standards - XForms, REST and XQuery, the term has been used by different people in different ways and has a spectrum of definitions from very broad to very specific.
One the most broad definitions, XRX can connote any web application architecture that stores XML on the client with REST interfaces to XML on the server. The only aspect that these architectures have in common is that they use a no-shredding system to store and edit records.
One of the most specific definitions and the definition that is used in this book is the use of XForms on the client, REST interfaces and XQuery on the server. This implies that many of the features of these standards are integral to the overall solution.
# Quantum Mechanics/Perturbation Theory
## Introduction

Perturbation Theory is an extremely important method of seeing how a Quantum System will be affected by a small change in the potential. And as such the Hamiltonian.
Perturbation Theory revolves around expressing the Potential as multiple (generally two) separate Potentials, then seeing how the second affects the system.
It allows us to get good approximations for systems where the Eigenstates are not all easily findable.
In the real life not many hamiltonians are exactly solvable. Most of the real life situations requires some approximation methods to solve their hamiltonians. Perturbation theory is one among them. Perturbation means small disturbance. Remember that the hamiltonian of a system is nothing but the total energy of that system. Some external factors can always affect the energy of the system and its behaviour. To analyse a system's energy, if we dont know the exact way of solution, then we can study the effects of external factors(perturbation) on the hamiltonian. For an example, we can study the elastic properties of a spring by the application of load on that. By this way, we can approximately predict the properties of the concerned systems. Perturbation applied to a system is of two types: time dependent and time independent and hence the theory. We have to split the hamiltonian into two parts. One part is a hamiltonian whose solution we know exactly and the other part is the perturbation term. By this way we can solve the problems with a very good approximation. For an example of this method in quantum mechanics, we can use the hamiltonian of the hydrogen atom to solve the problem of helium ion.

## First Order Corrections
### Energy

The first order correction to energy is given by
$\epsilon _{n}^{(1)}=\langle n|V|n\rangle =\int \Psi _{n}^{(0)^{\star }}(x){\hat {V}}(x)\Psi _{n}^{(0)}(x)dx$
The first order correction to the wave function is
$C_{k}^{(1)}={\frac {\langle k|V|n\rangle }{\epsilon _{n}^{(0)}-\epsilon _{k}^{(0)}}}$

### Proof

Let ${\hat {H}}_{0}$ be a Hamiltonian where all eigenstates are known.
${\hat {H}}_{0}\Psi _{n}^{(0)}=\epsilon _{n}^{(0)}\Psi _{n}^{(0)}\qquad n\in \mathbb {N} \qquad (1)$
$({\hat {H}}_{0}+{\hat {V}})\Psi =\epsilon \Psi \qquad (2)$
Now represent $\Psi$ as a linear combination of all its parts,
$\Psi =\sum c_{m}\Psi _{m}^{(0)}$
and put this into (2).
# Sport Innovation/Goal-line Technology/Arguments for the introduction of goal-line technology

Although there are some standalone arguments as to why goal-line technology should be implemented the majority are rebuttals to the arguments put forward against the introduction of goal-line technology. Below details the reasons GLT should be implemented:
The criteria developed by FIFA to test nine different systems clearly states that they must achieve a 100% accuracy in order to progress from phase to phase during the testing. Thus, inaccuracy of the technology should not be considered as a negative aspect of the technology implemented [1].
The thought that play will be stopped for extended periods of time or that the flow of the game will be disrupted because of the technology is incorrect. Again, the criterion used by FIFA to determine which technology to use has a strict rule concerning this. The criteria states that referees must receive an indication of whether a goal has been scored immediately and automatically confirmed within one second via the vibration and visual signal to the referees watch [1].
Supporters of technology in football say goal-line incidents are the only decisions that are entirely definitive and the answer can be provided to the referee within one second of the incident occurring [2]. The ball either crossed the line or it didn’t. The technology would not replace referees, it would help them [3] [4].
With the every increasing statistics regarding sports betting, there is simply just too much money involved in the sport to have incorrect decision made. An incorrect decision could also affect the outcome of the game being played. A disallowed goal could be the confidence breaker that allows a team to be beaten. A allowed non-goal could mean a losing team getting an advantage over the other.
Although there are cases both for and against the implementation of the technology, the stronger of the two and the future of sport seems to moving it towards the implementation of technology.

## References

[1] Conway, R. (2011, November 24). Premier League could use goal-line technology in 2012–13 season. Retrieved November 29, 2011, from http://news.bbc.co.uk/sport2/hi/football/15866363.stm
[2] Gibson, O. (2010, June 28). World Cup 2010: Stubborn FIFA rules out using Goal-line technology. The Guardian. Retrieved November 29, 2011, from http://www.guardian.co.uk/football/2010/jun/28/world-cup-2010-goal-line-technology-fifa
[3] Jackson, J. (2011, 24 November). Goal-line technology to be in place for 2012–13 Premier League season. The Guardian. Retrieved November 28, 2011, from http://www.guardian.co.uk/football/2011/nov/24/goalline-technology-fifa-ifab-rochdale?INTCMP=SRCH
[4] Michel Platini warns against introduction of goal-line technology. (2010, October 25). Retrieved November 28, 2011, from http://www.guardian.co.uk/football/2010/oct/25/michel-platini-goalline-technology-uefa?INTCMP=SRCH
# Introduction
## What is Swift?

Swift is an open source programming language, introduced by Apple in 2014. It is used for their operating systems macOS for desktops and iOS for mobile devices, for instance iPhone or iPad. Since 2015 Swift has also been available on Linux under the Apache Licence 2.0. It is a modern language that includes multiple paradigms like object orientation, functional- and imperative programming. Apple included useful features from other languages like C#, Ruby, Python or Haskell.

## Swift on Linux
### Install

Before you can start with Swift, you first have to download and install the compiler and other components. Those can be downloaded from Swift's download page. On Linux, clang has to be installed too using following command:

### IDE for Linux

Xcode is only available on macOS. However, there are alternatives which provide similar functionality:
* JetBrain's CLion offeres a Swift extension which features code highlighting and completion and also a debugger.
* The Open-Source Editor Atom offeres a plugin that works similar to Xcode's playground where you can easily try new code and immediately see what it does.

## First example

Before we dive right into Swift's basic and advanced features, let's have a look at the world-famous "Hello World" example.
Now, let's have a first glance at what variables and constants look like.

## References

* ↑ Apache Software Foundation [online][accessed: 18.09.2017] | https://www.apache.org/licenses/LICENSE-2.0
* ↑ Swift.org [online][accessed: 18.09.2017] | https://swift.org/getting-started/
* ↑ clang [online][accessed: 18.09.2017] | https://clang.llvm.org
* ↑ JetBrain Swift Plugin [online][accessed: 18.09.2017] | https://blog.jetbrains.com/clion/2015/12/swift-plugin-for-clion/
* ↑ Atom.io Swift Playground [online][accessed: 18.09.2017] | https://atom.io/packages/swift-playground
# Sport Innovation/Arguments for the introduction of goal-line technology

Although there are some standalone arguments as to why goal-line technology should be implemented the majority are rebuttals to the arguments put forward against the introduction of goal-line technology. Below details the reasons GLT should be implemented:
The criteria developed by FIFA to test nine different systems clearly states that they must achieve a 100% accuracy in order to progress from phase to phase during the testing. Thus, inaccuracy of the technology should not be considered as a negative aspect of the technology implemented.
The thought that play will be stopped for extended periods of time or that the flow of the game will be disrupted because of the technology is incorrect. Again, the criterion used by FIFA to determine which technology to use has a strict rule concerning this. The criteria states that referees must receive an indication of whether a goal has been scored immediately and automatically confirmed within one second via the vibration and visual signal to the referees watch.
Supporters of technology in football say goal-line incidents are the only decisions that are entirely definitive and the answer can be provided to the referee within one second of the incident occurring. The ball either crossed the line or it didn’t. The technology would not replace referees, it would help them.
With the every increasing statistics regarding sports betting, there is simply just too much money involved in the sport to have incorrect decision made. An incorrect decision could also affect the outcome of the game being played. A disallowed goal could be the confidence breaker that allows a team to be beaten. A allowed non-goal could mean a losing team getting an advantage over the other.
Although there are cases both for and against the implementation of the technology, the stronger of the two and the way sport seems to moving it towards the implementation of technology.
# Fried Green Tomatoes at the Whistle Stop Cafe/Chapter 68

The January 7, 1931 issue of The Valdosta Gazette reads, "Local Man Feared Dead." The search for Frank has ended. The police have searched as far as Tennessee and Georgia, but have found no sign of Frank. Smoote tells the reporter he has searched everywhere and Frank seems to have vanished.

## Analysis

Smoote gives up his search easily. Although he is sure he knows what happened to Frank, he doesn't share his suspicions, which is most likely to help protect Idgie & Ruth from being prersecuted. With the investigation brought to an end and the readers is given hope that Idgie and Ruth are safe from prosecution. However, the question of whether Idgie was involved in Frank's disappearance is still unanswered.
← Chapter 67 · Chapter 69 →
# Plastics Molding & Manufacturing/Plastic History
## Timeline of Plastic History

1865 - Alexander Parkes created first man made plastic by dissolving cellulose nitrate in alcohol and camphor containing ether. Resulting substance was transparent and easily molded when hot, but retained hardness while cold. He named this plastic compound called Parkesine.
1871 - John Wesley Hyatt working to discover as a substitute to ivory for billiard balls (possibly upon heard of $10,000 prices set by Phelan and Collender (later called Brunswick-Balke and Collender and now today simply called Brunswick), to anybody who could produce substitutes which retains the characteristic of ivory and cheaper to made. He discovered celluloid which are cellulose nitrate combined with camphor. The materials is easy to mold when under heat / high pressure and hardens when cooled.
1898 - Leo Hendrik Baekeland invented Velox, a photographic printing paper that did not require natural light (instead, a gas light) for development. Baekeland made a fortune in when George Eastman Kodak paid $750,000 for the invention.
1907 - Leo Hendrik Baekeland discovers Bakelite which are made from phenol and formaldehyde which are the first fully-synthetic plastic.His discovery coming from finding a replacement for shellac, which at that time was made from the shells of Asian lac beetles.
1913 - Fritz Klatte discovers vinyl chloride through the addition of hydrochloric acid and acetylene. The vinyl chloride is basis for the production of poly vinyl chloride (PVC) and its polymerisation.
1922 - Hermann Staudinger coined the terms "macro molecules". His research shows that the synthetic plastics are constructed out of macromolecules in a process he called "polymerisation". This discovery is the foundation of modern polymer chemistry and foundation of plastics.
1935 - Wallace Carothers, a DuPont researches develops nylons (polymerr fibers) as a substitute for silks as the Japan (a main source of silks) is closing their trade. The fibers is elastic when pulled out.
1937 - Otto Bayer discovers polyurethane from the diisocyanate polyaddition process
1938 - Roy Plunkett accidentally discovers PTFE while searching for refrigerant coolant substitute which are non-toxic and non-flammable. The white waxy materials have resistance to corrosion, low surface friction, and high heat resistance.
1948 - Fritz Stastny discovers polystyrene
1953 - Herrman Schnell discovers poly carbonate after synthesizes it from Bisphenol A and phosgene.
# Chess Opening Theory/1. d4/1...Nf6/2. b4

The Neo-Polish Variation is an variation of the Indian Defense in which White commits to a push of a flank pawn instead of a center pawn. This move is not seen much in the top level because it does not benefit White in most ways. As, Black will try to play e6 and attack the pawn or play g6 into a King's Indian type of setup, it is recommended that White plays very carefully and tries to play moves like Bb2 and a3, just like if they had played b4 on the first move.
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# Chess Opening Theory/1. d4/1...f5/2. h4

With this unusual but playable move, White aims to push their h-pawn all the way to h6 and weaken Black's kingside.

## Statistics

White win 22%, Draw 56%, Black win 22%
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# Chess Opening Theory/1. e4/1...d5/2. exd5/2...Qxd5/3. Nc3/3...Qa5/4. b4/4...Qxb4/5. Nb5/5...Qa5/6. Bc4/6...c6/7. Bxf7/7...Kxf7/8. Qh5

8. Qh5+
Black has been greedy with 6...c6 and will pay dearly for it. White will play 9. Nd6+ or 9. Nd4+, which is a discovered attack on Black's queen.
8...Kf6 keeps the queen, but after 9. Bb2+ Black might as well be down one, since White has a monstrous attack.
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# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bc4/3...Bc5/4. d4/4...Bxd4

This capture is the best way for Black to proceed without entering the Haxo Gambit. White has nothing better to do than 5. Nxd4 to open lines for their pieces.
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# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bc4/3...Bc5/4. d4/4...Bxd4/5. Nxd4

White captures the bishop to open up the d1-h4 diagonal for their queen. Black usually responds with 5...Nxd4 but 5...exd4 is a sound alternative.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5
## 2. d5

White attacks the knight. While 2...Ne5 is the most common reply, 2...Nb8 is slightly preferred by Stockfish on depth 55, even though it puts Black back into starting position.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5/2...Ne5
## 2...Ne5

This move provokes White to play 3. f4, attacking the knight, but 3. e4 is just as good and often transposes into the same line.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5/2...Ne5/3. f4
## 3. f4

Black has to retreat the knight by playing 3...Ng6.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5/2...Ne5/3. f4/3...Ng6
## 3...Ng6

White should take control of the center with 4. e4 or 4. Nf3.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5/2...Ne5/3. f4/3...Ng6/4. e4
## 4. e4

White has taken the center. Black should play 4...e5 or 4...e6.
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# Chess Opening Theory/1. d4/1...Nc6/2. d5/2...Ne5/3. f4/3...Ng6/4. e4/4...e5
## 4...e5!

Black undermines White's center. White needs to take en passant with 5. dxe6, otherwise Black's queen infiltrates on h4 with check.
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# Chess Opening Theory/1. d4/1...f5/2. h4/2...Nf6/3. h5/3...h6
### 3...h6

This move is a mistake. The white squares on the black kingside are weakened. White gets a good square g6 for his knight.
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# Chess Opening Theory/Organization
## Conventions for organization

Conventions for organization:
* Moves are organized in a directory-like structure.
* Moves from a given position are referenced as relative links from the current position. Thus, from the opening position at Chess Opening Theory, the position reached after 1. e4 is addressed as /1. e4. The position reached after 1. e4 e5 is addressed from the page at Chess Opening Theory as /1. e4/1...e5.
* White moves are referenced as move number, period, space, move (in short algebraic). The space prevents an unbroken string of characters in the page title becoming too wide for the page.
* Black moves are referenced as move number, 3 periods, move (in short algebraic). No spaces are used.
* Indications of check (+) or checkmate (#) are not included in the reference link because of Wikibooks restrictions on page titles, but should be indicated in the displayed text.
* Theory tables are optional, but very useful. The best way to include these is by copying the table from another position.
* Board diagrams should be included on each page – again, best copied from the previous position.
* If a Wikipedia entry exists for the position, reference the Wikipedia article and vice-versa.
* Do not copy analysis from copyrighted works. The moves themselves are not subject to copyright, but the analysis is.
* If you find a transposition (common) use a redirect to an existing position. An example of this is the transposition of the Blackmar-Diemer Gambit to the French Defence.

### Notation

* Castling is O-O as in PGN. Letter O, not number zero.
* There are no ! and ?s in the page names, nor indications of check/checkmate, nor e.p. though these may be included in the main text and hyperlinks.
* Position evaluations are optional. Stick with the Informator symbols: + − (Decisive advantage for White), +/− or ± (Clear plus for White), +/= or ⩲ (Slight plus for White), = (Equal), =/+ or ⩱ (Slight plus for Black), −/+ or ∓ (Clear plus for Black), − + (Decisive advantage for Black). Also: ∞ (Unclear), =/∞ (Compensation, whoever is down in material has compensation for it.).

### Style

* The words "white" and "black" are capitalized only when they are used as substitutes for player names, e.g. "White's rook" but "the black queen".
* Write "e4-square" and "f-file" instead of "e4 square" and "f file".
* Write "Defence" instead of "Defense".