OpenLLM-France/wikimedia huggingface dataset:
# Logic for Computer Scientists/Predicate Logic/Syntax
## Definition 1 (Syntax of predicate logic - Terms)

Assume a
* countable set of function symbols $\{f{_{i}}^{k}\mid i,k=1,2,3,\cdots \}$
* a countable set of $\{x_{i}\mid i=1,2,3,\cdots \}$
The set of terms is defined by the following induction:
* Variables $x_{i}$ are terms.
* If $t_{1},\cdots ,t_{k}$ are terms and $f_{i}^{k}$ is a function symbol, then $f_{i}^{k}(t_{1},\cdots ,t_{k})$ is a term.
Terms of type $f_{i}^{0}$ are special ones, they are called constants. In this case we omit the braces and denote them as $f_{i}^{0}$.
Terms are the syntactic counterpart of the above mentioned objects. Constants will denote the elements of the domain and function symbols will denote a way to refer to such objects.
The following definition introduces the formulae.

## Definition 2 (Syntax of predicate logic - Formulae)

Assume a countable set of predicate symbols$\{p_{i}^{k}\mid i=1,2,3,\cdots \}$. The set of (well-formed) formulae is defined by the following induction:
* If $t_{1},\cdots ,t_{k}$ are terms and $P_{i}^{k}$ is a predicate symbol, then $P_{i}^{k}(t_{1},\cdots ,t_{k})$ is a formula.
* If $F$ and $G$ are formulae, then $(F\land G)$ and $(F\lor G)$ are formulae.
* If $F$ is a formula, then $\lnot F$ is a formula.
* If $x$ is a variable and $F$ a formula, then $\forall xF$ and $\exists xF$ are formulae.
Formulae of type $P_{i}^{k}(t_{1},\cdots ,t_{k})$ are called atoms or atomic formulae.
Note that the concept of subformulae applies exactly like in the propositional case (Syntax (Propositional Logic)).
We introduce the following abbreviations, which will be used with indices as well:
| $u,v,w,x,z,$ | for variables |
| $a,b,c,\cdots$ | for constants |
| $f,g,h,\cdots$ | for function symbols |
| $p,q,r,\cdots$ | for predicate symbols |
Note the arity of function and predicate symbols is ommited in these abbreviations; we assume that it will be obvious from the context.
Example: Assume we want to represent the following equation, which holds for arbitrary elements in a field:
$x*(y+z)=x*y+x*z$
The two operators $*$ and $+$ are represented in a predicate logic formula as binary function symbols $f_{1}^{2}$ and $f_{2}^{2}$, the three variables are $x_{1},x_{2}$ and $x_{3}$, and the equality relation $=$ is the binary predicate symbol $P_{1}^{2}$. Altogether we have the following formula in predicate logic:
$\forall x_{1}\forall x_{2}\forall x_{3}(P_{1}^{2}(f_{2}^{2}(x_{1},f_{1}^{2}(x_{2},x_{3})),f_{1}^{2}(f_{2}^{2}(x_{1},x_{2}),f_{2}^{2}(x_{1},x_{3}))))$
In the following we will use the obvious and more liberal notation as in the propositional case.

## Definition 3: Bound and Free Occurrences of Variables

An occurrence of a variable $x$ in a formula $F$ is called bound, if it occurs in a subformula of $F$ which is of the form $\exists x\;G$ or $\forall x\;G$. Otherwise we call the occurrence free.
A formula, which does not contain a free occurrence of a variable is called closed.
Example: The following formula contains both free and bound occurrences of $x$ and $y$.
$\forall z(Q(z)\land \forall x(P(x,y))\lor \exists y(P(x,y)))$
# Logic for Computer Scientists/Propositional Logic
## Propositional Logic

This section introduces propositional logic. We will study syntax and model theoretic semantic of a language of classical propositional logic and we investigate various calculi for deciding certain properties of sentences in this language.
4.1 Preliminaries 4.2 Syntax 4.3 Semantics 4.4 Equivalence and Normal Forms 4.5 Horn clauses 4.6 Resolution 4.7 Analytic Tableaux
# Introduction to Mathematical Physics/N body problems and statistical equilibrium/Exercises

Exercice:
Paramagnetism. Consider a system constituted by $N$ atoms located at nodes of a lattice. Let $J_{i}$ be the total kinetic moment of atom number $i$ in its fundamental state. It is known that to such a kinetic moment is associated a magnetic moment given by:
$\mu _{i}=-g\mu _{B}J_{i}$
where $\mu _{B}$ is the Bohr magneton and $g$ is the Land\'e factor. $J_{i}$ can have only semi integer values.
Assume that the hamiltonian describing the system of $N$ atoms is:
$H=\sum -\mu _{i}B_{0}$
where $B_{0}$ is the external magnetic field. What sort of particles are the atoms in this system, discernables or undiscernables? Find the partition function of the system.
Exercice:
Study the Ising model at two dimension. Is it possible to envisage a direct method to calculate $Z$ ? Write a programm allowing to visualize the evolution of the spins with time, temperature being a parameter.
Exercice:
Consider a gas of independent fermions. Calculate the mean occupation number ${\bar {N}}_{\lambda }$ of a state $\lambda$. The law you'll obtained is called Fermi distribution.
Exercice:
Consider a gas of independent bosons. Calculate the mean occupation number ${\bar {N}}_{\lambda }$ of a state $\lambda$. The law you'll obtained is called Bose distribution.
Exercice:
Consider a semi--conductor metal. Free electrons of the metal are modelized by a gas of independent fermions. The states are assumed to be described by a sate density $\rho (\epsilon )$, $\epsilon$ being the nergy of a state. Give the expression of $\rho (epsilon)$. Find the expression binding electron number ${\bar {N}}$ to chemical potential. Give the expression of the potential when temperature is zero.
# Introduction to Mathematical Physics/N body problems and statistical equilibrium/N body problems

At chapter chapproncorps, $N$ body problem was treated in the quantum mechanics framework. In this chapter, the same problem is tackled using statistical physics. The number $N$ of body in interaction is assumed here to be large, of the order of Avogadro number. Such types of N body problems can be classified as follows:
* Particles are undiscernable. System is then typically a gas. In a classical approach, partition function has to be described using a corrective factor ${\frac {1}{N!}}$ (see section secdistclassi). Partition function can be factorized in two cases: particles are independent (one speaks about perfect gases)\index{perfect gas} Interactions are taken into account, but in the frame of a mean field approximation\index{mean field}. This allows to considerer particles as if they were independent (see van der Waals model at section secvanderwaals) In a quantum mechanics approach, Pauli principle can be included in the most natural way. The suitable description is the grand-canonical description: number of particles is supposed to fluctuate around a mean value. The Lagrange multiplier associated to the particles number variable is the chemical potential $\mu$. Several physical systems can be described by quantum perfect gases (that is a gas where interactions between particles are neglected): a fermions gas can modelize a semi--conductor. A boson gas can modelize helium and described its properties at low temperature. If bosons are photons (their chemical potential is zero), the black body radiation can be described.
* Particles are discernable. This is typically the case of particles on a lattice. Such systems are used to describe for instance magnetic properties of solids. Taking into account the interactions between particles, phase transitions like paramagnetic--ferromagnetic transition can be described . Adsorption phenomenom can be modelized by a set of independent particles in equilibrium with a particles reservoir (grand-canonical description). Those models are described in detail in [ph:physt:Diu89]. In this chapter, we recall the most important properties of some of them.
# Introduction to Mathematical Physics/N body problems and statistical equilibrium/Thermodynamical perfect gas

In this section, a perfect gas model is presented: all the particles are independent, without any interaction.
Remark:
A perfect gas corresponds to the case where the kinetic energy of the particles is large with respect to the typical interaction energy. Nevertheless, collisions between particles that can be neglected are necessary for the thermodynamical equilibrium to establish.
Classical approximation (see section secdistclassi) allows to replace the sum over the quantum states by an integral of the exponential of the classical hamiltonian $H(q_{i},p_{i})$. The price to pay is just to take into account a proportionality factor ${\frac {1}{2\pi \hbar }}$. Partition function $z$ associated to one particle is:
$z={\frac {1}{2\pi \hbar }}\int dq_{i}dp_{i}e^{-\beta H(p,q)}$
$z={\frac {1}{2\pi \hbar }}\int dq^{3}\int dp^{3}e^{-\beta H(p,q)}$
Partition function $z$ is thus proportional to $V$ :
$z=A(\beta )V$
Because particles are independent, partition function $Z$ for the whole system can be written as:
$Z={\frac {1}{N!}}z^{N}$
It is known that pressure (proportional to the Lagrange multiplier associated to the internal variable "volume") is related to the natural logarithm of $Z$; more precisely if one sets:
$F=-k_{B}T\log Z$
then
$p=-{\frac {\partial F}{\partial V}}.$
This last equation and the expression of $Z$ leads to the famous perfect gas state equation:
$pV=Nk_{B}T$
# Introduction to Mathematical Physics/N body problem in quantum mechanics/Introduction

In this chapter, $N$ body problems that quantum mechanics can treat are presented. In atoms, elements in interaction are the nucleus and the electrons. In molecules, several nuclei and electrons are in interaction. Cristals are characterized by a periodical arrangment of their atoms. In this chapter, only spectral properties of hamiltonians are presented. Thermodynamical properties of collections of atoms and molecules are presented at next chapter. From a mathematical point of view, this chapter is an application of the spectral method presented at section chapmethspec to study linear evolution problems.
# Introduction to Mathematical Physics/N body problems and statistical equilibrium/Spin glasses

Assume that a spin glass system \index{spin glass}(see section{secglassyspin}) has the energy:
$H=\sum J_{ij}S_{i}S_{j}$
Values of variable $S_{i}$ are $+1$ if the spin is up or $-1$ if the spin is down. Coefficient $J_{ij}$ is $+1$ if spins $i$ and $j$ tend to be oriented in the same direction or $-1$ if spins $i$ and $j$ tend to be oriented in opposite directions (according to the random position of the atoms carrying the spins). Energy is noted:
$H_{J}=\sum J_{ij}S_{i}S_{j}$
where $J$ in $H_{J}$ denotes the $J_{ij}$ distribution. Partitions function is:
$Z_{J}=\sum _{[s]}e^{-\beta H_{J}[s]}$
where $[s]$ is a spin configuration. We look for the mean ${\bar {f}}$ over $J_{ij}$ distributions of the energy:
${\bar {f}}=\sum _{J}P[J]f_{J}$
where $P[j]$ is the probability density function of configurations $[J]$, and where $f_{J}$ is:
$f_{J}=-\ln Z_{J}.$
This way to calculate means is not usual in statistical physics. Mean is done on the "chilled" $J$ variables, that is that they vary slowly with respect to the $S_{i}$'s. A more classical mean would consist to $\sum _{J}P[J]\sum _{[s]}e^{-\beta H_{J}[s]}$ (the $J$'s are then "annealed" variables). Consider a system $S_{j}^{n}$ compound by
$n$ replicas\index{replica} of the same system $S_{J}$. Its partition function $Z_{J}^{n}$ is simply:
$Z_{J}^{n}=(Z_{J})^{n}$
Let $f_{n}$ be the mean over $J$ defined by:
$f_{n}=-{\frac {1}{n}}\ln \sum _{J}P[J](Z_{J})^{n}$
As:
$\ln Z=\lim _{n\rightarrow 0}{\frac {Z^{n}-1}{n}}$
we have:
$\lim _{n\rightarrow 0}f_{n}=\lim _{n\rightarrow 0}\ln(\sum _{J}P[J][1+n\ln Z_{J}])$
Using $\sum _{J}P[J]=1$ and $\ln(1+x)=x+O(x)$ one has:
${\bar {f}}=\lim _{n\rightarrow 0}f_{n}.$
By using this trick we have replaced a mean over $\ln Z$ by a mean over $Z^{n}$; price to pay is an analytic prolongation in zero. Calculations are then greatly simplified [ph:sping:Mezard87].
Calculation of the equilibrium state of a frustrated system can be made by simulated annealing method .\index{simulated annealing} An numerical implementation can be done using the Metropolis algorithm\index{Metropolis}. This method can be applied to the travelling salesman problem (see [ma:compu:Press92] \index{travelling salesman problem}).
# Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Electron donors and acceptors

Proton donors and acceptors >> | Electrophiles and nucleophiles >>
The Lewis definition of acids and bases describes an acid as an electron acceptor and a base as an electron donator.
<< Foundational concepts | << Acids and bases | Proton donors and acceptors >> | Electron donors and acceptors | Electrophiles and nucleophiles >> | Alkanes >>
# Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Nucleus and electrons

<< Atomic structure | Shells and orbitals >>
* A simple model of a lithium atom. Not to scale!
Atoms are made up of a nucleus and electrons that orbit the nucleus. An atom in its natural, uncharged state has the same number of electrons as protons. If it gains or loses electrons, the atom is then referred to as an ion.

### The nucleus

The nucleus is made up of protons, which each have a positive charge, and neutrons, which have no charge. Neutrons and protons have about the same mass, and together account for most of the mass of the atom. Each of these particles is made up of even smaller particles, though the existence of these particles do not come into play at the energies and time spans in which most chemical reactions occur.

### Electrons

The electrons are negatively charged and fly around the nucleus of an atom at something like light speed. We cannot determine exactly where electrons are at any point in time, rather, we can only guess at the probability of finding an electron at a point in space relative to a nucleus at any point in time. The image depicts the Bohr model of the atom, in which the electrons inhabit discrete "orbitals" around the nucleus much like planets orbit the sun. Current models of the atomic structure hold that electrons occupy fuzzy clouds around the nucleus of specific shapes, some spherical, some dumbbell shaped, some with even more complex shapes. Even though the simpler Bohr model of atomic structure has been superseded, we still refer to these electron clouds as "orbitals". The number of electrons and the nature of the orbitals they occupy greatly influence the reactivity of atoms in organic chemistry.
<< Atomic structure | Nucleus and electrons | Shells and orbitals >>
# Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Proton donors and acceptors

<< Acids and bases | Electron donors and acceptors >>
The first and earliest definition of acids and bases was proposed in the 1800s by Swedish scientist Svante Arrhenius, who said that an acid was anything that dissolved in water to give up H+ ions. Such as hydrochloric acid and a base was anything that dissolved in water to give up OH- ions such a sodium hydroxide.
The Brønsted-Lowry definition of 1923 broadened this idea a bit:
    an acid is any proton donor, and a base is any proton acceptor.
The focus of this definition is on donating and accepting protons, and is not limited to aqueous solution.
The Brønsted-Lowry definition of acids and bases is one of two definitions we commonly use. The second definition deals not with protons but with electrons, and has a slightly different emphasis.
<< Foundational concepts | << Resonance | << Acids and bases | Proton donors and acceptors | Alkanes >>
# Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Hybridization

Hybridization refers to the combining of the orbitals of two or more covalently bonded atoms. Depending on how many free electrons a given atom has and how many bonds it is forming, it will hybridize in a certain manner. The electrons in the s and the p orbitals combine in certain manners to form the bonds. In particular, for most of the atoms of interest in organic chemistry (which form at most four bonds), the more free electrons an atom has, the less "p" character the hybridized bond will have, because the free electrons will take up some space of the p orbitals.
It is easy to determine the hybridization of an atom given a Lewis structure. First, you count the number of pairs of free electrons and the number of sigma bonds (single bonds). Do not count double bonds, since they do not affect the hybridization of the atom. Once the total of these two is determined, the hybridization pattern is as follows:
The pattern here is the same as that for the electron orbitals, which serves as a memory guide.
# Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/PKa and acidity

<< Electrophiles and nucleophiles | Alkanes >>
To measure acid strength of a compound, scientist typically use a quantity called pKa. It is defined as
$pKa=-log(Ka)$
where Ka is the acid dissociation constant. Stronger acids have small pKa values, where as weaker acids have higher pKa values.
<< Foundational concepts | << Acids and bases | Electrophiles and nucleophiles >> | pKa and acidity | Alkanes >>
# Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Molecular orbitals

In organic chemistry we look at the hybridization of electron orbitals in to something called molecular orbitals.
The s and p orbitals in a carbon atom combine into four hybridized orbitals that repel each other in a shape much like that of four balloons tied together.
Remember what a tetrahedron looks like ?
I knew you did ...
# Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Electrophiles and nucleophiles

<< Electron donors and acceptors | pKa and acidity >>
Electrophiles are "electron-lovers". (The suffix -phile means "lover of", as bibliophile means "lover of books"). Electrophiles seek electrons.
Nucleophiles, or "nucleus lovers", seek positively charged nuclei.
Electrophiles and nucleophiles are often ions, but sometimes not.
<< Foundational concepts | << Acids and bases | Electron donors and acceptors >> | Electrophiles and nucleophiles | pKa and acidity >> | Alkanes >>
# Mathematics Worksheet/Algebra

This section contains worksheets for algebra problems.
* Addition and subtraction of algebraic terms (Jun 26, 2009)
* Lowest Common Multiples of Algebra (Jun 26, 2009)
* Greatest Common Factor of Algebra (Jun 26, 2009)
* Algebraic exponents (Jun 27, 2009)
* Fractional algebra (Aug 07, 2009)
* Substituting (Jun 24, 2009)
* Factorization of algebraic terms (Jul 25, 2009)
* Pascal's Triangle (Jul 25, 2009)
* Polynomials (Jun 24, 2009)
# Mathematics Worksheet

The mathematics worksheet is a set of problems designed to allow practicing material that you have just learned. Each section in the worksheet will contain a brief description identifying the work to be done (although not necessarily how to do it), and a set of approximately 30 math problems to aid with your study.

## Table of Contents

* Addition and Subtraction of Integers (Aug 02, 2009)
* Multiplication and Division of Integers (Aug 12, 2009)
* Order of operations (Jun 26, 2009)
* Converting (Jun 21, 2009)
* Addition and Subtraction of Decimal Numbers (Jul 05, 2009)
* Multiplication and Division of Decimal Numbers (Jul 09, 2009)
* Greatest Commons Factors (Jul 05, 2009)
* Lowest Common Multiples (Jun 24, 2009)
* Ratio and Proportion (Jun 21, 2009)
* Algebra (Jul 25, 2009)
* Social Arithmetic (Jun 21, 2009)
* Area and Perimeter of 2D Shapes (Jul 18, 2009)
* Volume and Surface Area of 3D Shapes (Jul 04, 2009)
* Linear Equation One Variable (Jun 26, 2009)
* Inequalities with One Variable (Jun 21, 2009)
* Sets (Jun 21, 2009)
* Venn diagram (Jun 21, 2009)
* Line and angles (Jun 21, 2009)
* Equations of Straight Line (Jul 02, 2009)
* Relation and function (Jun 21, 2009)
* Linear Equation Two Variable (Jun 21, 2009)
* Pythagorean Theorem (Jun 21, 2009)
* Circle (Jul 02, 2009)
# Mathematics Worksheet/Addition and Subtraction of Integers

Follow the examples below when doing this worksheet:
* 22 + 42 = 64
* 42 + (-21) = 42 - 21 = 21
* 37 - 18 = 19
* 55 - (-21) = 55 + 21 = 76
* 31 + 5 = 36
* 36 - 5 = 31
* 36 - 55 = -19
# Mathematics Worksheet/Algebra/Factorization of algebraic terms

Factorize the following algebraic expressions below; Use ^ to write exponentiation! (9 questions)
# Pokémon/Pokédex/Agility

Agility is a Psychic-type no-damage move that greatly increases speed. This move is available in all versions.

## Type Matching

This move doesn't affect the opponent.

## Move effects

Base Damage: 0 Accuracy: N/A Special Effects: Template:DoubleRaiseSpeed

## Pokemon who learn the move

| Pokemon | Level | Version |
| Beedrill | 35 | Red, Blue, Yellow |
| Beedrill | 40 | Gold, Silver, Crystal, Ruby, Sapphire, Fire Red, Leaf Green |
| Dodrio | 47 | Gold, Silver, Crystal |
| Dodrio | 51 | Red, Blue, Yellow |
| Dodrio | 60 | Ruby, Sapphire, Fire Red, Leaf Green |
| Doduo | 37 | Gold, Silver, Crystal |
| Doduo | 44 | Red, Blue, Yellow |
| Doduo | 45 | Ruby, Sapphire, Fire Red, Leaf Green |
| Farfetch'd | 31 | Red, Blue, Yellow, Gold, Silver, Crystal |
| Farfetch'd | 36 | Ruby, Sapphire, Fire Red, Leaf Green |
| Fearow | 43 | Red, Blue, Yellow |
| Fearow | 47 | Gold, Silver, Crystal, Ruby, Sapphire, Fire Red, Leaf Green |
| Growlithe | 39 | Red, Blue, Yellow |
| Growlithe | 42 | Gold, Silver, Crystal |
| Growlithe | 43 | Ruby, Sapphire, Fire Red, Leaf Green |
| Hitmonchan | start | Red, Blue, Yellow |
| Pidgeot | 44 | Red, Blue, Yellow |
| Pidgeot | 46 | Gold, Silver, Crystal |
| Pidgeot | 48 | Ruby, Sapphire, Fire Red, Leaf Green |
| Pidgeotto | 40 | Red, Blue, Yellow |
| Pidgeotto | 43 | Gold, Silver, Crystal, Ruby, Sapphire, Fire Red, Leaf Green |
| Pidgey | 36 | Red, Blue, Yellow |
| Pidgey | 37 | Gold, Silver, Crystal |
| Pidgey | 39 | Ruby, Sapphire, Fire Red, Leaf Green |
| Pikachu | 33 | All |
| Ponyta | 38 | Ruby, Sapphire, Fire Red, Leaf Green |
| Ponyta | 43 | Gold, Silver, Crystal |
| Ponyta | 48 | Red, Blue, Yellow |
| Rapidash | 47 | Gold, Silver, Crystal |
| Rapidash | 55 | Red, Blue, Yellow |
| Spearow | 36 | Red, Blue, Yellow |
| Spearow | 43 | Gold, Silver, Crystal, Ruby, Sapphire, Fire Red, Leaf Green |

## Pokemon Contests

Appeal type: Tough Appeal level: 1 Jam: 0
# Pokémon/Pokédex/Acid Armor

Acid Armor is a Poison-type no-damage move. This move is available in all versions.

## Type Matching

None - this move doesn't affect the opponent.

## Move effects

Base Damage: 0
Accuracy: N/A
Special Effects: Template:DoubleRaiseDefense

## Pokemon who learn the move

| Pokemon | Level | Version |
| Grimer | 34 | Ruby, Sapphire, Emerald, Fire Red, Leaf Green |
| Grimer | 40 | Gold, Silver, Crystal |
| Grimer | 55 | Red, Blue, Yellow |
| Muk | 45 | Gold, Silver, Crystal |
| Muk | 60 | Red, Blue, Yellow |
| Vaporeon | 42 | Yellow, Gold, Silver, Crystal, Ruby, Sapphire, Emerald, Fire Red, Leaf Green |
| Vaporeon | 47 | Red, Blue |

## Egg Move

| Gulpin | Ruby, Sapphire, Fire Red, Leaf Green |
| Slugma | Gold, Silver, Crystal, Ruby, Sapphire, Fire Red, Leaf Green |
| Swalot | Ruby, Sapphire, Fire Red, Leaf Green |

## Pokemon Contests

Appeal type: Tough
Appeal level: 1
Jam: 0
# Pokémon/Pokédex/Aeroblast

Aeroblast is a Flying-type move. This move is available in all versions except Red/Blue/Yellow.

## Type Matching

Aeroblast is good on: Bug, Fighting, Grass Aeroblast is bad on: Rock, Electric, Steel

## Move effects

Base Damage: 100 Accuracy: 94.9% Special Effects: Template:HighCritical

## Pokemon who learn the move

| Pokemon | Level | Version |
| Lugia | ??? | Gold, Silver, Crystal |
| Lugia | 77 | Ruby, Sapphire, Fire Red, Leaf Green |

## Pokemon Contests

Appeal type: Cool Appeal level: 3 Jam: 0
# Pokémon/Pokédex/Alomomola
## Description

A fishlike Pokémon.

## Base Statistics

A Water type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Alomomola: grass, electric Moves that deal 50% of their damage to Alomomola: fire, water, ice, steel