OpenLLM-France/wikimedia huggingface dataset:
# High School Mathematics Extensions/Peano's Axioms

The Peano Axioms, also called Peano's Axioms, Dedekind-Peano Axioms, or the Peano postulates, are a series of axioms attempting to formalize the natural numbers ($\mathbb {N}$) using simple, intuitive axioms.

## Introduction

During the time of Peano, mathematical notation as a whole—especially mathematical logic—was a new field, and common notions of second-order logic did not exist. Peano's own notation for logic was unpopular, but he did popularize something similar to the common way of denoting elements of a set ($A\in B$), using $\epsilon$. While we will not be using Peano's old notation, we will be expressing the axioms in a common notation which you should be familiar with from the set theory chapters of this book.
The Peano axioms describe the properties of the natural set, commonly denoted N or in blackboard bold, $\mathbb {N}$. The first axiom states that 0 is a natural number.
    1. $0\in \mathbb {N}$. ("zero is a natural number")
The next four axioms describe the most important properties of the natural set, the equivalence relation.
    2. $\forall x\in \mathbb {N} :x=x$. ("all natural numbers equal themselves")
    3. $\forall x,y,z\in \mathbb {N} :x=y,y=z\implies x=z$. ("equality is transitive")
    4. $\forall x,y\in \mathbb {N} :x=y\implies y=x$. ("equality is symmetric")
    5. $\forall x\in \mathbb {N} :x=y\implies y\in \mathbb {N}$. ("if x is natural, and x equals y, y is natural")
After this, Peano introduces the idea of a successor function, commonly denoted $\mathbf {S}$, ${\widehat {S}}$, or $\mathrm {succ}$.
    6. $\forall x\in \mathbb {N} :\mathrm {succ} (x)\in \mathbb {N}$. ("the successor of any natural number is a natural number")
    7. $\forall x,y\in \mathbb {N} :\mathrm {succ} (x)=\mathrm {succ} (y)\iff x=y$. ("the successor function is an injection")
    8. $\mathrm {succ} (x)=0\implies \bot$. ("there is no x such that the successor of x is 0")
The obvious intuition is that we can compose the entire natural set by successively applying the successor function, but none of the axioms actually present this. Thus, there is motivation to also add the axiom of induction, which is commonly misattributed as part of Peano's axioms, but it was not described in Peano's original construction.
We can simply say that the natural set is constructible by successively applying the successor.
    9. The natural set, $\mathbb {N}$ is constructible by successively applying the successor function.
# High School Mathematics Extensions/Matrices/Problem Set/Solutions

* HSME
* Content
* Matrices
* Recurrence Relations
* Problems & Projects
* Problem Set
* Project
* Soultions
* Exercises Solutions
* Problem Set Solutions
* Misc.
* Definition Sheet
* Full Version

## Matrices Problem Set

At the moment, the main focus is on authoring the main content of each chapter. Therefore this exercise solutions section may be out of date and appear disorganised.
If you have a question please leave a comment in the "discussion section" or contact the author or any of the major contributors.
1.
    ${\begin{pmatrix}2&3\\3&5\end{pmatrix}}{\begin{pmatrix}?&?&?&?\\?&?&?&?\end{pmatrix}}={\begin{pmatrix}28&94&70&102\\44&153&112&163\end{pmatrix}}$
    ${\begin{pmatrix}2&3\\3&5\end{pmatrix}}^{-1}{\begin{pmatrix}2&3\\3&5\end{pmatrix}}{\begin{pmatrix}?&?&?&?\\?&?&?&?\end{pmatrix}}={\begin{pmatrix}2&3\\3&5\end{pmatrix}}^{-1}{\begin{pmatrix}28&94&70&102\\44&153&112&163\end{pmatrix}}$
    ${\begin{pmatrix}?&?&?&?\\?&?&?&?\end{pmatrix}}={\begin{pmatrix}2&3\\3&5\end{pmatrix}}^{-1}{\begin{pmatrix}28&94&70&102\\44&153&112&163\end{pmatrix}}$
    $={\frac {1}{2\times 5-3\times 3}}{\begin{pmatrix}5&-3\\-3&2\end{pmatrix}}{\begin{pmatrix}28&94&70&102\\44&153&112&163\end{pmatrix}}$
    $={\begin{pmatrix}(5\times 28+(-3)\times 44)&(5\times 94+(-3)\times 153)&(5\times 70+(-3)\times 112)&(5\times 102+(-3)\times 163)\\((-3)\times 28+2\times 44)&((-3)\times 94+2\times 153)&((-3)\times 70+2\times 112)&((-3)\times 102+2\times 163)\end{pmatrix}}$
    $={\begin{pmatrix}8&11&14&21\\4&24&14&20\end{pmatrix}}$
    Therefore the message is "iloveyou"
2.
    Combine the two matrices together, we have
    $A{\begin{pmatrix}1&3\\2&4\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}$
    $A{\begin{pmatrix}1&3\\2&4\end{pmatrix}}=I$
    Therefore the inverse of A is
    ${\begin{pmatrix}1&3\\2&4\end{pmatrix}}$
# High School Mathematics Extensions/Matrices/Project/Elementary Matrices

* HSME
* Content
* Matrices
* Recurrence Relations
* Problems & Projects
* Problem Set
* Project
* Soultions
* Exercises Solutions
* Problem Set Solutions
* Misc.
* Definition Sheet
* Full Version

## Project -- Elementary matrices

Throughout, $A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$
1. The matrices below are called elementary matrices. How are the matrices below different from the identity matrix I, describe each one.
* ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$
* ${\begin{pmatrix}1&f\\0&1\end{pmatrix}}$ where f is a scalar
* ${\begin{pmatrix}1&0\\f&1\end{pmatrix}}$ where f is a scalar
* ${\begin{pmatrix}f&0\\0&1\end{pmatrix}}$ where f is a scalar
* ${\begin{pmatrix}1&0\\0&f\end{pmatrix}}$ where f is a scalar
2. In each of the cases, compute B then describe how is B different from A
* $B={\begin{pmatrix}0&1\\1&0\end{pmatrix}}A$
* $B={\begin{pmatrix}1&f\\0&1\end{pmatrix}}A$ where f is a scalar
* $B={\begin{pmatrix}1&0\\f&1\end{pmatrix}}A$ where f is a scalar
* $B={\begin{pmatrix}f&0\\0&1\end{pmatrix}}A$ where f is a scalar
* $B={\begin{pmatrix}1&0\\0&f\end{pmatrix}}A$ where f is a scalar
3. The matrix ${\begin{pmatrix}1&2\\4&3\end{pmatrix}}$ has determinant not equal to zero. We can decompose the matrix into products of elementary matrices pre-multiplying the identity:
    ${\begin{pmatrix}1&2\\4&3\end{pmatrix}}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1&3\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\1&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&5\end{pmatrix}}{\begin{pmatrix}1&-3\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}$
Now suppose det(A) ≠ 0, can A be expressed as the product of elementary matrices and the identity?
4. a) Show that every elementary matrix has an inverse. Hint: use determinant.
b) Prove that every invertible matrix (a matrix that has an inverse) is the product of some elementary matrices pre-multiplying the identity.
5. A transpose of a matrix C is the matrix C$^T$ where the ith row of C is the ith column of C$^T$. Prove using elementary matrices that
    $(DE)^{T}=E^{T}D^{T}$
for arbitrary matrices D and E.
6. Show that every invertible matrix is also the product of some elementary matrices post-multiplying the identity.
7. How about non-invertible matrices? What can you say about them?
# High School Mathematics Extensions/Primes/Definition Sheet

* HSME
* Content
* Primes
* Modular Arithmetic
* Problems & Projects
* Problem Set
* Project
* Solutions
* Exercise Solutions
* Problem Set Solutions
* Misc.
* Definition Sheet
* Full Version
* PDF Version
About the definition sheet:
The definitions provided below may differ a little from the ones given in the chapter. If possible, print out the following definitions for easy reference.

### Definitions

Composite
    A composite is a whole number that is not a prime. The number 1 is not composite.
Coprimes
    Two numbers are coprimes if their greatest common divisor (gcd) equals 1.
Diophantine Equation (linear)
    An equation of the form ax + by = c. Where a, b and c are integer constants, and x and y are unknown integers.
Factorisation
    Alternatively spelt factorization. A process in which the prime factors of a natural number are found and the number expressed as the product of the individual factors.
gcd (greatest common divisor)
    The gcd of a and b is a number d, such that d divides a and d divides b; and that if e divides a and b, then e ≤ d.
Inverse
    In modular m arithmetic, the inverse of a is the number b such that
       $ab\equiv 1{\pmod {m}}\!$
    the inverse is unique. Not every number in every arithmetic have an inverse.
Modular arithmetic
    The arithmetic modulo m is the arithmetic where each number is represent by a number lying between 0 and m - 1. E.g. consider modulo 7 arithmetic, 11 is represented by 4; and -2 is represented by 5. We say 11 is equivalent to 4 mod 7; and -2 is equivalent to 5 mod 7. It is explained in more detail here.
Prime
    A prime number (or prime for short) is a whole number that can only be wholly divided by two different numbers, 1 and itself. The number 1 is thus not considered prime. We do not consider the negative numbers in this chapter.

### Theorems

Chinese Remainder Theoren
In a system of n congruencies
    $x\equiv a_{1}{\pmod {m_{1}}}\,\!$
    $x\equiv a_{2}{\pmod {m_{2}}}\,\!$
    $x\equiv a_{n}{\pmod {m_{n}}}\,\!$
, a solution exists if and only if for i and j with i ≠ j
    gcd(m$_i$,m$_j$) divides (a$_i$ - a$_j$)
Existence of inverse
    In modular m arithmetic, a has an inverse if and only if gcd(a,m) = 1.
Fundamental Theorem of Arithmetic
    Any integer (except for 1) can be expressed as the product of primes in one and only one way.
Infinitely many primes
    There are infinitely many primes.
# IB/Group 4/Computer Science/Databases
## Table of Contents

Common Core (SL & HL)
A.1 Basic concepts
A.2 The relational database model
A.3 Further aspects of database management
HL Extension
A.4 Further database models and database analysis
# IB/Group 4/Computer Science/Control
## Centralised control systems

7.1.1 Discuss a range of control systems.
7.1.2 Outline the uses of microprocessors and sensor input in control systems.
7.1.3 Evaluate different input devices for the collection of data in specified situations.
7.1.4 Explain the relationship between a sensor, the processor and an output transducer.
7.1.5 Describe the role of feedback in a control system.
7.1.6 Discuss the social impacts and ethical considerations associated with the use of embedded systems.

## Distributed systems

7.1.7 Compare a centrally controlled system with a distributed system.
7.1.8 Outline the role of autonomous agents acting within a larger system.
# IB/Group 4/Computer Science/Databases/The relational database model

Database management system (DBMS) and relational database management system (RDBMS) DBMS A database management system (DBMS) is system software for creating and managing databases. The DBMS provides users and programmers with a systematic way to create, retrieve, update and manage data. RDBMS
A Relational database management system (RDBMS) is a database management system (DBMS) that is based on the relational model as introduced by E. F. Codd. The data in an RDBMS is stored in database objects which are called as tables. Every table is broken up into smaller entities called fields. The fields in the CUSTOMERS table consist of ID, NAME, AGE, ADDRESS and SALARY. A record is also called as a row of data is each individual entry that exists in a table. A column is a vertical entity in a table that contains all information associated with a specific field in a table.
Outline the functions and tools of a DBMS
The functions are:
1. It Organizes Data
2. It integrates Data
3. It Separates Data
4. It Controls Data
5. It Retrieves Data
6. It Protects Data.

#### 1. Data are organized or structured according to the specifications of the data definition language.

The database administrator introduces these specifications at the time the database is established, and may be reintroduced as the data base configuration changes. Data are organized in the manner most suitable to each application.

#### 4. It Controls Data:

A data base management system appears to an application programmer to be an extension of the operating system software. As it receives data storage requests from host programs, it controls how and where data are physically stored. On data retrievals, it locates and returns requested elements of data to the programs.

#### 5. A record of data can be obtained via a data base management system:

(1) Serially (in its physically stored sequence),
(2) Sequentially, according to the value of a user-specified key,
(3) Randomly by key,
(4) Randomly by address, and
(5) By structural link.
All or any portion of the data record can be returned to the user.

#### 6. A database management system protects and secures both the content of a data base and the relationships of data elements.

   Data are protected against access by unauthorized users, physical damage, operating system failure, simultaneous updating, and certain interruptions initiated by a host program.
ACCESS CONTROL
Authorization Authentication
data locking is locking the variable which is required by you and using it shared and exclusive lock 2phase locking
Define the term schema- A database schema is the skeleton structure that represents the logical view of the entire database. It defines how the data is organized and how the relations among them are associated. It formulates all the constraints that are to be applied on the data. A database schema defines its entities and the relationship among them.
# IB Music/Prescribed Works-2014
## Yellow River Concerto by Chen Yung
### 1st Movement
##### Rhythm

* triplets
* multimeter changes between compound and duple meter
* repetitive ostinato in bar 6 in violins
* unexpected pulses in places like in bar 25 in the piano motif "the fisherman's call" there are 13 16th notes when there should be 12

##### Melody

* main themes are ABA'B'ABcoda

##### Harmony

* Major Pentatonic scale with a tonal center based on D

##### Style
#### 2nd Movement
##### Style
#### 3rd Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual

#### 4th Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual

## Classical Symphony by Sergei Prokofiev
#### 1st Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual

#### 2nd Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual

#### 3rd Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual

#### 4th Movement

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style

## Comparisons between Overall

* Rhythm
* Melody
* Harmony
* Structure
* Texture
* Style
* Contextual
# IB Music/Prescribed Works-2014/Harmony
#### Prokofiev

* 1) Dmaj transfers to some keys that aren't related which is a 20th century technique like the diminished 7th (d-->c)
* 2) Amaj which is the dominant of D, it then transfers to the mediant and into the minor submediant which is a 20th c technique
* 3) Dmaj
* 4)

#### Chen Yung

* 1st movement is in Dmaj
* 2nd movement is in Bb Maj
* 3rd movement is in Eb maj
* 4th movement is in C maj
* based on the major pentatonic scale of D
# IB Physics

Printable Version

## Old syllabus table of contents
### Standard and Higher Level Core

* Topic 1 - Physics and Physical Measurement
* Topic 2 - Mechanics
* Topic 3 - Thermodynamics
* Topic 3 - Thermal Physics
* Topic 4 - Oscillations and Waves
* Topic 5 - Electric Currents
* Topic 6 - Fields and Forces
* Topic 7 - Atomic and Nuclear Physics
* Topic 8 - Energy Power and Climate Change

### Higher Level Core only

* Topic 9 - Motion in Fields
* Topic 10 - Thermal Physics
* Topic 11 - Wave Phenomena
* Topic 12 - Electromagnetic Induction
* Topic 13 - Quantum and Nuclear Physics (SL option B)
* Topic 14 - Digital Technology (Part of SL option C)

### Option topics

* Option C - Digital Technology Option
* Option E - History and Development of Physics SL
* Option E - History and Development of Physics HL
* Option F - Astrophysics SL
* Option F - Astrophysics HL
* Option G - Relativity (HL)
* Option G - 2010 Electromagnetic waves (HL)
* Option H - Optics SL
* Option H - Optics HL
* Option I - Biomedical Physics (HL)

## 2016 Syllabus table of contents

* Core
** Measurements and Uncertainties
*** Measurements in Physics
*** Uncertainties and Errors
*** Vectors and Scalars
** Mechanics
** Thermal Physics
** Waves
** Electricity and Magnetism
** Circular Motion and Gravitation
** Atomic, Nuclear and Particle Physics
** Energy Production
* Additional Higher Level
** Wave Phenomena
** Fields
** Electromagnetic Induction
** Quantum and Nuclear Physics
* Options
** Relativity
** Engineering Physics
** Imaging
** Astrophysics
# IB Music/Prescribed Works-2014/Contextual
#### Prokofiev

* written in 1917, during the time of the Russian Revolution, this meant that that everything was in chaos as there was such fight for power. The economy collapsed causing composers either stay or leave. Prokofiev decided to leave. He went to France and then the US, where he wasn't all that successful.

#### Chen Yung

* written in 1967, during the Cultural Revolution, this meant that anything that wasn't Chinese was to be eliminated because they didn't want any influence from other cultures, under Mao's orders. In this period people weren't allowed to play western instruments like the piano, violin, to the point where if they were caught their fingers would be crushed. However, there were exceptions, which is where Chen Yung came in, as he played an arrangement of the "Yellow River Cantata," which was a highly favoured piece, as it was written for the revolution. It was known as a propaganda piece written to promote communism and recall the atrocities of the Japanese invasion.
# IB Music/Prescribed Works-2014/Style
#### Prokofiev

* neoclassism

#### Chen Yung

* work for piano
# IB Music/Music Theory/Scales
## Keys

Learn to recognise the key that any given piece of music begins in. This starting key is called the tonic key, and can be found easily by looking at the key signature, and checking for the possible raised seventh note (leading note) which shows if the key is minor. This may require you to learn the "three semitones down" rule for finding the relative minor key, or the circle of 5ths. If you play an instrument you will find that you can learn key signatures pretty quickly by practicing recognition on each piece of music that you play.
Also learn to recognise if the piece has modulated into a different key - this can be seen by accidentals that make the notes of a different scale. Be careful here that the piece isn't simply chromatic, or that it hasn't become modal.

## Modes

Learn to recognise when a piece of music becomes modal - 'sounds like medieval or church music' is probably the best description. There are different modes, although it's not necessary to be able to name the mode that the segment is in (it might help with points though)
# IB Music/Prescribed Works-2014/Texture

(texture can refer to orchestration or the use of that instrumentation)

#### Prokofiev

* As he is imitating the classical period, as his piece is neoclassical, he uses instrumentation of the classical period meaning that the texture can be seen as polyphonic with intervals of complex homophony and imitation.

#### Chen Yung

* Thinner texture as the piano is emphasized and the orchestra is more for support. Sometimes there is a conversation (call and response) between the piano and the orchestra. The texture can be seen as homophonic as the piano is being featured, while the orchestra is playing more of an accompanying role.
# IB Music/Prescribed Works-2014/Rhythm

1) triplets
* multimeter changes between compound and duple meter
* repetitive ostinato in bar 6 in violins
* unexpected pulses in places like in bar 25 in the piano motif "the fisherman's call" there are 13 16th notes when there should be 12
# IB Music/Prescribed Works-2014/Structure
#### Prokofiev

4 movements
* 1) fast (allegro)
* 2) slow (larghetto)
* 3) fast but not too fast (non troppo allegro)
* 4) lively, fast tempo (molto vivace)

#### Chen Yung

4 movements
* 1) fast (allegro molto agitato)
* 2) Slow (adagio maestoso (slow majestic)
* 3) medium fast (andantino grazioso)
* 4) fast (allegro)
# Electronic Properties of Materials

As of its conception, this book is explicitly broken into three distinct sections. In the first section, Quantum Mechanics for Engineers, the fundamental concepts and mathematics needed for the engineer to operate within a quantum framework are presented. The second section, Band Theory of Solids, expands upon the quantum mechanical concepts from the first to explain bonding and the quantum electronic structure of solids. The final section, Electronic Properties, explains how the electronic properties of materials come from the electronic structure derived from quantum mechanics.

## Table of Contents
### Section 1: Quantum Mechanics for Engineers (Sept 1, 2021)

Chapter 1: Quantum Mechanics Overview (Sept 1, 2021)
Chapter 2: The Stern-Gerlach Experiment (Sept 1, 2021) Chapter 3: The Fundamental Postulates (Sept 1, 2021) Chapter 4: Particle in a Box (Sept 1, 2021) Chapter 5: Momentum Velocity and Position (Sept 1, 2021) Chapter 6: Degeneracy (Sept 1, 2021) Chapter 7: Hydrogen (Sept 1, 2021) Chapter 8: Variational Methods (Sept 1, 2021) Chapter 9: Perturbation Methods (Sept 1, 2021) Chapter 10: Many Electron Atoms (Sept 1, 2021) Chapter 11: Density Functional Theory (Sept 1, 2021)
# Electronic Properties of Materials/Quantum Mechanics for Engineers/Density Functional Theory

This is the eleventh chapter of the first section of the book Electronic Properties of Materials.
# Electronic Properties of Materials/Quantum Mechanics for Engineers

This is a section in the book Electronic Properties of Materials
Within this section there are 11 chapters planned.
# Electromagnetic radiation/X-rays

Electromagnetic radiation; 10. X-rays
These have a wavelength of 0.01 to 10 nm, hence photon energies of 1keV to 1000keV. They were discovered by Wilhelm Roentgen in 1895.
Often, X-rays are divided into hard X-rays (wavelength of 0.01 to 0.1 nm, photon energies of 10keV to 1000keV) and soft X-rays (wavelength of 0.1 to 10 nm, photon energies of 0.1keV to 10keV). In general, shorter wavelength X-rays are said to be harder than longer wavelength ones.
X-rays are widely used in medicine and technology because they can penetrate many materials hence reveal internal structures. Excessive exposure of living beings to X-rays, especially very hard ones, can be damaging and even fatal. Long-term exposure to even moderate levels can cause cancer in humans.
# ID Board Review/Human Immunodeficiency Virus (HIV) Infection/HIV Treatment Regimens

* When to Start HIV Therapy
* Antiretroviral Therapies by Classes
* Selection of Optimal Initial HIV Regimen
* Treatment-Experienced HIV Patients
* Adverse Effects of Treatment (Table)
* Drug-Drug Interactions (Table)
* HIV Laboratory Monitoring (Table)
# ID Board Review/Human Immunodeficiency Virus (HIV) Infection/Complications of HIV/Opportunistic Infections (OIs)/Immune reconstitution inflammatory syndrome (IRIS)
## Definition

* Dramatic inflammatory response to OI as immune response improves with initiation of ART
* Immune system reconstitution “unmasks” active infections: Most commonly Mycobacterium avium, TB, or fungal infections
* CMV retinitis, and crytpococcal meningitis can occur too
* Typically occurs weeks-months after starting ART, w HIV VL falling sharply and CD4 rising

## Clinical Diagnosis

* High level of IRIS suspicion based on timing
* Must rule out other causes and other infections

## Management

* Continue treating OIs
* Continue ART if possible
** May need to hold for a few days if Sx get severe
* Treat symptomatically
* May resort to steroids if IRIS is extremely severe

## Prevention

* In Pts w TB + HIV, start on TB treatment for at least 2 weeks
** Then start ART
* In Pts w cryptococcal meningitis + HIV, Tx the Crypto until intracranial pressures normalize
** DO NOT START TX SIMULTANEOUSLY
# ID Board Review/Human Immunodeficiency Virus (HIV) Infection/HIV Treatment Regimens/Antiretroviral Therapies by Classes/Non-Nucleoside Reverse Transcriptase Inhibitors (NNRTIs)

* Rilpivirine
* Efavirenz
* Etravirine
* Nevirapine
# ID Board Review/Human Immunodeficiency Virus (HIV) Infection/HIV Treatment Regimens/Antiretroviral Therapies by Classes/Protease Inhibitors (PIs)

* Atazanavir
* Darunavir
* Indinavir
* Lopinavir/ritonavir
* Ritonavir
* Saquinavir