OpenLLM-France/wikimedia huggingface dataset:
# A First Course to Network Coding/Probability: Concepts and Useful Equations
## Random Variable

In probability and statistics, a random variable, aleatory variable or stochastic variable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).

## Mutual Independence

In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

## I.I.D

In w:probability theory and w:statistics, a w:sequence or other collection of random variables is independent and identically distributed (i.i.d.) if each random variable has the same w:probability distribution as the others and all are mutually independent.
# A-level Physics (Advancing Physics)/Vectors/Trigonometry

Trigonometry is the study of the ratios between sides on a right-angled triangle. Consider the following diagram:
Depending on the value of the angle BAC, the ratios between the three sides of the triangle will vary. These three sides are generally given the names 'hypotenuse' (H), 'opposite' (O) and 'adjacent' (A). The opposite side is opposite the angle being considered. The hypotenuse is the side opposite the right angle, and the adjacent side is the remaining side, which is also next to the angle being considered.
The three basic trigonometric ratios, in relation to an acute angle θ, are the sine, cosine and tangent ratios, as follows:
$\sin {\theta }={\frac {O}{H}}$
$\cos {\theta }={\frac {A}{H}}$
$\tan {\theta }={\frac {O}{A}}$
It is often helpful to remember them by using the mnemonic 'SOH-CAH-TOA'.
These ratios can be calculated using the appropriate buttons on a scientific calculator. In addition, if you already know the ratio between two sides, and wish to calculate the angle from it, you can use the sin$^{−1}$, cos$^{−1}$ and tan$^{−1}$ buttons to use the inverse function. These functions are technically known as arcsin, arccosin and arctan.
# A-level Physics (Advancing Physics)/Voltage/Worked Solutions

1. A battery has an EMF of 5V. What is the total potential difference across all the components in the circuit?
They are the same! The voltage decreases across the circuit from the voltage at one end of the battery to the other.
2. The voltages (relative to the voltage of the battery) on either side of a resistor are -6V and -5V. What is the potential difference across the resistor?
-5 - -6 = -5 + 6 = 1V
3. At a given point in a circuit, 5C of charge have 10 kJ of potential energy. What is the voltage at this point?
$V={\frac {10\times 10^{3}}{5}}=2{\mbox{ kV}}$
4. Why do the electrons move to a point 1 cm further along the wire?
The point further down the wire has a lower voltage (i.e. less energy per. coulomb) - the electrons will be at a lower potential energy at this point, and fall to it, just as objects fall to lower potential energies due to gravity.
# A-level Physics (Advancing Physics)/Young's Slits/Worked Solutions

1. A 2-slit experiment is set up in which the slits are 0.03 m apart. A bright fringe is observed at an angle 10° from the normal. What sort of electromagnetic radiation was being used?
$\lambda =d\sin {\theta }=0.03\times \sin {10}\approx 5.21\times 10^{-3}{\mbox{ m}}$
So, microwaves were being used.
2. Light, with a wavelength of 500 nm, is shone through 2 slits, which are 0.05 m apart. What are the angles to the normal of the first three dark fringes?
$n\lambda =2d\sin {\theta }\,$
$n\times 500\times 10^{-9}=2\times 0.05\times \sin {\theta }$, where n is an odd integer.
$\theta =\arcsin {\frac {n\times 500\times 10^{-9}}{0.05\times 2}}=\arcsin {(n\times 500\times 10^{-8})}$
Then, substitute in n = 1,3 and 5 to gain corresponding values of θ:
| n | θ (°) |
| 1 | 2.9 x 10$^{−4}$ |
| 3 | 8.6 x 10$^{−4}$ |
| 5 | 1.4 x 10$^{−3}$ |
3. Some X-rays, with wavelength 1 nm, are shone through a diffraction grating in which the slits are 50 μm apart. A screen is placed 1.5 m from the grating. How far are the first three light fringes from the point at which the normal intercepts the screen?
$n\times 10^{-9}={\frac {50\times 10^{-6}\times x}{1.5}}\approx 33.3\times 10^{-6}\times x$, where n is an integer.
$x={\frac {n\times 10^{-9}}{33.3\times 10^{-6}}}=30\times 10^{-6}\times n$
Then, substitute in n = 1,2 and 3 to gain corresponding values of x:
| n | x (m) |
| 1 | 30 x 10$^{−6}$ |
| 2 | 60 x 10$^{−6}$ |
| 3 | 90 x 10$^{−6}$ |
# A-level Physics (Advancing Physics)/Vectors/Worked Solutions

1. Which of the following are vectors?
* 20 cm is scalar, not a vector - it does not have a directional component.
* 9.81 ms$^{−2}$ towards the centre of the earth is a vector.
* 5 km south-east is a vector.
* 500 ms$^{−1}$ on a bearing of 285.3° is a vector.
2. A displacement vector a is the resultant vector of two other vectors, 5 m north and 10 m south-east. What does a equal, as a displacement and a bearing?
| Displacement (m) | Bearing (°) | Horizontal Displacement (m) | Vertical Displacement (m)' |
| 5 | 000 | 0 | 5 |
| 10 | 135 | $\sin {135}={\frac {s}{10}}$, so $s={\frac {10}{\sqrt {2}}}$ | $\cos {135}={\frac {s}{10}}$, so $s=-{\frac {10}{\sqrt {2}}}$ |
So, for the resultant vector:
${\mbox{Horizontal displacement }}=0+{\frac {10}{\sqrt {2}}}\approx 7.07{\mbox{ m}}$
${\mbox{Vertical displacement }}=5-{\frac {10}{\sqrt {2}}}\approx -2.07{\mbox{ m}}$
This gives us a right-angled triangle. So:
$\theta =\arctan {\frac {2.07}{7.07}}\approx 16.3$°, so the bearing equals 90° + 16.3° = 106.3°.
${\mbox{Resultant displacement}}={\sqrt {7.07^{2}+(-2.07)^{2}}}\approx 7.4{\mbox{ m}}$
3. If I travel at a velocity of 10 ms$^{−1}$ on a bearing of 030°, at what velocity am I travelling north and east?
${\mbox{Vertical velocity (North)}}=10\times \cos {30}=5{\sqrt {3}}{\mbox{ ms}}^{-1}\approx 8.66{\mbox{ ms}}^{-1}$
${\mbox{Horizontal velocity (East)}}=10\times \sin {30}=5{\mbox{ ms}}^{-1}$
4. An alternative method of writing vectors is in a column, as follows:
$\mathbf {a} ={\binom {x}{y}}$,
where x and y are the vertical and horizontal components of the vector respectively. Express |a| and the angle between a and ${\tbinom {1}{0}}$ in terms of x and y.
By Pythagoras' theorem:
$(|\mathbf {a} |)^{2}=x^{2}+y^{2}$
$|\mathbf {a} |={\sqrt {x^{2}+y^{2}}}$
Let θ be the angle between a and ${\tbinom {1}{0}}$.
$\tan {\theta }={\frac {y}{x}}$
$\theta =\arctan {\frac {y}{x}}$
This angle θ is known as the argument of a.
5. A more accurate method of modelling the trajectory of a ball is to include air resistance as a constant force F. How would this be achieved?
Once the arrow representing the acceleration due to gravity has been added on, add an horizontal arrow pushing against the motion of the ball. Since F = ma, the magnitude of this acceleration is F divided by m.
Note that this model is still not perfect. In fact, F is not constant - it depends on the horizontal component of the ball's velocity.
# A-level Physics (Advancing Physics)/Voltage

Charge moves through a circuit, losing potential energy as it goes. This means that the charge travels as an electric current. Voltage is defined as the difference in potential energy per. unit charge, i.e.
$V={\frac {E}{Q}}$
where V is voltage (in volts), E is the difference in potential energy (in joules) and Q is charge (in coulombs).
There are two electrical properties which are both measured in volts (commonly abbreviated V), and so both are known under the somewhat vague title of 'voltage'. Both are so called because they change the potential energy of the charge.

## Electromotive Force (EMF)

Keep in mind, that EMF as the name suggests is not an electrical force, it is basically the potential difference across the terminals when the key is open i.e. when no current is drawn from the cell. EMF is named so by the scientists who performed faulty experiments and named it so, hence, just a tribute to their contribution to physics it is still called EMF but the definition has changed with time.

## Potential Difference

As charge travels around a circuit, each coulomb of charge has less potential energy, so the voltage (relative to the power source) decreases. The difference between the electric potential at two points in a circuit is known as potential difference, and can be measured with a voltmeter.

### Series Circuits

In a series circuit, the total voltage (EMF) is divided across the components, as each component causes the voltage to decrease, so each one has a potential difference. The sum of the potential differences across all the components is equal to the total voltage (but batteries have their own 'internal resistances', which complicates things slightly, as we will see).

### Parallel Circuits

In a parallel circuit, the potential difference across each branch of the circuit is equal to the EMF, as the same 'force' is pushing along each path of the circuit. The number of charge carriers (current) differs, but the 'force' pushing them (voltage) does not.

## Questions

1. A battery has an EMF of 5V. What is the total potential difference across all the components in the circuit?
2. The voltages (relative to the voltage of the battery) on either side of a resistor are -6V and -5V. What is the potential difference across the resistor?
3. At a given point in a circuit, 5C of charge have 10 kJ of potential energy. What is the voltage at this point?
4. Why do the electrons move to a point 1 cm further along the wire?
Worked Solutions
# A-level Physics (Advancing Physics)/What is a wave?/Worked Solutions

1. Through what medium are sound waves propagated?
Usually the air, but most material objects will carry sound.
2. What aspects of the behaviour of light make it look like a wave?
Superposition and diffraction (and others).
3. What aspects of the behaviour of light make it look like a particle?
Photoelectric effect
4. Consider the diagram on the right. White light is partially reflected by the transparent material. Some of the light, however, is refracted into the transparent material and reflected back by the opaque material. The result is two waves travelling in the same place at the same time at the same polarisation (the light is not a single beam). Why does, say, the red light disappear?
If the refractive index and width of the transparent material are correct, then the two waveforms leaving it will be half a wavelength apart, as one will have travelled further. Destructive interference will cause the red light to become 'invisible'.
5. What is the wavelength of green light?
500 nm
6. The lowest frequency sound wave humans can hear has a frequency of approximately 20 Hz. Given that the speed of sound in air is 343ms$^{−1}$, what is the wavelength of the lowest frequency human-audible sound?
$\lambda ={\frac {v}{f}}={\frac {343}{20}}=17.15{\mbox{m}}$
# ABC graphs
## Authors

Vladislav Ivaništšev

## Introduction

ABC graphs is a general name for diagrams showing Atom symbols, Bonding paths, atomic Charges, lone electrons (with Dots), and empty orbitals (with Ellipsis). Specific information shown on a given diagram is reflected in its name, so one may refer to AB, ABC, ABCD or ABCDE graphs.
An AB graph is a molecular graph analogues to a ball-and-stick model, yet it shows atoms with symbols and bonding paths with single lines (instead of balls and multiple sticks).
For all graphs more complicated than AB, let us agree that each drawn line corresponds to a σ-bond sharing of a pair of electrons. Then, to draw a structure one should use simple rules for determining the atomic charges, number of lone electrons, and empty orbitals.

## Table of Contents

* Acronym and definitions
* Rules for drawing ABC graphs

### Drawing examples

* AB graph drawing examples
* ABC graph drawing examples
* ABCD graph drawing examples
* ABD graph drawing examples
* ABCDE graph drawing examples

### Structure–property relations

* ABC-type graphs and acid-base chemistry
* ABC-type graphs and redox chemistry

### Chemistry of elements

* Sulfur chemistry
# Annotated Republic of China Laws/Copyright Act/Article 95

| ←Article 94 | Copyright Act Republic of China (Taiwan) Law Article 95 | Article 96-1→ |

## Promulgated amendment on 2003-07-09
### Article 95

    A person who violates any provision of Article 112 shall be punished by imprisonment for not more than one year, detention, or, in lieu thereof or in addition thereto, a fine of not less than twenty thousand and not more than two hundred and fifty thousand New Taiwan Dollars.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
# Annotated Republic of China Laws/Copyright Act/Article 96-1

| ←Article 95 | Copyright Act Republic of China (Taiwan) Law Article 96-1 | Article 96-2→ |

## Newly adopted and promulgated on 2003-07-09
### Article 96-1

    For a violation of Article 80bis, a sentence of up to one year imprisonment or detention shall be imposed, and in lieu thereof or in addition thereto a fine of not less than twenty thousand and not more than two hundred and fifty thousand New Taiwan Dollars may also be imposed.

## Promulgated amendment on 2004-09-01
### Article 96-1

    In any of the following circumstances a sentence of up to one year imprisonment or detention shall be imposed, or in lieu thereof or in addition thereto, a fine of not less than twenty thousand and not more than two hundred and fifty thousand New Taiwan Dollars:
    1. Violation of Article 80bis.
    2. Violation of paragraph 2 of Article 80ter.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
* ↑ "Copyright Act: Article 96-1". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-24.
# Annotated Republic of China Laws/Copyright Act/Article 91

| ←Article 90-12 | Copyright Act Republic of China (Taiwan) Law Article 91 | Article 91-1→ |

## Promulgated amendment on 2003-07-09
### Article 91

    A person who infringes on the economic rights of another person by means of reproducing the work with intent to profit shall be punished by imprisonment for not more than five years, detention, or in addition thereto a fine of not less than two hundred thousand and not more than two million New Taiwan Dollars.
    A person who infringes on the economic rights of another person by means of reproducing the work without the intent to profit, where the number of copies reproduced exceeds five, or where the total amount of infringement calculated by the market value of lawful copies of the work at the time of seizure exceeds thirty thousand New Taiwan Dollars, shall be punished by imprisonment for not more than three years, detention, or in lieu thereof or in addition thereto a fine of not more than seven hundred and fifty thousand New Taiwan Dollars.
    A person who commits the offense in paragraph 1 by means of reproducing onto an optical disk shall be punished by imprisonment for not more than five years, detention, or in addition thereto a fine of not less than five hundred thousand and not more than five million New Taiwan Dollars.

## Promulgated amendment on 2004-09-01

Article 91
    A person who infringes on the economic rights of another person by means of reproducing the work without authorization shall be punished by imprisonment for not more than three years, detention, or in lieu thereof or in addition thereto a fine not more than seven hundred and fifty thousand New Taiwan Dollars.
    A person who infringes on the economic rights of another person by means of reproducing the work without authorization with the intent to sell or rent shall be imprisoned not less than six months and not more than five years, and in addition thereto, may be fined not less than two hundred thousand and not more than two million New Taiwan Dollars.
    A person who commits the offense in the preceding paragraph by means of reproducing onto an optical disk shall be imprisoned not less than six months and not more than five years, and in addition thereto, may be fined not less than five hundred thousand and not more than five million New Taiwan Dollars.
    A work only for personal reference or fair use of a work does not constitute infringement of copyright.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
* ↑ "Copyright Act: Article 91". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
# Annotated Republic of China Laws/Copyright Act/Article 93

| ←Article 92 | Copyright Act Republic of China (Taiwan) Law Article 93 | Article 94→ |

## Promulgated amendment on 2003-07-09
### Article 93

    In any of the following circumstances, where there is intent to profit, a sentence of up to two years imprisonment or detention shall be imposed, and in addition thereto a fine of not more than five hundred thousand New Taiwan Dollars may also be imposed:
    1. Where the provisions of Article 70 have been violated.
    2. Where another person's economic rights have been infringed by any of the means specified in subparagraphs 2, 3, 5, or 6 of Article 87.
    A person who commits the crime in the preceding paragraph without intent to profit, where the number of copies exceeds five, or the damage incurred by the copyright owner exceeds thirty thousand New Taiwan Dollars, shall be punished by imprisonment for not more than one year, detention, or in lieu thereof or in addition thereto a fine of not more than two hundred fifty thousand New Taiwan Dollars.

## Promulgated amendment on 2004-09-01
### Article 93

    In any of the following circumstances, a sentence of up to two years imprisonment or detention shall be imposed, or in lieu thereof or in addition thereto, a fine of not more than five hundred thousand New Taiwan Dollars:
    1. Infringement of the author's moral rights as set forth in the provisions of articles 15 through 17.
    2. Violations of the provisions of Article 70.
    3. Infringement of another person's copyright by any of the means specified in subparagraphs 1, 3, 5, or 6 of article 87; provided this shall not apply to offenses as referred to in paragraph 2 or paragraph 3 of Article 91bis.

## Promulgated amendment on 2007-07-11
### Article 93

    In any of the following circumstances, a sentence of up to two years imprisonment or detention shall be imposed, or in lieu thereof or in addition thereto, a fine of not more than five hundred thousand New Taiwan Dollars:
    1.Infringement of the author's moral rights as set forth in the provisions of articles 15 through 17.
    2.Violations of the provisions of Article 70.
    3.Infringement of another person's copyright by any of the means specified in paragraph 1, subparagraphs 1, 3, 5, or 6 of Article 87, provided this shall not apply to offenses as referred to in paragraph 2 or paragraph 3 of Article 91bis.
    4.Violations of subparagraph 7 of paragraph 1 of Article 87.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
* ↑ Derived from 2007 text.
* ↑ "Copyright Act: Article 93". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2014-01-22. Retrieved 2016-11-24.
# Annotated Republic of China Laws/Copyright Act/Article 96-2

| ←Article 96-1 | Copyright Act Republic of China (Taiwan) Law Article 96-2 | Article 98→ |

## Newly adopted and promulgated on 2003-07-09
### Article 96-2

    If a fine is to be imposed pursuant to the provisions of this Chapter, the financial ability of the offender and the benefit he/she has obtained through commission of the offense shall be taken into account. If the benefit obtained exceeds the maximum fine, such fine may be increased within the limit of the obtained benefit.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
# Annotated Republic of China Laws/Copyright Act/Article 98

| ←Article 96-2 | Copyright Act Republic of China (Taiwan) Law Article 98 | Article 98-1→ |

## Promulgated full amendment on 1998-01-21
### Article 98

(deleted)

## Newly adopted and promulgated on 2003-07-09
### Article 98

    An article used in the commission of an offense or acquired through the commission of an offense set forth in Article 91 through Article96bis may be confiscated; provided, for an offense set forth in paragraph 3 of Article 91 or paragraph 3 of Article 91bis, articles which may be confiscated are not limited to those belonging to the offender.

## Promulgated amendment on 2006-05-30 effective on 2006-07-01
### Article 98

    An article used in the commission of an offense or acquired through the commission of an offense set forth in Article 91 through Article 93, Article 95 through Article 96bis may be confiscated; provided, for an offense set forth in paragraph 3 of Article 91 or paragraph 3 of Article 91bis, articles which may be confiscated are not limited to those belonging to the offender.

## Promulgated amendment on 2016-11-30
### Article 98

    In an offense set forth in paragraph 3 of Article 91 or paragraph 3 of Article 91bis, an article used in the commission of the offense, or in preparation for the commission of the offense, or an article derived from the commission of the offense, may be confiscated, regardless of whether it belongs to the offender.

## Annotated by Wikibooks

* ↑ "Copyright Act". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2003-07-09. Retrieved 2016-11-23.
* ↑ Derived from 2003 text.
* ↑ "Copyright Act: Article 98". Laws & Regulations Database of The Republic of China. Ministry of Justice (Taiwan). 2006-11-30. Retrieved 2016-12-09.