OpenLLM-France/wikimedia huggingface dataset:
# Supplementary mathematics/Riemannian geometry

Riemannian geometry is a branch of differential geometry that investigates and studies the contents of Riemannian manifolds, i.e. smooth manifolds equipped with Riemannian metric, this manifold structure is equipped with inner multiplication on the tangent space at any point, so that from one point to The other point changes smoothly. Also, this structure specifically acquires local concepts such as angle, bend length, surface area and volume. From these, some other global quantities can be obtained by integration.
# Statistics/Distributions/F
### F Distribution

Named after Sir Ronald Fisher, who developed the F distribution for use in determining ANOVA critical values. The cutoff values in an F table are found using three variables- ANOVA numerator degrees of freedom, ANOVA denominator degrees of freedom, and significance level.
ANOVA is an abbreviation of analysis of variance. It compares the size of the variance between two different samples. This is done by dividing the larger variance over the smaller variance. The formula of the F statistic is:
$F(r_{1},r_{2})={\frac {\chi _{r_{1}}^{2}/r_{1}}{\chi _{r_{2}}^{2}/r_{2}}}$
where $\chi _{r_{1}}^{2}$ and $\chi _{r_{2}}^{2}$ are the chi-square statistics of sample one and two respectively, and $r_{1}$ and $r_{2}$ are their degrees of freedom, i.e. the number of observations.
One example could be if you want to compare apples that look alike but are from different trees and have different sizes. You want to investigate whether they have the same variance of the weight on average.
There are three apples from the first tree that weigh 110, 121 and 143 grams respectively, and four from the other which weigh 88, 93, 105 and 124 grams respectively. The mean and variance of the first sample are 124.67 and 16.80 respectively, and of the second sample 102.50 and 16.01. The chi-square statistic of the first sample is
$({\frac {110-124.67}{16.80}})^{2}+({\frac {121-124.67}{16.80}})^{2}+({\frac {143-124.67}{16.80}})^{2}=2.00$,
and for the second sample
$({\frac {88-102.50}{16.01}})^{2}+({\frac {93-102.50}{16.01}})^{2}+({\frac {105-102.50}{16.01}})^{2}+({\frac {124-102.50}{16.01}})^{2}=3.00$.
The F statistic is now F = ${\frac {3/4}{2/3}}=1.125$. The Chi-square statistic divided by degrees of freedom appears on the nominator for the second sample because it was larger than that of the first sample.
The critical value of the F distribution for 4 degrees of freedom in the nominator and 3 degrees of freedom in the denominator, i.e. F(f1=4, f2=3) is 9.12 at a 5% level of confidence. Since the test statistic 1.125 is smaller than the critical value, we cannot reject the null hypothesis that they have the same variance. The conclusion is that they have the same variance.

#### External links

* Interactive F Distribution Web Applet (Java)
# Statistics/Distributions/Chi-square
### Chi-square Distribution

Chi-square distribution is related to normal distribution. A chi-square statistic is the sum of a number of independent and standard normal random variables.
Assume that we have n number of random variables Z, that are normally distributed. Therefore, we can write $Z\sim N(0,1)$. If we square Z such that $Z^{2}$, then we get the chi-square distribution $Z^{2}\sim \chi _{1}^{2}$. If we sum n number of $\chi _{1}^{2}$, we can write
$Y=Z_{1}^{2}+Z_{2}^{2}+...+Z_{n}^{2}\sim \chi _{n}^{2}$.
One example could be that we want to know whether the weight of a set of eight apples is normally distributed. Chi-square distribution can be used to test for this. Assume that the apples weigh 88, 93, 110, 76, 78, 121, 92 and 86 grams, and we have knowledge of the mean and the standard deviation weight of all apples. We obtain the normally distributed Z values by subtracting the mean weight (93) and divide by the standard deviation (15.41). For example, the first apple has Z-score $Z_{1}={\frac {88-93}{15.41}}=-0.3245$ using four decimal points. Square all the Z values, then taking the sum yields a Chi-squared distributed random variable with mean 8 and variance 16.
Now when we have the value of the chi-square statistic Y, we compare it to the critical value of the chi-square distribution at n = 8 degrees of freedom and 95% level of significance which can found in a Chi-square statistical table. The null hypothesis is that the sample of apples is normally distributed. It is rejected if the value of the test statistic is higher than the critical value.
The chi-square distribution is a special case of the gamma distribution, where a=2 and p=k/2. The probability density function is:
    ${\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\quad x\geq 0,\,k\in [1,2,...]$

#### Summary statistics

The mean of a chi-squared is $k$
The variance of a chi-squared is $2k$
For the proof of these, see the gamma distribution.

#### External links

* Interactive Chi-Square Distribution Web Applet (Java)
# Statistics/Distributions/Discrete

'Discrete' data are data that assume certain discrete and quantized values. For example, true-false answers are discrete, because there are only two possible choices. Valve settings such as 'high/medium/low' can be considered as discrete values. As a general rule, if data can be counted in a practical manner, then they can be considered to be discrete.
To demonstrate this, let us consider the population of the world. It is a discrete number because the number of civilians is theoretically countable. But since this is not practicable, statisticians often treat this data as continuous. That is, we think of population as within a range of numbers rather than a single point.
For the curious, the world population is 6,533,596,139 as of August 9, 2006. Please note that statisticians did not arrive at this figure by counting individual residents. They used much smaller samples of the population to estimate the whole. Going back to Chapter 1, this is a great reason to learn statistics - we need only a smaller sample of data to make intelligent descriptions of the entire population!
Discrete distributions result from plotting the frequency distribution of data which is discrete in nature.

### General Properties
#### Cumulative Distribution Function

A discrete random variable has a cumulative distribution function that describes the probability that the random variable is below the point. The cumulative distribution must increase towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.
ń====Probability Mass Function====
A discrete random variable has a probability mass function that describes how likely the random variable is to be at a certain point. The probability mass function must have a total of 1, and sums to the cdf. The pmf is represented by the lowercase f.

#### Special Values

The expected value of a discrete variable is $\sum _{n_{min}}^{n_{max}}x_{i}f(x_{i})$
The expected value of any function of a discrete variable g($X$) is $\sum _{n_{min}}^{n_{max}}g(x_{i})f(x_{i})$
The variance is equal to $E((X-E(X))^{2})$
External Links
Simulating binomial, hypergeometric, and the Poisson distribution: Discrete Distributions
# Statistics/Distributions/Discrete Uniform
### Discrete Uniform Distribution

The discrete uniform distribution (not to be confused with the continuous uniform distribution) is where the probability of equally spaced possible values is equal. Mathematically this means that the probability density function is identical for a finite set of evenly spaced points. An example of would be rolling a fair 6-sided die. In this case there are six, equally like probabilities.
One common normalization is to restrict the possible values to be integers and the spacing between possibilities to be 1. In this setup, the only two parameters of the function are the minimum value (a), the maximum value (b). (Some even normalize it more, setting a=1.) Let n=b-a+1 be the number of possibilities. The probability density function is then
$f\colon \{a,a+1,\ldots ,b-1,b\}\to \mathbb {R}$
$f\left(x\right)={\frac {1}{n}}$

#### Mean

Let $S=\{a,a+1,\ldots ,b-1,b\}$. The mean (notated as $\operatorname {E} [X]$) can then be derived as follows:
    $\operatorname {E} [X]=\sum _{x\in S}xf(x)=\sum _{i=0}^{n-1}\left({\frac {1}{n}}(a+i)\right)$
    $\operatorname {E} [X]={1 \over n}\left(\sum _{i=0}^{n-1}a+\sum _{i=0}^{n-1}i\right)$
Remember that in $\sum _{i=0}^{m}i=(m^{2}+m)/2$
    $\operatorname {E} [X]={1 \over n}\left(na+{(n-1)^{2}+(n-1) \over 2}\right)$
    $\operatorname {E} [X]={2na+n^{2}-2n+1+n-1 \over 2n}$
    $\operatorname {E} [X]={2a+n-1 \over 2}$
    $\operatorname {E} [X]={a+b \over 2}$

#### Variance

The variance ($E[(X-EX)^{2}]$) can be derived:
    $\operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\sum _{x\in S}f(x)(x-E[X])^{2}=\sum _{i=0}^{n-1}\left({\frac {1}{n}}\left((a+i)-{a+b \over 2}\right)^{2}\right)$
    $\operatorname {Var} (X)={1 \over n}\sum _{i=0}^{n-1}\left({a+2i-b \over 2}\right)^{2}$
    $\operatorname {Var} (X)={1 \over 4n}\sum _{i=0}^{n-1}(a^{2}+4ai-2ab+4i^{2}-4ib+b^{2})$
    $\operatorname {Var} (X)={1 \over 4n}\left[\sum _{i=0}^{n-1}(a^{2}-2ab+b^{2})+\sum _{i=0}^{n-1}(4ai-4ib)+\sum _{i=0}^{n-1}4i^{2}\right]$
    $\operatorname {Var} (X)={1 \over 4n}\left[n(a^{2}-2ab+b^{2})+4(a-b)\sum _{i=0}^{n-1}i+4\sum _{i=0}^{n-1}i^{2}\right]$
Remember that in $\sum _{i=0}^{m}i^{2}=m(m+1)(2m+1)/6$
    $\operatorname {Var} (X)={1 \over 4n}\left[n(b-a)^{2}+4(a-b)[(n-1)n/2]+4[(n-1)n(2n-1)/6]\right]$
    $\operatorname {Var} (X)={1 \over 4n}\left[n(n-1)^{2}-2(n-1)(n-1)n+2(n-1)n(2n-1)/3\right]$
    $\operatorname {Var} (X)={1 \over 4}\left[-(n-1)^{2}+2(n-1)(2n-1)/3\right]$
    $\operatorname {Var} (X)={1 \over 12}\left[-3(n-1)^{2}+2(n-1)(2n-1)\right]$
    $\operatorname {Var} (X)={1 \over 12}\left[-3(n^{2}-2n+1)+2(2n^{2}-3n+1)\right]$
    $\operatorname {Var} (X)={n^{2}-1 \over 12}$
# Statistics/Distributions/Exponential
### Exponential Distribution

Exponential distribution refers to a statistical distribution used to model the time between independent events that happen at a constant average rate λ. Some examples of this distribution are:
* The distance between one car passing by after the previous one.
* The rate at which radioactive particles decay.
For the stochastic variable X, probability distribution function of it is:
$f_{x}(x)={\begin{cases}\lambda e^{-\lambda x},&{\mbox{if }}x\geq 0\\0,&{\mbox{if }}x<0\end{cases}}$
and the cumulative distribution function is:
$F_{x}(x)={\begin{cases}0,&{\mbox{if }}x<0\\{1-e^{-\lambda x}},&{\mbox{if }}x\geq 0\end{cases}}$
Exponential distribution is denoted as $X\in {\mbox{Exp(m)}}$, where m is the average number of events within a given time period. So if m=3 per minute, i.e. there are three events per minute, then λ=1/3, i.e. one event is expected on average to take place every 20 seconds.

#### Mean

We derive the mean as follows.
    $\operatorname {E} [X]=\int _{-\infty }^{\infty }x\cdot f(x)dx$
    $\operatorname {E} [X]=\int _{0}^{\infty }x\lambda e^{-\lambda x}dx$
    $\operatorname {E} [X]=\int _{0}^{\infty }(-x)(-\lambda e^{-\lambda x})dx$
We will use integration by parts with u=−x and v=e$^{−λx}$. We see that du=-1 and dv=−λe$^{−λx}$.
    $\operatorname {E} [X]=\left[-x\cdot e^{-\lambda x}\right]_{0}^{\infty }-\int _{0}^{\infty }(e^{-\lambda x})(-1)dx$
    $\operatorname {E} [X]=[0-0]+\left[{-1 \over \lambda }(e^{-\lambda x})\right]_{0}^{\infty }$
    $\operatorname {E} [X]=\left[0-{-1 \over \lambda }\right]$
    $\operatorname {E} [X]={1 \over \lambda }$

#### Variance

We use the following formula for the variance.
    $\operatorname {Var} (X)=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}$
    $\operatorname {Var} (X)=\int _{-\infty }^{\infty }x^{2}\cdot f(x)dx-\left({2}\right)^{2}$
    $\operatorname {Var} (X)=\int _{0}^{\infty }x^{2}e^{-2x}dx-{2}$
We'll use integration by parts with $u=-x^{2}$ and $v=e^{-2x}$. From this we have $du=-2x$ and $v=-2e^{-2x}$.
    $\operatorname {Var} (X)=\left\{\left[-x^{2}\cdot e^{-\lambda x}\right]_{0}^{\infty }-\int _{0}^{\infty }(e^{-\lambda x})(-2x)dx\right\}-{1 \over \lambda ^{2}}$
    $\operatorname {Var} (X)=[0-0]+{2 \over \lambda }\int _{0}^{\infty }x\lambda e^{-\lambda x}dx-{1 \over \lambda ^{2}}$
We see that the integral is just $\operatorname {E} [X]$ which we solved for above.
    $\operatorname {Var} (X)={2 \over \lambda }{1 \over \lambda }-{1 \over \lambda ^{2}}$
    $\operatorname {Var} (X)={1 \over \lambda ^{2}}$

### External links

* Interactive Exponential Distribution Web Applet (Java)
# Statistics/Distributions/Continuous

A continuous statistic is a random variable that does not have any points at which there is any distinct probability that the variable will be the corresponding number.

### General Properties
#### Cumulative Distribution Function

A continuous random variable, like a discrete random variable, has a cumulative distribution function. Like the one for a discrete random variable, it also increases towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.

#### Probability Distribution Function

Unlike a discrete random variable, a continuous random variable has a probability density function instead of a probability mass function. The difference is that the former must integrate to 1, while the latter must have a total value of 1. The two are very similar, otherwise. The pdf is represented by a lowercase f.

#### Special Values

Let R be the set of points of the distribution.
The expected value for a continuous variable X with probability density function f is defined as $\int _{R}xf(x)dx$.
More generally, the expected value of any continuously transformed variable g(X) with probability density function f is defined as $\int _{R}g(x)f(x)dx$.
The mean of a continuous or discrete distribution is defined as $E[X]$.
The variance of a continuous or discrete distribution is defined as $E[(X-E[X]^{2})]$.
Expectations can also be derived by producing the Moment Generating Function for the distribution in question. This is done by finding the expected value $E[\exp(tX)]$. Once the Moment Generating Function has been created, each derivative of the function gives a different piece of information about the distribution function.
${\frac {dE[\exp(tX)]}{dt}}$ = mean ${\frac {d^{2}E[\exp(tX)]}{dt^{2}}}$ = variance ${\frac {d^{3}E[\exp(tX)]}{dt^{3}}}$ = skewness ${\frac {d^{4}E[\exp(tX)]}{dt^{4}}}$ = kurtosis
# Salut Jonathan!/Prefaco

Salut Jonathan! (Hello, Jonathan!) is the story of a young Englishman who goes to visit his friend in another European country...or so it would seem.
It is written entirely in the language Neo, a language created by the Arturo Alfandari that was originally published in 1961 and updated in 1965/66. The book Salute, Jonathan! was originally written in Interlingue by Mithridates and published on Wikibooks in 2019.
Salut Jonathan! begins with very simple language. The first words in the book are Un vir starar in un civ (a man stands in a city), and it continues from there. The words the reader knows are repeated and new words are added slowly and in context, so that no dictionary is required to read it. Neo was created to be an international language, with a very simple and straightforward grammar. Thanks to its simplicity and regularity, you will know how to read and use it by the end of Salut, Jonathan!.
The method in the book is based on the same used in books such as English by the Nature Method, Le français par la méthode nature, and L'italiano secondo il metodo natura, as well as Hans Ørberg's Lingua Latina per se Illustrata.
Curious about the true nature of the story in the book? You will begin to get clues as to the true story around Chapter 4 as the language becomes more and more complex and an actual story begins to form. Salut, Jonathan! is in fact a full translation of a classic book that you are likely very familiar with. Read on to find out!
# Sidux/Useable applications/Image editors

    Using graphic editors you can create or edit graphic files.
    At previous chapter " Graphic viewers" I showed you a few application for watching, organizing
    and basic editing graphic files.
    This chapter is designed to applications for graphic files editors.

## GIMP

    GIMP (The GNU Image Manipulation Program) is a popular application for creating and editing raster graphics.
    Using it you can do for example:
       a. create graphics and logos.
       b. change file sizes, cut theme off, etc.
       c. change colors.
       d. connect files using layers.
       e. convert files to many different forms.
    GIMP supports graphic forms like that: JPEG, BMP, PNG, XPM, TIFF, TGA, MPEG, PS, PDF, PCX, GIF...
    1. Installation:
    2. Run it from: Menu-> Graphics-> GNU Image Manipulation Program.
    GIMP home page: http://www.gimp.org/
    On Wikipedia: http://en.wikipedia.org/wiki/GIMP

## Inkscape

    Inkscape is a vector graphic editor application.
    1. Installation:
    2. Run if from: Menu-> Graphics-> Inkscape.
    Inkscape home page: http://www.inkscape.org/
    On Wikipedia: http://en.wikipedia.org/wiki/Inkscape
    And on Wikibooks: http://en.wikibooks.org/wiki/Inkscape

## Blender

    Blender is a 3D graphics application and can be used for creating and editing interactive 3D applications
    such as video games, animated films or visual effects.
    1. It is available in Debian's repositories:
    2. Run it from: Menu-> Graphics-> Blender.
    Blender home page: http://www.blender.org/
    And on Wikipedia: http://en.wikipedia.org/wiki/Blender_%28software%29

## Fotoxx

    Fotoxx is a small application for basic graphics editing including panoramas making.
    1. Installation:
    2. Run it: Menu-> Graphics-> fotoxx.
    Fotoxx home page: http://kornelix.squarespace.com/fotoxx/

## Others

* AC3D - 3D graphics maker: http://www.inivis.com/
* Art of Illusion - 3D graphics editor: http://www.artofillusion.org/
* Gestalter - vector graphics maker/editor: http://www.linotux.ch/gestalter/
* Image2mpeg - slide shows movie maker using photos: http://www.gromeck.de/index.php?id=image2mpeg
* K-3D - 3D graphics editor: http://www.k-3d.org/
* Tuxpaint - drawing applications for kids: http://www.tuxpaint.org/
* ImageMagick - bitmaps maker/editor: http://www.imagemagick.org/script/index.php
* Cenon - graphics editor: http://www.cenon.info/frame_gb.html
* Hugin - panoramas maker: http://hugin.sourceforge.net/
# Sidux/Useable applications/File managers

    A file manager is an application which lets you work on files and folders such as creating, moving,
    delating or name changing.
    Thunar is a default file manager for XFCE environment but there is not any problem to work on any other.

## Thunar

    1. If you work on different graphic environment than XFCE, you can install Thunar:
    2. Run it in Terminal with the command:
    or run it from: Menu-> Accessories or System-> Thunar File Manager.
    Thunar home page: http://thunar.xfce.org/
    And on Wikipedia: http://en.wikipedia.org/wiki/Thunar

## Nautilus

    A default file manager for GNOME environment is Nautilus.
    It has some more functions than Thunar but it's a little slower.
    1. Installation:
    2. Run it from: Menu-> Accessories-> File Manager or in Terminal:
    Nautilus home page: http://live.gnome.org/Nautilus
    On Wikipedia: http://en.wikipedia.org/wiki/Nautilus_%28file_manager%29

## Dolphin

    Dolphin is a default file manager for KDE for a while, it replaced previous file manager and web browser Konqueror.
    1. Installation:
    2. Run it: Menu-> System-> Dolphin.
    Dolphin home page: http://dolphin.kde.org/index.html
    And on Wikipedia: http://en.wikipedia.org/wiki/Dolphin_%28software%29

## PCMan

    PCMan is a small and quick file manager designed for LXDE environment.
    1. Install it:
    2. And run it from: Menu-> Accessories or System-> PCMan File Manager.
    PCMan home page: http://pcmanfm.sourceforge.net/
    On Wikipedia: http://en.wikipedia.org/wiki/PCMan_File_Manager

## Midnight Commander

    One of text console file manager is Midnight Commander.
    1. Installation:
    2. Run it in Terminal as an user:
    or as root:
    The basic operations on files are here: Sidux/System and packages management/Midnight Commander.
    Home page: http://www.midnight-commander.org/
    And on Wikipedia: http://en.wikipedia.org/wiki/Midnight_Commander

## Others

* Konqueror: http://www.konqueror.org/, Wikipedia: http://en.wikipedia.org/wiki/Konqueror
* Krusader: http://krusader.org/, Wikipedia: http://en.wikipedia.org/wiki/Krusader
# Sidux/Useable applications/Java virtual machine

    Java is a programming language known as Java Virtual Machine lets runninig applets
    (applications) on computers or web browsers.
    It's indispensable element for running applications such as OpenOffice.org or Jalbum.

## Installation from repositories

    Installation of proprietary Java version with it's web browser plug-in:
    When installator finish downloading it'll display the licence, just scroll it down to the end of it and agree typing "yes".

## Installation from package

    1. If you don't want to install Java from repositories. then:
       a. Download "self-extracting file" from: java.com.
       b. Move the package to: "/usr/lib/jvm" folder.
       c. Go to "/usr/lib/jvm" folder using Terminal as root:
       d. Check is the package there:
       e. Install it typing "yes" on the end of the licence:
       f. Create Java plug-in symbolic link to web browser plug-in folder:
       g. You can make symbolic link using Midnight Commander, just mark "libjavaplugin_oji.so" and then choose
       File tab-> SymLink to "/usr/lib/mozilla/plugins" folder.

## OpenJava

    Open source Java version installation:
    and it's web browser plug-in:

## Verification

    After installation you can check is your Java installed correctly or is the latest version: java.com
    Java home page: http://www.oracle.com/technetwork/java/index.html
    And on Wikipedia: http://en.wikipedia.org/wiki/Java_%28programming_language%29