OpenLLM-France/wikimedia huggingface dataset:
# Conic Sections/Ellipse

The geometric definition of an ellipse is the locus of a point which moves in a plane such that the sum of its distances from the two points called foci add up to a constant(greater than the distance between the said foci). It can also be defined as a conic where the eccentricity is less than one. Ellipses have two directrices, one on each side.

## Graphing an Ellipse

The general equation for an ellipse where its major, or longer, axis is horizontal is : ${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$. Where the major axis is vertical, it is: ${\frac {(y-k)^{2}}{a^{2}}}+{\frac {(x-h)^{2}}{b^{2}}}=1$
* The center is located at $(h,k)$.
* The foci are found at a distance of $c$ from the centre along the major axis, where $c={\sqrt {a^{2}-b^{2}}}$.
* The eccentricity of the ellipse can be found from the formula: $b^{2}=a^{2}(1-e^{2})$ where e is eccentricity. A higher eccentricity makes the curve appear more 'squashed', whereas an eccentricity of 0 makes the ellipse a circle.
* The directrices are the lines $x=\pm {\frac {a}{e}}$
* The major axis has a length of $2a$ and the minor one $2b$.
* The sum of the distance from each point to each of the foci is $2a$.

## Applications

* Assuming a perfectly reflective perfect ellipse, any light emitted from a focus of an ellipse will be reflected to intercept the other focus. This property is exploited in "Whispering Galleries" where 2 people stand at the foci of an arbitrarily large elliptical room and can clearly hear each other whispering.
* The reflective property is also useful in medicine, where it is used to treat gall stones. A machine emits sonic waves at one of the foci in an elliptical bowl, and the area of the patient where the gall stones are located is placed at the other focus. The waves are reflected and focused on the stones, causing them to be broken down and become small enough to pass from the body.
* The orbit of every planet is an ellipse with the Sun at one of the two foci (Kepler's first law). The orbits of objects in space are mostly elliptical in shape.

## References

* http://www.purplemath.com/modules/ellipse.htm
# Conic Sections/General Conic Sections

The general form of any conic section is $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$. In most cases, B=0 because any other value would cause the conic to be rotated and makes analysis much more difficult. For the rest of this page we will assume that B=0. The general equation can be algebraically manipulated to become the standard form of the specific conic it describes. However, it would first be useful to know what that conic is. That can be determined simply by comparing A and C. If either A or C, but not both, equals 0, the equation describes a parabola. If A=C, it describes a circle. If A and C are both positive or both negative, but not equal, it is an ellipse. If A and C have opposite signs, it is a hyperbola. If A=C=0, it's obviously just a line.
# Conic Sections/Types of Conic Sections

A conic section is defined as the locus of all points where the distance from the point to the focus and the point to a straight line known as the directrix is a constant ratio. This ratio is known as the eccentricity of the section.
* If the ratio is 1, the section is known as a parabola.
* If the ratio is less than one, the section is an ellipse. A circle is a special case of an ellipse.
* If the ratio is greater than one, the section is hyperbola.
All conic sections can be formed by taking the intersection of a cone and a plane, hence the name.
# Conic Sections/Rotation of Axes
## Introduction

A conic is a second degree polynomial. When it is expanded, you get an equation of the form: $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$. If the value of $B$ is zero then the conic is not rotated and sits on the x- and y- axes. If $B$ is non-zero, then the conic is rotated about the axes, with the rotation centred on the origin.

## Graphing
### Graphing a Rotated Conic

If you are asked to graph a rotated conic in the form $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$, it is first necessary to transform it to an equation for an identical, non-rotated conic. This is then plotted onto new axes which are drawn onto the graph. The equation for the nonrotated conic can be found by: $AX^{2}+BXY+CY^{2}+DX+EY+F=0$. Note how capital letters are used for the pronumerals. This signifies that they represent different values to the original equation.
To determine the values of X and Y, you use the formulae:
* $\tan 2\theta ={\frac {B}{A-C}}$
* $x=X\cos \theta -Y\sin \theta$
* $y=X\sin \theta +Y\cos \theta$
By substituting the new values of $x$ and $y$ into the original equation, a new one can be obtained which represents a non-rotated conic which can be plotted on a set of axes rotated at $\theta$ (anti-clockwise) to the original x- and y- axes. When you do this, however, it will still be necessary to determine the new rotated location of points such as the vertex, foci and directrixes. This can be done using the following formulae:
* $X=x\cos \theta +y\sin \theta$
* $Y=-x\sin \theta +y\cos \theta$
Where (X,Y) are the new rotated coordinates of the original point (x,y).
The formula: $B^{2}-4AC$ can be used to determine the type of conic from the original equation before you start graphing:
* $B^{2}-4AC=0$: Parabola
* $B^{2}-4AC<0$: Ellipse
* $B^{2}-4AC>0$: Hyperbola

### Rotating a Conic

If you wish to rotate a conic by a certain angle, $\theta$, it is relatively simple. All you do is make the following substitution from the previous section:
* $x=X\cos \theta -Y\sin \theta$
* $y=X\sin \theta +Y\cos \theta$
Replacing the $x$ and $y$ values from the function with these new ones. Then simplify the answer, and it will be a function for the same conic rotated by $\theta$ counter-clockwise about the origin.

## References

* http://www.mecca.org/~halfacre/MATH/rotation.htm
* http://www.sparknotes.com/math/precalc/conicsections/section5.rhtml
# Conic Sections/Lines and Points

Lines and points are referred to as "degenerate conic sections," and are not commonly associated with conics. They are still conic sections, though, for the reason that they can be formed from plane intersections with a conic—but the difference with these conics is that the conic section shares a center with the conic itself.
2 Intersecting Lines
Picture a hyperbolic conic section. Now move that hyperbola over until the center of the hyperbola coincides with the center of the conic. The conic section formed will be a pair of intersecting lines.
1 Line
Picture a parabolic conic section. Now move the conic section over until the vertex of the parabola coincides with the center of the conic. The resultant conic section will be a single line.
1 Point
The aforementioned degenerate conic section is unique in that it can be formed from two normal conic sections. Picture either an elliptical or a circular conic section. Now slide the conic section over until the center of the section coincides with the center of the conic. The degenerate conic section that results from this movement is a single point—the center of the conic.
# Conlang/Advanced

The Advanced level of this book is for those familiar with the internal structure of the elements of language, who want to learn how to create sophisticated, unusual, and even really exotic conlangs. It aims to introduce you to language features and techniques that can be used to create conlangs with extra flair.

## At the edge

At the Beginning level, everything was fresh and exciting. We had something of an overview of the subject, because we didn't have enough details for us to lose track of the forest amidst the trees. At the Intermediate level, as we were learning the names for features of the landscape, the big picture got harder to see, and we were fighting a tendency for things to seem overly cut-and-dried.
Now we're at the Advanced level, and the character of this level is different again. While we can never fully recapture that sense of wholeness from the Beginner level, at this level things not only interconnect, but often seem to start bleeding into each other. There are folks who present a neatly ordered, cut-and-dried picture at this level, and they may explain — truthfully — that all their close colleagues agree on these things; but if so, it's because all their close colleagues belong to a close-knit school sharing that view. You may find other folks presenting ordered pictures inconsistent with that one; and the different schools may use different sets of advanced jargon, with similar-sounding names that sometimes mean very different things, or very different-sounding names for substantially the same thing.
This is the realm of controversies.
For conlangers looking to innovate, this can also be a treasure trove. Exactly because of the controversies, and the blurring of distinctions, there's wide-open scope for modifying, remixing, and repurposing features, breaking out of some molds altogether while creating fresh takes on familiar themes. It's certainly possible for an expert in a subject to get hide-bound, rigid in their thinking; but this is also where you can improvise new approaches that are not just different, but persuasively different.

## What's in this level?

Each section describes features and techniques in a different aspect of language design. The sections are:
* Phonology: Unusual ways to make speech sounds.
* Grammar: Unusual ways to form and arrange words.
* The link between a Language and its Culture.
* Metaphor and its applications for Conlanging.
# Conic Sections/Hyperbola

A hyperbola can be defined as the locus of all points such that the difference between the distance of each point from two points known as the foci is constant. This is similar to an ellipse, however whereas in an ellipse the distances sum to a constant, here they subtract to one. A hyperbola can also be defined as the locus of all points such that the distance of the point to the closest focus divided by the distance to the directrix, a ratio known as the eccentricity, is a constant greater than one.

## Graphing Hyperbolae

The general form of a hyperbola with a horizontal transverse axis, i.e. the axis on which the foci lie, is: ${\frac {(x-h)^{2}}{a^{2}}}-{\frac {(y-k)^{2}}{b^{2}}}=1$ A hyperbola with a vertical transverse axis, on the other hand, is of the form: ${\frac {(y-k)^{2}}{a^{2}}}-{\frac {(x-h)^{2}}{b^{2}}}=1$.
* The centre of the hyperbola is $(h,k)$.
* The two foci are found at a distance of $c$ from the centre in each direction along the transverse axis, where $c={\sqrt {a^{2}+b^{2}}}$.
* The eccentricity of the hyperbola can be found by ${\frac {c}{a}}$. A higher eccentricity makes the hyperbola 'steeper', whereas a smaller one makes it more 'curvy'.
* The vertices are a distance of $a$ from the centre of the hyperbola in each direction along the transverse axis.
* The difference between the distances of any point from each of the foci is equal to $2a$.
* Hyperbola also have two asymptotes. An asymptote is a line that points on the graph continuously approach yet never reach. If you extended the graph vast distances along each axis, you would observe it getting closer and closer to the asymptote, however it would never actually reach it. The asymptotes can be found by $y=k\pm {\frac {b}{a}}(x-h)$ for a horizontal hyperbola, or $y=k\pm {\frac {a}{b}}(x-h)$ for a vertical one.

## References

* http://www.sparknotes.com/math/precalc/conicsections/section4.rhtml
* http://www.khanacademy.com
* http://www.purplemath.com/modules/hyperbola.htm
# Econometric Theory/A Moving-Average (MA) Process

A moving average (MA) process is essentially created by lagging the residual term of the time series. For example a time series x with an autoregressive (AR) process of length 1 and a moving average process of length 2 can be expressed as x(t) = x(t-1) + e(t) + e(t-1) + e(t-2). In general, a time series can be represented by an ARMA(p,q) process, where p denotes the length of the autoregressive process, while q denotes the length of the moving average process.
# Econometric Theory/Definitions
## Identity Matrix

The identity matrix, with a size of n, is an n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is commonly denoted as $I_{n}$, or simply by I if the size is immaterial or can be easily determined by the context.
    $I_{1}=[1]\quad I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\quad I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}$
The most important property of the identity matrix is that, when multiplied by another matrix, A, the result will be A
    $AI_{n}=A\,$ and $I_{n}A=A\,$.
# Econometric Theory/F-Test

An F-test involves the computation of an F-statistic, which is then compared to the critical values of an F-distribution for a given significance and numerator and denominator degrees-of-freedom.
An F-statistic is calculated by dividing a chi-squared distribution divided by its degrees-of-freedom by another (independent) chi-squared distribution by its degrees-of-freedom. The resulting F-statistic has two degrees-of-freedom parameters, one each for the numerator and the denominator.
Therefore, the F-statistic for ${\hat {\beta _{1}}}$ would be:
We know (somehow) that $[Z(0,1)]^{2}=\chi ^{2}[1]$, therefore we set the numerator equal to:
$Z({\hat {\beta _{1}}})^{2}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}(\sum X_{i}^{2})}{\sigma ^{2}}}\sim \chi ^{2}[1]={\frac {\chi ^{2}[1]}{1}}$
From the same implication of the last assumption of the CLRM as used by the t-test explanation,
${\frac {\chi ^{2}[N-2]}{N-2}}\sim {\frac {\hat {\sigma ^{2}}}{\sigma ^{2}}}$
Therefore, putting it all together gives us: $F({\hat {\beta _{1}}})={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}(\sum X_{i}^{2})/\sigma ^{2}}{{\hat {\sigma ^{2}}}/\sigma ^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {\sigma ^{2}}}/\sum X_{i}^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {Var}}({\hat {\beta _{1}}})}}\sim F[1,N-2]$
# False Friends of the Slavist/Map vermę

Semasiological map for *vermę
R. время • Blr. Ø • Ukr. верем'я • Pol. Ø • Ka. Ø • Slk. Ø • Sln. vreme • Cr. vrijeme • Bo. vrijeme • Sb. време • Mk. време • Bg. време •
| *vermę | Meanings: ‘time’ ‘weather’ ‘tense (grammar)’   no such word (For areas without shading there is no information yet.) |
| *vermę | Meanings | Meanings | Meanings |
| *vermę | time | weather | tense (grammar) |
| R. время | + | + | + |
| Blr. Ø | | | |
| Ukr. верем'я | | + | |
| Pol. Ø | | | |
| Ka. Ø | | | |
| Slk. Ø | | | |
| Sln. vreme | | + | |
| Cr. vrijeme | + | + | + |
| Bo. vrijeme | + | + | + |
| Sb. време | + | + | + |
| Mk. време | | + | + |
| Bg. време | + | + | + |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map von

Semasiological map for *vonь(je)
R. вонь • Blr. Ø • Ukr. Ø • Pol. woń • Ka. wòniô • LSo. wóń • USo. wóń • Cz. vůně • Slk. vôňa • Sln. vonj • Bg. воня •
| *vonь(je) | Meanings: ‘stench, stink’ ‘scent, fragrance’ ‘unpleasant odour’   no such word (For areas without shading there is no information yet.) |
| *vonь(je) | Meanings | Meanings | Meanings |
| *vonь(je) | stench, stink | scent, fragrance | unpleasant odour |
| R. вонь | + | | |
| Blr. Ø | | | |
| Ukr. Ø | | | |
| Pol. woń | | + | |
| Ka. wòniô | | + | |
| LSo. wóń | | + | + |
| USo. wóń | | + | |
| Cz. vůně | | + | |
| Slk. vôňa | | + | |
| Sln. vonj | | + | + |
| Bg. воня | + | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vozъ

Semasiological map for *vozъ
R. воз • Blr. воз • Ukr. віз(1), візок(4) • Pol. wóz(1), wózek(4) • Ka. wòz (1), wòzyk (4) • Cz. vůz • Slk. voz • Sln. voz • Cr. voz • Bo. voz • Sb. воз • Mk. воз • Bg. ∅ •
| *vozъ | Meanings: ‘cart’ ‘wagon’ ‘carriage (horse-drawn)’ ‘pram (baby carriage)’ ‘car (automobile)’ ‘train’   no such word (For areas without shading there is no information yet.) |
| *vozъ | Meanings | Meanings | Meanings | Meanings | Meanings | Meanings |
| *vozъ | cart | wagon | carriage (horse-drawn) | pram (baby carriage) | car (automobile) | train |
| R. воз | + | | | | | |
| Blr. воз | + | | | | | |
| Ukr. віз(1), візок(4) | + | | | + | | |
| Pol. wóz(1), wózek(4) | + | | | + | | |
| Ka. wòz (1), wòzyk (4) | + | | | + | | |
| Cz. vůz | | | + | | + | |
| Slk. voz | | + | | | + | |
| Sln. voz | + | | | | | |
| Cr. voz | + | + | | | | |
| Bo. voz | | | | | | + |
| Sb. воз | | | | | | + |
| Mk. воз | | | | | | + |
| Bg. ∅ | | | | | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vek

Semasiological map for *věkъ
R. век • Blr. век • Ukr. вік • Pol. wiek • Ka. wiek • LSo. wěk • USo. wěk • Cz. věk • Slk. vek • Sln. vek • Cr. vijek • Bo. vijek • Sb. век • Mk. век • Bg. век •
| *věkъ | Meanings: ‘age (of a person)’ ‘age, epoch’ ‘century’ ‘lifetime’ ‘eternity’   no such word (For areas without shading there is no information yet.) |
| *věkъ | Meanings | Meanings | Meanings | Meanings | Meanings |
| *věkъ | age (of a person) | age, epoch | century | lifetime | eternity |
| R. век | | + | + | + | |
| Blr. век | + | + | | | |
| Ukr. вік | + | + | + | + | + |
| Pol. wiek | + | + | + | | |
| Ka. wiek | + | + | | | |
| LSo. wěk | | | + | | + |
| USo. wěk | | + | + | | |
| Cz. věk | + | + | + | | |
| Slk. vek | + | + | | | |
| Sln. vek | | + | | | |
| Cr. vijek | | + | + | + | |
| Bo. vijek | | + | + | + | |
| Sb. век | | + | + | + | |
| Mk. век | | + | + | + | |
| Bg. век | | + | + | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vychod

Semasiological map for *vychodъ
R. выход • Blr. выхад • Ukr. вихід • Pol. Ø • Ka. wëchód • LSo. wuchod • USo. wuchod • Cz. východ • Slk. východ • Sln. Ø • Cr. Ø • Bo. Ø • Sb. Ø • Mk. Ø • Bg. Ø •
| *vychodъ | Meanings: ‘east’ ‘exit’ ‘low tide’ ‘river mouth’   no such word (For areas without shading there is no information yet.) |
| *vychodъ | Meanings | Meanings |
| *vychodъ | east | exit | low tide | river mouth |
| R. выход | | + | | |
| Blr. выхад | | + | | |
| Ukr. вихід | | + | | |
| Pol. Ø | | | | |
| Ka. wëchód | | + | + | + |
| LSo. wuchod | | + | | |
| USo. wuchod | + | + | | |
| Cz. východ | + | + | | |
| Slk. východ | + | + | | |
| Sln. Ø | | | | |
| Cr. Ø | | | | |
| Bo. Ø | | | | |
| Sb. Ø | | | | |
| Mk. Ø | | | | |
| Bg. Ø | | | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map za

Semasiological map for *za
R. за(1,2,4), из-за(3) • Blr. за(1,2,4), з-за(3) • Ukr. за • Pol. za • Ka. za • LSo. za • USo. za • Cz. za • Slk. za • Sln. za • Cr. za • Bg. за •
| *za | Meanings: ‘behind’ ‘for’ ‘because of’ ‘during’ ‘after’ ‘like, as’ ‘in (time)’   no such word (For areas without shading there is no information yet.) |
| *za | Meanings | Meanings | Meanings | Meanings | Meanings | Meanings | Meanings |
| *za | behind | for | because of | during | after | like, as | in (time) |
| R. за(1,2,4), из-за(3) | + | + | + | + | | | |
| Blr. за(1,2,4), з-за(3) | + | + | + | + | | | |
| Ukr. за | + | + | | + | | | + |
| Pol. za | + | | | + | | | + |
| Ka. za | + | + | | | + | + | + |
| LSo. za | + | | | | | + | + |
| USo. za | + | | | | | | + |
| Cz. za | + | + | | + | | | + |
| Slk. za | + | + | | + | | | + |
| Sln. za | + | + | | | + | | |
| Cr. za | | + | + | + | + | | |
| Bg. за | | + | | | + | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vorg

Semasiological map for *vorgъ
R. враг • Blr. вораг • Ukr. ворог • Pol. wróg • Ka. warg • Cz. vrah • Slk. vrah • Sln. vrag • Cr. vrag • Bo. vrag • Sb. враг • Mk. враг • Bg. враг •
| *vorgъ | Meanings: ‘enemy’ ‘devil, demon’ ‘murderer’ ‘slyboots, smartie’   no such word (For areas without shading there is no information yet.) |
| *vorgъ | Meanings | Meanings | Meanings | Meanings |
| *vorgъ | enemy | devil, demon | murderer | slyboots, smartie |
| R. враг | + | | | |
| Blr. вораг | + | | | |
| Ukr. ворог | + | | | |
| Pol. wróg | + | | | |
| Ka. warg | + | | | |
| Cz. vrah | | | + | |
| Slk. vrah | | | + | |
| Sln. vrag | | + | | |
| Cr. vrag | | + | | |
| Bo. vrag | | + | | |
| Sb. враг | | + | | + |
| Mk. враг | + | + | | + |
| Bg. враг | + | | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vredny

Semasiological map for *verdьnъ
R. вредный • Blr. Ø • Ukr. Ø • Pol. wredny • Cz. Ø • Slk. Ø • Sln. vreden • Cr. vrijedan • Bo. vrijedan • Sb. вредан • Mk. вреден • Bg. вреден •
| *verdьnъ | Meanings: ‘harmful’ ‘precious’ ‘able, clever’ ‘nasty, repulsive’ ‘insidious, deceitful’ ‘hard-working’ ‘worth’   no such word (For areas without shading there is no information yet.) |
| *verdьnъ | Meanings | Meanings | Meanings | Meanings | Meanings | Meanings | Meanings |
| *verdьnъ | harmful | precious | able, clever | nasty, repulsive | insidious, deceitful | hard-working | worth |
| R. вредный | + | | | | | | |
| Blr. Ø | | | | | | | |
| Ukr. Ø | | | | | | | |
| Pol. wredny | | | | + | + | | |
| Cz. Ø | | | | | | | |
| Slk. Ø | | | | | | | |
| Sln. vreden | | | | | | | + |
| Cr. vrijedan | | + | | | | + | |
| Bo. vrijedan | | + | | | | + | |
| Sb. вредан | | + | | | | + | |
| Mk. вреден | | | + | | | | |
| Bg. вреден | + | | | | | | |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map vygoda

Semasiological map for
R. выгода • Blr. выгода • Ukr. виго́да(1), ви́года(2) • Pol. wygoda • Ka. wigòda • Cz. výhoda • Slk. výhoda • Bg. изгода •
* Meanings:
* ‘comfort’
* ‘profit, advantage’
* no such word
* (For areas without shading there is no information yet.)
| | Meanings |
| | comfort | profit, advantage |
| R. выгода | | + |
| Blr. выгода | + | |
| Ukr. виго́да(1), ви́года(2) | + | + |
| Pol. wygoda | + | |
| Ka. wigòda | + | |
| Cz. výhoda | | + |
| Slk. výhoda | | + |
| Bg. изгода | | + |
| < Index of semasiological maps | < False Friends of the Slavist Homepage |
# False Friends of the Slavist/Map veselьje

Semasiological map for *veselьje
R. веселье • Blr. вяселле • Ukr. весілля • Pol. wesele • Cz. veselí • Slk. veselie • Sln. veselje • Cr. veselje • Bo. veselje • Sb. весеље • Mk. веселие • Bg. веселие •
| *veselьje | Meanings: ‘fun, joy, happiness’ ‘wedding’   no such word (For areas without shading there is no information yet.) |
| *veselьje | Meanings | Meanings |
| *veselьje | fun, joy, happiness | wedding |
| R. веселье | + | |
| Blr. вяселле | | + |
| Ukr. весілля | | + |
| Pol. wesele | | + |
| Cz. veselí | + | |
| Slk. veselie | + | + |
| Sln. veselje | + | |
| Cr. veselje | + | |
| Bo. veselje | + | |
| Sb. весеље | + | |
| Mk. веселие | + | |
| Bg. веселие | + | |
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