OpenLLM-France/wikimedia huggingface dataset:
# Permutations and partitions in the OEIS

Permutations of a finite set are often sorted lexicographically, but reverse colexicographical order gives an infinite order of finite permutations. (In this order the inversion vector of the n-th permutation interpreted as a factorial number is n.) The finite permutations are stored as rows of a triangle in A055089.
Partitions of a finite set and partitions of an integer are also often ordered lexicographically, but the sequences A231428 and A194602 define an infinite order.
These infinite orderings make it possible to define some mappings as sequences:
Each permutation corresponds to a set partition (A249618), and more importantly to an integer partition (A198380) called its cycle type. (A set partition corresponds only to a finite number of permutations, but each cycle type > 0 corresponds to an infinite number of permutations.) Each set partition corresponds to an integer partition (A249617) - and each integer partition > 0 to an infinite number of set partitions.
Furthermore each permutation and partition can be assigned a number that for simplicity's sake is called "blocks" in the diagram: For permutations these are the cycles except fixed points (A055090), for set partitions the non-singleton blocks (A249615) and for integer partitions the non-one addends (A194548).
And of course they can be assigned the total number of elements in these "blocks": A055093, A249616 and a staircase like sequence that is too boring to be in the OEIS.
| perm. | set part. | int. part. | Mappings between OEIS sequences related to permutations, set partitions and integer partitions |
| A198380 cycle type of the n-th permutation A055089 n-th permutation as triangle row A249618 set partition of the n-th permutation A231428 n-th set partition interpreted as binary number A249617 integer partition of the n-th set partition A194602 n-th integer partition interpreted as binary number A055090 number of cycles in the n-th permutation A055093 number of elements moved by the n-th permutation A249615 number of non-singleton blocks in the n-th set partition A249616 number of non-singleton elements in the n-th set partition A194548 number of non-one addends in the n-th integer partition a(0)=0; p=A000041; n≥1: p(i−1)≤n<p(i) ⇒ a(n)=i sum of non-one addends in the n-th integer partition ||||

## Equalities

| pe2ip = sp2ip( pe2sp ) | A198380 = A249617( A249618 ) |
| pe2bl = sp2bl( pe2sp ) = ip2bl( sp2ip( pe2sp ) ) | A055090 = A249615( A249618 ) = A194548( A249617( A249618 ) ) |
| sp2bl = ip2bl( sp2ip ) | A249615 = A194548( A249617 ) |
| pe2el = sp2el( pe2sp ) = ip2el( sp2ip( pe2sp ) ) | A055093 = A249616( A249618 ) = ip2el( A249617( A249618 ) ) |
| sp2el = ip2el( sp2ip ) | A249616 = ip2el( A249617 ) |
For simplicity's sake it is assumed that A231428 and A194548 have index 0 like all the other sequences, although at the moment they have index 1 in the OEIS.
# Perquin's Life Purposes

Every person in the world has set a purpose for their life. According to the theory of S. Perquin, life purposes can be roughly divided into four domains: mental-individualistic, mental-collectivistic, physical-collectivistic and physical-individualistic.
* Mental-individualists are motivated by a desire for self-fulfillment through cognitive and emotional enrichment, seeking to enhance their inner lives, expand their knowledge and achieve personal growth.
* Mental-collectivists are motivated by the aspiration to improve the lives of others through mental or emotional support, often pursuing goals that involve teaching, mentoring or guiding others to help them grow and reach their potential.
* Physical-collectivists are motivated by a desire to enhance the well-being of others through practical, tangible means, working to provide basic needs, improve living conditions or contribute to the physical health and safety of their communities.
* Physical-individualists are motivated by a drive to achieve tangible success and material comfort, focusing on attaining wealth, physical health and personal accomplishments that reflect their success in the external world.
# Personal computers

This is a lesson in in the course, Introduction to Computer Science, which is a part of The School of Computer Science

## Objective

Personal computers revolutionized the world in a multitude of ways. After this lesson about the history of personal computers, students should be able to answer these questions:
* What is a personal computer?
* How is it different from computers of the previous generation?
* How have our lives and the world at large been influenced by the PC?

## Contents

To quote Wikipedia, "A personal computer (PC) is a computer whose original sales price, size, and capabilities make it useful for individuals."
Personal computers are computers that are used by one person at a time. Most computers in existence today are personal computers. Personal computers are most commonly used at home or in an office, but they can be found elsewhere, especially if the personal computer in question is a laptop. The concept of a personal computer began in the 1970s with important individuals involved in academics or research owning a personal computer for use in the workplace. Prior to this, computers were massive and extremely expensive machines that were only used by large companies, universities, and the government. These computers, obviously, were too expensive for most people to purchase, but all that changed in 1975 with the invention of the microprocessor. This product allowed computers to be much smaller and produced at a much cheaper cost, however these first computers with microprocessors were not very easy to use. The use of personal computers continued to grow throughout the 1980s with more development making computers cheaper, smaller, and easier to use. In the 1990s, computers became even more powerful causing them to be mainstream in many homes. Today, computers are better than ever and are used by people for work, games, information, and other things.

## See also:

* Personal Computers
# Permutation group/Number/2/Introduction/Section
## Definition

For a set $M$, we call the set
    $\operatorname {Aut} \,(M)=\operatorname {Perm} \,(M)={\left\{\varphi :M\longrightarrow M\mid \varphi {\text{ bijective}}\right\}}\,$
of all bijective mappings
$M$
$M$
The operation is the composition of mappings, therefore, it is associative, the identity is the neutral element. The inverse element for a bijective mapping is just the inverse mapping. Hence, this is a group. A bijective mapping $\varphi \colon M\rightarrow M$ is also called a permutation.
For the finite set $I=\{1,\ldots ,n\}$, we also write
    $S_{n}=\operatorname {Perm} \,(I)\,.$
A permutation of a finite set can be described with a (complete) value table or with an arrow diagram.

## Definition

Let $M$ be a finite set and let $\pi$ be a permutation on $M$. Then $\pi$ is called a cycle of order $r$, if there exists a subset $Z\subseteq M$, containing $r$ elements and such that $\pi$ is on $M\setminus Z$ the identity and such that $\pi$ commutes the elements of $Z$ in a cyclic way. If $Z=\{z,\pi (z),\,\pi ^{2}(z),\ldots ,\pi ^{r-1}(z)\}$, then we write
    $\pi =\langle z,\pi (z),\,\pi ^{2}(z),\ldots ,\pi ^{r-1}(z)\rangle \,.$

## Example

We consider the permutation
We can write this as the product of the two cycles $\langle 1,2\rangle$ and $\langle 3,5,4\rangle$.
An element $x\in M$ with $\pi (x)=x$ is called a fixed point of the permutation. The (action) scope of a permutation is the set of points from $M$ which are not fixed points. For a cycle, the set $Z$ is the scope. We mention without proof that every permutation is a product of cycles. Such a product representation is called a cycle representation.

## Definition

For a natural number $n$, one puts
    $n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,$
$n$

## Lemma

Let $M$ be a finite set with $n$ elements. Then the permutation group
    $\operatorname {Perm} \,(M)\cong S_{n}\,$
$n!$

### Proof

Let $M=\{1,\ldots ,n\}$. For $1$, there are $n$ possible images, for $2$, there are $n-1$ possible images remaining, for $3$, there are $n-2$ possible images remaining, etc. Therefore, there are altogether
    $n(n-1)(n-2)\cdots 2\cdot 1=n!\,$
possible permutations.
$\Box$
# Personal development

Personal development covers activities that improve awareness and identity, develop talents and potential, build human capital and facilitate employability, enhance quality of life and contribute to the realization of dreams and aspirations. Not limited to self-help, the concept involves formal and informal activities for developing others in roles such as teacher, guide, counselor, manager, life coach or mentor. When personal development takes place in the context of institutions, it refers to the methods, programs, tools, techniques, and assessment systems that support human development at the individual level in organizations.

## Readings

* Wikipedia: Personal development

## Resources

* Personal Development Planning: A Summary of Research and Evaluation
* Landmark Education
* Hoffman Institute
# Trigonometry/Polar

While SohCahToa is a popular mnemonic, it is better to define three essential trig functions using polar coordinates. The figure shows how the sine and cosine functions can be defined for all θ, i.e., angles outside the first quadrant (0<θ<π/2).
| $x=r\cos \theta$ | $y=r\sin \theta$ |
This calculates x and y if r and θ are known.
       Example: Suppose θ is in the second quadrant, e.g., θ=3π/4, we see that x<0, and hence cos(3π/4)<0. On the other hand, sin(3π/4)>0 in the second quadrant, since y>0 for this value of θ.
The following calculates r and θ if x and y are known:
| $r={\sqrt {x^{2}+y^{2}}}$ | $\theta =\tan ^{-1}{\frac {y}{x}}$ |
The tangent and its inverse function are defined by
| $\tan \theta ={\frac {y}{x}}$ | $\tan ^{-1}\left({\frac {y}{x}}\right)=\theta$ |
This simple relation between the tangent and its inverse holds only in the first quadrant (0<θ<π/2) because the inverse function is not well defined for all angles. Also,it must be emphasized that the exponent $-1$ on the $\tan$ function does NOT represent the multiplicative inverse:
    $\tan ^{-1}\theta \neq {\frac {1}{\tan \theta }}$
For this reason, some authors avoid the superscript "-1" and instead write the arctangent as arctan:
    $\theta =\tan ^{-1}{\frac {y}{x}}\equiv \arctan {\frac {y}{x}}$
Just as lines of constant x and y are used to illustrate a Cartesian coordinate system contours of constant r and θ are used to depict polar coordinates as shown above and to the right.
In physics these equations are often used to describe the components of a vector. If A$_x$ and A$_y$ are the components of the vector A:
| $A_{x}=A\cos \theta$ | $A_{y}=A\sin \theta$ |
| $A={\sqrt {A_{x}^{2}+A_{y}^{2}}}$ | $\theta =\tan ^{-1}{\frac {A_{y}}{A_{x}}}\Leftrightarrow \tan \theta ={\frac {A_{y}}{A_{x}}}$ |
# Trigonometry/Spherical

Spherical trigonometry relates to spherical triangles, and trihedral angles. A spherical triangle is a triangle projected on a sphere. Spherical trigonometry has applications in navigation, and astronomy.

## Definitions and explanations

A lune is a shape on a sphere from one angle merged to an angle on the opposite side of the sphere. It is formed by two circles on a sphere. The area between two lines of longitude on one side of a globe are an example of a lune. A digon or a bi-angle are other terms for a lune.
A lune is composed of two spherical triangles, when split in half equidistant from each angle. These two congruent spherical triangles are called colunar triangles.
Unlike plane trigonometry, in spherical trigonometry it is possible for a spherical triangle to have up to three obtuse or right angles.
A birectangular triangle has two right angles.
A trirectangular triangle has three right angles.
Polar triangle are triangles with angles at the center of the sphere, and they are in relation to spherical triangles.
# Trombone

The trombone is a brass instrument that, like many other instruments, comes in several variations. The most common types are variations of the tenor and the bass trombones.
Types of Trombones:
* Tenor Trombone
* Bass Trombone
* Contrabass Trombone
* Alto Trombone
* Soprano Trombone
* Sopranino and Piccolo Trombones
* Valve Trombone
The trombone is an instrument where the trombonist moves the slide between positions, each of which progressively lengthens the air column, and produces a lower pitch on the same partial. Positions range from first (almost all the way pulled in, but not all the way) to seventh, with first being the higher within each partial. The basic tenor trombone's fundamental note is Bb.
The trombone is mostly a non-transposing instrument with music written in the bass, tenor, alto, and treble clefs.

## Technique

Playing on the trombone is similar to other brass instruments in the method of moving from partial to partial. The trombonist moves their lips tighter and allows air pressure to allow the instrument to increase in partials, the opposite is done to play down partials.
Sometimes music will require the use of a mute. Mutes will alter the sound of the note by somehow attaching to the bell of the trombone. There are a variety of mutes available, and it is important to ensure you purchase the correct type.
# Polynomials/K/Product of linear polynomials and zeroes/Fact/Proof/Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Show that every polynomial $P\in K[X]$, $P\neq 0$, can be decomposed as a product
    $P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,$
where $\mu _{j}\geq 1$ and $Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers $\lambda _{1},\ldots ,\lambda _{k}$ and the exponents $\mu _{1},\ldots ,\mu _{k}$ are uniquely determined apart from the order.
# Polynomials/Zero in -2,3/Degree up to 4/Dimension/Exercise

Show that the set of real polynomials of degree $\leq 4$ which have a zero at $-2$ and a zero at $3$ is a finite dimensional subspace of $\mathbb {R} [X]$. Determine the dimension of this vector space.
# Pomology/Introduction and data sources

< Wikiversity
Part of the Department of Pomology, Tropical and Subtropical Pomology I.
* Last lesson:
* Next lesson: Fruit and its importance
This lesson is in preparation --Juan 18:19, 6 December 2006 (UTC)

## Discussion Record
### Questions and misunderstandings
### Theme discussion

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# Pomology

Pomological research is mainly focused on the development, cultivation and physiological studies of fruit trees. The goals of fruit tree improvement include enhancement of fruit quality, regulation of production periods, and reduction of production cost.

## Resources

* Pomology Teaching Tools
* Botany Ink Drawing - how to make botanical drawings
* Tropical and Subtropical Pomology I - contain lessons with the most important fruits of the world=under-graduate level.
* Tropical and Subtropical Pomology II - contain lessons with important fruits to the regional markets=under-graduate level.
* Tropical and Subtropical Pomology III - contain lessons with less important fruits of tropics and subtropics=under-graduate level.
* Special Tropical and Subtropical Pomology - it contains the knowledge of Tropical and Subtropical Pomology I-III, but going deeper (there are more projects). On the site of theoretical studying participants can also study practical things by their own or with a teacher. They are also doing a primary research (=literature survey)=graduate level.
* Ethnobotany of Fruit Crops - this subject tries to cover the fruit crops with a local importance and focuses on their ethnobotany. Students are working upon research ethnobotanical projects (firstly needs to establish Wikiversity research policy!) according their interests. Opening a new project is free and also the working teams
* Banana Production - highly specialized subject on the most consumed fruit of the world.
* Citriculture - highly specialized subject on the most consumed fruit of the subtropics.

## Wanted Subjects/Lessons

Temperate Tree Fruit Pomology
# Pomology/Fruit and its importance

< Wikiversity Part of the Department of Pomology, Tropical and Subtropical Pomology I.
* Last lesson: Introduction and data sources
* Next lesson: Methods of propagations and treatments to fruit crops
This lesson is in preparation --Juan 18:19, 6 December 2006 (UTC)

## Fruits and their importance

The dried fruit of Berberis vulgaris (barberry) is used in herbal medicine. The chemical constituents include isoquinolone alkaloids, especially berberine, with a full list of phytochemicals compiled.
"Carotenoids, which are generally present in leaf, flower, fruit, and shoot of plants, play an important role in the stabilization of lipid membranes, the photosynthesis, and the protecion against strong radiation and photooxidative processes. Experiments with coffee species also showed that the transcript levels of enzymes involved in the synthesis of carotenoids increased under stress conditions [34]."
"Coffee seeds are rich in biologically active substances and polyphenols such as kaempherol, quercetin, ferulic, sinapic, nicotinic, quinolic, tannic, and pyrogallic acids which possess antioxidant, hepatoprotective, antibacterial, antiviral, anti-inflammatory, and hypolipidaemic effects [41], [42], [43], [44], [45], [46], [47], [48], [49]. Besides the cis-isomers of chlorogenic acid in Arabic coffee [50], caffeic, chlorogenic, p-coumaric, ferulic, and sinapic acids, as well as rutin, quercetin, kaempferol, and isoquercitrine were detected in its fruit and that of Bengal coffee [51]."
Fennel is widely cultivated, both in its native range and elsewhere, for its edible, strongly flavored leaves and fruits. Its aniseed or liquorice flavor comes from anethole, an aromatic compound also found in anise and star anise, and its taste and aroma are similar to theirs, though usually not as strong.
The aromatic character of fennel fruits derives from essential oil (volatile oils) imparting mixed aromas, including trans-anethole and estragole (resembling liquorice), fenchone (mint and camphor), limonene, 1-octen-3-ol (mushroom). Other phytochemicals found in fennel fruits include polyphenols, such as rosmarinic acid and luteolin, among others in minor content.
Prunus cerasus (sour cherry, tart cherry, or dwarf cherry) a species of Prunus in the subgenus Prunus subg. Cerasus (cherries), native to much of Europe and southwest Asia is closely related to the sweet cherry (Prunus avium), but has a fruit that is more acidic. Its sour pulp is edible.

## Discussion Record
### Questions and missunerstandings
### Theme Discussion

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# Video conferencing/Education
## Learning Tasks

* Identify the role of video conferencing tools for education and capacity building!
* how can you use Open Educational Resources in conjunction with Video conferencing?
# Video conferencing/Handwriting and Annotations

The aim of this learning unit is to be able to deal with the handwritten development of documents and the idea of handwritten solutions in video conferencing.

## Discuss student solution on paper

Analyze the use of LibreOffice-Draw to create diagrams that are discussed collectively. In addition, handwritten documents can be drawn ideas developed within the video conferencing software. These collaboratively developed ideas can then be integrated into the office software via screenshot and transferred as a copy to the LibreOffice-Draw file for further processing.

## Digital pens for life notation and annotation of solutions

You can use digital mouse pens as input media or tablets with digital pens to develop collaborative solutions of a drawing nature. Analyze the possibilities of
* pen mice and graphic tablets with pen for collaborative work.
* Offline OpenSource annotation software OpenSource Tools like Xournal++ for annotation of student solutions provide as PDF documents. PDF documents can be shared as a moderator with annotations in BigBlueButton video sessions.

## Learning Task

* (Comparison Digital Whiteboards for Collaboration) Compare handwriting and annotations of PDF documents with digital OpenSource white boards (see Youtube Demo by Marlon Lückert 2020 or GitHub-Repository Whiteboard VR). What are the differences what are common features? Explain which Open Source tool would you use for different use cases!
* (Annotated PDF Offline and share VidConf) Compare annotation of PDF documents in BigBlueButton with other OpenSource Tools like Xournal++ for the annotation of documents. What are similarities and what are differences? (e.g. the possibility to save annotated documents in a video conference session, collaborative development of documents, ...)
* (Hardware and Driver Check) See Basic Hardware and Driver Check to test hardware in Firefox on Linux, Windows, MacOSX if the hardware support the basic features of mouse movement and emulation of drawing with pressing of the left mouse button. This is just a basic feature hardware. In generell processing the pressure on the stylus/digital pen, that is tranfered into link thickness, brightness or transparency of the lines is not required. So it is recommended to perform a basic hardware and browser test on your device and operating system before you buy the hardware for your educational unit.
* (BigBlueButton Video conferencing) Explain how collaboratively notes and diagrams can be developed in a Video conferencing environment.
# Video conferencing/Improvements
## Annotations

* PDF documents can be uploaded by moderators and can be annotated by students and teachers. Saving with or without annotations can be a helpful feature for students joining the learning session in BBB.

## Rotating/Cyclic Image Update (CIU)

* (Cyclic Image Update) Videostreams can be submitted to BBB server, but instead of broadcasting the streams to all clients it could be helpful to update a still image in a cyle. Especially when a lot of students join a learning session, it is helpful to see a face from the student, to have a visual impression who is talking. To have a dynamic videostream during the session might not be as relevant as a still image.
* Define a default image in the profile of students and teacher will show that still image, even if no webcam is switched on.
* The rotating image update could use the bandwidth of a single videostream to update the still image of the participants. Large numbers of participants will increase the time span in which the images are updated but it will not increase the consumption of bandwidth. Especially old computers could benefit from such a video session, with visual impression of participants and low bandwidth consumption especially in rural areas, with poor connectivity to the internet.