OpenLLM-France/wikimedia huggingface dataset:
# Affine spaces/Intersection/Dimension/Exercise

Let $E$ be an affine space of dimension $d$, and let $F,G\subseteq E$ be affine subspaces of dimension $r$ and $s$, respectively. Show that $F\cap G=\emptyset$ is either empty, or that its dimension is at least $r+s-n$.
# Jordan normal form/Diagonalizable part/Same eigenvalues/Exercise

Let $\varphi \colon V\rightarrow V$ be a trigonalizable linear mapping, together with its canonical additive decomposition
    $\varphi =\varphi _{\rm {diag}}+\varphi _{\rm {nil}}\,$
in the sense of fact. Show that the eigenvalues of $\varphi$ and of $\varphi _{\rm {diag}}$ coincide. Does this also hold for their algebraic multiplicity? Does this also hold for their geometric multiplicity?
# MDLD/finite order

Start=finite order Target=Group theory/Order of an element/Definition
# MDLD/finite order (group element)

Start=finite order (group element) Target=Group theory/Order of an element/Definition
# Nilpotent matrix/Exponential mapping/Properties/Exercise

Let $n\in \mathbb {N} _{+}$, and let a field $K$ of characteristic $0$ be fixed. For a nilpotent $n\times n$-matrix $M$, let $\exp M$ be defined by
    $\exp M=\sum _{k=0}^{n}{\frac {1}{k!}}M^{k}\,.$
a) Show that for commuting nilpotent matrices $M,N$, the equality
    $\exp {\left(M+N\right)}=\exp M\circ \exp N\,$
holds.
b) Show that for a nilpotent matrix $M$, the matrix $\exp M$ is invertible.
c) Show that for a nilpotent matrix $M$, the matrix $\exp M$ is unipotent.
# MDLD/affine subspace (translated)

Start=affine subspace (translated) Target=Affine subspace/Translated linear subspace/Definition
# MDLD/connecting arrow

Start=connecting arrow Target=Affine space/Introduction/Section
# MDLD/dimension (affine space)

Start=dimension (affine space) Target=Affine space/Dimension/Basis number/Definition
# Studies of Euler diagrams

dummy
This resource shows the Euler diagrams for many Boolean functions, and aims to find a systematic way to draw them. This is part of the documentation of a software, that is not yet published.
* criteria for good Euler diagrams
* examples (list of named Boolean functions)
* terminology used in this project
* sequences

## splits

A split is a generalization of a set without the notion of inside and outside. It just splits the universe in two sides.

## decompose

decomposition into bundles, i.e. parts of the Euler diagram that are connected by crossing circles

## ternary labels

While the cells can be labeled with binary numbers, all segments (including edges and vertices) can be labeled with balanced ternary numbers.

## To do

* How are Euler diagrams related to hypergraphs?
# Studies of Euler diagrams/blighted

dummy
A Boolean function is blighted, if it is bloated or blotted.
It is bloated, if some arguments are equal or complementary to each other. It is blotted, if some arguments are equal or complementary to the universe (or the empty set).
A blighted BF can be reduced to one with a smaller valency. The blight is expressed as a set partition extended with complements between blocks. The remaining blightless BF can be expressed in an Euler diagram.

## bloated

| tokosi is a 5-ary function whose bloatless part is 3-ary. | bareto is a 4-ary function whose bloatless part is 2-ary. |

## blotted

| Set C in fobope is empty. | Set H in lapava is the universe. |
# MDLD/dimension (affine)

Start=dimension (affine) Target=Affine space/Dimension/Basis number/Definition
# MDLD/affine mapping

Start=affine mapping Target=Affine space/Affine-linear mapping/Definition
# MDLD/affine generating system

Start=affine generating system Target=Affine subspace/Affine generating system/Definition
# MDLD/polynomials

Start=polynomials See=polynomial MDLD/polynomial Target=Polynomial ring/Field/Polynomial function/Introduction/Section
# MDLD/dimensions (affine space)

Start=dimensions (affine space) See=dimension (affine space) MDLD/dimension (affine space) Target=Affine space/Dimension/Basis number/Definition
# MDLD/affine mappings

Start=affine mappings See=affine mapping MDLD/affine mapping Target=Affine space/Affine-linear mapping/Definition
# MDLD/linear part (affine)

Start=linear part (affine) Target=Affine mapping/Introduction/Section
# Affine mapping/Graph/Factorization/Exercise

Let
    $\varphi \colon E\longrightarrow F$
be an affine-linear mapping between the affine spaces $E$ and $F$ over $K$.
a) Show that the graph $G$ of $\varphi$ is an affine subspace of the product space $E\times F$.
b) Show that the mapping
    $\psi \colon E\longrightarrow G,P\longmapsto (P,\varphi (P)),$
is an isomorphism of affine spaces.
c) Show that
    $\varphi =p_{2}\circ \psi \,$
holds, where $p_{2}$ is the projection onto $F$.
# MDLD/isomorphism (affine)

Start=isomorphism (affine) Target=Affine-linear mapping/Affine isomorphism/Definition
# Adolescence

Adolescence (from the Latin adulescere which means, growing up) is a transition phase between childhood and adulthood, marked by significant biological, cognitive and psychosocial changes. Several organizations, such as the National Academies of Science, Engineering, and Medicine (NASEM) and the Society for Adolescent Health Medicine (SAHM), define adolescence as the period from age 10 to 24. In contrast, the Mass Cultural Council, aligning with Erik Erikson's work on the stages of psychosocial development, defines it as ages 12 to 23. This broader definition highlights the importance of adolescent development for lifelong health and well-being, and recognizes that this phase provides significant changes and opportunities for self-discovery, forming social bonds, and exploring identity.

## Resources

* Drinking and Driving
* Teen-Impaired Driving
* Teen pregnancy
* Teenage pregnancy and prenatal care
* Teen Problems
# Informal learning activities/Rs

Rsa
# Homeschooling/Primary school/Resources
## Primary school homeschool resources
### Science

* Science for kindergarten
* Solar System still under construction

### Math

* Primary school mathematics department

### Other

* Elementary typing

## Primary school homeschooling streams

Warning: Work in progress! We are attempting to create a guide for essential knowledge for each "grade" of homeschooling. Understanding that homeschoolers are generally more free to take their time than public-schooled children, we have organized the courses on a more or less pick-and-choose basis.
* Kindergarten homeschooling stream plan (5-6 year olds)
# WikiJournal of Science/Volume 7 Issue 1

WikiJournal of Science Open access • Publication charge free • Public peer review • Wikipedia-integrated
Volume 7(1)
Previous Volume 1(1) Volume 1(2) Volume 2(1) Volume 3(1) Volume 4(1) Volume 5(1) Volume 6(1)
Editorial guidelines Ethics statement Bylaws Financials Calendar
WikiJournal of Science is an open-access, free-to-publish, Wikipedia-integrated academic journal for science, mathematics, engineering and technology topics. WJS WikiJSci Wiki.J.Sci. WikiJSci WikiSci WikiScience Wikiscience Wikijournal of Science Wikiversity Journal of Science WikiJournal Science Wikipedia Science Wikipedia science journal STEM Science Mathematics Engineering Technology Free to publish Open access Open-access Non-profit online journal Public peer review
# Hosts file

The hosts file is a plain text file used to map host names to IP addresses.
On Windows, it is located in the C:\Windows\System32\drivers\etc folder.
In Linux it is located in: http//192.168.1.24.24
/hosts

## Actions

* View the Hosts File
* Edit the Hosts File

## See Also

* Internet Protocol Analysis - Name Resolution
# Open Data/Understand nutrition and food labels

This page is about a learning project to teach about nutrition, so a person can understand what to do differently if they want to go on a diet, and than teaching how to apply that knowledge to reading food labels to ensure a person can meet there nutritional goals.
The learning resource is following the Open Community Approach and therefore the learner should be able to learn with Open Source software and have access to open data.

## Learning Task

* (Open Food Facts) Open Food Facts is a food products database that is created in collaborative way by everyone and for everyone to use and cross-check the data in the database. Explain, what are the benefits and challenges of an open food fact database independent from industry. Analyze scientific results that address the ability to discriminate nutritional quality with a simplified score with colored labels. Discuss challenges and benefits of an approach.
* (Human nutrition) Perform an course about Nutrition according to your own knowledge or from a national perspective to mitigate Health risks.
** (Open University) - Course about Nutrition - Basics
** (National Perspective) OpenWHO - National Perspective on Global Nutrition Targets
Explain how food facts can support to improve human nutrition. What are the requirements and constraints of a Digital Learning Environment for this topic?
* (Collborative Knowledge Base) Wikipedia and Wikiversity are maintained by a community and the knowledge base or learning resources are shared with community. What are the similarities and differences between Open Food Facts, Wikiversity, Wikipedia and how do they relate to Open Course for Capacity Building?
* (Food Facts in Learning Resources) Health related topics must be handled with great care and responsibility. What are key elements that should be learned about food nutrition facts and the diet the learner wants to select?
* (Health and Food Production) Identify grand challenges that are related to food, agriculture and climate change in different regions of the world. How can local food production be organized, so that people can survive economically with agricultural activity and the local food production can cover all required nutrition facts for a healthy diet? How can food facts include also regional and local food production? What are the main challenges for integration of nutrition facts for local and regional food production that are dependent on the geographical location of the customer who wants to be informed about the food facts?
* (Barcode) For industrial food the barcode is a unique identifier (Database - primary key) for product. Explain how a unique identifier helps to link products with nutrition facts. What is the benefit of a barcode or QR code, to find nutrition facts?
* (Footprint of Food Production) Identify food facts beyond nutrition fact, e.g. fresh water consumption and the carbon footprint to produce specific food.
# Matrix/Staircase form by elementary manipulations/Fact

Let $K$ be a field, and let $M$ denote an $m\times n$-matrix over $K$.
elementary row operations, and a (new) numbering of the columns
    $j_{1},\,j_{2},\ldots ,j_{n},$
and an $r\leq n$ such that, in the new matrix, the columns have the form
    $s_{j_{k}}={\begin{pmatrix}b_{1,j_{k}}\\\vdots \\b_{k,j_{k}}\\0\\\vdots \\0\end{pmatrix}}{\text{ with }}b_{k,j_{k}}\neq 0{\text{ for }}k\leq r$
and
    $s_{j_{k}}={\begin{pmatrix}b_{1,j_{k}}\\\vdots \\b_{r,j_{k}}\\0\\\vdots \\0\end{pmatrix}}{\text{ for }}k>r.$
By further elementary row operations, and by swapping of columns, the matrix can be brought to the form
    $[[:Template:Matrixinstaircaseform]]$
with
$d_{ii}\neq 0$
# Invertible matrix/Staircase form/Identity matrix/Fact

Let $K$ be a field, and let $M$ denote an invertible $n\times n$-matrix over $K$.
elementary row operations such that, after these manipulations, a matrix of the form
    ${\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}$
with $d_{i}\neq 0$ arises. By further elementary row operation, one can also obtain the
# Matrix/K/General linear group/Definition

For a field $K$ and $n\in \mathbb {N} _{+}$, the set of all invertible $n\times n$-matrices with entries in $K$ is called the general linear group over $K$. It is denoted by $\operatorname {GL} _{n}\!{\left(K\right)}$.
# Matrix/Row rank and column rank/Fact/Proof

In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. The row rank of $M$ equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank $r$, since the first $r$ rows are linearly independent, and, apart from this, this, there are only zero rows. It has also column rank $r$, since the $r$ columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these $r$ columns. By exercise, the column rank is preserved by elementary row manipulations.
# Sets/Introduction/Section

A set is a collection of distinct objects, which are called the elements of the set. By distinct, we mean that it is clear which objects are considered to be equal, and which are considered to be different. The containment of an element $x$ to a set $M$ is expressed by
    $x\in M\,,$
the noncontainment by
    $x\notin M\,.$
For every element, exactly one of these possibilities holds.
An important principle for sets is the principle of extensionality, i.e., a set is determined by the elements it contains; beyond that, it bears no further information. In particular, two sets coincide if they contain the same elements.
The set that does not contain any element is called the empty set, and is denoted by
    $\emptyset .$
A set $N$ is called a subset of a set $M$ if every element from $N$ does also belong to $M$. For this relation, we write $N\subseteq M$ (some people write $N\subset M$ for this). One also says that the inclusion $N\subseteq M$ holds. The subset relation $N\subseteq M$ is a statement using "for all“, as it makes a claim about all elements from $N$. If we want to show $N\subseteq M$, then we have to show for an arbitrary element $x\in N$ that also the containment $x\in M$ holds . In order to show this, we are only allowed to use the property $x\in N$.
Due to the principle of extensionality, we have the following important equality principle for sets, saying that
    $M=N{\text{ if and only if }}N\subseteq M{\text{ and }}M\subseteq N$
holds. In mathematical praxis, this means that the equality of two sets is established by proving the two inclusions (in two independent steps). This also has the cognitive advantage that the reasoning gets a direction; it is always clear which conditions can be used and where to go. This principle is analogous to the principle from propositional logic that an equivalence between two statements means both implications, and is best shown by proving the two implications.
# Sets/Descriptions/Introduction/Section

There are several ways to describe a set. The easiest one is to just list the elements of the set, here the order of the listing is not important. For finite sets, this is possible; however, for infinite sets, one has to describe a "construction rule“ for the elements.
The most important set given by an infinite listing is the set of natural numbers
    $\mathbb {N} =\{0,1,2,3,\ldots \}\,.$
Here, a certain set of numbers is described by the first elements in the hope that this indicates how the listing goes on and which numbers belong to the set. An important point is that $\mathbb {N}$ is not a set of certain digits, but the set of values represented by these digits or sequences of digits. For a natural number, there are many possibilities to represent it, the decimal expansion is just one of them.
We discuss now the description of sets by properties. Let a set $M$ and a certain property $E$ (a predicate) be given, such that the property can be applied to the elements of $M$. Hence, for the property $E$, we have in $M$ the subset consisting of all the elements from $M$ which fulfil this property. We write for this subset given by $E$
    ${\left\{x\in M\mid E(x)\right\}}={\left\{x\in M\mid x{\text{ fulfils property }}E\right\}}\,.$
This only works for such properties for which the statement $E(x)$ is well-defined for every $x\in M$. If one introduces such a subset, then one gives a name to it, which often reflects the name of the property, like
    $E={\left\{x\in \mathbb {N} \mid x{\text{ is even}}\right\}}\,,$
    $O={\left\{x\in \mathbb {N} \mid x{\text{ is odd}}\right\}}\,,$
    $S={\left\{x\in \mathbb {N} \mid x{\text{ is a square number}}\right\}}\,,$
    ${\mathbb {P} }={\left\{x\in \mathbb {N} \mid x{\text{ is a prime number}}\right\}}\,.$
For the sets occurring in mathematics, a multitude of mathematical properties is relevant, and, therefore, there is a multitude of relevant subsets. But also in the sets of everyday life like the set $C$ of the students in a course, there are many important properties which determine certain subsets, like
    $O={\left\{x\in C\mid x{\text{ lives in Osnabr}}{\ddot {\rm {u}}}{\text{ck}}\right\}}\,,$
    $P={\left\{x\in C\mid x{\text{ studies physics}}\right\}}\,,$
    $D={\left\{x\in C\mid x{\text{ has birthday in December}}\right\}}\,.$
The set $C$ itself is also given by a property, since
    $C={\left\{x\mid x{\text{ is a student in the course}}\right\}}\,.$
# Space/Plane equation/Example for sets/Example

We consider the set
    $E={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in \mathbb {R} ^{3}\mid 5x-y+3z=0\right\}}\,.$
This is the subset inside $\mathbb {R} ^{3}$ containing all those points with coordinates ${\begin{pmatrix}x\\y\\z\end{pmatrix}}$ fulfilling the condition
    $5x-y+3z=0\,.$
This condition has a clear meaning for every point ${\begin{pmatrix}x\\y\\z\end{pmatrix}}$, it can be true or false. Hence, this is a well-defined subset. For example, the points ${\begin{pmatrix}0\\0\\0\end{pmatrix}}$ and ${\begin{pmatrix}2\\-3\\-{\frac {13}{3}}\end{pmatrix}}$ belong to the set, the point ${\begin{pmatrix}2\\4\\0\end{pmatrix}}$ does not belong to the set. If we want to check for a point ${\begin{pmatrix}x\\y\\z\end{pmatrix}}$ whether it belongs to $E$, we just have to check the condition. In this respect, the given description of $E$ is very good. If instead we would like to have a good overview of $E$ as a whole, then this description is not so convincing. We claim that $E$ coincides with the set
    $E'={\left\{r{\begin{pmatrix}3\\0\\-5\end{pmatrix}}+s{\begin{pmatrix}0\\3\\1\end{pmatrix}}\mid r,s\in \mathbb {R} \right\}}\,.$
This second description presents the set as the set of all elements that can be built in a certain way, namely as a linear combination of the points ${\begin{pmatrix}3\\0\\-5\end{pmatrix}}$ and ${\begin{pmatrix}0\\3\\1\end{pmatrix}}$ with arbitrary real coefficients. The advantage of this description is that one gets immediately an overview of all its elements. For example, it is clear that it contains infinitely many elements. However, in this description, it is more difficult to decide whether a given element belongs to the set or not.
In order to show that both sets are identical, we have to show $E\subseteq E'$ and $E'\subseteq E$. For the first inclusion, let $P={\begin{pmatrix}x\\y\\z\end{pmatrix}}\in E$. Then
    ${\begin{pmatrix}x\\y\\z\end{pmatrix}}={\frac {x}{3}}{\begin{pmatrix}3\\0\\-5\end{pmatrix}}+{\frac {y}{3}}{\begin{pmatrix}0\\3\\1\end{pmatrix}}\,.$
Here, the equality in the first and in the second component is clear, and the equality in the third component is a reformulation of the starting equation
    $5x-y+3z=0\,.$
Taking $r={\frac {x}{3}}$ and $s={\frac {y}{3}}$, we see that $P\in E'$. Now suppose that $P={\begin{pmatrix}x\\y\\z\end{pmatrix}}\in E'$. This means that there is a representation
    $P={\begin{pmatrix}x\\y\\z\end{pmatrix}}=r{\begin{pmatrix}3\\0\\-5\end{pmatrix}}+s{\begin{pmatrix}0\\3\\1\end{pmatrix}}={\begin{pmatrix}3r\\3s\\-5r+s\end{pmatrix}}\,$
with some real numbers $r,s\in \mathbb {R}$. In order to show that this point belongs to $E$, we have to show that it fulfills the defining equation of $E$. But this is clear because of
    $5x-y+3z=5(3r)-3s+3(-5r+s)=0\,.$
# Sets/Operations/Introduction/Section

Similar to the construction of new statements from given statements by connecting them with logical connectives, there are operations to make new sets from old ones. The most important operations are the following:
* Union
    $A\cup B:={\left\{x\mid x\in A{\text{ or }}x\in B\right\}}\,.$
* Intersection
    $A\cap B:={\left\{x\mid x\in A{\text{ and }}x\in B\right\}}\,.$
* Difference set
    $A\setminus B:={\left\{x\mid x\in A{\text{ and }}x\notin B\right\}}\,.$
For these operations to make sense, the sets need to be subsets of a common basic set. This ensures that we are talking about the same elements. Quite often, this basic set is not mentioned explicitly and has to be understood from the context. A special case of the difference set is the complement of a subset $A\subseteq G$ in a given set $G$, also denoted as
    $\complement A:=G\setminus A={\left\{x\in G\mid x\not \in A\right\}}\,.$
If two sets have an empty intersection, meaning $A\cap B=\emptyset$, we also say that they are disjoint.
# Space/Plane equation/Example intersection/Example

We consider the sets
    $E={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in K^{3}\mid 5x-y+3z=0\right\}}\,$
(from example) and
    $F={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in K^{3}\mid 4x+2y-7z=0\right\}}\,,$
and we are interested in the intersection
    $G:=E\cap F={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in K^{3}\mid 5x-y+3z=0{\text{ and }}4x+2y-7z=0\right\}}\,.$
A point lies in this intersection if and only if it fulfills both conditions, that is, both equations (let us call them $I$ and $II$). Does there exist a "simpler“ description of this intersection set? A point that fulfills both equations does also fulfill the equation that arises when we add the equations together, or when we multiply the equation with a number $s\in K$. Such a linear combination of the equations is, for example,
    $4I-5II=-14y+47z=0\,.$
Therefore,
    ${\begin{aligned}G&={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in K^{3}\mid 5x-y+3z=0{\text{ and }}4x+2y-7z=0\right\}}\\&={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in K^{3}\mid 5x-y+3z=0{\text{ and }}-14y+47z=0\right\}},\end{aligned}}$
since we can reconstruct the original second equation from the new second equation and vice versa. Hence, the conditions are equivalent. The advantage of the second description is that the variable $x$ does not occur in the new second equation; it has been eliminated. Therefore, we can resolve with respect to $y$, and we obtain
    $y={\frac {47}{14}}z\,.$
For $x$, we must have
    $x={\frac {1}{5}}y-{\frac {3}{5}}z={\frac {1}{5}}\cdot {\frac {47}{14}}z-{\frac {3}{5}}z={\frac {47}{70}}z-{\frac {42}{70}}z={\frac {1}{14}}z\,.$
Also, these two resolved equations are together equivalent with the original equations, and, therefore, we have
    $G={\left\{{\begin{pmatrix}{\frac {1}{14}}z\\{\frac {47}{14}}z\\z\end{pmatrix}}\mid z\in K\right\}}\,.$
This description yields a more explicit overview of the set $G$.
# Product set/First and last name/Example

Let $F$ be the set of all first names, and $L$ be the set of all last names. Then
    $F\times L$
is the set of all names. The elements of this set are, in pair notation, $({\text{Heinz}},{\text{Miller}})$, $({\text{Petra}},{\text{Miller}})$ and $({\text{Lucy}},{\text{Sonnenschein}})$. From a name, one can easily deduce the first name and the last name by looking at the first or the second component. Even if all first names and all last names do really occur, not every combination of a first name and a last name does occur. For the product set, all possible combinations are allowed.
# Product set/Chess board/Example

A chess board (meaning the set of squares of a chess board where a chess piece may stand) is the product set
    $\{a,b,c,d,e,f,g,h\}\times \{1,2,3,4,5,6,7,8\}.$
Every square is a pair, e.g., $(a,1),(d,4),(c,7)$. Because the two component sets are different, one may write instead of pair notation simply $a1,d4,c7$. This notation is the starting point to describe chess positions and complete chess games.