OpenLLM-France/wikimedia huggingface dataset:
# Vector space/Generating system/Introduction/Section
## Definition

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{1},\ldots ,v_{n}$ denote a family of vectors in $V$. Then the vector
    $s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}{\text{ with }}s_{i}\in K$
is called a linear combination of this vectors
$(s_{1},\ldots ,s_{n})$
Two different coefficient tuples can define the same vector.

## Definition

Let $K$ be a field, and let $V$ be a $K$-vector space. A family $v_{i}\in V$, $i\in I$, is called a generating system (or spanning system) of $V$, if every vector $v\in V$ can be written as
    $v=\sum _{j\in J}s_{j}v_{j}\,,$
with a finite subfamily $J\subseteq I$, and with
$s_{j}\in K$
In $K^{n}$, the standard vectors $e_{i}$, $1\leq i\leq n$, form a generating system. In the polynomial ring $K[X]$, the powers $X^{n}$, $n\in \mathbb {N}$, form an (infinite) generating system.

## Definition

Let $K$ be a field, and let $V$ be a $K$-vector space. For a family $v_{i}$, $i\in I$, we set
    $\langle v_{i},\,i\in I\rangle ={\left\{\sum _{i\in J}s_{i}v_{i}\mid s_{i}\in K,\,J\subseteq I{\text{ finite subset}}\right\}}\,,$
The empty set generates the null space. The null space is also generated by the element $0$. A single vector $v$ spans the space $Kv={\left\{sv\mid s\in K\right\}}$. For $v\neq 0$, this is a line, a term we will make more precise in the framework of dimension theory. For two vectors $v$ and $w$, the "form“ of the spanned space depends on how the two vectors are related to each other. If they both lie on a line, say $w=sv$, then $w$ is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by $v$. If this is not the case (and $v$ and $w$ are not $0$), then the two vectors span a "plane“.
We list some simple properties for generating systems and linear subspaces.

## Lemma

Let $K$ be a field and let $V$ be a
$K$
* Let $U_{j}$, $j\in J$, be a family of subspaces of $V$. Then the intersection is a linear subspace.
    $U=\bigcap _{j\in J}U_{j}\,$
* Let $v_{i}$, $i\in I$, be a family of elements of $V$ and consider the subset $W$ of $V$ which is given by all linear combinations of these elements. Then $W$ is a linear subspace of $V$.
* The family $v_{i}$, $i\in I$, is a system of generators of $V$ if and only if
    $\langle v_{i},\,i\in I\rangle =V\,.$

### Proof

$\Box$
* ↑ This follows from the definition, if we use the convention that the empty sum equals $0$.
# Vector space/Generating system and spanning subspace/Fact

Let $K$ be a field, and let $V$ be a $K$-vector space. Then the following hold.
* Let $U_{j}$, $j\in J$, be a family of linear subspaces of $V$. Then the intersection is a linear subspace.
    $U=\bigcap _{j\in J}U_{j}\,$
* Let $v_{i}$, $i\in I$, be a family of elements of $V$, and consider the subset $W$ of $V$ which is given by all linear combinations of these elements. Then $W$ is a linear subspace of $V$.
* The family $v_{i}$, $i\in I$, is a system of generators of $V$ if and only if
    $\langle v_{i},\,i\in I\rangle =V\,.$
# Digital Media Concepts/ "Can't Help Myself" Robot
## What is it?

In 2016, two artists named, Sun Yuan and Peng Yu, created a robot out of stainless steel and rubber enclosed in acrylic walls. Surrounding the robot was a deep red liquid and the only job that this robot had was to attempt and keep the red liquid contained as much as possible. Attached to this robot were sensors that detected anytime the red liquid strayed too far and the arm on this robot would swing over from wherever it was to shovel the liquid back towards the robot to attempt to contain it.
This robot is programmed to continuously sweep in at the red paint desperately trying to keep the mess contained. This piece was created to represent the overworked and robot-like human society that is today.

## Reflection of Society

Today, humans are under an absurd amount of stress and pressure that many people struggle with on a day-to-day basis. They try to keep up with it but overtime it creates a mess on themselves and the world around them. People are held to a certain standard that does not allow for them to have much room for mistakes, which then overtime eventually eats away at a person.

## Endless Labor

Society has become their own robots in their own lives. From the constant pressure and stress that comes from their everyday jobs and everyday lives, society encloses themselves to an unhealthy routine that does not allow for free-time or de-stressors. In February 2023, a survey was released asking American workers where most of their stress comes from. The results came out as 83% of workers suffer from work-related stress and 25% saying that their job was their biggest stressor in their lives. Just like the robot, society is programmed to their one job, a job that overcomes them.

## Obsession and Repetitiveness

As much as this robot and society are both laborers to their everyday jobs, the constant repetitiveness can also become an obsession. The robot represents the society and how blind they are to what their jobs have made of them. Because of the constant sweeping and swinging around of the arm, the red liquid that is supposedly getting "cleaned" up is slowly overtime making a mess of it. But since the robot is programmed to do that one job, it overtime gives off an energy that it is obsessed with it to keep its area clean. Just like society, unhealthy relationships whether that being towards their job or other things going on in their lives, an obsession can overcome them and their own messes can be made.
# WikiJournal Preprints/社交媒体热搜操控与网络暴力防范策略 研究

WikiJournal Preprints Open access • Publication charge free • Public peer review
WikiJournal of Science
WikiJournal of Humanities
WikiJournal of PPB
Editorial guidelines Ethics statement Bylaws Indexing Financials and reports 1-year/3-year plan Proposed platform Meeting minutes Calendar
WikiJournal User Group is a publishing group of open-access, free-to-publish, Wikipedia-integrated academic journals. <seo title=" Wikiversity Journal User Group, WikiJournal Free to publish, Open access, Open-access, Non-profit, online journal, Public peer review "/>
This article is an unpublished pre-print not yet undergoing peer review.
To submit this article for peer review, please: Submit authorship declaration form
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## Article information

Author: Firstname Lastname
* ↑ email@address.com

## First Heading

Manuscript text goes re

### Subheading
## Additional information
### Acknowledgements

Any people, organisations, or funding sources that you would like to thank.

### Competing interests

Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

### Ethics statement

An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
# BCP/Fallopia japonica

Fallopia japonica
| Flowering branch Wikipedia • Commons • Wikibooks (horticulture) • Wikibooks (subject) • Wikispecies • Fruit and seed clock | Profile for Fallopia japonica (Japanese Knotweed) Identifying Characteristics Flowers:The flowers are small, creamy white, produced in erect racemes 6–15 cm long in late summer and early autumn.Foliage:The leaves are broad oval with a truncated base, 7–14 cm long and 5–12 cm broad, with an entire margin.Stem:hollow stems with distinct raised nodes that give it the appearance of bambooLife Cycle: Perennial Recent Logs Jomegat 23:52, 23 August 2008 (UTC) --SB_Johnny 08:23, 24 August 2008 (UTC) --SB_Johnny 09:24, 31 August 2008 (UTC) Jomegat 12:40, 2 September 2008 (UTC) Jomegat 22:46, 10 September 2008 (UTC) (view all logs) |
Global data: Temperate zone season(s): Late Summer, Early Fall
Regional data: Southeastern Pennsylvania: (invasive or naturalized) (cultivated) August, September New Hampshire: (invasive or naturalized) August , September
Additional images:
# Adjugate matrix/Definition

For a square matrix $M\in \operatorname {Mat} _{n}(K)$, we call
    $M^{\operatorname {adj} }=(b_{ij}){\text{ with }}b_{ij}=(-1)^{i+j}\det M_{ji}$
the adjugate matrix of $M$, where $M_{ji}$ arises from $M$ by deleting the $j$-th row and the $i$-th column.
# Matrix/Adjugate matrix/Formula with determinant/Fact/Proof

Let $M={\left(a_{ij}\right)}_{ij}$. Let the coefficients of the adjugate matrix be denoted by
    $b_{ik}=(-1)^{i+k}\det M_{ki}\,.$
The coefficients of the product $(M^{\operatorname {adj} })\cdot M$ are
    $c_{ij}=[[:Template:Sumk1n]]=[[:Template:Sumk1n]]\,.$
In case $j=i$, this is $\det M$, as this sum is the expansion of the determinant with respect to the $j$-th column. So let $j\neq i$, and let $N$ denote the matrix that arises from $M$ by replacing in $M$ the $i$-th column by the $j$-th column. If we expand $N$ with respect to the $i$-th column, then we get
    $0=\det N=[[:Template:Sumk1n]]=c_{ij}\,.$
Therefore, these coefficients are $0$, and the first equation holds. The second equation is proved similarly, where we use now the expansion of the determinant with respect to the rows.
# Matrix/Adjugate matrix/Formula with determinant/Fact

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix over $K$.
    $(M^{\operatorname {adj} })\cdot M=(\det M)E_{n}=M\cdot (M^{\operatorname {adj} })\,$
If $M$ is invertible, then
    $M^{-1}={\frac {1}{\det M}}\cdot M^{\operatorname {adj} }\,.$
# Net (command)/User

Net is a Windows command used to view and modify network settings. These activities will show you how to use the net user command.
Note: To complete this activity, you must have an administrative user account or know the username and password of an administrator account you can enter when prompted.

## Preparation

To prepare for this activity:
* Start Windows.
* Log in if necessary.

## View Users

To view user accounts on your computer:
* Open an elevated/administrator command prompt.
* Type net user and press Enter.
* Observe the list of user accounts on your computer.

## Add a User Account(fr works)

To add a user account to your computer:
* Open cmd with Administrator and Type net user username password /add, where username is the name of the new user and password is the password for the new user account. For example, if the username is Bill and the password is Passw0rd, you would type net user Bill Passw0rd /add. Then press Enter.
* Type net user and press Enter to confirm the user account has been added.
Please note that passwords with a dollar-sign ($) are not allowed.

## Delete a User Account

To delete a user account from your computer:
* Type net user and press Enter to view user accounts on your computer.
* Type net user username /delete, where username is the name of the user you wish to delete. For example, if the username is Bill, you would type net user Bill /delete. Then press Enter.
** If you see "syntax of this command is:", check spelling and spacing
* Type net user and press Enter to confirm the user account has been deleted.
* Close the command prompt to complete this activity.

## Readings

* Wikipedia: Net User
# Binomial formula/R/Distributive law/Exercise

Formulate the binomial formula for two real numbers, and prove it using the distributive law.
# Natural numbers/Taking powers/Neither commutative nor associative/Exercise

Show that taking powers of positive natural numbers, i.e., the assignment
    $\mathbb {N} \times \mathbb {N} \longrightarrow \mathbb {N} ,(a,b)\longmapsto a^{b},$
is neither commutative nor associative. Does this operation have a neutral element?
# Group/Inverse/Self inverse/Exercise

Let $G$ be a group. Show that
    ${\left(x^{-1}\right)}^{-1}=x\,$
holds for all $x\in G$.
# Ring/(x^2-3yzy-2zy^2+4xy^2)(2xy^3x-z^2xyx)(1-3zyxz^2y)/Compute/Exercise

Let $R$ be a ring, and let $x,y$ and $z$ denote elements in $R$. Compute the product
    ${\left(x^{2}-3yzy-2zy^{2}+4xy^{2}\right)}{\left(2xy^{3}x-z^{2}xyx\right)}{\left(1-3zyxz^{2}y\right)}.$
What is the result, when the ring is commutative?
# Field/Rules for fractions/Exercise

Let $x,y,z,w$ be elements in a field, and suppose that $z$ and $w$ are not zero. Prove the following fraction rules.
    ${\frac {x}{1}}=x\,,$
    ${\frac {1}{z}}=z^{-1}\,,$
    ${\frac {1}{-1}}=-1\,,$
    ${\frac {0}{z}}=0\,,$
    ${\frac {z}{z}}=1\,,$
    ${\frac {x}{z}}={\frac {xw}{zw}}\,$
    ${\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$
    ${\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$
Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is
    $(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?$
Show that the popular formula
    ${\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,$
does not hold.
# Field/Reverse distributive law/Exercise

Show that, in a field, the "reversed distributive law“, that is,
    $a+(bc)=(a+b)\cdot (a+c)\,,$
does not hold.
# Modular arithmetic/Even and odd numbers/Calculative rules/Exercise

Describe and prove rules for the addition and the multiplication of even and odd integer numbers. Define on the set with two elements
    $\{E,O\}$
an "addition“ and a "multiplication“ that "represent“ these rules.
# Field/2 not 0/Product with squares/Explicit/Exercise

Let $K$ be a field with $2\neq 0$. Show that for $f,g\in K$, the relation
    $fg={\frac {1}{4}}{\left({\left(f+g\right)}^{2}-{\left(f-g\right)}^{2}\right)}\,$
holds.
# Real non negative numbers/Maximum/Structural properties/Exercise

Discuss the operation
    $\mathbb {R} _{\geq 0}\times \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},(x,y)\longmapsto \operatorname {max} \,(x,y),$
looking at associative law, commutative law, existence of a neutral element and existence of inverse element.
# Power set/Ring structure with symmetric difference/Exercise

Let $M$ be a set. Show that the power set ${\mathfrak {P}}\,(M)$ is a commutative ring, if we consider the intersection $\cap$ as multiplication and the symmetric difference
    $A\triangle B={\left(A\setminus B\right)}\cup {\left(B\setminus A\right)}\,$
as addition (what are the neutral elements?).
# Field/Bijectivity of one-sided operations/Exercise

Show that in a field $K$, the following properties hold.
(1) For every $a\in K$, the mapping
    $\alpha _{a}\colon K\longrightarrow K,x\longmapsto x+a,$
is bijective.
(2) For every $b\in K$, $b\neq 0$, the mapping
    $\mu _{b}\colon K\longrightarrow K,x\longmapsto bx,$
is bijective.
# Field/General distributive law/Exercise

Prove the general distributive law for a field.
# Field/Basic properties/Fact/Proof2

* We have $a0=a(0+0)=a0+a0$. Subtracting $a0$ (meaning addition with the negative of $a0$) on both sides gives the claim.
* See exercise.
* See exercise.
* See exercise.
* We prove this by contradiction, so we assume that $a$ and $b$ are both not $0$. Then there exist inverse elements $a^{-1}$ and $b^{-1}$ and hence $(ab){\left(b^{-1}a^{-1}\right)}=1$. On the other hand, we have $ab=0$ by the premise and so the annulation rule gives hence $0=1$, which contradicts the field properties.
    $(ab){\left(b^{-1}a^{-1}\right)}=0{\left(b^{-1}a^{-1}\right)}=0\,,$
* This follows with a double induction, see exercise.
# System of linear equations/Set of variables/Equivalent system/Manipulations/Fact

Let $K$ be a field, and let
    ${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&c_{2}\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&c_{m}\end{matrix}}$
be an inhomogeneous system of linear equations over $K$.
equivalent system.
* Swapping two equations.
* The multiplication of an equation by a scalar $s\neq 0$.
* The omitting of an equation, if it occurs twice.
* The duplication of an equation (in the sense to write down the equation again).
* The omitting or adding of a zero row (zero equation).
* The replacement of an equation $H$ by the equation that arises if we add to $H$ another equation $G$ of the system.
# System of linear equations/Set of variables/Equivalent system/Manipulations/Fact/Proof

Most statements are immediately clear. (2) follows from the fact that if
    $\sum _{i=1}^{n}a_{i}\xi _{i}=c\,$
holds, then also
    $\sum _{i=1}^{n}(sa_{i})\xi _{i}=sc\,$
holds for every $s\in K$. If $s\neq 0$, then this implication can be reversed by multiplication with $s^{-1}$.
(6). Let $G$ be the equation
    $\sum _{i=1}^{n}a_{i}x_{i}=c\,,$
and $H$ be the equation
    $\sum _{i=1}^{n}b_{i}x_{i}=d\,.$
If a tuple $(\xi _{1},\ldots ,\xi _{n})\in K^{n}$ satisfies both equations, then it also satisfies the equation $H'=G+H$. And if the tuple satisfies the equations $G$ and $H'$, then it also satisfies the equation $G$ and $H=H'-G$.
# Linear system/Elimination lemma/Fact

Let $K$ denote a field, and let $S$ denote an (inhomogeneous) system of linear equations over $K$ in the variables $x_{1},\ldots ,x_{n}$. Suppose that $x$ is a variable which occurs in at least one equation $G$ with a coefficient $a\neq 0$.
$H$
$G$
can be replaced by an equation $H'$, in which $x$ does not occur any more, and such that the new system of equations $S'$ that consists of $G$ and the equations $H'$, is equivalent
$S$
# Linear system/Elimination lemma/Fact/Proof

Changing the numbering, we may assume $x=x_{1}$. Let $G$ be the equation
    $ax_{1}+\sum _{i=2}^{n}a_{i}x_{i}=b\,$
(with $a\neq 0$), and let $H$ be the equation
    $cx_{1}+\sum _{i=2}^{n}c_{i}x_{i}=d\,.$
Then the equation
    $H'=H-{\frac {c}{a}}G\,$
has the form
    $\sum _{i=2}^{n}{\left(c_{i}-{\frac {c}{a}}a_{i}\right)}x_{i}=d-{\frac {c}{a}}b\,,$
and $x_{1}$ does not occur in it. Because of $H=H'+{\frac {c}{a}}G$, the systems are equivalent.
# Linear inhomogeneous system/Elimination/Echelon form/Fact/Proof

This follows directly from the elimination lemma, by eliminating successively variables. Elimination is applied firstly to the first variable (in the given ordering), say $x_{s_{1}}$, which occurs in at least one equation with a coefficient $\neq 0$ (if it only occurs in one equation, then this elimination step is already done). This elimination process is applied as long as the new subsystem (without the working equation used in the elimination step before) has at least one equation with a coefficient for one variable different from $0$. After this, we have in the end only equations without variables, and they are either only zero equations, or there is no solution.
# Linear inhomogeneous system/Elimination/Echelon form and triangular form/Fact/Proof

This follows directly from the elimination lemma, by eliminating successively variables. Elimination is applied firstly to the first variable (in the given ordering), say $x_{s_{1}}$, which occurs in at least one equation with a coefficient $\neq 0$ (if it only occurs in one equation, then this elimination step is already done). This elimination process is applied as long as the new subsystem (without the working equation used in the elimination step before) has at least one equation with a coefficient for one variable different from $0$. After this, we have in the end only equations without variables, and they are either only zero equations, or there is no solution.
When we set $y_{1}=x_{s_{1}},y_{2}=x_{s_{2}},\ldots ,y_{m}=x_{s_{m}}$, and denote the other variables with $y_{m+1},\ldots ,y_{n}$, then we obtain the described system in triangular form.
# Linear inhomogeneous system/Strictly triangular/Solution/Fact/Proof

This is clear, as when the tuple $(x_{m+1},\ldots ,x_{n})$ is given, the rows determine successively the other variables from bottom to top.
# Linear system/Matrix/Manipulations/Remark

A linear system can be written briefly as
    $Ax=c\,$
with an $m\times n$-matrix $A$ and an $m$-tuple $c$. The manipulations at the equations that we do in the elimination procedure, can be performed directly for the matrix, or for the extended matrix that arises when we extend $A$ by the column $c$. Essentially, we replace a row by the sum of the row with a multiple of another row. This has the advantage that we do not have to write down the variables. However, one should then not swap the variables. At the end, the arising matrix in echelon form can be interpreted again as a linear system.
# Open problems in 2d CFT

Two-dimensional conformal field theory has been an active topic of research since the 1980s, with applications to statistical physics and quantum gravity.

## Big questions
### CFT description of some particular systems

1. Is diffusion-limited aggregation in two dimensions conformally invariant? If yes, which CFT describes it?
2. Which CFT describes KPZ surface growth?
3. Which CFT describes the infrared limit of $N$ coupled $Q$-state Potts models in 2d, coupled by the energy, with $N\neq 1,2$ and $2<Q<4$?

### Space of CFTs

4. For compact unitary Virasoro-CFTs with $c>1$, which values of the central charge $c$ are possible?
5. Are there compact unitary CFTs with $c\in \mathbb {Q}$ that are neither rational, nor exactly marginal deformations of rational CFTs?
6. Are there exactly solvable non-diagonal Virasoro-CFTs with $c\geq 25$? See Virasoro CFTs with a large central charge.

## Technical problems
### Algebraic structures and conformal blocks

* Virasoro conformal blocks at rational central charge.
* Prove Zamolodchikov's recursive representation of w:Virasoro conformal blocks. (See Section 4.3 of ref.) Related issue: prove the convergence of Virasoro conformal blocks.
* Define an interchiral algebra that leads to the known interchiral representations and interchiral conformal blocks. (See Section 4.2 of ref.)
* Compute the fusion product of representations of the affine Lie algebra ${\hat {\mathfrak {s\ell }}}_{2}$, including the representations that appear in the $SL_{2}(\mathbb {R} )$ w:Wess-Zumino-Witten model, and the degenerate representations needed to bootstrap that model. (See Section 4.4.3 of ref. and ref.)

### Solving specific CFTs

* Boundary Liouville theory with $c\leq 1$.
* Conformal Toda theory.
* Generalized D-series minimal models, i.e. non-rational limits of D-series minimal models, see Solvable non-diagonal 2d CFTs.
* Solving the SL(2,R) WZW model.
* Logarithmic minimal models are believed to exist at rational central charges, with primary fields in the extended Kac table. They might be constructed as limits of generalized minimal models.

### Various issues

* Computing entanglement entropy in 2d CFTs.