OpenLLM-France/wikimedia huggingface dataset:
# MyOpenMath/Physics images

* Lens and wavefronts.gif
* Partial transmittance.gif
* Snells law wavefronts.gif
* Two sources interference.gif
* Wave equation 1D fixed endpoints.gif
* Exp series.gif
* Atomic-orbital-clouds spd m0.png
* Bohr atom model (mul).svg
* Hydrogen Density Plots.png
* AirShower.svg
* Aurore australe - Aurora australis.jpg
* Alfa beta gamma radiation penetration.svg
* Rays_of_light_through_a_different_medium_(labelled).svg
* Coupled oscillators.gif
* Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz.gif
* Simple harmonic motion animation 1.gif
* Heisenberg microscope with wavefronts and electron scatter.svg
* Wave-particle duality.gif
* Circlestrafing animation.gif
* Rolling Racers - Moment of inertia.gif
* Model of the initiation of termination of a Rayleigh-Taylor instability in 2D.gif
* Karman Vortex street ani.gif
* Mars Loop.gif
* Zirkumpolar ani.gif
* Tuned circuit animation 3 400ms.gif
* Dipole.gif
* Brachistochrone.gif
* Halley's Comet animation.gif
* Galilean moon Laplace resonance animation 2.gif
* Surface chemical diffusion.gif
* InfiniteSquareWellAnimation.gif
* Karman trefftz.gif
* Propagation du tsunami en profondeur variable.gif
* Propagation du tsunami en profondeur variable.gif
* Vortex-street-animation.gif
* Maxwell's demon.svg
* Ergodic hypothesis w reflecting rays.jpg
* Proving Carnot Theorem.svg
* Fizeau experiment schematic.svg
* Stellar aberration illustration.svg
* Entropy flip 2 coins.jpg
* Earth-moon-field.svg
* Van de graaf generator.svg
* WestinghouseAtomSmasher.jpg
* Electric Bell animation.gif
* Faraday cage.gif
* Leaderlightnig.gif
* wikibooks: Electronics/Photocopier
* File:Simple harmonic motion animation 2.gif
# Determinant/Recursively/Alternating/Fact

Let $K$ be a field and $n\in \mathbb {N} _{+}$.
determinant
    $\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$
is
# Determinant/Recursively/Multilinear/Fact/Proof

Let
    $M:={\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,,M':={\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\w\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}{\text{ and }}{\tilde {M}}:={\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u+w\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}},$
where we denote the entries and the matrices arising from deleting a row in an analogous way. In particular, $u=\left(a_{k1},\,\ldots ,\,a_{kn}\right)$ and $w=\left(a_{k1}',\,\ldots ,\,a_{kn}'\right)$. We prove the statement by induction over $n$, For $i\neq k$, we have ${\tilde {a}}_{i1}=a_{i1}=a'_{i1}$ and
    $\det {\tilde {M}}_{i}=\det M_{i}+\det M'_{i}\,$
due to the induction hypothesis. For $i=k$, we have $M_{k}=M_{k}'={\tilde {M}}_{k}$ and ${\tilde {a}}_{k1}=a_{k1}+a'_{k1}$. Altogether, we get
    ${\begin{aligned}\det {\tilde {M}}&=\sum _{i=1}^{n}(-1)^{i+1}{\tilde {a}}_{i1}\det {\tilde {M}}_{i}\\&=\sum _{i=1,\,i\neq k}^{n}(-1)^{i+1}a_{i1}(\det {M}_{i}+\det {M}'_{i})+(-1)^{k+1}(a_{k1}+a'_{k1})(\det {\tilde {M}}_{k})\\&=\sum _{i=1,\,i\neq k}^{n}(-1)^{i+1}a_{i1}\det {M}_{i}+\sum _{i=1,\,i\neq k}^{n}(-1)^{i+1}a_{i1}\det {M}'_{i}+(-1)^{k+1}a_{k1}\det M_{k}+(-1)^{k+1}a'_{k1}\det M_{k}\\&=\sum _{i=1}^{n}(-1)^{i+1}a_{i1}\det {M}_{i}+\sum _{i=1,\,i\neq k,\,}^{n}(-1)^{i+1}a_{i1}\det {M}'_{i}+(-1)^{k+1}a'_{k1}\det M_{k}\\&=\sum _{i=1}^{n}(-1)^{i+1}a_{i1}\det {M}_{i}+\sum _{i=1}^{n}(-1)^{i+1}a'_{i1}\det {M}'_{i}\\&=\det M+\det M'.\end{aligned}}$
The compatibility with the scalar multiplication is proved in a similar way, see exercise.
# Calculus/Introduction

Calculus is the math pertaining to slopes (or 'steepness') and areas under curves. I will define that a slope of, say 2 means that if you move 1 unit forwards, you move 2 units up, and that the same applies for any other number

## Prerequisites

* The four basic operations (Addition(+), Subtraction(-), Multiplication(×), Division(÷))
* Maybe algebra (subtle introduction in Section 2)

## The Derivative

The derivative is the math name for the math of slopes. To give an example, the word 'parabola' means that, for example, if you walk 4 meters away from it, the parabola is 4*4=16 meters above your head, and this holds true if you walked any other distance from it (and replaced the 4 with the actual distance!). Now, if you got a friend to walk up the parabola directly overhead, they would feel a slope of 2*4=8, and the same would be true if you replace 4 with anything else, and you kept the 2 because you think that 2 matters more than 8 to you...
To write this down, we say: ${dy \over dx}=2x$, and in this case, ${dy \over dx}$ means 'slope' and x means how far you walked away from the center of the parabola.
Another way of thinking of derivatives is in terms of change, as shown in the above example, when the distance from the centre changes so does the distance of the point on the parabola from your head. Derivative is essentially the study of this relation of changes in values.

## The integral

The integral is the math name for the math of areas under curves. To give an example, if you gave me a number, which I'll call x, I can give you the integral of y=x (the height, y is the same as the horizontal distance, x), which is the area of the blue triangle. Notice that the blue triangle is half of a square with side length x. This implies that the blue triangle's area is x*x/2, and thus, the integral of x is x*x/2.
To write this down, we say: ${\frac {x*x}{2}}=\int xdx$. $\int dx$ means that you're taking the area under a curve and the x between the sign and the dx represents the y value. Had y=x*x, it would have looked like this: $\int x*xdx$, with x*x representing the y value.
# Latin II/At work Lesson 1

Salvēte omnēs! Welcome back to Latin for Wikiversity. Today we'll learn about different jobs and where people work. After this lesson, you'll be able to talk about what you and others do for work.

## New Grammar

In Latin, we often use the preposition "in" with the ablative case to say where someone works: - Mārcus in tabernā labōrat = Marcus works in a shop - in scholā labōrō = I work in a school

## New Vocabulary

| Latin | English | Notes |
| medicus, -ī (m.) | doctor | Root of English "medical" |
| mercātor, -ōris (m.) | merchant | Root of English "mercantile" |
| magister, -trī (m.) | teacher | Root of English "master" |
| architectus, -ī (m.) | architect | Direct source of English word |
| taberna, -ae (f.) | shop | Root of English "tavern" |
| officium, -ī (n.) | office | Root of English "office" |
| labōrō, -āre | work | Root of English "labor" |
| negōtium, -ī (n.) | business, occupation | Root of "negotiate" |

## New Sentences

| Latin | English | Notes |
| Quid agis? | What do you do? | Common way to ask about profession |
| Medicus sum. | I am a doctor. | Simple identification |
| In officīō labōrō. | I work in an office. | Place where + labōrō |
| Mārcus est mercātor. | Marcus is a merchant. | Simple identification |
| Ubi labōrās? | Where do you work? | Question about workplace |
| In tabernā labōrō. | I work in a shop. | Answer about workplace |
| Magistra in scholā docet. | The teacher teaches in school. | Workplace + specific verb |
| Architectus novās domōs facit. | The architect makes new houses. | Job + activity |
| Pater meus negōtium habet. | My father has a business. | Possession |
| In urbe labōrāmus. | We work in the city. | Plural form |
| Paula medica est et in hospitiō labōrat. | Paula is a doctor and works in a hospital. | Combined sentences |
| Amīcus meus mercātor est; in magnā tabernā labōrat. | My friend is a merchant; he works in a large shop. | Complex sentence |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Basic Professions Vocabulary |
| Step two | Next try learning and writing the sentences using this: Basic Professions Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
In our next lesson, we'll learn how to talk about work schedules and when people work. For now, practice asking and answering "Quid agis?" and "Ubi labōrās?" Valēte!
# Latin II/At work Lesson 3

Salvēte omnēs! Welcome back to Latin for Wikiversity. In our previous lessons, we learned about jobs and work schedules. Today we'll learn how to talk about workplace relationships and duties.

## New Grammar

In Latin, we use some special patterns to talk about duties and working relationships:
* pro + ablative = "for" (someone)
    For example: pro dominō labōrō = I work for the boss
* opus est + ablative = "there is need of"
    For example: opus est pecūniā = there is need of money

## New Vocabulary

| Latin | English | Notes |
| dominus, -ī (m.) | boss, master | Root of "dominate" |
| collēga, -ae (m.) | colleague | Direct source of English word |
| officium, -ī (n.) | duty | Already know meaning "office" |
| mercēs, -ēdis (f.) | wages, pay | Root of "merchandise" |
| praestō, -āre | perform, carry out | |
| dēbeō, -ēre | ought, must | Root of "debt" |
| parō, -āre | prepare, arrange | Root of "prepare" |
| iubeō, -ēre | order, command | |

## New Sentences

| Latin | English | Notes |
| Pro dominō labōrō. | I work for the boss. | Working relationship |
| Cum collēgīs labōrō. | I work with colleagues. | Using cum + ablative |
| Officium praestāre dēbeō. | I must perform my duty. | Obligation |
| Dominus mercēdem dat. | The boss gives wages. | Direct object |
| Collēga opus parat. | The colleague prepares the work. | Simple action |
| Dominus mē iubet labōrāre. | The boss orders me to work. | With infinitive |
| Opus est pecūniā. | There is need of money. | Impersonal expression |
| Bene cum collēgīs labōrāmus. | We work well with colleagues. | With adverb |
| Mārcus officium cotīdiē praestat. | Marcus performs his duty daily. | Time expression |
| Paula pro magnō officīō labōrat. | Paula works for a large office. | Complex meaning |
| Dominus novum opus parat. | The boss prepares new work. | With adjective |
| Collēgae mercēdem accipere dēbent. | The colleagues must receive pay. | Complex sentence |

## Practice

Practice and learn the words and phrases in this lesson :
* This content is not currently available on Memrise
* Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |
This completes our series on work and professions. You can now talk about jobs, schedules, and workplace relationships! In our next unit, we'll learn about city locations and buildings. Valēte!
# Latin II/At work Lesson 2

Salvēte omnēs! Welcome back to Latin for Wikiversity. Last time we learned about different jobs and workplaces. Today we'll learn how to talk about when people work and for how long.

## New Grammar

In Latin, we use some special patterns to talk about time:
* ā/ab + ablative ... ad + accusative = "from ... to ..."
    For example: ā secundā hōrā ad octāvam = from 2 o'clock to 8
* The accusative case shows duration of time
    For example: trēs hōrās = for three hours

## New Vocabulary

| Latin | English | Notes |
| hōra, -ae (f.) | hour | Root of English "hour" |
| cotīdiē | daily, every day | Common time expression |
| māne (indecl.) | in the morning | |
| vesperī | in the evening | Locative form |
| tempus, -oris (n.) | time | Root of "temporal" |
| diēs, -ēī (m.) | day | Root of "diurnal" |
| incipiō, -ere | begin | Root of "incipient" |
| fīniō, -īre | finish, end | Root of "finite" |

## New Sentences

| Latin | English | Notes |
| Cotīdiē labōrō. | I work every day. | Simple frequency |
| Quando labōrās? | When do you work? | Time question |
| Māne incipiō. | I begin in the morning. | Time of day |
| Vesperī fīniō. | I finish in the evening. | Time of day |
| Octō hōrās labōrō. | I work for eight hours. | Duration |
| Ā secundā hōrā ad decimam labōrat. | He works from 2 o'clock to 10. | Time span |
| Cotīdiē in tabernā quattuor hōrās labōrat. | Every day she works in the shop for four hours. | Combined time expressions |
| Māne incipiō et vesperī fīniō. | I begin in the morning and finish in the evening. | Combined sentence |
| Quandō incipis labōrāre? | When do you begin to work? | With infinitive |
| Brevī tempore labōrāmus. | We work for a short time. | Time expression |
| Mārcus longum diem labōrat. | Marcus works a long day. | Duration |
| Paula multās hōrās in officīō labōrat. | Paula works many hours in the office. | Complex sentence |

## Practice

Practice and learn the words and phrases in this lesson :
* This content is not currently available on Memrise
* Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |
In our next lesson, we'll learn how to talk about workplace relationships and duties. Until then, practice asking and answering questions about when people work! Valēte!
# Latin II/In the city Lesson 1

Salvēte omnēs! Welcome back to Latin for Wikiversity. Today we'll learn about important places in a Roman city. We'll focus on public buildings and spaces that you might visit during your day.

## New Grammar

In Latin, we use different prepositions to talk about location: - in + ablative = "in/at" (location where) - ad + accusative = "to" (motion towards) - ā/ab + ablative = "from" (motion away from)

## New Vocabulary

| Latin | English | Notes |
| forum, -ī (n.) | forum, marketplace | Center of Roman public life |
| templum, -ī (n.) | temple | Root of English "temple" |
| basilica, -ae (f.) | public hall, basilica | Used for business and law |
| theātrum, -ī (n.) | theater | Root of English "theater" |
| bibliothēca, -ae (f.) | library | From Greek "book storage" |
| via, -ae (f.) | street, road | Root of English "via" |
| platēa, -ae (f.) | plaza, square | Root of "place" |
| propinquus, -a, -um | nearby, close | Used with dative |

## New Sentences

| Latin | English | Notes |
| Forum in mediā urbe est. | The forum is in the middle of the city. | Location |
| Ad templum eō. | I go to the temple. | Motion towards |
| Ā basilicā veniō. | I come from the basilica. | Motion from |
| Theātrum templō propinquum est. | The theater is close to the temple. | Using propinquus |
| In bibliothēcā librōs legō. | I read books in the library. | Activity + location |
| Per viam ambulāmus. | We walk along the street. | Movement along |
| In plateā stāmus. | We stand in the square. | Simple location |
| Via ā forō ad templum dūcit. | The road leads from the forum to the temple. | Complex direction |
| Ubī est bibliothēca? | Where is the library? | Question |
| Templum nōn longē abest. | The temple is not far away. | Distance |
| Per multās viās errāmus. | We wander through many streets. | Multiple locations |
| In basilicā iūdex sententiam dīcit. | In the basilica the judge gives his verdict. | Complex sentence |

## Practice

Practice and learn the words and phrases in this lesson :
* This content is not currently available on Memrise
* Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |
In our next lesson, we'll learn about more city locations, including shops and private buildings. Until then, practice describing where places are in your city! Valēte!
# Latin II/In the city Lesson 3

Salvēte omnēs! Welcome back to Latin for Wikiversity. After learning about public buildings and shops, today we'll learn how to describe neighborhoods and give directions in a Roman city.

## New Grammar

To give directions in Latin, we use several prepositions with different cases:
* inter + accusative = between
* trāns + accusative = across
* circum + accusative = around
* prope + accusative = near
We also use the commands:
* ī/īte = go! (singular/plural)
* verte/vertite = turn! (singular/plural)

## New Vocabulary

| Latin | English | Notes |
| vīcus, -ī (m.) | neighborhood, street | Root of "vicinity" |
| angulus, -ī (m.) | corner | Root of "angle" |
| regiō, -ōnis (f.) | district, region | Root of "region" |
| pōns, pontis (m.) | bridge | Root of "pontiff" |
| vertō, -ere | turn | Root of "convert" |
| dexter, -tra, -trum | right | Root of "dexterous" |
| sinister, -tra, -trum | left | Root of "sinister" |
| rēctus, -a, -um | straight | Root of "direct" |

## New Sentences

| Latin | English | Notes |
| In hōc vīcō habitō. | I live in this neighborhood. | Location |
| Ad angulum verte! | Turn at the corner! | Command |
| Haec regiō templum habet. | This district has a temple. | Description |
| Trāns pontem ī! | Go across the bridge! | Direction |
| Ad dextram verte! | Turn right! | Direction |
| Rēctā viā ī! | Go straight ahead! | Direction |
| Inter templum et basilicam est. | It is between the temple and basilica. | Location |
| Circum forum ambulāmus. | We walk around the forum. | Movement |
| Quōmodo ad bibliothēcam eō? | How do I get to the library? | Asking directions |
| Ad sinistram verte, deinde rēctā ī. | Turn left, then go straight. | Complex directions |
| Prope pontem caupōna est. | There is an inn near the bridge. | Location |
| Trāns viam et circum angulum īte! | Go across the street and around the corner! | Multiple directions |

## Practice

Practice and learn the words and phrases in this lesson :
* This content is not currently available on Memrise
* Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |
This completes our series on city locations and buildings. You can now describe where places are and give directions in Latin! Next time we'll begin learning about health and medicine. Valēte!
# Latin II/In the city Lesson 2

Salvēte omnēs! Welcome back to Latin for Wikiversity. Last time we learned about public buildings. Today we'll learn about shops and private buildings in a Roman city.

## New Grammar

To describe what kind of shop something is, Latin often uses a genitive:
* taberna librāria = bookshop (literally "shop of books")
* taberna vīnāria = wine shop
We can also use the suffix -ārius/a to describe shop types or shopkeepers:
* librārius = bookseller
* vīnārius = wine merchant

## New Vocabulary

| Latin | English | Notes |
| taberna, -ae (f.) | shop | Already know as "shop/workshop" |
| caupōna, -ae (f.) | inn, restaurant | Root of "copious" |
| vīlla, -ae (f.) | house, villa | Root of English "villa" |
| horreum, -ī (n.) | storehouse, warehouse | |
| librārius, -ī (m.) | bookseller | From liber (book) |
| vīnārius, -ī (m.) | wine merchant | From vīnum (wine) |
| pistor, -ōris (m.) | baker | Root of "pistol" (pounding) |
| vendō, -ere | sell | Root of "vend" |

## New Sentences

| Latin | English | Notes |
| In tabernā librāriā sum. | I am in the bookshop. | Type of shop |
| Caupōna vīnum bonum habet. | The inn has good wine. | Simple possession |
| Mārcus in magnā vīllā habitat. | Marcus lives in a large villa. | Location |
| Frūmentum in horreō est. | The grain is in the warehouse. | Storage |
| Librārius multōs librōs vendit. | The bookseller sells many books. | Occupation |
| Ad pistōrem īmus. | We go to the baker. | Motion towards |
| Vīnārius in forō vīnum vendit. | The wine merchant sells wine in the forum. | Complex sentence |
| Ubi est taberna pistōria? | Where is the bakery? | Question |
| Propē caupōnam habitāmus. | We live near the inn. | Location |
| In horreō labōrant. | They work in the warehouse. | Work location |
| Māne ad tabernām eō. | In the morning I go to the shop. | Time + motion |
| Vīlla templō propinqua est. | The villa is close to the temple. | Using propinquus |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Shops and Houses Vocabulary |
| Step two | Next try learning and writing the sentences using this: Shops and Houses Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
In our next lesson, we'll learn about city neighborhoods and how to give directions. Until then, practice describing the shops in your area! Valēte!
# Latin II/Sport and leisure Lesson 2

Salvēte omnēs! Welcome back to Latin for Wikiversity. Today we'll learn more vocabulary related to sports and games, focusing on team activities and competition.

## New Words

| Latin | English | Audio (Classical) | Notes |
| victōria, -ae (f.) | victory | File:La-cls-victoria.ogg | Related to vincō (conquer) |
| certāmen, -inis (n.) | contest, match | File:La-cls-certamen.ogg | From certō (compete) |
| turma, -ae (f.) | team | File:La-cls-turma.ogg | Originally cavalry unit |
| adversārius, -ī (m.) | opponent | File:La-cls-adversarius.ogg | From versus (against) |
| campus, -ī (m.) | field | File:La-cls-campus.ogg | Source of English "campus" |
| arbiter, -trī (m.) | referee | File:La-cls-arbiter.ogg | Also means "judge" |
| corōna, -ae (f.) | crown, prize | File:La-cls-corona.ogg | Award for victory |
| mēta, -ae (f.) | goal, turning post | File:La-cls-meta.ogg | Finish line in races |

## Competition Verbs

| Latin | English | Notes |
| certō, -āre | compete | Regular 1st conjugation |
| vincō, -ere | win, conquer | From previous lessons |
| laudō, -āre | praise, cheer | Regular 1st conjugation |

## New Sentences

| Latin | English | Notes |
| Turma nostra victōriam cupīvit. | Our team desired victory. | Perfect tense |
| In campō certāmus. | We compete on the field. | Location with in + ablative |
| Arbiter regulās explicat. | The referee explains the rules. | Present tense |
| Adversāriī fortiter lūdunt. | The opponents play bravely. | Using adverb |
| Ad mētam currēbant. | They were running to the goal. | Imperfect tense |
| Spectātōrēs turmam laudant. | The spectators cheer the team. | Direct object |
| Corōnam victor accēpit. | The winner received the crown. | Perfect tense |
| Contrā adversāriōs certābimus. | We will compete against opponents. | Future tense |
| Certāmen māgnum erit. | It will be a great contest. | Future of sum |
| Turma victrix corōnam habet. | The winning team has the crown. | Present tense |

## Common Phrases

| Latin | English | Notes |
| Fortiter certā! | Compete bravely! | Command |
| Euge! | Well done! | Interjection |
| Ad mētam! | To the goal! | Common cheer |
| Vincēmus! | We will win! | Future tense |

## Practice

Practice and learn the words and phrases in this lesson :
* Not currently availabl on Memrise
* Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) |
Now you can describe more complex sporting activities and competitions in Latin! Next time we'll move on to new topics. Valēte!
# Nilpotent endomorphism/Minimal polynomial/Fact

Let $K$ denote a field, and let $V$ denote a $K$-vector space of finite dimension. Let
    $\varphi \colon V\longrightarrow V$
be a linear mapping. Then the following statements are equivalent.
* $\varphi$ is nilpotent.
* The minimal polynomial of $\varphi$ is a power of $X$.
* The characteristic polynomial of $\varphi$ is a power of $X$.