OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/OCR2 Proofreading Test

necessary to consider the second bundle. The curvature form of our connection is a tensorial quadratic differential form in $M$, of type $ad(G^{\prime })$ and with values in the Lie Algebra $L(G^{\prime })$ of $G^{\prime }$. Since the Lie algebra $L(G)$ of $G$ is a subalgebra of $L(G^{\prime })$, there is a natural projection of $L(G^{\prime })$ into the quotient space $L(G^{\prime })/L(G)$. The image of the curvature form under this projection will be called the torsion form or the torsion tensor. If the forms $\pi ^{\rho }$ in (13) define a $G$-connection, the vanishing of the torsion form is expressed analytically by the conditions $(22)==\quad c_{j^{\prime \prime }k^{\prime \prime }}^{i^{\prime \prime }}=0.$ \quad We proceed to derive the analytical formulas for the theory of a $G$-connection without torsion in the tangent bundle. In general we will consider such formulas in $B_{G}$. The fact that the G-connection has no torsion simplifies (13) into the form $(23)==\quad d\omega ^{i}=\Sigma _{\rho ,k}a_{\rho k}^{i}\pi ^{\rho }\wedge \omega ^{k}.$ By taking the exterior derivative of (23) and using (18), we get $(24)==\quad \Sigma _{\rho ,k}a_{\rho k}^{i}\Pi ^{\rho }\wedge \omega ^{k}=0,$ where we put $(25)==\quad \Pi ^{\rho }=d\pi ^{\rho }+{\frac {1}{2}}\Sigma _{\sigma .\tau }\gamma _{\sigma \tau }^{\rho }\pi ^{\sigma }\wedge \pi ^{\tau }.$ For a fixed value of $k$ we multiply the above equation by $\omega ^{1}==\wedge .=..=\wedge ==\omega ^{k-1}==\wedge ==\omega ^{k+1}\ldots ==\wedge ==\omega ^{n},$ getting $\sum _{\rho }a_{\rho k}^{i}{\Pi ^{\rho }}==\wedge ==\omega ^{1}==\wedge .=..=\wedge ==\omega ^{n}=0,$ or $\Sigma _{\rho }a_{\rho k}^{i}{\Pi ^{\rho }}\equiv 0,\ \mathrm {m} \mathrm {o} \mathrm {d} \ \omega ^{j}.$
\noindent Since the infinitesimal transformations $X_{\rho }$ are linearly independent, this implies that $\Pi ^{\rho }\equiv 0,==\mathrm {m} \mathrm {o} \mathrm {d} \ \omega ^{j}.$ It follows that $\Pi ^{\rho }$ is of the form $\Pi ^{\rho }=\Sigma _{j}\phi _{j}^{\rho }\wedge \omega ^{j}$ where $\phi _{j}^{\rho }$ are Pfaffian forms. Substituting these expressions into (24), we get $\Sigma _{\rho ,j,k}(a_{\rho k}^{i}\phi _{j}^{\rho }-a_{\rho j}^{i}\phi _{k}^{\rho })\wedge \omega ^{j}\wedge \omega ^{k}=0.$ It follows that $\Sigma _{\rho }(a_{\rho k}^{i}\phi _{j}^{\rho }-a_{\rho j}^{i}\phi _{k}^{\rho })\equiv 0,==\mathrm {m} \mathrm {o} \mathrm {d} \ \omega ^{k}.$ Since $G$ has the property $(C)$, the above equations imply that $\phi _{j}^{\rho }\equiv 0,==\mathrm {m} \mathrm {o} \mathrm {d} \ \omega ^{k}.$
# PlanetPhysics/Non Abelian Theory

A non-Abelian theory is one that does not satisfy one, several, or all of the axioms of an Abelian theory, such as, for example, those for an Abelian category theory.

### Examples

ETAC and ETAS axiom interpretations that do not satisfy--in addition to the ETAC or ETAS axioms-- the $Ab1$ to $Ab6$ axioms for an abelian category are all examples on non-Abelian categories; a more detailed list is also presented next.
In a general sense, any Abelian category (or abelian category ) can be regarded as a `good' model for the category of Abelian, or commutative, groups. Furthermore, in an Abelian category $Ab$ every class, or set, of morphisms $Hom_{Ab}(-,-)$ forms an Abelian (or commutative) group. There are several strict definitions of Abelian categories involving 3, 4 or up to 6 axioms defining the Abelian character of a category. To illustrate non-Abelian theories it is useful to consider non-Abelian structures so that specific properties determined by the non-Abelian set of axioms become `transparent' in terms of the properties of objects for example for concrete categories that have objects; such examples are presented separately as non-Abelian structures .

### Further examples of non-Abelian theories

The following is only a short list of non-Abelian theories:
* non-Abelian algebraic topology, including also non-Abelian homological algebra;
non-Abelian algebraic topology overview and R. Brown 2008 preprint,.\\ (See also the [2008 http://planetmath.org/?op=getobj&from=lec&id=75 recent book exposition] with the title "Nonabelian Algebraic Topology" vol. 1 by Brown and Sivera,(respectively, vol. 2 with Higgins, in preparation ).
* Non-Abelian Quantum Algebraic Topology;
* Non-Abelian gauge field theory (in Quantum Physics);
* noncommutative geometry;
* The axiomatic theory of supercategories (ETAS);
* higher dimensional algebra (HDA)
* $LM_{n}$ Logic algebras;
* Non-Abelian categorical ontology.

### Remarks

The following alternative definition by Barry Mitchell of an Abelian category should also be mentioned as "an exact additive category with finite products." .
He also published in his textbook the following theorem: (Theorem 20.1 , on p.33 of Barry Mitchell in "Theory of Catgeories", 1965, Academic Press: New York and London):
\begin{theorem} The following statements are equivalent :
* (a) $Ab$ is an abelian category;
* (b) $Ab$ has kernels, cokernels, finite products, finite coproducts, and is both normal and conormal;
* (c) $Ab$ has pushouts and pullbacks and is both normal and conormal.
\end{theorem}
# PlanetPhysics/Non Commutative Structure

Let $(C,\circ )$ be a structure consisting of a class , $C$, together with a binary operation $\circ$ defined for pairs of objects in $C$ (or elements of $C$ when the latter is a small class, i.e., a set). The structure-- and the operation $\circ$-- are said to be noncommutative if
   ${\begin{matrix}a\circ b\neq b\circ a\end{matrix}}$
for either at least some or all of the $a,b$ pairs in $C$ for which the operation is defined.
A structure that is noncommutative is also called sometimes a \htmladdnormallink{non-Abelian {http://planetphysics.us/encyclopedia/AbelianCategory3.html} structure}, although the latter term is, in general, more often used to specify non-Abelian theories. A binary operation that is not commutative is said to be non-commutative (or noncommutative ). Thus, a noncommutative structure can be alternatively defined as any structure whose binary operation is not commutative (that is, in the commutative case one has
   ${\begin{matrix}a\circ b=b\circ a\end{matrix}}$
for all $a,b$ pairs in $C$, and also that the operation $\circ$ is defined for all pairs in $C$).
An example of a commutative structure is the field of real numbers-- with two commutative operations in this case-- which are the addition and multiplication over the reals.
A commutative group is also called Abelian , whereas a category with structure that has commutative diagrams is not necessarily Abelian --unless it does satisfy the Ab1 to Ab6 axioms that define an abelian category (or equivalently, if it has the properties specified in Mitchell's alternative definition of an Abelian category .)
An example of a non-commutative operation is the multiplication over $n\times n$ matrices. Another example of a noncommutative algebra is a general Clifford algebra, which is of fundamental importance in the algebraic theory of observable quantum operators and also in Quantum Algebraic Topology.
# Materials Science and Engineering/List of Topics/Statistical Mechanics
## Statistical Mechanics
### Probability

General Defintions

### Classical Statistical Mechanics

Clasical Mechanics
Equations of Motion
Langrangian Formulation
Microscopic State of a System
Equilibrium Ensembles

### Quantum Statistical Mechanics

Vibrations of a Solid
# Materials Science and Engineering/List of Topics/Statistical Mechanics/Classical Mechanics

Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science and technology.
Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light. Furthermore, general relativity is employed to handle gravitation at a deeper level. In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies. The other sub-field is quantum mechanics.
The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century workers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "relativistic physics" from that category. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form. The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. This is further described in the following sections. More abstract, and general methods include Lagrangian mechanics and Hamiltonian mechanics. While the terms classical mechanics and Newtonian mechanics are usually considered equivalent (if relativity is excluded), much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton.
# Materials Science and Engineering/List of Topics/Statistical Mechanics/Probability/General Definitions
## Mathematical treatment

In mathematics a probability of an event, A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A). An impossible event has a probability of 0, and a certain event has a probability of 1. However, the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".
The chance of the opposite of an event P(not A) = 1-P(A), for example the chance of not rolling a six on a six-sided die is 1 - chance of rolling a six = ${1}-{\tfrac {1}{6}}={\tfrac {5}{6}}$. If two events, A and B are independent then the joint probability is
    $P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B),\,$
for example if two coins are flipped the chance of both being heads is ${\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}$.
If two events are mutually exclusive then the probability of either occurring is
    $P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B).$
For example, the chance of rolling a 1 or 2 on a six-sided die is $P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}$.
If the events are not mutually exclusive then
    $\mathrm {P} \left(A{\hbox{ or }}B\right)=\mathrm {P} \left(A\right)+\mathrm {P} \left(B\right)-\mathrm {P} \left(A{\mbox{ and }}B\right)$.
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B". It is defined by
    $P(A\mid B)={\frac {P(A\cap B)}{P(B)}}.\,$
If $P(B)=0$ then $P(A\mid B)$ is undefined.
Summary of probabilities :
| Event | Probability |
| A | $P(A)\in [0,1]\,$ |
| not A | $P(\sim A)=1-P(A)\,$ |
| A or B | ${\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}$ |
| A and B | ${\begin{aligned}P(A\cap B)&=P(A|B)P(B)\\&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}$ |
| A given B | $P(A|B)\,$ |
# The necessities in computer architecture

This course belongs to the Electrical & Computer Engineering Studies

## MIPS Assembly

* Assembly (pdf)

## ALU Subsystem
### Endian

* Endian (pdf)

### Floating Pointer Numbers

* Rounding (pdf)

## Controller Subsystem
#### MCU (Memory Control Unit)
#### IO Controller

* GPIO (6A.pdf)

#### Interrupt Controller
## ADC & DAC

* ADC / DAC Overview (pdf)
# Poetry/Practice/Eternity

The Spirits:
Lie still. Relax. You’ve done too much, -Ambitious creature!- dreamed too much… And made yourself ill, with this to show? Look at yourself. You’re doomed, alive.
Look… and see… remember well That which you have forsaken Is eternity so bare That you would bear mortality?
For here, you could remain with us And here, could lie forever
And yet, you’ve chosen to die. For that is the cost of life in all its incarnations; With no absolute, no base but balance. Nothing can live unchanging, And nothing eternal live past an instant
And to think of it! The shadow of Death overbearing, always the knowledge your time will end. Not knowing how, or where, or when, or why… That is the cost of being alive.
The Newborn:
But therein, brethren, lies the greatness,
That life could change, yet somehow persist. And would always be something of challenge, unique
While amidst the familiar, the ages past
See in that Death, the greatest choice. That we could sustain our lives as we pleased… or encourage fate, and resk adventure.
That in the Valley of the Shadow of Death,
    Each Light, for it,
       Would be all the brighter
# Poetry/Practice

This page is a project to see how collaborative poetry can evolve. It is also a practice space to create and write poetry to refine your writing skills. If you would like to have stable versions of your work please place in the user namespace (i.e. as a subpage to your userpage).

### Poems

* Trust
* Feuer
* The Fool
* Eternity
* You Disturbed Me
* Universal Language of Absolutes

### Essays/Reports/Rants

* Perfection
* On Hatred
* Privacy, Security, and Implied Mutual Exclusion
# Poetry/Practice/Feuer

A man, shaking softly, chanced upon embers.
Shivering, haltingly, rifled the corpse.
He found it with breath, garnered its Spark
from which he raised Fire. Raised Life.
He carried it with him, shy from the rain
And in shelter of wind, breathed for its life
He scavenged its tinder, foraged its food...
And basked in the heat of its warmth and its light.
But in his time with the Fire... He'd lost his good fear, let it grow into maelstrom for beauty within.
And in his time with the Fire... He'd forgotten his burns, drawn closer... closer...
He was Consumed
# Poetry/Practice/Trust

Trust is isolated and alone, the last relic of humanity in the face of civilization. It still breathes, still lives, and for being forgotten, is strong.
"Friends"?
    Meaningless.
"Sworn"?
    Upon lies.
"I love you"
    Why? >In tandem
    What do you want.>In tandem
"I need you to trust me." Then you don't understand...
"I trust you."
Now listen and see
    the inumerable friends and colleages... my acquaintainces.
       They know as little of me as I of them. We mingle, we chatter and whine, but at heart are aloof. They are reliable only insofar as their habits, and thus far valued. Here are those that, were they to know my true countenance, would shrink away, or seek my demise. Here are creatures neither loved nor disdained, merely... incompatible.
    those who may say they know me
       and are with me at ease. These are kin, in that they are capable of kindness. Trusted? Maybe, the definition seems tenous enough. With them, at least, I am not at guard. The blinds may stand, the masks remain, but we feel our collective ways past them delicately, not daring to trod upon the wounds they hide. There is only familiarity here, and trust extends only so far.
But you, friend...
    "I trust you"
Thou art my nemesis and my friend, for you can see... me... Every instinct I have screams to hold you at bay, but one of us has stripped them away. You are either my saviour... or my betrayer, and as I shed my armour I show you something greater than love, than hatred. I am trusting you to look upon me, to judge me, and to choose.
I have trusted you... That you may destroy me. So choosing to grind me into earth. That you may save me, if you think me of worth. Or you can leave as you came, both condemning me and sparing me in as deft a motion.
    I have given you the key.
What I need to know... Do you trust me?
# Poetry/Practice/The Fool

~Mephisto, to Me and Us~
Look before me, to the shining soul and see that for which my mind evades, For which it seeks divorced of heart, And in the shield of logic, finds a certain... selfish.... safety.
Look in that soul, look in thine self. Is even your heart so soon forsaken? Or in the trap that breeds but weakness, Is there, maybe, strength?
For flawless logic failed to account,
Failed to take in mind. A thing both wise and widely known, To those who know their kind:
To know that for all his Faults, the Fool can stand And aid in the aid of a helping Hand
Good Fool... Remember: “Omnia vincit amor”
# Piano/Gymnopédie 1 by Erik Satie

This score is played as ABAC, where A is 32 bars long while B and C are each 8 bars.

#### B & C
## Chords

Just the chords, each one presented only one time, in order of initial appearance.
I am unable to increase the size of the staff.

#### A chords
##### back

The backup file permits me to experiment with ways to increase the size of the staff.
# Piazza

Piazza is a Q&A web service. It has been described as a mixture between a wiki and a forum that can be used with learning management systems. The word Piazza comes from the Italian word for plaza—a common city square where people can come together to share ideas and knowledge.

## Readings

* Wikipedia: Piazza (web service)

## Multimedia

* YouTube: Piazza

## Activities

* Review the Piazza reading and/or multimedia links above.
* As a student, join an existing Piazza course, or as an instructor, create a new Piazza course.
* Review available resources, including Course Information, Staff, Resources, and Groups.
* Review existing Q&A discussion posts and respond to any that interest you.
* Create a New Post as a Note, introducing yourself to the class.
* Create a New Post as a Question to see the difference between questions and notes.
* Create a New Post as a Poll to see the difference between notes and polls.

## See Also

* Educational Technology
# Picard/Grammar1

In grammar, we distinguish the name, the article, the adjective, the adverb, the verb, the pronoun...

## Names and articles

The name is a class of words which indicates the categories of beings, things and the concepts.

### male name

             ch'gardin (the garden) , ech mon.nieu (the sparrow) , echl'éfant (the boy)
             un mon.nieu (a sparrow) ,

### female name

             el vake , l'vake , la vake , (the cow) , chole feume (the woman)
             ène vake , un.ne vake , eune vake , in.ne vake (a cow)

### plurals of names

             chés garchons (the guys) , chés éreilles , echs éreilles (ears)

### Partitive article du

             J’ai du chuke dins min sanke. (I have some sugar in the blood)

## Numeral adjectives

* Un, Deus, Troés, Quate, Chonc, ... ( 1 , 2 , 3 , 4 , 5, ...)
* deus foé quate font uit. ( 2 x 4 = 8 )
* prumié, sgond, trojième, quatième, chinquième, ... (first, segond, third, 4th, ... )

## Possessive adjectives

* min fiu , em bouke, mes fius, mes vakes (My son, my mouth, my sons, my cows)
* emn éfant, emn éreille, mes éreilles, ems éreilles (My child, my ear, my ears, my ears)

## Verbs

In picard language, we have 3 groups:
       - first group = canter, warder (to sing , to keep)
       - second group = finir, bolir (to finish, to boil)
       - 3th group = mintir, rinde, boère, mette, etc ... (to lie, to make, to drink, to put, ...)

## Negative forms
### n' ... poin

       i n'pérle poin picard. (he does not speak picard)

### non

       Ch'est un carpintié ? - Non, ch'est un émnusié. (It is a carpenter? No, it is a joiner)

### mie

       Ch'est mie cho ! (It is not this!)

## Interrogative forms

* ch'est coér loin d'ichi ? (It is still far from here?)
* varos-tu ? (Will you come?)
* Cha vo-ti? (Are you all right?)
* Te viens-jou ? (Did you come from?)

## Inquiring adjectives

* Quo qu'té bos ? (What are you drinking?)
* Qué lives qu'os volez ? (What books do you want?)
* Conbin d'peumes éq vos volez ? (How many apples do you want?)

## Pronoun interrogatives

* Té pérle éd quoué ? (What are you talking about?)
| Prumier nivieu | Vocabulary |
# Picard/Picard bases

* Introduction to Picard Picard bases
* Your first steps in Picard
* Here are the very first sentences of Picard in this course :
* Who is it ?
* Ch'est qui ?
* It's John
* Ch'est Jean
* Hello, you are fine ?
* Bojour, os allez bin ?
* Really good, thank you very much !
* Vraiment bin, ej vos armérchie !
* Goodbye John
* Adé Jean

### Basic vocabulary

| English | Picard | French |
| Hello! | Bojour ! or Bojour mes gins ! (formal) or Salut ti z’aute ! (informal) | Bonjour (Literal translation: Bonjour mes gens ! or Salut vous autres !) |
| Good evening! | Bonsoèr ! | Bonsoir |
| Good night! | La boinne nuit ! | Bonne nuit ! |
| Goodbye! | À l’arvoïure ! or À t’ervir ! | Au revoir ! |
| Have a nice day! | Eune boinne jornée ! | Bonne journée ! |
| Please/if you please | Sins vos komander (formal) or Sins t' komander (informal) | S'il vous plaît (lit: sans vous commander) |
| Thank you | Merchi | Merci |
| I am sorry | Pardon or Échtchusez-mi | Pardon or Excusez-moi |
| What is your name? | Kmint qu’os vos aplez ? | Comment vous appelez-vous ? |
| How much? | Combin qu’cha coûte ? | Combien ça coute ? |
| I do not understand. | Éj n'comprinds poin. | Je ne comprends pas. |
| Yes, I understand. | Oui, j' comprinds. | Oui, je comprends. |
| Help! | À la rescousse ! | À l'aide (lit.: À la rescousse !) |
| Can you help me please? | Povez-vos m’aider, sins vos komander ? | Pouvez-vous m'aider, s'il vous plaît ? |
| Where are the toilets? | D'ousqu'il est ech tchioér ? | Où sont les toillettes ? (Slang: Où sont les chiottes ?) |
| Do you speak English? | Parlez-vos inglé ? | Parlez-vous anglais ? |
| I do not speak Picard. | Éj n’pérle poin picard. | Je ne parle pas picard. |
| I do not know. | Éj n’sais mie. | Je ne sais pas. |
| I know. | Éj sais. | Je sais. |
| I am thirsty. | J’ai soé. (literally, "I have thirst") | J'ai soif. |
| I am hungry. | J’ai fan. (literally, "I have hunger") | J'ai faim. |
| How are you? / How are things going? / How is everything? | Comint qu’i va ? (formal) or Cha va t’i ? | Comment vas-tu ? or Ça va ? |
| I am fine. | Cha va fin bien. | Ça va bien. |
| English | Inglé | Anglais |
| Sugar | Chuque | Sucre |
| Crybaby | Brayou | Pleurnicheur (lit: brailleur) |
| previous lesson | previous page | next page | lesson intro | next lesson | Department | Division |
# Piano/For beginners
## letsplaykidsmusic.com

* This section contains links to www.letsplaykidsmusic.com and 150+ Free Kids Songs
* Not for absolute beginners, but this piece from S.T.A.Y looks easy. It's available commercially at https://musescore.com/user/33687032/scores/7115513 . They don't spell out the financial obligations.

### Pick your first song

* Twinkle on YouTube
       Simple instructions
* Jingle Bells on YouTube
* Happy Birthday on YouTube
* Mary Had a Little Lamb on YouTube

### Twinkle
## Footnotes

* ↑ See also https://www.letsplaykidsmusic.com/wp-content/uploads/2014/01/twinkle.pdf and https://www.letsplaykidsmusic.com/twinkle-twinkle-little-star-easy-piano-music/
* ↑ See also http://www.letsplaykidsmusic.com/wp-content/uploads/2013/11/Jingle-Bells.pdf and https://www.letsplaykidsmusic.com/jingle-bells-very-easy-piano-sheet-music/
* ↑ See also http://www.letsplaykidsmusic.com/wp-content/uploads/2013/10/happy-birthday-chord-two-hands.pdf and https://www.letsplaykidsmusic.com/happy-birthday-easy-piano-music/
* ↑ See also https://makingmusicfun.net/public/assets/pdf/sheet_music/mary-had-a-little-lamb-piano.pdf and https://makingmusicfun.net/htm/f_printit_free_printable_sheet_music/mary_had_a_little_lamb_piano