OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Groupoid Categories

simply by considering a groupoid as a category {Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G}_1} } with all invertible morphisms, and objects defined by the groupoid class or set of groupoid elements; then, the groupoid category, Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _2} }, is defined as the $2$-category whose objects are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathsf{\G''' _1} } categories (groupoids), and whose morphisms are functors of Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G''' _1} } categories consistent with the definition of groupoid homomorphisms, or in the case of topological groupoids, consistent as well with topological groupoid homeomorphisms. The 2-category of groupoids Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _2} }, plays a central role in the generalised, categorical Galois theory involving fundamental groupoid functors.
# PlanetPhysics/Groupoid Representation Theorem

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

### Groupoid representation theorem

We shall briefly consider a main result due to Hahn (1978) that relates groupoid and associated groupoid algebra representations:
\begin{theorem} {\rm (source:.)}~ Any representation of a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} with Haar measure </math>(\nu, \mu)$inaseparableHilbertspace$\H$inducesa*-algebrarepresentation<math>f\mapsto X_{f}$ of the associated groupoid algebra Failed to parse (unknown function "\grp"): {\displaystyle \Pi \grp, \nu)} in Failed to parse (unknown function "\grp"): {\displaystyle L^2 (U_{\grp} , \mu, \mathbb{H} )} with the following properties:
(cf. p. 50 of Hahn, 1978). \end{theorem}

### Remarks

Furthermore, according to Seda (1986, on p.116) the continuity of a Haar system is equivalent to the continuity of the convolution product $f*g$ for any pair $f,g$ of continuous functions with compact support. One may thus conjecture that similar results could be obtained for functions with locally compact support in dealing with convolution products of either locally compact groupoids or quantum groupoids. Seda's result also implies that the convolution algebra Failed to parse (unknown function "\grp"): {\displaystyle C_{conv}(\grp)} of a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is closed with respect to the convolution {*} if and only if the fixed Haar system associated with the measured groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is continuous (Buneci, 2003).
In the case of groupoid algebras of transitive groupoids, Buneci (2003) showed that representations of a measured groupoid Failed to parse (unknown function "\grp"): {\displaystyle ({\grp, [\int \nu ^u d \tilde{ \lambda}(u)]=[\lambda]})} on a separable Hilbert space $\mathbb {H}$ induce non-degenerate $*$--representations $f\mapsto X_{f}$ of the associated groupoid algebra Failed to parse (unknown function "\grp"): {\displaystyle \Pi (\grp, \nu,\tilde{\lambda})} with properties formally similar to (1) and (2) above. Moreover, as in the case of groups, \textit{there is a correspondence between the unitary representations of a groupoid and its associated C*--convolution algebra representations} (p.182 of Buneci, 2003), the latter involving however fiber bundles of Hilbert spaces instead of single Hilbert spaces. Therefore, groupoid representations appear as the natural construct for Algebraic Quantum Field Theories (AQFT) in which nets of local observable operators in Hilbert space fiber bundles were introduced by Rovelli (1998).
# PlanetPhysics/Growth of Exponential Function

Lemma. $\lim _{x\to \infty }{\frac {x^{a}}{e^{x}}}=0$ for all constant values of $a$.
Proof. \, Let $\varepsilon$ be any positive number.\, Then we get:
$0<{\frac {x^{a}}{e^{x}}}\leqq {\frac {x^{\lceil a\rceil }}{e^{x}}}<{\frac {x^{\lceil a\rceil }}{\frac {x^{\lceil a\rceil +1}}{(\lceil a\rceil +1)!}}}={\frac {(\lceil a\rceil +1)!}{x}}<\varepsilon$ as soon as\, $x>\max\{1,{\frac {(\lceil a\rceil +1)!}{\varepsilon }}\}$.\, Here, $\lceil \cdot \rceil$ means the ceiling function;\, $e^{x}$ has been estimated downwards by taking only one of the all positive terms of the series expansion $e^{x}=1+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}+\cdots$\\
\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.} The growth of the real exponential function\,\, $x\mapsto b^{x}$\,\, exceeds all power functions, i.e. $\lim _{x\to \infty }{\frac {x^{a}}{b^{x}}}=0$ with $a$ and $b$ any constants,\, $b>1$.
Proof. \, Since\, $\ln b>0$,\, we obtain by using the lemma the result $\lim _{x\to \infty }{\frac {x^{a}}{b^{x}}}=\lim _{x\to \infty }\left({\frac {x^{\frac {a}{\ln b}}}{e^{x}}}\right)^{\ln b}=0^{\ln b}=0.$\\
Corollary 1. \, $\lim _{x\to 0+}x\ln {x}=0.$
Proof. \, According to the lemma we get $0=\lim _{u\to \infty }{\frac {-u}{e^{u}}}=\lim _{x\to 0+}{\frac {-\ln {\frac {1}{x}}}{\frac {1}{x}}}=\lim _{x\to 0+}x\ln {x}.$\\
Corollary 2. \, $\lim _{x\to \infty }{\frac {\ln {x}}{x}}=0.$
Proof. \, Change in the lemma\, $x$\, to\, $\ln {x}$.\\
Corollary 3. \, $\lim _{x\to \infty }x^{\frac {1}{x}}=1.$ \, (Cf. limit of nth root of n.)
Proof. \, By corollary 2, we can write:\, $x^{\frac {1}{x}}=e^{\frac {\ln {x}}{x}}\longrightarrow e^{0}=1$\, as\, $x\to \infty$ (see also theorem 2 in limit rules of functions).
# PlanetPhysics/Groupoid Action

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
* $\pi (xg)=s(g),$
* $xu(\pi (x))=x,$ and
* $(xg)h=x(gh),$ whenever the operations are defined.
{\mathbf Note:} The groupoid action generalizes the concept of group action in a non-trivial way
# PlanetPhysics/Groupoid C Dynamical Systems

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
is a triple Failed to parse (unknown function "\grp"): {\displaystyle (A, \grp_{lc}, \rho )} , where: $A$ is a C*-algebra, and Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a locally compact (topological) groupoid with a countable basis for which there exists an associated continuous Haar system and a continuous groupoid (homo) morphism Failed to parse (unknown function "\grp"): {\displaystyle \rho: \grp_{lc} \longrightarrow Aut(A)} defined by the assignment $x\mapsto \rho _{x}(a)$ (from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to $A$) which is continuous for any $a\in A$; moreover, one considers the norm topology on $A$ in defining Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . (Definition introduced in ref..)
A groupoid C*-dynamical system can be regarded as an extension of the ordinary concept of dynamical system. Thus, it can also be utilized to represent a quantum dynamical system upon further specification of the C*-algebra as a von Neumann algebra, and also of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} as a quantum groupoid; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).
# PlanetPhysics/Groupoid Representations Induced by Measure

A groupoid representation induced by measure can be defined as measure induced operators or as operators induced by a measure preserving map in the context of Haar systems with measure associated with locally compact groupoids, $\mathbf {G_{lc}}$. Thus, let us consider a locally compact groupoid $\mathbf {G_{lc}}$ endowed with an associated Haar system $\nu =\left\{\nu ^{u},u\in U_{\mathbf {G_{lc}} }\right\}$, and $\mu$ a quasi-invariant measure on $U_{\mathbf {G_{lc}} }$. Moreover, let $(X_{1},{\mathfrak {B}}_{1},\mu _{1})$ and $(X_{2},{\mathfrak {B}}_{2},\mu _{2})$ be measure spaces and denote by $L^{0}(X_{1})$ and $L^{0}(X_{2})$ the corresponding spaces of measurable functions (with values in $\mathbb {C}$). Let us also recall that with a measure-preserving transformation $T:X_{1}\longrightarrow X_{2}$ one can define an operator induced by a measure preserving map , $U_{T}:L^{0}(X_{2})\longrightarrow L^{0}(X_{1})$ as follows.
\begin{displaymath} (U_T f)(x):=f(Tx)\,, \qquad\qquad f \in L^0(X_2),\; x \in X_1 \end{displaymath}
Next, let us define $\nu =\int \nu ^{u}d\mu (u)$ and also define $\nu ^{-1}$ as the mapping $x\mapsto x^{-1}$. With $f\in C_{c}(\mathbf {G_{lc}} )$, one can now define the measure induced operator $'''Ind'''\mu (f)$ as an operator being defined on $L^{2}(\nu ^{-1})$ by the formula: $'''Ind'''\mu (f)\xi (x)=\int f(y)\xi (y^{-1}x)d\nu ^{r(x)}(y)=f*\xi (x)$
Remark:
One can readily verify that :
$\left\|'''Ind'''\mu (f)\right\|\leq \left\|f\right\|_{1}$,
and also that $'''Ind'''\mu$ is a proper representation of $C_{c}(\mathbf {G_{lc}} )$, in the sense that the latter is usually defined for groupoids.
# PlanetPhysics/Groupoid Homomorphism

Let ${\mathsf {G}}_{1}$ and ${\mathsf {G}}_{2}$ be two groupoids considered as two distinct categories with all invertible morphisms between their objects (or 'elements'), respectively, $x\in Ob({\mathsf {G}}_{1})={{\mathsf {G}}_{0}}^{1}$ and $y\in Ob({\mathsf {G}}_{2})={{\mathsf {G}}_{0}}^{2}$. A groupoid homomorphism is then defined as a functor $h:{\mathsf {G}}_{1}\longrightarrow {\mathsf {G}}_{2}$.
A composition of groupoid homomorphisms is naturally a homomorphism, and natural transformations of groupoid homomorphisms (as defined above by groupoid functors) preserve groupoid structure(s), i.e., both the algebraic and the topological structure of groupoids. Thus, in the case of topological groupoids, ${\mathsf {G}}$, one also has the associated topological space homeomorphisms trivial groupoid that naturally preserve topological structure.
Remark: Note that the morphisms in the category of groupoids, $Grpd$, are, of course, groupoid homomorphisms, and that groupoid homomorphisms also form (groupoid) functor categories defined in the standard manner for categories.
# PlanetPhysics/Groupoids

This is a contributed topic on groupoids and their applications.
Groupoids are a key concept in modern topology, and especially in algebraic topology; they may be considered as one of the simplest, special types of categories.

## Introduction

Several classes of groupoids and large groupoids shall be considered in this topic with pertinent examples that illustrate the construction of groupoids through several extensions of the much simpler (and global) group symmetry to both higher order symmetries and dimensions, as well as internal (or local, partial) plus external symmetry. Considered as powerful tools for investigating both Abelian and non-Abelian structures, groupoids are now essential for understanding topology, and are one of the important--if not the most important -- concepts in algebraic topology

## Groupoids

Def.
* a "magma: a set with a total binary operation" or
* a "set with a partial binary operation that is associative and has inverses and identities"
is called a groupoid.
"A groupoid is a category in which every morphism is an isomorphism."

## Topology

Def.
* a "branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms", or
* a "collection τ of subsets of a set X such that the empty set and X are both members of τ and τ is closed under arbitrary unions and finite intersections"
is called a topology.
Def. a "branch of topology that associates objects from abstract algebra to topological spaces" is called an algebraic topology.

## Brief Description

Groupoids are generalizations or extensions of the concept of group, supergroup, `virtual group', and paragroup, in several ways; one may simply extend the notion of a group viewed as an one-object category to a many-object category with group-like elements and all invertible morphisms. Another enrichment of the notion of a group--as in the case of topological groups-- is the concept of topological groupoid ${\mathsf {G}}$. One can also think of a groupoid as a class of linked groups, and further extend the latter groupoid definition to higher dimensions through `geometric'-algebraic constructions, for example, to double groupoids, cubic groupoids, ..., groupoid categories, groupoid supercategories, and so on. Crossed modules of groups and crossed complexes also correspond to such extended groupoids.
For precise definitions of specific classes of groupoids, see also groupoid and topological groupoid definitions, as well as those entries listed next as examples.

## Additional examples

These are examples of major classes of groupoids defining the several extensions and enrichment possibilities of the notions of group and group symmetry introduced in the above definition are the subject of several other entries:
* 2-groupoids (please see groupoid categories )
* Double groupoids; homotopy double groupoid of a Hausdorff space #higher homotopy groupoids and the higher dimensional, generalized Van Kampen theorems #Groupoid category
* Crossed complexes
* higher dimensional algebra (HDA)
* Groupoid super-categories ($n$-categories, etc.)
* Groupoid supercategories
# Review:Audacity

Filmmaking Software Review
Narrative Film Production:
Free Software for Filmmaking
Audacity :
| Audacity This is a simple sound editing program. You need this to convert sound files to the format used at Wikiversity which is the OGG file format. | Uses at WikiU Film School For dialog recording For film scoring |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:Disk 2 - "24 Unedited Scenes"

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
Disk #2 - "24 Unedited Scenes" DVD-Data :
| Disk #2 - "24 Unedited Scenes" This is behind-the-scenes look at filming 24 different scenes. Only about 12 of the 24 scenes are interesting. Still this is the cheapest way to tour the movie sets of Hollywood. The price (not listed on their website) is $8 US plus shipping. (You must be a film student to purchase this disk. Fortunately, if you have completed an assignment here, you qualify for 30 days.) If you have never worked in Hollywood, you should purchase this disk to gain an understanding of what goes on during the filming of a scene. Highly recommended to young filmmakers and actors. | Uses at WikiU Film School Film editing Introduction to filmmaking |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:Disk 9 - "Hearing Voices" Editing Workshop

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
"Hearing Voices" Editing Workshop DVD-Video :
| "Hearing Voices" Editing Workshop DVD-Video This is behind-the-scenes look at filming and editing of one scene. A brief introduction to "L-Cuts". DVD-Video. The price (not listed on their website) is $8 US plus shipping. (You must be a film student to purchase this disk. Fortunately, if you have completed an assignment here, you qualify for 30 days.) This disk is one of the few required disks for the film editing project. I have not seen anything else which is as good. | Uses at WikiU Film School Film editing Introduction to filmmaking |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:Finale NotePad

Filmmaking Software Review
Narrative Film Production:
Free Software for Filmmaking
Finale NotePad :
| Finale NotePad I will test this program shortly. It is totally free and simple to use so it should be very useful. | Hello, I'm Jon. Finale NotePad is an excellent product. The license key costs $10, and you can't beat that. I only needed a fairly simple means to take some rather sloppy handwritten sheet music and transcribe it into printed sheet music. NotePad did just that. NotePad was very intuitive and easy to use. I did not need an expensive program interfaced with MIDI, lots of bells and whistles, etc., just a basic graphics program that would allow me to manually compose sheet music like a word processor, using mostly just my computer mouse. Before I found NotePad, I was trying to make sheet music with Photoshop, which as you can imagine, was extremely tedious in trying to paste each note as a layered graphic on a blank sheet music graphic. For $10, this program is an incredible value! The Folks at Finale saved me days of hard work in Photoshop, and it only took me a couple of hours to produce two pages of sheet music after opening up and playing with NotePad. Now that I am familiar with NotePad, it will go even quicker next time. I hope this helps any buying decision you make. Regards, Jon Uses at WikiU Film School Film scoring |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:ArtRage2 Free

Filmmaking Software Review
Narrative Film Production:
Free Software for Filmmaking
ArtRage 2 Free from Ambient Designs :
| ArtRage 2 Free This is a fun program! And the free version is fully working (but not as powerful and the full version.) This is not a demo program. This is the real thing! Since you will occasionally need a painting program while you are making your motion pictures, immediately download this and play with it. You will enjoy it and it will be useful. | Uses at WikiU Film School For thumbnail storyboards For creating the movie prop poster |
# Review:Jam Pack: Symphony Orchestra

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
Jam Pack:Symphony Orchestra :
| Apple's Jam Pack:Sympony Orchestra GarageBand comes free with each new Macintosh computer but Jam Pack:Sympony Orchestra is not free. When combined together, they produce fantastic musical sound effects. Ideal for people with no musical ability. Jam Pack:Symphony Orchestra costs less than MOTU Symphonic Instrument but MOTU Symphonic Instrument has a boy's chorus. But MOTU Symphonic Instrument is awkward to use and does not work well with GarageBand. Your instructor does NOT recommend MOTU Symphonic Instrument. The only Windows alternative that I know of is Tascam's GigaStudio 3.0 - Orchestra. | Uses at WikiU Film School Film scoring Other Uses See Mad Max's Course in Film Scoring for Motion Pictures |

## Contact your instructor

    Your instructor for this filmmaking class is Robert Elliott. You can email me by clicking here.
# Review:Disk 1 - "Follow Dave?" Editing Workshop

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
"Follow Dave?" Editing Workshop CD-ROM :
| "Follow Dave?" Editing Workshop CD-ROM This is behind-the-scenes look at filming and editing of one scene. Includes a 10 minute movie which explains how to edit dialog. The price (not listed on their website) is $8 US plus shipping. (You must be a film student to purchase this disk. Fortunately, if you have completed an assignment here, you qualify for 30 days.) This disk is one of the few required disks for the film editing project. I have not seen anything else which is as good. As of November 2007, the Star Movie Shop is not open to the public. For students in the courses, the instructor will loan you a copy of this disk if you promise to return it in one month. | Uses at WikiU Film School Film editing Introduction to filmmaking |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:FrameForge 3D Studio 2 Demo Version

Filmmaking Software Review
Narrative Film Production:
Free Software for Filmmaking
FrameForge 3D Studio 2 demo version :
| FrameForge 3D Studio - demo version This is an amazingly educational program! The demo version can only be used 20 times and has other limitations. However, that is enough for you to learn a tremendous amount about cinematography. A wonderful opportunity. Anyone interested in filmmaking should immediately download this program. Easy to use but very powerful. | Uses at WikiU Film School For 3D storyboards |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# Review:MOTU Symphonic Instrument

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
MOTU Symphonic Instrument :
| MOTU Symphonic Instrument MOTU Symphonic Instrument is an alternative to Jam Pack:Sympony Orchestra. Your instructor does NOT recommend this product. Sound Great The literature for this product sounds great. But if you have ever used GarageBand with Jam Pack:Symphony Orchestra, you might be very disappointed. MOTU Symphonic Instrument is not easy to set up. It is slow. It requires huge amounts of memory. And it does NOT work well with GarageBand. Does not work with GarageBand 2. The only good thing I have found about this product is the voices which are not available with any of the Jam Packs. | Uses at WikiU Film School Film scoring Other Uses See Mad Max's Course in Film Scoring for Motion Pictures |

## Contact your instructor

    Your instructor for this filmmaking class is Robert Elliott. You can email me by clicking here.
# Review:"Fireworks Display" Editing Workshop

Filmmaking Software Review
Narrative Film Production:
Software for Filmmaking (not free)
"Hearing Voices" Editing Workshop DVD-Video :
| "Hearing Voices" Editing Workshop DVD-Video This is behind-the-scenes look at filming and editing of one scene. An extremely look at "L-Cuts". DVD-Video. The price (not listed on their website) is $8 US plus shipping. (You must be a film student to purchase this disk. Fortunately, if you have completed an assignment here, you qualify for 30 days.) Even though this disk is rather boring, it is a good disk for advanced film editors. Not required but recommended. | Uses at WikiU Film School Film editing Introduction to filmmaking |

## Contact Your Instructor

* Your instructor for these courses is Robert Elliott. To email me, you simply Click Here.
# War Seminar

    Part of the School of History.
The War Seminar learning project is devoted to discussions about the history of war as a cultural practice and current wars in the world.
This learning project offers learnings activities that encourage the participation of Wikiversity editors who are interested in war as a cultural phenomenon. Concepts to learn include: the causes of war and the history of attempts to limit the number and destructiveness of wars.

## Call for contributions

Wikiversity needs quality essays on the question of war.
Here are some questions to ponder as you select your topic: What mechanism causes people to live peacefully with their neighbors but choose war with those living in distant lands? What problems do political boundaries cause? What problems do political boundaries solve? What is the purpose of dividing nations by political and geographic boundaries? How do nations enact laws? What are different types of governments that have the most political power in various nations?
To begin your draft, enter title that is not already on the list below and write. If you wish to change your title, or have any other questions, visit us at Wikiversity:Colloquium.

### List of contributions

* Art and War
* Learning materials
* Roadmaps to peace
* Terrorism and war
* War is in our DNA
* Winning the War on Terror

## Outstanding Project and a request for someone to judge one essay

* Art and War is especially moving.
* Winning the War on Terror was too long for this curator to read. If you don't feel like writing an essay, perhaps you might wish to judge this one and leave comments at Talk:War Seminar.