OpenLLM-France/wikimedia huggingface dataset:
# NEES

The George E. Brown, Jr. Network for Earthquake Engineering Simulation or NEES is created by the National Science Foundation to give researchers the tools to learn how earthquakes and tsunamis impact buildings, bridges, utility systems and other critical components of civil infrastructure.
NEES is a network of 15 large-scale, experimental sites that incorporate shake-tables, centrifuges that simulate earthquake effects, unique laboratories, a tsunami wave basin and field-testing equipment. All are linked to a centralized data pool and earthquake simulation software, bridged together by the high-speed Internet. The new NEEShub system, a communications web that uses collaborative tools and tele-presence technologies, allows off-site researchers to interact in real time with any of the networked sites.
With these tools, engineers and students from all parts of the country can collaborate on multi-site experiments using simulators that generate earthquake effects strong enough to bring down full-sized buildings.
From that knowledge will come a new set of rules from which engineers can design structures and materials that will better withstand earthquake forces.
# Linear mapping/Matrix/Composition/Fact

correspondence between linear mappings and matrices, the composition of linear mappings corresponds to the
More precisely: let $U,V,W$ denote vector spaces over a field $K$ with bases
    ${\mathfrak {u}}=u_{1},\ldots ,u_{p},\,{\mathfrak {v}}=v_{1},\ldots ,v_{n}{\text{ and }}{\mathfrak {w}}=w_{1},\ldots ,w_{m}.$
Let
    $\psi :U\longrightarrow V{\text{ and }}\varphi :V\longrightarrow W$
denote linear mappings. Then, for the describing matrix of $\psi ,\,\varphi$, and of the composition $\varphi \circ \psi$, the relation
    $M_{\mathfrak {w}}^{\mathfrak {u}}(\varphi \circ \psi )=(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))\circ (M_{\mathfrak {v}}^{\mathfrak {u}}(\psi ))\,$
holds.
# Linear mapping/Matrix/Composition/Fact/Proof2

We consider the chain of mappings
    $U{\stackrel {\psi }{\longrightarrow }}V{\stackrel {\varphi }{\longrightarrow }}W.$
Suppose that $\psi$ is described by the $n\times p$-matrix $B=(b_{jk})_{jk}$, and that $\varphi$ is described by the $m\times n$-matrix $A={\left(a_{ij}\right)}_{ij}$ (with respect to the bases). The composition $\varphi \circ \psi$ has the following effect on the base vector $u_{k}$.
    ${\begin{aligned}{\left(\varphi \circ \psi \right)}{\left(u_{k}\right)}&=\varphi {\left(\psi {\left(u_{k}\right)}\right)}\\&=\varphi {\left(\sum _{j=1}^{n}b_{jk}v_{j}\right)}\\&=\sum _{j=1}^{n}b_{jk}\varphi (v_{j})\\&=\sum _{j=1}^{n}b_{jk}{\left(\sum _{i=1}^{m}a_{ij}w_{i}\right)}\\&=\sum _{i=1}^{m}{\left(\sum _{j=1}^{n}a_{ij}b_{jk}\right)}w_{i}\\&=\sum _{i=1}^{m}c_{ik}w_{i}.\end{aligned}}$
Here, these coefficients $c_{ik}=\sum _{j=1}^{n}a_{ij}b_{jk}$ are just the entries of the product matrix $A\circ B$.
# Linear mapping/Matrix/Relation/Section

Due to fact, a linear mapping
    $\varphi \colon K^{n}\longrightarrow K^{m}$
is determined by the images $\varphi (e_{j})$, $j=1,\ldots ,n$, of the standard vectors. Every $\varphi (e_{j})$ is a linear combination
    $\varphi (e_{j})=\sum _{i=1}^{m}a_{ij}e_{i}\,,$
and therefore the linear mapping is determined by the elements $a_{ij}$. So, such a linear map is determined by the $mn$ elements $a_{ij}$, $1\leq i\leq m$, $1\leq j\leq n$, from the field. We can write such a data set as a matrix. Because of the determination theorem, this holds for linear maps in general, as soon as in both vector spaces bases are fixed.

## Definition

Let $K$ denote a field, and let $V$ be an $n$-dimensional vector space with a basis ${\mathfrak {v}}=v_{1},\ldots ,v_{n}$, and let $W$ be an $m$-dimensional vector space with a basis ${\mathfrak {w}}=w_{1},\ldots ,w_{m}$.
For a linear mapping
    $\varphi \colon V\longrightarrow W,$
the matrix
    $M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=(a_{ij})_{ij}\,,$
where $a_{ij}$ is the $i$-th coordinate of $\varphi (v_{j})$ with respect to the basis ${\mathfrak {w}}$, is called the describing matrix for $\varphi$ with respect to the bases.
For a matrix $M=(a_{ij})_{ij}\in \operatorname {Mat} _{m\times n}(K)$, the linear mapping $\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)$ determined by
    $v_{j}\longmapsto \sum _{i=1}^{m}a_{ij}w_{i}$
in the sense of fact,
$M$
For a linear mapping $\varphi \colon K^{n}\rightarrow K^{m}$, we always assume that everything is with respect to the standard bases, unless otherwise stated. For a linear mapping $\varphi \colon V\rightarrow V$ from a vector space in itself (what is called an endomorphism), one usually takes the same bases on both sides. The identity on a vector space of dimension $n$ is described by the identity matrix, with respect to every basis.

## Theorem

Let $K$ be a field, and let $V$ be an $n$-dimensional vector space with a basis ${\mathfrak {v}}=v_{1},\ldots ,v_{n}$, and let $W$ be an $m$-dimensional vector space with a basis ${\mathfrak {w}}=w_{1},\ldots ,w_{m}$. Then the mappings
    $\varphi \longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ){\text{ and }}M\longmapsto \varphi _{\mathfrak {w}}^{\mathfrak {v}}(M),$
defined in definition, are inverse

### Proof

$\Box$

## Example

A linear mapping
    $\varphi \colon K^{n}\longrightarrow K^{m}$
is usually described by the matrix $M$ with respect to the standard bases on the left and on the right. The result of the matrix multiplication
    ${\begin{pmatrix}y_{1}\\\vdots \\y_{m}\end{pmatrix}}=M{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\,$
can be interpreted directly as a point in $K^{m}$. The $j$-th column of $M$ is the image of the $j$-th standard vector $e_{j}$.
# Linear mapping/Matrix and basis/Rank/Fact/Proof/Exercise

Let $K$ be a field, and let $V$ and $W$ be vector spaces over $K$ of dimensions $n$ and $m$. Let
    $\varphi \colon V\longrightarrow W$
be a linear map, described by the matrix $M\in \operatorname {Mat} _{m\times n}(K)$ with respect to two bases. Prove that
    $\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,.$
# Linear mapping/Matrix for basis/Surjective and column generating system/Exercise

Let $K$ be a field and let $V$ and $W$ be vector spaces over $K$ of dimensions $n$ and $m$. Let
    $\varphi \colon V\longrightarrow W$
be a linear map, described by the matrix $M\in \operatorname {Mat} _{m\times n}(K)$ with respect to two bases. Prove that $\varphi$ is surjective if and only if the columns of the matrix form a system of generators for $K^{m}$.
# Linear mapping/Matrix/Standard spaces/Example

A linear mapping
    $\varphi \colon K^{n}\longrightarrow K^{m}$
is usually described by the matrix $M$ with respect to the standard bases on the left and on the right. The result of the matrix multiplication
    ${\begin{pmatrix}y_{1}\\\vdots \\y_{m}\end{pmatrix}}=M{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\,$
can be interpreted directly as a point in $K^{m}$. The $j$-th column of $M$ is the image of the $j$-th standard vector $e_{j}$.
# Linear mapping/Matrix for basis/Surjective and column generating system/Exercise/Solution

We consider the commutative diagramm
    ${\begin{matrix}K^{n}&{\stackrel {\Psi _{\mathfrak {v}}}{\longrightarrow }}&V&\\\!\!\!\!\!M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\downarrow &&\downarrow \varphi \!\!\!\!\!&\\K^{m}&{\stackrel {\Psi _{\mathfrak {w}}}{\longrightarrow }}&W.&\!\!\!\!\!\\\end{matrix}}$
Since the coordinate mappings are bijective, the map $\varphi$ is surjective if and only if $M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$ is surjective. The image vector of the $j$th standard vector $e_{j}$ under $M$ is the $j$th column of $M$ and the image space for $M$ is the space generated by the columnns. Therefore surjectivity is equivalent with the property that the columns form a generating system of $K^{m}$.
# Literature/2002/Huddleston

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

| w: Rodney Huddleston w: Geoffrey Pullum | w: Systemic functional grammar w: Systemic functional linguistics w: Word grammar w: Dependency grammar w: Phrase structure grammar |
# Literature/2003

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/2003/Barkun $^{[^]}$
* Literature/2003/Miller $^{[^]}$
* Literature/2003/Norway $^{[^]}$
* Literature/2003/Perlman $^{[^]}$
* Rees, Martin (2003). Our Final Hour: A Scientist's Warning: How Terror, Error, and Environmental Disaster Threaten Humankind's Future in This Century -- On Earth and Beyond. Basic Books. $^{[^]}$
* Literature/2003/Samuels $^{[^]}$
* Literature/2003/Todd $^{[^]}$
* Literature/2003/Umpleby $^{[^]}$
# Literature/2002/Cleveland

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Author

From 1991 to 2000, he served as president of the w: World Academy of Art and Science.
* Literature/1993/Cleveland $^{[^]}$
* Cleveland, Harlan (1985). The Knowledge Executive: Leadership in an Information Society. New York: Truman Tally Books. $^{[^]}$

## Excerpts

* The key to the management of complexity was the division of labor. The benefits of modernization were available only to societies that educated most of their people to function as specialists in a division-of-labor economy. Thus there came to pass, late the the second millennium A.D., slaveless societies that responded to a technological imperative by giving citizenship to all their citizens. Thomas Jefferson foresaw this macrotrend as early as 1813. "An insurrection has ... begun," he wrote to John Adams, "of science, talents, and courage, against rank and birth, which have fallen into contempt." He was spending his postpresidential years building the University of Virginia and promoting education and scholarship from his Monticello home. (p. 3)

## Wikimedia

* w: DIKW
* w: World Academy of Art and Science
# Literature/2003/Rees

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

| w: Wikipedia | w: World Encyclopedia |

## Chronology

* Reagle Jr., Joseph Michael (2010). Good Faith Collaboration: The Culture of Wikipedia. MIT Press. $^{[^]}$
* Gorman, Michael (2004). "Google and God's Mind: The problem is, information isn't knowledge." (Commentary) Los Angeles Times, December 17, 2004. $^{[^]}$
* Rees, Martin (2003). Our Final Hour: A Scientist's Warning: How Terror, Error, and Environmental Disaster Threaten Humankind's Future in This Century -- On Earth and Beyond. Basic Books. $^{[^]}$

## Reviews

* In order to avoid Human extinction, Rees advocates control of scientific research worldwide, and control of open access to such research (such as Wikipedia). He states that in the 1990s Aum Shinrikyo tried unsuccessfully to obtain an Ebola virus sample, which they could now create in their Mount Fuji lab, using ingredients and instructions from the Internet.
# Literature/2004/Crystal

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Reviews

http://books.google.com/books?id=dE8dAQAAIAAJ
* Leading British linguist Crystal (Shakespeare's Words) immediately distinguishes his pluralistic study of English's evolution from the standard, narrowly focused histories by describing not only how it evolved on an isolated island example from a Germanic language to the standard English we know today., but also on marginalized regional dialects, vernaculars and other "nonstandard" examples, beginning with the origins of Old English. He shows, for example, how even Chaucer and Shakespeare embraced dialects in The Canterbury Tales and Henry V. There are also lighter moments, such as Crystal's examination of the Anglo-Saxon intonations of Yoda in Star Wars and of Tolkien's Middle Earth idioms. Writing of the 18th century, the author contrasts the proscriptions of Dr. Johnson and others regarding spelling, grammar and pronunciation with the efforts of Americans such as Noah Webster to differentiate American from British English. (Regional and ethnic variations elsewhere in the British Empire receive more cursory treatment.) However, Crystal glosses over the current status struggle in the U.K. between more "authentic" dialects, such as the northern Liverpudlian, and newer ones, such as the suburban Estuary English. As for the language's future, Crystal wishes to see Standard English taught alongside familiarization with the varieties of dialects. Although he doesn't spell out how to accomplish this, his well-informed and appealing book makes a good case for the importance of dialects.
# Literature/2004

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Crystal, David (2004). The Stories of English. Allen Lane. ISBN-978-1585677191 $^{[^]}$
* Literature/2004/Feenberg $^{[^]}$
* Foster, Allen (2004). "A Nonlinear Model of Information-seeking Behavior," Journal of the American Society for Information Science and Technology, vol. 55 no. 3 (1 February 2004) pp. 228–237. $^{[^]}$
* Gorman, Michael (2004). "Google and God's Mind: The problem is, information isn't knowledge." (Commentary) Los Angeles Times, December 17, 2004. $^{[^]}$
* Literature/2004/Green $^{[^]}$
* Literature/2004/Harris $^{[^]}$
* Maniura, Robert (2004). Pilgrimage to Images in the Fifteenth Century: The Origins of the Cult of Our Lady of Częstochowa. Boydell Press. $^{[^]}$
* Osterwalder A. (2004). The Business Model Ontology - A Proposition In A Design Science Approach, Lausanne: Université de Lausanne $^{[^]}$
* Literature/2004/Putnam $^{[^]}$
* Literature/2004/Zuckerberg $^{[^]}$
# Literature/2004/Gorman

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Author

* Gorman was dean of library services at w: California State University, Fresno and 2004 president-elect of the w: American Library Association.

## Excerpts

* The books in great libraries are much more than the sum of their parts. They are designed to be read sequentially and cumulatively, so that the reader gains knowledge in the reading.
* The nub of the matter lies in the distinction between information (data, facts, images, quotes and brief texts that can be used out of context) and recorded knowledge (the cumulative exposition found in scholarly and literary texts and in popular nonfiction). When it comes to information, a snippet from Page 142 might be useful. When it comes to recorded knowledge, a snippet from Page 142 must be understood in the light of pages 1 through 141 or the text was not worth writing and publishing in the first place.
# Literature/2004/Liker

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Authors

* w:University of Michigan professor of industrial engineering

## Wikimedia

| w: The Toyota Way w: kaizen w: Genchi Genbutsu w: Toyota Production System w: Lean manufacturing w: The India Way | w: second-order cybernetics w: human-computer interaction w: organizational learning w: Möbius strip w: Hopf link w: Yin and yang |

## Chronology

* Liker, Jeffrey K. (2004). The Toyota Way: 14 Management Principles from the World's Greatest Manufacturer. McGraw-Hill. ISBN 978-0-07-139231-0. $^{[^]}$
* Literature/2001/Toyota $^{[^]}$
* Foerster, Heinz von (1990). "Ethics and Second-Order Cybernetics," SEHR, volume 4, issue 2 (1995) Constructions of the Mind $^{[^]}$
* Argyris, Chris and Schön, Donald (1978). Organizational Learning: A Theory of Action Perspective. New York: Addison-Wesley, 1978. ISBN 0-201-00174-8 $^{[^]}$
* Bateson, Gregory & Mead, Margaret (1976). "For God's Sake, Margaret: Conversation with Gregory Bateson and Margaret Mead." CoEvolutionary Quarterly (Summer 1976) no. 10, pp. 22-44. $^{[^]}$
* 1975/March/J $^{[^]}$
* Ogden, C. K. & I. A. Richards (1923). The Meaning of Meaning: A Study of the Influence of Language upon Thought and of the Science of Symbolism. London: Routledge & Kegan Paul Ltd. $^{[^]}$
# New Testament (Greek)/Γράψτε κι εσείς

Δοκιμάστε να γράψετε μερικές προτάσεις από το κείμενο σε απλά ελληνικά. Ή μπορείτε να ξαναγράψετε την ιστορία ή ένα μέρος του με απλά λόγια. Ή αν προτιμάτε, μπορείτε να γράψετε μια άλλη ιστορία ή παραβολή από την Καινή Διαθήκη.
Πώς διαφέρει η γραμματική των δικών σας προτάσεων από τις προτάσεις στο κείμενο; Τι σας έκανε ιδιαίτερη εντύπωση;
Αν ενδιαφέρεστε, μπορείτε να μάθετε περισσότερο για τις γραμματικές διαφορές ανάμεσα στην ελληνιστική κοινή και την νεοελληνική εδώ: (στα τμήματα "Basic Morphology" και "Basic Syntax").
* float Γράψτε το κείμενό σας και τις σκέψεις σας.
* float Διαβάστε τα σχόλια και τις ιδέες άλλων. Μπορείτε επίσης να τα σχολιάσετε εσείς.
* float Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Γράψτε κι εσείς/Ideas

Εδώ μπορείτε να συζητήσετε το κείμενο, τις ερωτήσεις, τα σχόλια των συμμαθητών σας, κλπ. Μπορείτε επίσης να θέσετε τις δικές σας ερωτήσεις και να προσφέρετε βοήθεια σε άλλους.
# New Testament (Greek)/Δεύτερη ανάγνωση

Τώρα μπορείτε να ξαναδιαβάσετε το κείμενο και να προσέχετε τις άγνωστες λέξεις και άλλες λεπτομέρειες. Εκτός αν το θέλετε, δεν χρειάζεται να ψάξετε τις λέξεις στο λεξικό, ούτε να έχετε πλήρη κατανόηση της γραμματικής. Κοιτάξτε τις παρακάτω ερωτήσεις και διαβάστε το κείμενο έχοντας αυτές στο νου.
* Ποιες λέξεις φαίνονται να είναι ίδιες ή παρόμοιες με λέξεις της νεοελληνικής που ξέρετε; Σημειώστε μερικές από αυτές.
* Αναγνωρίζετε όλες τις μορφές των παραπάνω λέξεων στο κείμενο; Αν κάτι σας προβληματίζει ιδιαίτερα, μπορείτε να το ερευνήσετε στην ιστοσελίδα Perseus. Δείτε τους χρήσιμους συνδέσμους για πληροφορίες σχετικά με το Perseus.
* float Γράψτε τις σκέψεις και τα σχόλια σας.
* float Διαβάστε τα σχόλια και τις ιδέες άλλων. Μπορείτε επίσης να τα σχολιάσετε εσείς.
* float Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Δεύτερη ανάγνωση/Ideas

Εδώ μπορείτε να συζητήσετε το κείμενο, τις ερωτήσεις, τα σχόλια των συμμαθητών σας, κλπ. Μπορείτε επίσης να θέσετε τις δικές σας ερωτήσεις και να προσφέρετε βοήθεια σε άλλους.
# New Testament (Greek)/Λεξιλόγιο

Ορίστε το πρώτο μέρος του κειμένου μας με συνδέσμους προς μερικές λέξεις της σύγχρονης γλώσσας.
Οι σύνδεσμοι οδηγούν προς το Βικιλεξικό (Wiktionary), ένα πολυγλωσσικό λεξικό που απευθύνεται σε ελληνόφωνους. (Βρίσκεται υπό κατασκευή και συντάσσεται από μη ειδικούς ανθρώπους. Όπως κάθε wiki, μπορεί να περιέχει ανακρίβειες.)
Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Μερικές ερωτήσεις/Ideas

Εδώ μπορείτε να συζητήσετε το κείμενο, τις ερωτήσεις, τα σχόλια των συμμαθητών σας, κλπ. Μπορείτε επίσης να θέσετε τις δικές σας ερωτήσεις και να προσφέρετε βοήθεια σε άλλους.
# New Testament (Greek)/Απόλυτα άσχετο

Φορέσατε ποτέ σας παλτό από καμηλόμαλλο; (Εγώ πάντως όχι.)
Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Εισαγωγή
### Τι κάνουμε;

Για όσους μαθαίνουν την νεοελληνική γλώσσα κια ενδιαφέρονται για την κατανόηση του πρωτότυπου κειμένου της Καινής Διαθήκης, ας διασκεδάσουμε λίγο με μία ιστοριούλα από το Ευαγγέλιο Κατά Μάρκον. Σκοπός αυτών των σελίδων είναι να καταλάβετε καλύτερα τι σας βοηθά στην κατανόηση όταν διαβάζετε ένα γραπτό κείμενο. Επίσης μπορείτε να συνειδητοποιήσετε μερικά χρήσιμα εργαλεία που σας είναι ήδη διαθέσιμα.

### Πώς το κάνουμε;

Θα διαβάσετε το κείμενο δύο φορές. Την πρώτη φορά, δεν είναι αναγκαίο να καταλάβετε κάθε λεπτομέρεια. Βγάλτε απλώς το γενικό νόημα του κειμένου. Ένας τρόπος για να καταφέρετε αυτό είναι να μην δώσετε πολλή προσοχή στις άγνωστες λέξεις ή τα ίσως περίεργα γραμματικά στοιχεία. Έχετε άλλες ιδέες στο νου για το πώς να αντιμετωπίσετε το κείμενο; Δοκιμάστε τες και ύστερα γράψτε τις εντυπώσεις σας!
Στις ακόλουθες σελίδες θα έχετε την ευκαιρία να κρατήσετε σημειώσεις, να θέσετε τις δικές σας ερωτήσεις, και να προτείνετε άλλα θέματα για έρευνα. Επιπλέον, αν θέλετε μπορείτε να επεξεργαστείτε τις σελίδες τις ίδιες και να προσθέσετε ιδέες, ερωτήσεις, και οτιδήποτε άλλο σας φαίνεται χρήσιμο για τον επόμενο αναγνώστη. Ας αρχίσουμε!
Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Κείμενο

Ορίστε το απόσπασμα από το Ευαγγέλιο Κατά Μάρκον:
Πηγή:Κατά_Μάρκον (Mark 1:1-8)
Μπορείτε να ακούσετε το ίδιο κείμενο εδώ (αρχείο mp3).
Γυρίστε πίσω στην αρχή.
# New Testament (Greek)/Μερικές ερωτήσεις

* Τι κάνατε για να βγάλετε νόημα από το κείμενο; Πώς διαβάσατε το κείμενο, με ποια διαδικασία;
* Αναγνωρίσατε την ιστορία; Αν ναι, με ποιον τρόπο, και πώς σας βοήθησε αυτό;
* Υπάρχουν συγκεκριμένες λέξεις ή φράσεις που σας βοήθησαν να καταλάβετε τη γενική ιδέα; Ποιες είναι μερικές από αυτές;
* Όταν διαβάσετε ένα δύσκολο κείμενο περισσότερες από μία φορές, μάλλον καταλαβαίνετε καλύτερα το νόημα. Γιατί; Τι κάνετε κατά τη δεύτερη ανάγνωση που σας βοηθά;
* Τι προτιμήσατε να παρακολουθήσετε, το γραπτό κείμενο ή το προφορικό; Για ποιο λόγο; Γενικά προτιμάτε να ακούτε ή να διαβάζετε άγνωστες λέξεις και φράσεις στα ελληνικά; Ποια είναι η προτίμησή σας όσον αφορά καινούργιες αγγλικές λέξεις και φράσεις;
* float Γράψτε τις σημειώσεις, σκέψεις και απορίες σας.
* float Διαβάστε τα σχόλια και τις ιδέες άλλων. Μπορείτε επίσης να τα σχολιάσετε εσείς.
* float Γυρίστε πίσω στην αρχή.
# PlanetPhysics/Supergroupoid

an involutive automorphism $\gamma :Id{\mathcal {G}}\to Id{\mathcal {G}}$, that is, $\gamma \circ \gamma =Id\circ Id{\mathcal {G}}.$
# PlanetPhysics/Supersymmetry

Supersymmetry or Poincar\'e, (extended) quantum symmetry is usually defined as an extension of ordinary spacetime symmetries obtained by adjoining $N$ spinorial generators $Q$ whose anticommutator yields a translation generator: $\left\{Q,Q\right\}=\left\{P\right\}$.
As further explained in ref.:
   "This (super) symmetry...(of the superspace)... can be realized on ordinary fields (that are defined as certain functions of physical spacetime(s)) by transformations that mix bosons and fermions. \emph{Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as cumbersome and inconvenient as doing vector calculus component by component. A compact alternative to this `component field' approach is given by the superspace--superfield approach}", which is defined next.
Quantum superspace, or superspacetimes , can be defined as an extension(s) of ordinary spacetime(s) to include additional anticommuting coordinates, for example, in the form of $N$ two-component Weyl spinors $\theta$.
(Quantum) superfields $\Psi (x,\theta )$ are functions defined over such superspaces, or superspacetimes. Taylor series expansions of the superfield functions can be then performed with respect to the anticommuting coordinates $\theta$; this Taylor series has only a finite number of terms and the series expansion coefficients obtained in this manner are the ordinary `component fields' specified above.
Remarks: Supersymmetry is expected to be manifested, or observable, in such superspaces, that is, the supersymmetry algebras are represented by translations and rotations involving both the spacetime and the anticommuting coordinates. Then, the transformations of the `component fields' can be computed from the Taylor expansion of the translated and rotated superfields . Especially important are those transformations that mix boson and fermion symmetries; further details are found in ref..
# PlanetPhysics/Superfluid2

A superfluid is described as a macroscopically coherent quantum system $S_{F}$ that exhibits `unusual' properties such as negligible losses or no resistance to flows. Therefore, it is defined as a phase that exhibits most unusual effects, previously considered to be anomalous from a Thermodynamic viewpoint, that can be however explained in terms of quantum statistical theory/quantum statistical mechanics and long-range coherence of coupled boson systems. Such theories are, for example, L. D. Landau's phenomenological/ microscopic theory of superfluidity in liquid $^{4}$He (the 1962 Nobel Prize in Physics), and Sir Anthony Leggett's quantum statistical theory of superfluidity in liquid $^{3}$He (the 2003 Nobel Prize in Physics shared with Alexei A. Abrikosov).
As an example, in a superconductor the long-ranged coupled Cooper electron pairs form a superfluid that sustains very high electric currents without any significant heating of the metal. Liquid $^{3}$He is another example of a superfluid where the liquid flow can take place upwards in the absence of externally applied pressure and no apparent resistance to flow--no viscosity.