OpenLLM-France/wikimedia huggingface dataset:
# General Relativity/Comma derivative

< General Relativity
In General Relativity we write our (4-dimensional) coordinates as $(x^{0},x^{1},x^{2},x^{3})$. The flat Minkowski spacetime coordinates ("Local Lorentz frame") are $x^{0}=ct$, $x^{1}=x$, $x^{2}=y$, and $x^{3}=z$, where $c$ is the speed of light, $t$ is time, and $x$, $y$, and $z$ are the usual 3-dimensional Cartesian space coordinates.
A comma derivative is just a convenient notation for a partial derivative with respect to one of the coordinates. Here are some examples:
1. $T_{\ \beta ,\gamma }^{\alpha }={\frac {\partial T_{\ \beta }^{\alpha }}{\partial x^{\gamma }}}$
2. $f_{,\mu }={\frac {\partial f}{\partial x^{\mu }}}$
3. $w_{\ ,\nu }^{\mu }={\frac {\partial w^{\mu }}{\partial x^{\nu }}}$
4. $\Gamma _{\ \beta \gamma ,\mu }^{\alpha }={\frac {\partial \Gamma _{\ \beta \gamma }^{\alpha }}{\partial x^{\mu }}}$
If several indices appear after the comma, they are all taken to be part of the differentiation. Here are some examples:
1. $S_{\alpha \ ,\mu \nu }^{\ \beta }=\left(S_{\alpha \ ,\mu }^{\ \beta }\right)_{,\nu }={\frac {\partial }{\partial x^{\nu }}}\left({\frac {\partial S_{\alpha }^{\ \beta }}{\partial x^{\mu }}}\right)={\frac {\partial ^{2}S_{\alpha }^{\ \beta }}{\partial x^{\nu }\partial x^{\mu }}}$
2. $f_{,\alpha \beta \beta }=\left[\left(f_{,\alpha }\right)_{,\beta }\right]_{,\beta }={\frac {\partial ^{3}f}{\partial ^{2}x^{\beta }\partial x^{\alpha }}}$
Now, we change coordinate systems via the Jacobian $x_{\ ,\nu }^{\mu }$. The transformation rule is $x^{\bar {\mu }}=x^{\mu }x_{\ ,\mu }^{\bar {\mu }}$.
Finally, we present the following important theorem:
Theorem: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\delta _{\beta }^{\alpha }$
Proof: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\sum _{\mu =0}^{3}{\frac {\partial x^{\alpha }}{\partial x^{\mu }}}{\frac {\partial x^{\mu }}{\partial x^{\beta }}}$, which by the chain rule is ${\frac {\partial x^{\alpha }}{\partial x^{\beta }}}$, which is of course $\delta _{\beta }^{\alpha }$. $\square$
So, as a matrix, the matrix inverse of $\left(x_{\ ,\mu }^{\alpha }\right)$ is $\left(x_{\ ,\beta }^{\mu }\right)$.
# General Relativity/Coordinate systems and the comma derivative

<General Relativity
In General Relativity we write our (4-dimensional) coordinates as $(x^{0},x^{1},x^{2},x^{3})$. The flat Minkowski spacetime coordinates ("Local Lorentz frame") are $x^{0}=ct$, $x^{1}=x$, $x^{2}=y$, and $x^{3}=z$, where $c$ is the speed of light, $t$ is time, and $x$, $y$, and $z$ are the usual 3-dimensional Cartesian space coordinates.
A comma derivative is just a convenient notation for a partial derivative with respect to one of the coordinates. Here are some examples:
1. $T_{\ \beta ,\gamma }^{\alpha }={\frac {\partial T_{\ \beta }^{\alpha }}{\partial x^{\gamma }}}$
2. $f_{,\mu }={\frac {\partial f}{\partial x^{\mu }}}$
3. $w_{\ ,\nu }^{\mu }={\frac {\partial w^{\mu }}{\partial x^{\nu }}}$
4. $\Gamma _{\ \beta \gamma ,\mu }^{\alpha }={\frac {\partial \Gamma _{\ \beta \gamma }^{\alpha }}{\partial x^{\mu }}}$
If several indices appear after the comma, they are all taken to be part of the differentiation. Here are some examples:
1. $S_{\alpha \ ,\mu \nu }^{\ \beta }=\left(S_{\alpha \ ,\mu }^{\ \beta }\right)_{,\nu }={\frac {\partial }{\partial x^{\nu }}}\left({\frac {\partial S_{\alpha }^{\ \beta }}{\partial x^{\mu }}}\right)={\frac {\partial ^{2}S_{\alpha }^{\ \beta }}{\partial x^{\nu }\partial x^{\mu }}}$
2. $f_{,\alpha \beta \beta }=\left[\left(f_{,\alpha }\right)_{,\beta }\right]_{,\beta }={\frac {\partial ^{3}f}{\partial ^{2}x^{\beta }\partial x^{\alpha }}}$
Now, we change coordinate systems via the Jacobian $x_{\ ,\nu }^{\mu }$. The transformation rule is $x^{\bar {\mu }}=x^{\mu }x_{\ ,\mu }^{\bar {\mu }}$.
Finally, we present the following important theorem:
Theorem: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\delta _{\beta }^{\alpha }$
Proof: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\sum _{\mu =0}^{3}{\frac {\partial x^{\alpha }}{\partial x^{\mu }}}{\frac {\partial x^{\mu }}{\partial x^{\beta }}}$, which by the chain rule is ${\frac {\partial x^{\alpha }}{\partial x^{\beta }}}$, which is of course $\delta _{\beta }^{\alpha }$. $\square$
# General Relativity/Cosmology

<General Relativity
Relativistic cosmology is based on the following three assumptions:
* the cosmological principle
* Weyl's postulate
* general relativity.
# General Relativity/Covariant Differentiation

<General Relativity
Now that we have established some of the basic of tensor algebra and curved space, lets try to do something within that curved space. We start by taking a derivative. In Einstein notation, taking a derivative looks like
${\frac {\partial }{\partial x^{j}}}f(x^{i})$
In the spirit of seeing how things work when we transform coordinates, we convert the coordinates from $x^{i}$ to $x^{\prime i}$ by a tensor transformation $x^{\prime i}=A_{k}^{i}x^{k}$
So let's figure out what our derivatives look like in our new coordinate system.
${\frac {\partial }{\partial x^{\prime j}}}f(x^{\prime i})=$
${\frac {\partial }{\partial x^{k}}}f(x^{\prime i}){\frac {\partial x^{k}}{\partial x^{\prime j}}}=$
${\frac {\partial }{\partial x^{k}}}f(x^{l}){\frac {\partial x^{l}}{\partial x^{\prime i}}}{\frac {\partial x^{k}}{\partial x^{\prime j}}}$
So if the transform is a constant we get a very nice result.....
The result is much less nice if the transform changes with location, that is to say instead of transform $A_{k}^{i}$ we use the transform $A_{k}^{i}(x^{j})$
What we have here is a nice part, and a not so nice part. If only there was a way to get rid of the not so nice part. At this point we do an interesting trick and that is to redefine the notion of derivative. Instead of defining derivative as simply the way we do in Euclidean space, we create a new type of derivative called the covariant derivative. The covariant derivative is like the normal derivative, except that we add a "fudge factor" to get rid of the not nice parts of the equation so that the result transforms nicely.
# General Relativity/Differentiable manifolds

<General Relativity
A smooth $n$-dimensional manifold $\mathrm {M} ^{n}$ is a set together with a collection of subsets $\{O_{\alpha }\}$ with the following properties:
* Each $p\in \mathrm {M}$ lies in at least one $O_{\alpha }$, that is $\mathrm {M} =\cup _{\alpha }O_{\alpha }$.
* For each $\alpha$, there is a bijection $\psi _{\alpha }:O_{\alpha }\longrightarrow U_{\alpha }$, where $U_{\alpha }$ is an open subset of $\mathbb {R} ^{n}$
* If $O_{\alpha }\cap O_{\beta }$ is non-empty, then the map $\psi _{\alpha }\circ \psi _{\beta }^{-1}:\psi _{\beta }[O_{\alpha }\cap O_{\beta }]\longrightarrow \psi _{\alpha }[O_{\alpha }\cap O_{\beta }]$ is smooth.
The bijections are called charts or coordinate systems. The collection of charts is called an atlas. The atlas induces a topology on M such that the charts are continuous. The domains $O_{\alpha }$ of the charts are called coordinate regions.

## Examples

* Euclidean space, $\mathbb {R} ^{n}$ with a single chart ($O=\mathbb {R} ^{n},\psi =$ identity map) is a trivial example of a manifold.
* 2-sphere $S^{2}=\{(x,y,z)\in \mathbb {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}$.
    Notice that $S^{2}$ is not an open subset of $\mathbb {R} ^{3}$. The identity map on $\mathbb {R} ^{3}$ restricted to $S^{2}$ does not satisfy the requirements of a chart since its range is not open in $\mathbb {R} ^{3}.$
    The usual spherical coordinates map $S^{2}$ to a region in $\mathbb {R} ^{2}$, but again the range is not open in $\mathbb {R} ^{2}.$ Instead, one can define two charts each defined on a subset of $S^{2}$ that omits a half-circle. If these two half-circles do not intersect, the union of the domains of the two charts is all of $S^{2}$. With these two charts, $S^{2}$ becomes a 2-dimensional manifold. It can be shown that no single chart can possibly cover all of $S^{2}$ if the topology of $S^{2}$ is to be the usual one.
# Guitar/Blues Exercises
## Exercise 1

The exercise below is a blues in C using seventh chords.
| C7 | C7 | C7 | C7 |
| F7 | F7 | C7 | C7 |
| G7 | F7 | C7 | C7 |
The partial seventh chords below are to be played as stab chords. Play these partial chords on the back beat (play the chord on the snare) using a down-stroke to add to the rhythmic drive. Note that the second finger remains on the same string. This allows easy movement between the chords and fast tempos can be achieved with ease.
* C7
* F7
* G7

## Exercise 2

Here is a blues box in Dm
| Dm7 | Dm7 | Dm7 | Dm7 |
| Gm7 | Gm7 | Dm7 | Dm7 |
| Am7 | Gm7 | Dm7 | Am7 |
You can play a blues using only minor seventh chords. In most cases where you form a half-barre the first finger will be used but for the Gm7 and Am7 below you can use the third finger. The choice is yours though I think you'll find in this instance using the third finger is more comfortable.
* Dm7
* Gm7
* Am7
In this variation you will be using a single chord shape to play the twelve bar blues. Fingering has been given for the Dm7.
* Dm7
* Gm7
* Am7

## Exercise 3

Here is a blues exercise in A using sevenths.
For a soul feel try jumping from the first note to the fifth note in each bar when starting the riff.

## Exercise 4

Here is a typical blues rhythm.
# Guitar/Arpeggios and Sweep Picking
## Introduction

The word arpeggio (ar-peh-jee-oh) is Italian for "like a harp". It is a common technique for playing chords on the harp. To play an arpeggiated chord on the guitar, pick each note of the chord slowly, one string at a time. You can play arpeggios with a plectrum or fingerstyle.

### Exercise 1

Below is a simple arpeggio study using these chords:
* A major
* D major
* E major

## Sweep picking

Sweep picking is a more specialized technique, occurring most often in metal. It involves playing a fast arpeggio with a special technique: when switching from one string to the next, mute the note currently ringing by lifting the fretting finger. A sweep can become a rake if notes are muted incorrectly. Rakes can sound nice, but they are not sweeps. Remember only one note can ring out at a time or it won't sound good. It takes practice and it helps to start slow and build up speed.
Below is example tablature of sweep picking:
This is not the only way to notate sweeps. Small sweeps can be indicated with grace notes or even the arpeggio notation with the word "sweep" (or, less correctly, "rake") written above.
In a more classical approach, arpeggios must follow a distinct pattern of notes depending on the chord/scale we're playing. This is similar to playing chords note-by-note on a piano (not on a guitar).
The basic chords (the major and minor triads) are composed of three tones: the first, the third and the fifth note of the scale (major or minor, depending on the chord type).
For instance, the C major scale is: C D E F G A B. So, according to the 1-3-5 principle, the C major triad consists of C, E and G. Note that the C major chord on a guitar also consists only of these three notes but they are not always in the 1-3-5 order. Now, while playing "classical arpeggios", you would not just pick around the chord randomly but you would play C, E, G, then C, E, G an octave higher, etc. This is what is called an arpeggio scale. You can play around it, up and down with complete freedom or just use the 1-3-5 pattern as a bass line. This method can also be used with more complex chords (sus4, maj7, etc.) but then it follows a pattern different from 1-3-5 structure, depending on the chord type. In all, this is a very simple but effective method for composing.
While playing guitar, this might not appear as interesting as picking "full" six-string chords but it can be used to give your music a classical edge. It also has a more lead quality to it than using full chords and requires more skill. Playing fast arpeggios like these is sometimes used in metal music with very satisfactory results. The "classical arpeggios" are in no way better than the "harp like chords" and it is ultimately up to the player/composer to choose what is best for the song in question.
# Guitar/Authors

This book has no authors other than the public: it is open for anyone and everybody to improve. Therefore, this is more properly a list of acknowledgements of contributors than a list of authors. Whoever we are, this is where we get to brag about our accomplishments in writing this book.

## List of major contributors

* Kef Li Eric Marcus X-Schecter is the esoteric pen name of the former primary author of (er, contributor to) this book. He is, in his own words, not entirely qualified to write this book: he is learning as much as the readers are! He plays the electric guitar in the fingerpicking style exclusively, and has written a small number of songs. He currently enjoys writing and transcribing guitar tablature for the Power Tab Archive.
* Sluffs, added chord diagram images to the Chord Reference section
* Daniel made various minor contributions.
* GABaker, who has had a lot trouble tuning cheap guitars, contributed to the section on tuning.
* NickPenguin helped redesign many sections, helped translate content from the German guitar wikibook and created the Guitar template.
* Michael Hoffman explained classic techniques for shaping distortion tones and controlling distortion independently from listening volume, on the Tone and Volume page.
* Meemo created and added much content to the different types of guitars section and added the paragraph on stringing guitars, which seemed important until he read the list of external tutorials...
* Sameer Kale Did a good bit of the Chords section a while ago, started off the Rhythm section, along with adding a few bits to ther places. He doesn't know why such a page as this exists, but does not want to be left out.
* Kiefer.Wolfowitz edited Alternate Tunings, attempting to incorporate his material from English Wikipedia (not from w:Guitar tunings but rather from w:Regular tunings, w:Major-thirds tuning, and w:Ralph Patt).
* Others (add your name and description if you made a major contribution)
* Various anonymous persons.
# Guitar/Blanks

The purpose of this page is to provide blank tab and chord templates for anyone writing a lesson or an exercise. If you find something useful that can save time when typing out tabs, please add it to this section
When you type the numbers in the tab, please space them nicely.
When you write a chord, an x means don't play, a 0 is open and the frets are numbered. If you are writing chords on double digit frets, used something similar to the 2nd chord template.
This template is intended for easy creating of scale and arpeggio charts. Please use the numbers provided to indicate notes.
# Handbook of Management Scales/Information sharing
## Information sharing (alpha = 0.7486)
### Items

* Our supply chain members practice Electronic Data Interchange, either vie VAN or Internet.
* Our supply chain members regularly (at least once a quarter) exchange supply and demand forecasts with each other.
* Our supply chain members frequently (at least once a month) exchange demand change information with each other to facilitate operational plans and reduce reliance on second-guesses.

### Source

* Min/Mentzer (2004): Developing and measuring supply chain management concepts. Journal of Business Logistics, Vol. 25, No. 1, pp. 63-99.

### Related Scales

* Buyer communication
* Information integration
* Internal quality information usage
* Information sharing quality
# Handbook of Management Scales/Ingratiation
## Ingratiation (alpha = 0.83)
### Items

* Compliment your colleagues so they will see you as likeable.
* Take an interest in your colleagues’ personal lives to show them that you are friendly.
* Praise your colleagues for their accomplishments so they will consider you a nice person.
* Do personal favors for your colleagues to show them that you are friendly.

### Source

* Bolino/Turnley (1999): Measuring Impression Management in Organizations: A Scale Development Based on the Jones and Pittman Taxonomy. Organizational Research Methods, Vol. 2, No. 2, pp. 187-206.
# Handbook of Management Scales/Integration
## Integration (alpha = 0.80, CR = 0.80)
### Description

Three relational competencies were identified: communication, cooperation and integration.

### Definition

Integration refers to the process of combining efforts to integrate supplier and customer information and inputs into internal planning (Swink et al., 2007).

### Items

To what extent do the statements apply to the relationship of your company with your suppliers and customers? (1 – strongly disagree; 7 – strongly agree):
* We have full access to joint planning systems. (0.80)
* We synchronize our production plans. (0.67)
* We carry out joint electronic data interchange. (0.62)
* We have knowledge of inventory mix/levels. (0.77)

### Source

* Wieland/Wallenburg (2013): The Influence of Relational Competencies on Supply Chain Resilience: A Relational View. International Journal of Physical Distribution & Logistics Management, Vol. 42, No. 4, pp. 300-320. Adapted from Frohlich/Westbrook (2001).

### Related Scales

* Communication
* Cooperation

## Integration (alpha = 0.90)
### Description

Based on learning theory and multilevel thinking, measures were developed and validated for four types of team learning processes, as “team learning originates in individual intuition, is amplified through interpretation and integration, and manifests itself at the team level via the codification of collective cognition and action”.

### Definition

Integration is defined as “the process of developing shared understanding among individuals and of taking coordinated action through mutual adjustment” (Crossan et al., 1999, p. 525).

### Items

* In our team, we managed to agree upon the actions and activities that we should perform. (0.86)
* In our team, we shared information, ideas and results while performing project tasks. (0.96)
* During the project, our team developed and experimented with a number of different implementation scenarios, project deliverables, prototypes or production designs. (0.96)

### Source

* Kostopoulos et al. (2013): Structure and Function of Team Learning Emergence: A Multilevel Empirical Validation. Journal of Management, Vol. 39, No. 6, pp. 1430–1461
# Handbook of Management Scales/Information sharing quality
## Information sharing quality (alpha = 0.828, CR = 0.838, AVE = 0.635)
### Description

The article examines three factors of intellectual capital: Worker expertise, information sharing quality, and team psychological safety.
The 5-point Likert-type scale for information sharing quality was developed from previous literature in order to assure content validity. Items from Earley/Mosakowski (2000) and Moenaert/Souder (1996) were modified.

### Definition

Both the quality of an organization’s information and the value of its information systems reflect the organization’s information sharing potential.

### Items

* The implementation team members talked openly and freely. (dropped after confirmatory factor analysis)
* Implementation team members did not seem to understand what one another was saying during their discussion. (reverse coded; dropped after confirmatory factor analysis)
* When implementation team members shared information, it was likely to be used.
* Team members shared information in useful formats.
* Team members often shared valuable information.

### Source

* Lee et al. (2011): The roles of worker expertise, information sharing quality, and psychological safety in manufacturing process innovation: An intellectual capital perspective. Production and Operations Management, Vol. 20, No. 4, pp. 556–570

### Related Scales

* Information sharing
* Internal quality information usage
# Handbook of Management Scales/Informational justice
## Informational justice (alpha = 0.90/0.93)
### Description

Four dimensions of organizational justice were measured with a scale developed and validated by Colquitt (2001): procedural, distributive, interpersonal, and informational justice. The scale ranged from 1 (to a very small extent) to 5 (to a very large extent).

### Definition

Organizational justice refers to perceptions of fairness in decision-making and resource allocation environments.
Informational justice reflects the degree of justification and truthfulness offered during procedures.

### Items

* Has your supervisor been candid when communicating with you?
* Has your supervisor explained decision-making procedures thoroughly?
* Were your supervisor’s explanations regarding procedures reasonable?
* Has your supervisor communicated details in a timely manner?
* Has your supervisor tailored communications to meet individuals’ needs?

### Source

* Colquitt/Rodell (2011): Justice, trust, and trustworthiness: A longitudinal analysis integrating three theoretical perspectives. Academy of Management Journal, Vol. 54, No. 6, pp. 1183–1206

### Related Scales

* Distributive justice
* Interpersonal justice
* Procedural justice
# Handbook of Management Scales/Instrumental support
## Instrumental support (alpha = 0.74)
### Description

Drawing on social support theory and using data from two samples of lower wage workers, the authors develop and validate a measure of family supportive supervisor behaviors (FSSB). FSSB is conceptualized as a multidimensional superordinate construct with four subordinate dimensions: emotional support, instrumental support, role modeling behaviors, and creative work-family management. Results from multilevel confirmatory factor analyses and multilevel regression analyses provide evidence of construct, criterion-related, and incremental validity.

### Definition

Family supportive supervisor behaviors are defined as behaviors exhibited by supervisors that are supportive of families.
Instrumental support is reactive and pertains to supervisor support as he or she responds to an individual employee’s work and family needs in the form of day-to-day management transactions.

### Items

* I can depend on my supervisor to help me with scheduling conflicts if I need it.
* I can rely on my supervisor to make sure my work responsibilities are handled when I have unanticipated nonwork demands.
* My supervisor works effectively with workers to creatively solve conflicts between work and nonwork.

### Source

* Hammer et al. (2009): Development and Validation of a Multidimensional Measure of Family Supportive Supervisor Behaviors (FSSB). Journal of Management, Vol. 35, No. 4, pp. 837-856.