OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Transformation From Rectangular to Generalized Coordinates

We take a system with a total of $3N\equiv n$ Cartesian coordinates of which $\nu$ are independent. We denote Cartesian coordinates by the same letter $x_{i}$, understanding by this symbol all the coordinates $x,y,z$; this means that $i$ varies from $1$ to $3N$, that is, from $1$ to $n$. The generalized coordinates we denote by $q_{\alpha }$ $(l\leq \alpha \leq \nu )$. Since the generalized coordinates completely specify the position of their system, $x_{i}$ are their unique functions:
$x_{i}=x_{i}(q_{1},q_{2},\dots q_{\alpha },\dots ,q_{v})$
From this it is easy to obtain an expression for the Cartesian components of velocity. Differentiating the function of many variables $x_{i}(\dots q_{\alpha })$ with respect to time, we have
   ${\frac {dx_{i}}{dt}}=\sum _{\alpha =1}^{\nu }{\frac {\partial x_{i}}{\partial q_{\alpha }}}{\frac {dq_{\alpha }}{dt}}$
In the subsequent derivation we shall often have to perform summations with respect to all the generalized coordinates $q_{\alpha }$, and double and triple sums will be encountered. In order to save space we will use Einstein summation.
The total derivative with respect to time is usually denoted by a dot over the corresponding variable:
${\frac {dx_{i}}{dt}}={\dot {x_{i}}};\,\,\,{\frac {dq_{\alpha }}{dt}}={\dot {q_{\alpha }}}$
In this notation, the velocity (1) in abbreviated form becomes:
   ${\dot {x_{i}}}={\frac {\partial x_{i}}{\partial q_{\alpha }}}{\dot {q_{\alpha }}}$
Differentiating this with respect to time again, we obtain an expression for the Cartesian components of acceleration:
${\ddot {x_{i}}}={\frac {d}{dt}}\left({\frac {\partial x_{i}}{\partial q_{\alpha }}}\right){\dot {q_{\alpha }}}+{\frac {\partial x_{i}}{\partial q_{\alpha }}}{\ddot {q_{\alpha }}}$
The total derivative in the first term is written as usual:
${\frac {d}{dt}}\left({\frac {\partial x_{i}}{\partial q_{\alpha }}}\right)={\frac {\partial ^{2}x_{i}}{\partial q_{\beta }\partial q_{\alpha }}}{\dot {q_{\beta }}}$
The Greek symbol over which the summation is performed is deonted by the letter $\beta$ to avoid confusion with the symbol $\alpha$, which denotes the summation in the expression for velocity (2). Thus we obtain the desired expression for ${\ddot {x_{i}}}$:
   ${\ddot {x_{i}}}={\frac {\partial ^{2}x_{i}}{\partial q_{\beta }\partial q_{\alpha }}}{\dot {q_{\beta }}}{\dot {q_{\alpha }}}+{\frac {\partial x_{i}}{\partial q_{\alpha }}}{\ddot {q_{\alpha }}}$
The first term on the right-hand side contains a double summation with respect to $\alpha$ and $\beta$.
# PlanetPhysics/Using Convolution to Find Laplace Transforms

We start from the relations (see the table of Laplace transforms)
   ${\begin{matrix}e^{\alpha t}\;\curvearrowleft \;{\frac {1}{s\!-\!\alpha }},\quad {\frac {1}{\sqrt {t}}}\;\curvearrowleft \;{\sqrt {\frac {\pi }{s}}}\qquad (s>\alpha )\end{matrix}}$
where the curved arrows point from the Laplace-transformed functions to the original functions.\, Setting\, $\alpha =a^{2}$\, and dividing by ${\sqrt {\pi }}$ in (1), the convolution property of Laplace transform yields ${\frac {1}{(s\!-\!a^{2}){\sqrt {s}}}}\;\;\curvearrowright \;\;e^{a^{2}t}*{\frac {1}{\sqrt {\pi t}}}\;=\;\int _{0}^{t}\!e^{a^{2}(t-u)}{\frac {1}{\sqrt {\pi u}}}\,du.$ The substitution \,$a^{2}u=x^{2}$\, then gives ${\frac {1}{(s\!-\!a^{2}){\sqrt {s}}}}\;\curvearrowright \;{\frac {e^{a^{2}t}}{\sqrt {pi}}}\int _{0}^{a{\sqrt {t}}}\!e^{-x^{2}}\!\cdot \!{\frac {a}{x}}\!\cdot \!{\frac {2x}{a^{2}}}\,dx\;=\;{\frac {e^{a^{2}t}}{a}}\!\cdot \!{\frac {2}{\sqrt {\pi }}}\int _{0}^{a{\sqrt {t}}}\!e^{-x^{2}}\,dx\;=\;{\frac {e^{a^{2}t}}{a}}\,{\rm {erf}}\,a{\sqrt {t}}.$
Thus we may write the formula
   ${\begin{matrix}{\mathcal {L}}\{e^{a^{2}t}\,{\rm {erf}}\,a{\sqrt {t}}\}\;=\;{\frac {a}{(s\!-\!a^{2}){\sqrt {s}}}}\qquad (s>a^{2}).\end{matrix}}$
Moreover, we obtain ${\frac {1}{({\sqrt {s}}\!+\!a){\sqrt {s}}}}\;=\;{\frac {{\sqrt {s}}\!-\!a}{(s\!-\!a^{2}){\sqrt {s}}}}\;=\,{\frac {1}{s-a^{2}}}-{\frac {a}{(s-a^{2}){\sqrt {s}}}}\;\curvearrowright \;e^{a^{2}t}-e^{a^{2}t}\,{\rm {erf}}\,a{\sqrt {t}}\;=\;e^{a^{2}t}(1-{\rm {erf}}\,a{\sqrt {t}}),$ whence we have the other formula
   ${\begin{matrix}{\mathcal {L}}\{e^{a^{2}t}\,{\rm {erfc}}\,a{\sqrt {t}}\}\;=\;{\frac {1}{(a\!+\!{\sqrt {s}}){\sqrt {s}}}}.\end{matrix}}$

### An improper integral

One can utilise the formula (3) for evaluating the improper integral $\int _{0}^{\infty }{\frac {e^{-x^{2}}}{a^{2}\!+\!x^{2}}}\,dx.$ We have $e^{-tx^{2}}\;\curvearrowleft \;{\frac {1}{s\!+\!x^{2}}}$ (see the table of Laplace transforms).\, Dividing this by $a^{2}\!+\!x^{2}$ and integrating from 0 to $\infty$, we can continue as follows:
   Failed to parse (unknown function "\sijoitus"): {\displaystyle \begin{matrix} \int_0^\infty\frac{e^{-tx^2}}{a^2\!+\!x^2}\,dx & \;\curvearrowleft\; \int_0^\infty\frac{dx}{(a^2\!+\!x^2)(s\!+\!x^2)} \;=\; \frac{1}{s\!-\!a^2}\int_0^\infty\left(\frac{1}{a^2\!+\!x^2}-\frac{1}{s\!+\!x^2}\right)dx\\ & \;=\; \frac{1}{s\!-\!a^2}\sijoitus{x=0}{\quad\infty}\left(\frac{1}{a}\arctan\frac{x}{a}-\frac{1}{\sqrt{s}}\arctan\frac{x}{\sqrt{s}}\right)\\ & \;=\; \frac{1}{s\!-\!a^2}\!\cdot\!\frac{\pi}{2}\left(\frac{1}{a}-\frac{1}{\sqrt{s}}\right) \;=\; \frac{\pi}{2a}\!\cdot\!\frac{1}{(a\!+\!\sqrt{s})\sqrt{s}}\\ & \;\curvearrowright\; \frac{\pi}{2a}e^{a^2t}\,{\rm erfc}\,a\sqrt{t} \end{matrix}}
Consequently, $\int _{0}^{\infty }{\frac {e^{-tx^{2}}}{a^{2}\!+\!x^{2}}}\,dx\;=\;{\frac {\pi }{2a}}e^{a^{2}t}\,{\rm {erfc}}\,a{\sqrt {t}},$ and especially $\int _{0}^{\infty }{\frac {e^{-x^{2}}}{a^{2}\!+\!x^{2}}}\,dx\;=\;{\frac {\pi }{2a}}e^{a^{2}}\,{\rm {erfc}}\,a.$
# PlanetPhysics/Transformation Between Cartesian Basis Vectors and Polar Basis Vectors

From the definition of a covariant vector (covariant tensor of rank 1)
   ${\bar {T}}_{i}=T_{j}{\frac {\partial x^{j}}{\partial {\bar {x}}^{i}}}$
the corresponding transformation matrix is
   $A_{ij}={\frac {\partial x^{j}}{\partial {\bar {x}}^{i}}}$
In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have
$r={\sqrt {x^{2}+y^{2}}}$ $\theta =tan^{-1}\left({\frac {y}{x}}\right)$
and for polar to cartesian
$x=r\cos \theta$ $y=r\sin \theta$
So if we designate $({\hat {e}}_{x},{\hat {e}}_{y})$ as the bar coordinates, then the transformation components from a polar basis vector $({\hat {e}}_{r},{\hat {e}}_{\theta })$ to a cartesian basis vector $({\hat {e}}_{x},{\hat {e}}_{y})$ is calculted as
$A_{11}={\frac {\partial {x}^{1}}{\partial {\bar {x}}^{1}}}={\frac {\partial r}{\partial x}}={\frac {x}{\sqrt {x^{2}+y^{2}}}}$
$A_{12}={\frac {\partial {x}^{2}}{\partial {\bar {x}}^{1}}}={\frac {\partial \theta }{\partial x}}=-{\frac {y}{x^{2}+y^{2}}}$
$A_{21}={\frac {\partial {x}^{1}}{\partial {\bar {x}}^{2}}}={\frac {\partial r}{\partial y}}={\frac {y}{\sqrt {x^{2}+y^{2}}}}$
$A_{22}={\frac {\partial {x}^{2}}{\partial {\bar {x}}^{2}}}={\frac {\partial \theta }{\partial y}}={\frac {x}{x^{2}+y^{2}}}$
The components of cartesian basis vectors to polar basis vectors transform the same way, but now the polar coordinates have the bar
$B_{11}={\frac {\partial {x}^{1}}{\partial {\bar {x}}^{1}}}={\frac {\partial x}{\partial r}}=\cos \theta$
$B_{12}={\frac {\partial {x}^{2}}{\partial {\bar {x}}^{1}}}={\frac {\partial y}{\partial r}}=\sin \theta$
$B_{21}={\frac {\partial {x}^{1}}{\partial {\bar {x}}^{2}}}={\frac {\partial x}{\partial \theta }}=-r\sin \theta$
$B_{22}={\frac {\partial {x}^{2}}{\partial {\bar {x}}^{2}}}={\frac {\partial y}{\partial \theta }}=r\cos \theta$
In summary, the {\mathbf components of covariant basis vectors} in cartesian coordinates and polar coordinates transform between each other according to
$\left[{\begin{matrix}{\hat {e}}_{x}\\{\hat {e}}_{y}\end{matrix}}\right]=\left[{\begin{matrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&-{\frac {y}{x^{2}+y^{2}}}\\{\frac {y}{\sqrt {x^{2}+y^{2}}}}&{\frac {x}{x^{2}+y^{2}}}\end{matrix}}\right]\left[{\begin{matrix}{\hat {e}}_{r}\\{\hat {e}}_{\theta }\end{matrix}}\right]$
$\left[{\begin{matrix}{\hat {e}}_{r}\\{\hat {e}}_{\theta }\end{matrix}}\right]=\left[{\begin{matrix}\cos \theta &\sin \theta \\-r\sin \theta &r\cos \theta \end{matrix}}\right]\left[{\begin{matrix}{\hat {e}}_{x}\\{\hat {e}}_{y}\end{matrix}}\right]$
# PlanetPhysics/Universe

The physical Universe is understood as the global spacetime with all its physical contents such as: matter, energy, dark matter, dark energy, gravitational fields, electromagnetic, electroweak and nuclear fields, and so on (Higgs bosons ?). Some quantum theories assume the existence of mutliple physical universes or ""multi-verses .
# PlanetPhysics/Universe of Discourse

Boole (1854) defines universe of discourse in the following manner:
   In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. $\ldots$ Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. (Boole 1854/1958, p. 42).
# PlanetPhysics/Trivial Groupoid

"At the opposite extreme to varieties determined by their finite members are those which have only one finite member, the trivial groupoid."

## Category

"Categorically any setoid is a trivial groupoid, ie a category where every morphism is an isomorphism."
# PlanetPhysics/Uc Locally Compact Quantum Groupoids
### Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider locally compact quantum groupoids ($LCQGn$) defined as \htmladdnormallink{locally compact groupoids {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a Haar system}, $\nu$, Failed to parse (unknown function "\G"): {\displaystyle (\G,\nu):= ([\grp, G_2, \mu], \nu)} , or as derived from a (\htmladdnormallink{non-commutative {http://planetphysics.us/encyclopedia/AbelianCategory3.html}) weak Hopf algebra} (WHA), with the additional condition of uniform continuity over Failed to parse (unknown function "\grp"): {\displaystyle \grp} defined as follows . Let us also consider a space Failed to parse (unknown function "\grp"): {\displaystyle LUC(\grp)} of left uniformly continuous elements in 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 L^{\infty}(\grp)} defined over $G_{2}$, which is endowed with the induced product topology from the subset $G^{2}$ of composable pairs in the topological groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} . This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from `quantum groupoids' derived from a weak Hopf algebra.

#### C*-algebra Comparison and Example

Consider $LCG$ to be a locally compact quantum group. Then consider the space $LUC(G)$ of left uniformly continuous elements in $L^{\infty }(G)$ introduced in ref.. (The definition according to V. Runde (\em loc. cit.) covers both the space of left uniformly continuous functions on a locally compact group and (Granirer's) uniformly continuous functionals on the Fourier algebra.) Also consider $LUC(G)$ which is then an \htmladdnormallink{operator {http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} system containing the C*-algebra $C_{o}(G)$}. One may compare the groupoid C*-convolution algebra, $G_{CA}$ -- obtained in the general case-- with the C*-algebra $C_{o}(G)$ obtained from $LUC(G)$ in the particular case of uniform continuity over a locally compact group.
# Web Design/Using the Internet as a Learning Tool

Web Design → Using the Internet as a Learning Tool
This page is part of the Web Design project. :
It is impossible for anyone to ignore the incredible learning opportunities that the internet brings! Although this course focuses on website design, we'll also be exploring ways to leverage the incredible information and tools that are freely available on the net so that we can keep on learning web design after completing the course and continue on for the rest of our lives! Sorry if that sounds cliche, but it is exciting!

## Module Aim

Some of the things that you might learn in this module:
* putting down in writing a topic or area that you would like to investigate further
* finding useful resources that may facilitate your learning
* interacting with others who are currently involved in the topic or area
* Reflecting on your learning as you go in a way that allows other interested people to participate
* Building a list of RSS feeds that will help keep you up-to-date in different areas
Bear in mind that the skills that you practise in this module are life-long learning skills! Ideally you'll demonstrate these skills on an ongoing basis.

## Suggested Activities

* If you don't already have your own blog, then setting up your own and starting to write regularly would be a great way to start learning!
* Discover the history of the Web - An interactive group activity exploring the history of the web.
* Starting to build own set of RSS feeds for web design news (you can start by seeing what other web-designers read, for eg: HTMLDogs blog roll)
* Contributing to the learnings of others in the course
* Contributing to a Wikipedia article of your choice.
* del.icio.us - Starting your own social bookmarks collection!
* Have a look through the Networked learning entry here in wikipedia.

## Your learning resources

* Getting started with Blogging in Education - A gentle introduction to blogging, from "what is a blog" and creating your own blog through to considering how blogging can be used to learn in education.
* Blogger - If you want to get straight in there and create your own blog, this is a great place to start!
* Bloglines - As you find other blogs that are of interest to you, you might want to use an easy tool like Bloglines to help you track those blogs.
* Category:Blogging
* http://www.alistapart.com/ - An online magazine for people who make websites. (The magazine which coined 'Responsive Web Design')
* http://www.smashingmagazine.com/ - An online magazine for inspiration and learning.

## Related Qualifications

See Using the Internet as a Learning Tool:Quals.
# Web Design/Using the Internet as a Learning Tool:Quals

The module Using the Internet as a Learning Tool is related to qualifications in the following countries:

## Australia

The following units of competency from the ICA40305 Certificate IV in Information Technology (Websites) are related to this module:
* BSBCMN304A Contribute to personal skill development and learning
# Web Design/Sample Timetable

Web Design → Sample Timetable
This page is part of the Web Design project. :
This week-by-week timetable through the web course is intended only as a sample. All of us work at different paces and have different skills when we start. Feel free to update with your own feedback and experience!
Note: This is being replaced by the new structure ideas.
# Web Design/URL format

URLs make resources such as online posts addressable, allowing people to bookmark and share content. Like titles, they can facilitate finding resources again.
Social media platforms use various URL formats. Some include the posting account's name while others only contain an ID. Blogging platforms' URL format may contain a year and a month number.

## Case studies

YouTube (/watch?v=[ID]) and Instagram (/p/[ID]) only use the post ID, whereas Twitter puts user name ahead of ID (/[username]/status/[ID]).
The Question & Answer site Quora just uses question titles with words separated by dashes and without numerical IDs, whereas sites of the StackExchange network (e.g. SuperUser, AskUbuntu, StackOverflow) use numerical IDs followed by keywords from the question title as URL slug which can be changed without breaking the link.
Internet forum software typically allows linking to subforums, threads (or "topics"), and individual posts through a numerical identifier. Threads' pages may be linked with page numbers or post count offsets. Links to individual posts may show the individiual post, or the post within a thread for context. The default URL may contain redundant parameters with numerical identifiers for subforum, thread, and post ID, to allow falling back in that order should an item be unavailable.
# Social psychology (psychology)/Tutorials/Australian zeitgeist/Tutor notes
## Timings

* Introduction 20-25 mins
* Audio 50-55 mins
* Wishlist (self) - 2 mins
* Group solutions - 15 mins
* Presentations - 20 mins
# Social psychology (psychology)/Tutorials/Cross-cultural training/Cultural map development
## Cultural map development by students

* Ask students (in pairs) to prepare a cultural map for a skill in a culture with which they are familiar but with which others may be unfamiliar (Assume an audience who isn't familiar with the cultural scenario). (~20 minutes)
* Create a map on an overhead for projection or whiteboard or butcher's paper (~10 minutes)
* Develop a 2-3 minute role-play which explicitly and clearly demonstrates cultural competence in achieving a task in a novel cultural setting. (~10 minutes)
# Social psychology (psychology)/Tutorials/Cross-cultural training/Cultural identity
## Cultural identity

Purpose: This exercise is designed to help individuals identify cultural influences on their personal identity.
* What is cultural identity?
* Ask individuals to draw a concept map of their cultural identity e.g.,
** indicate group memberships,
** use size/colour to highlight the overlay between one's cultural groupings/shadings.
* Cultural groupings may fall along ethnic/racial lines, but also encourage students to consider other cultures/sub-cultures which they may identify with, e.g.,
*** youth culture vs. older people's culture
*** online vs. offline culture
*** rural vs. city
*** skateboarding culture
*** church culture, etc.
* Allow 5 to 10 minutes for individual concept mapping
* In small groups (3 to 4), allow 10 to 15 minutes for participants to share, explain, and possibly modify their maps by sharing them with one another.
* Optional: Ask students to consider what "saliency" is and how it influences cultural identity.
** This should hopefully lead to discussion about fluidity/dynamic nature of cultural identity, i.e., that it changes with situational and cognitive circumstance.
* Note: Relates to Blog 1 Social Self
# Social psychology (psychology)/Tutorials/Cross-cultural training/Cultural map role-plays
## Cultural map role-plays

Each pair (~10 minutes each):
* Presents cultural map
* Role-plays effective behaviour
* Allow audience to ask questions and to offer feedback, e.g.,
** Where the ABCD phases evident?
** Was it easy to understand?
# WikiJournal of PPB/Volume 1 Issue 1

WikiJournal of Psychology, Psychiatry and Behavioral Sciences Open access • Publication charge free • Public peer review • Wikipedia-integrated
Volume 1(1)
Previous
Editorial guidelines Ethics statement Bylaws Financials Calendar
WikiJournal WikiJournal of Psychology, Psychiatry and Behavioral Sciences is an open-access, free-to-publish, Wikipedia-integrated academic journal for Psychologial health and wellbreing topics. <seo title=" WJP, WikiJPPB, Wiki.J.PPB., WikiJPPB, Wikiversity Journal PPB, WikiJournal of PPB, WikiJournal PPB, Wikipedia PPB, Wikipedia PPB journal, WikiPPB, Wikipsychology, Psychology, Psychiatry, Behavioral Sciences Free to publish, Open access, Open-access, Non-profit, online journal, Public peer review "/>
# WikiJournal of Psychology, Psychiatry and Behavioural Sciences/Editorial board/Application
## Editorial board application of

Areas of expertise:
Relevant experience:
    Professional -
    Publishing -
    Open -
I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. ~~

## Use

Above is a draft that preloads when someone wishes to apply to be on the Editorial board. It can be invoked with the link below:
   https://en.wikiversity.org/w/index.php?title=Special:NewSection/Talk:WikiJournal_of_Psychology,_Psychiatry_and_Behavioural_Sciences/Editorial_board/Editorial_board&preload=WikiJournal_of_Psychology,_Psychiatry_and_Behavioural_Sciences/Editorial_board%2FEditorial_board%2FApplication
# WikiJournal of Science

WikiJournal of Science Open access • Publication charge free • Public peer review • Wikipedia-integrated
Volume 7(1)
Previous Volume 1(1) Volume 1(2) Volume 2(1) Volume 3(1) Volume 4(1) Volume 5(1) Volume 6(1)
Editorial guidelines Ethics statement Bylaws Financials Calendar
WikiJournal of Science is an open-access, free-to-publish, Wikipedia-integrated academic journal for science, mathematics, engineering and technology topics. WJS WikiJSci Wiki.J.Sci. WikiJSci WikiSci WikiScience Wikiscience Wikijournal of Science Wikiversity Journal of Science WikiJournal Science Wikipedia Science Wikipedia science journal STEM Science Mathematics Engineering Technology Free to publish Open access Open-access Non-profit online journal Public peer review
# Progress and Prospects in Parkinson's Research/Causes/Deficiencies/Norepinephrine
## Background

Norepinephrine is also known as noradrenline and more commonly as the 'fright and flight' compound.
Norepinephrine is synthesized from dopamine by dopamine β-hydroxylase. It is released from the adrenal medulla into the blood as a hormone, and is also a neurotransmitter in the central nervous system and sympathetic nervous system where it is released from noradrenergic neurons in the locus coeruleus. The actions of norepinephrine are carried out via the binding to adrenergic receptors.
A sudden shock produces an autonomic reaction in the sympathetic nervous system, generating a pulse of norepinephrine, which speeds up the action of the heart and charges the limbs with fresh, oxygenated blood, ready to run or take other reactive action.

## Research

2002
The National Institute of Neurological Disorders and Stroke – NINDS gives an account of the research of Dr. David Goldstein Chief of the Clinical Neurocardiology Section at NINDS. Using positron emission tomography (PET) scans, researchers at the National Institute of Neurological Disorders and Stroke (NINDS) found that nearly all of the 29 Parkinson's disease patients they studied had decreased numbers of norepinephrine-producing nerve endings in the heart. This can lead to a sudden drop in blood pressure termed ‘ orthostatic hypotension’.

## Related Pages

Causes > Deficiencies
Sub Pages:
    Blood - Melatonin - Norepinephrine - Vitamin_D - Zinc