OpenLLM-France/wikimedia huggingface dataset:
# Numerical Analysis/Vandermonde example

We'll find the interpolating polynomial passing through the three points $(1,-6)$, $(2,2)$, $(4,12)$, using the Vandermonde matrix.
For our polynomial, we'll take $(1,-6)=(x_{0},y_{0})$, $(2,2)=(x_{1},y_{1})$, and $(4,12)=(x_{2},y_{2})$.
Since we have 3 points, we can expect degree 2 polynomial.
So define our interpolating polynomial as:
$p(x)=a_{2}x^{2}+a_{1}x+a_{0}$.
So, to find the coefficients of our polynomial, we solve the system $p(x_{i})=y_{i}$, $i\in \{0,1,2\}$.
    $\left({\begin{array}{ccc}x_{0}^{2}&x_{0}&1\\x_{1}^{2}&x_{1}&1\\x_{2}^{2}&x_{2}&1\\\end{array}}\right)*\left({\begin{array}{c}a_{2}\\a_{1}\\a_{0}\end{array}}\right)=\left({\begin{array}{c}y_{0}\\y_{1}\\y_{2}\end{array}}\right)$
In order to solve the system, we will use an augmented matrix based on the Vandermonde matrix, and solve for the coefficients using Gaussian elimination. Substituting in our $x$ and $y$ values, our augmented matrix is:
$\left({\begin{array}{ccc|c}1^{2}&1&1&-6\\2^{2}&2&1&2\\4^{2}&4&1&12\end{array}}\right)$
Then, using Gaussian elimination,
$\left({\begin{array}{ccc|c}1&1&1&-6\\4&2&1&2\\16&4&1&12\end{array}}\right)\Rightarrow \left({\begin{array}{ccc|c}1&1&1&-6\\0&-2&-3&26\\0&-12&-15&108\end{array}}\right)\Rightarrow \left({\begin{array}{ccc|c}1&1&1&-6\\0&-2&-3&26\\0&0&3&-48\end{array}}\right)\Rightarrow \left({\begin{array}{ccc|c}1&1&0&10\\0&2&0&-22\\0&0&1&-16\end{array}}\right)\Rightarrow \left({\begin{array}{ccc|c}1&0&0&-1\\0&1&0&11\\0&0&1&-16\end{array}}\right)$
Our coefficients are $a_{2}=-1$, $a_{1}=11$, and $a_{0}=-16$. So, the interpolating polynomial is
$p(x)=-x^{2}+11x-16$.

## Adding a point

Now we add a point, $(3,-10)=(x_{3},y_{3})$, to our data set and find a new interpolation polynomial with this method.
Since we have 4 points, we will have degree 3 polynomial.
Thus our polynomial is $p(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$,
and we get the coefficients by solving the system $p(x_{i})=y_{i}$.
Constructing our augmented matrix as before and using Gaussian elimination, we get:
$\left({\begin{array}{cccc|c}1^{3}&1^{2}&1&1&-6\\2^{3}&2^{2}&2&1&2\\4^{3}&4^{2}&4&1&12\\3^{3}&3^{2}&3&1&-10\end{array}}\right)\Rightarrow \left({\begin{array}{cccc|c}1&1&1&1&-6\\0&-4&-6&-7&50\\0&-48&-60&-63&396\\0&-18&-24&-26&152\end{array}}\right)\Rightarrow \left({\begin{array}{cccc|c}1&1&1&1&-6\\0&-4&-6&-7&50\\0&0&12&21&-204\\0&0&3&{\frac {11}{12}}&-73\end{array}}\right)$
$\Rightarrow \left({\begin{array}{cccc|c}1&1&1&1&-6\\0&-4&-6&-7&50\\0&0&12&21&-204\\0&0&0&{\frac {1}{4}}&-22\end{array}}\right)\Rightarrow \left({\begin{array}{cccc|c}1&1&1&0&82\\0&-4&-6&0&-566\\0&0&12&0&1644\\0&0&0&1&-88\end{array}}\right)\Rightarrow \left({\begin{array}{cccc|c}1&1&0&0&-55\\0&-4&0&0&256\\0&0&1&0&137\\0&0&0&1&-88\end{array}}\right)$
$\Rightarrow \left({\begin{array}{cccc|c}1&0&0&0&9\\0&1&0&0&-64\\0&0&1&0&137\\0&0&0&1&-88\end{array}}\right)$
Therefore, our polynomial is:
$p(x)=9x^{3}-64x^{2}+137x-88$.
# Numerical Analysis/Romberg's method

Romberg's method approximates a definite integral by applying Richardson extrapolation to the results of either the trapezoid rule or the midpoint rule.
The initial approximations $R_{k,0}$ are obtained by applying either the trapezoid or midpoint rule with $2^{k}+1$ points. In the case of the trapezoid rule on $\left[a,b\right]$,
$R_{k,0}=h_{k}\left({\frac {f(a)+f(b)}{2}}+\sum _{i=1}^{2^{k}-1}f(a+h_{k}i)\right)$
where
$h_{k}={\frac {b-a}{2^{k}}}$
For $k>0$, we can reduce the number of places the function is evaluated by using our previously obtained approximations instead of re-sampling. For the trapezoid rule, this improvement gives
$R_{0,0}=\left({\frac {b-a}{2}}\right)[f(a)+f(b)]$
and
$R(k,0)={\frac {1}{2}}R(k-1,0)+h_{k}\sum _{i=1}^{2^{k-1}}f(a+(2i-1)h_{k})$
Richardson's extrapolation is then applied recursively, giving
$R_{k,j}={\frac {4^{j}R_{k,j-1}-R_{k-1,j-1}}{4^{j}-1}}$
Each successive level of improvement increases the order of error term from $O(h^{2j})$ to $O(h^{2j+2})$ at the expense of doubling the number of places the function is evaluated.
# PlanetPhysics/R Supercategory

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One notes also that R-diagrams can be defined as functors according to the standard Barry Mitchell's construction.
# PlanetPhysics/Radial Equation

The radial equation is
$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}u}{dr^{2}}}+\left[V+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}\right]u=Eu$
It's derivation is straight forward using separation of variables with the time independent Schr\"odinger equation in spherical coordinates.
# PlanetPhysics/Recursive Function 2

Intuitively, a recursive function may be defined as an integer-valued function of one or more integer variables which may be computed by a definite algorithm. In order to produce a rigorous definition, one may proceed long at least two approaches; one may define the notion of algorithm rigorously in order to complete the intuitive definition given above or one may proceed inductively, first declaring certain functions to be recursive and then specifying definite procedures by which one may construct any other recursive function starting from the initial set. In this entry, we shall concentrate on the latter approach, only making a few brief remarks regarding the former approach towards the end.
* The constant function $c:\mathbb {Z} _{+}\to \mathbb {Z} _{+}$ defined by $c(x)=1$ for all $x\in \mathbb {Z} _{+}$ is a recursive function.
* The addition function $+:\mathbb {Z} _{+}^{2}\to \mathbb {Z} _{+}$ and the multiplication function $\times :\mathbb {Z} _{+}^{2}\to \mathbb {Z} _{+}$ are recursive function.
* The projection functions $I_{m}^{n}\colon \mathbb {Z} _{+}^{n}\to \mathbb {Z} _{+}$ with $1\leq m\leq n$ defined as $I_{m}^{n}(x_{1},\ldots ,x_{n})=x_{m}$ are recursive functions.
* {\it (Closure under composition)} If $f\colon \mathbb {Z} _{+}^{n}\to \mathbb {Z} _{+}$ is a recursive function and $g_{i}\colon \mathbb {Z} _{+}^{m}\to \mathbb {Z} _{+}$ with $i=1,\ldots n$ are recursive functions, then $h\colon \mathbb {Z} _{+}^{n}\to \mathbb {Z} _{+}$, defined by $h(x_{1},\ldots ,x_{n})=f(g_{1}(x_{1},\ldots ,x_{m}),\ldots ,g_{n}(x_{1},\ldots ,x_{m}))$ is a recursive function.
* {\it (Closure under primitive recursion)}If $f\colon \mathbb {Z} _{+}^{n}\to \mathbb {Z} _{+}$ and $g\colon \mathbb {Z} _{+}^{n+2}\to \mathbb {Z} _{+}$ are recursive function, then $h\colon \mathbb {Z} _{+}^{n+1}\to \mathbb {Z} _{+}$, defined by the recursion
$h(n+1,x_{1},\ldots ,x_{k})=g(h(n,x_{1},\ldots ,x_{k}),n,x_{1},\ldots ,x_{k})$ with the initial condition $h(0,x_{1},\ldots ,x_{k})=f(x_{1},\ldots ,x_{k})$ is a recursive function.
* $f(0,x_{1},\ldots ,x_{n}),f(1,x_{1},\ldots ,x_{n}),\ldots ,f(y,x_{1},\ldots ,x_{n})$ are all defined, #
* $f(z,x_{1},\ldots ,x_{n})=0$ when $1\leq z<y$, and #
* $f(y,x_{1},\ldots ,x_{n})=0$, otherwise $g(x_{1},\ldots ,x_{n})$ is undefined.
# PlanetPhysics/Regular Measure

A regular measure $\mu _{R}$ on a topological space $X$ is a measure on $X$ such that for each $A\in {\mathcal {B}}(X)$ , with $\mu _{R}(A)<\infty$), and each $\varepsilon >0$ there exist a compact subset $K$ of $X$ and an open subset $G$ of $X$ with $K\subset A\subset G$, such that for all sets $A'\in {\mathcal {B}}(X)$ with $A'\subset G-K$, one has $\mu _{R}(A')<\varepsilon$.
# PlanetPhysics/Rayleigh Ritz Method

The Rayleigh-Ritz method is an algorithm for obtaining approximate solutions to eigenvalue ODEs. It can be neatly summarized as follows:
* Choose an approximate form for the eigenfunction with the lowest eigenvalue (the ground state wavefunction, in the language of quantum mechanics). Include one or more free parameters.
* Find the expectation value of the eigenvalue with respect to the trial eigenfunction.
* Minimize the resulting equation with respect to the free parameter(s), hence finding a value for the free parameter.
* Substitute this new eigenfunction back into the expectation value.
* The expectation value obtained is an upper bound for the actual eigenvalue of the true eigenfunction.

## Example

Consider the time independent Schr\"odinger equation for a one-dimensional harmonic oscillator potential:
   $\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {1}{2}}m^{2}\omega ^{2}\right)\psi =E\psi$
where $m$ is the mass of the particle in the well, and $\omega$ is the angular velocity a classical particle would move with in the well. This equation can be solved exactly using Frobenius' method, and leads to eigenfunctions of the form of Hermite polynomials multiplied by Gaussians, and half-integer eigenvalues of the form $E_{n}=(n+1/2)\hbar \omega$. Since the solutions are known, it is a good test case. We choose the ground state wavefunction of the infinite potential well as our trial eigenfunction:
   $\psi ={\frac {\cos({\frac {\pi x}{2a}})}{\sqrt {a}}}$
with $a$ as our free parameter. We now find the expectation value:
   $\langle E\rangle =\langle \psi |{\hat {H}}|\psi \rangle =\int _{-a}^{a}\psi ^{*}{\hat {H}}\psi \,dx$
Evaluating the integral, we find
   $\langle E\rangle ={\frac {\hbar ^{2}\pi ^{2}}{8ma^{2}}}+m\omega ^{2}a^{2}\left({\frac {1}{6}}-{\frac {1}{pi^{2}}}\right)$
We now minimise this with respect to $a$ to obtain:
   $2m\omega ^{2}a\left({\frac {1}{6}}-{\frac {1}{\pi ^{2}}}\right)={\frac {\hbar ^{2}\pi ^{2}}{4ma^{2}}}$
Hence:
   $a=\pi \left({\frac {3}{4(\pi ^{2}-6)}}\right)^{\frac {1}{4}}\left({\frac {\hbar }{m\omega }}\right)^{\frac {1}{2}}$
Substituting this into the expecation value $\langle E\rangle$ we obtain
   $\langle E\rangle ={\frac {1}{2}}\left({\frac {\pi ^{2}-6}{3}}\right)^{\frac {1}{2}}\hbar \omega$
   $\langle E\rangle \approx 0.568\hbar \omega$
The analytical value is of course $0.5\hbar \omega$. Considering the crudeness of the approximation used, the result is impressive.
# Open interviews

This is a project which includes/describes a type of Interview which is done in a transparent way. Further development of this project can include instructions on how to conduct interviews in various ways that have qualities of transparency and interaction between interviewer (Q46034607) and interviewee (Q55534929).
This project is also about interacting with communities. ie. making polls of to learn more about communities, culture and subcultures.

## Introduction

Ask me anything may be a trade marked term. Reddit hosts ask me anythings. Reddit can be utilized for learning.
Barack Obama has done ask me anything on Reddit.
This is meant to be a spot where individuals can offer interviews. These interviews should be related to learning, teaching, or research.
What do you want to learn about? What can you teach others about? We can probably all learn from one another if we just take the time to listen and be curious.
Note: Your offer of an open interview may be deleted without discussion if it is construed as promotional or not directly related to the Wikiversity:Mission.

### Examples on Reddit

These are examples of open interviews on Reddit (they are called AMA's or Ask Me Anything on Reddit). These are examples of educational and learning related AMA's.
* Science AMA Series: I'm Greg Lowry, Investigator and Deputy Director with the Center for the Environmental Implications of NanoTechnology. We do research on the behavior of nanoparticles in the environment and their implications, and am here today to talk about it. AMA!
* I am Chris Roser, a professor studying the past, present, and future of manufacturing, and just published my first book. AMA!
* I’m Steve Horwitz, Professor of Economics at St. Lawrence University. My article, “There is No Such Thing as Trickle-Down Economics” recently made the front page of Reddit. AMA!
* ACS AMA: I'm Kyle Bradbury, Managing Director of the Energy Data Analytics Lab at the Duke University Energy Initiative. Ask me anything about energy storage systems and energy data analytics.

### Open Polls

As a sub project this is a system of interacting with communities through "Open Polls" by providing polls and results there-of.
The first example will be about interacting with gaming communities to find out more things about the "gamer identity". If work is done here then a project might(or might not be) started regarding the subject.
/Open Polls System
# Open file format

An open file format is a published specification for storing digital data, usually maintained by a standards organization. Basically this means that an electronic file can be used and implemented by anyone. For example, an open file format can be used by proprietary and free and open source software. In contrast, proprietary formats are controlled and defined by private interests.

## Examples of open file formats
### Multimedia

* JPEG – an image format standardized by ISO/IEC
* PNG (Portable Network Graphics) – a raster image format standardized by ISO/IEC
* SVG (Scalable Vector Graphics) – a vector image format standardized by W3C
* VRML/X3D – realtime 3D data formats standardized by ISO/IEC
* Ogg, container for Vorbis, FLAC, Speex (audio formats) & Theora (a video format)
* mkv (Matroska), container for all type of multimedia formats (audio, video, images, subtitles)

### Text

* HTML/XHTML – a markup language standardized by W3C
* ODF (OpenDocument) – a suite of office document formats standardized by OASIS and ISO/IEC
* PDF – a document image format standardized by ISO

### Other

* XML – a markup language standardized by W3C
# Open pedagogy

This Education resource is a "stub". Learn how you can help Wikiversity to develop it.
The concept of open pedagogy can be traced back to at least the 1960s (Derosa & Jhangiani, 2018; Johnston & Stewart, 2020; Morgan, 2016). It seems there might be two histories of the term. One (the older stream) focuses on the holistic practice of openness in the context of education, whilst the more recent stream focuses more narrowly on the creation and use of open educational resources (Curtis, 2017).
A hybrid conceptualisation of open pedagogy is that it describes teaching and learning techniques that draw upon open and participatory technologies and high-quality open educational resources (OER) in order to facilitate collaborative and flexible learning (Wiley, 2013).
Morgan (2016, para. 6) traces the origins of open pedagogy back to Paquette (1979), who describes three sets of foundational values for the practice:
* autonomy and interdependence
* freedom and responsibility
* democracy and participation
# Open education with Moodle and Wikiversity/Proposal
## Proposal

This presentation will discuss how Moodle can be used in conjunction with Wikiversity as a platform for open education, with consideration of challenges, opportunities, pros and cons. Principles of open academia will be described, with demonstration of sites including:
* Survey research and design in psychology (Moodle/Wikiversity)
* Motivation and emotion (Moodle/Wikiversity)
* Psychology 102 (Moodle/Wikiversity)
* Social psychology (Wikiversity)
# Open educational resources - open space event

The open educational resources open space event will be held July 17th 2008 in Alexandria, Egypt. The more precise time and space will be announced later.
Everyone invloved in open educational resources and free content in education and related topics is welcome. The event will be and mini open space conference (2 hours) with demonstrations and peer-to-peer talks. Everyone attentding is free to present their work for others and get feedback.
The OER open conference event is organized within the Wikimania conference.
In the Wikimedia conference the topic of open education reseouces is also discussed in the Open educational resources and globalization -panel, on 18th 2008.

## Location

Library of Alexandria, Egypt

## Participants

--Teemu Leinonen 01:39, 4 June 2008 (UTC)
(2024-10-11UTC16:51:54): "Open educational resources - open space event" Location: (linkback://en.wikiversity.org/wiki/Open_educational_resources_-_open_space_event)
# Open marriage

This psychology resource is a "stub". Learn how you can help Wikiversity to develop it.
Open marriage - what shortfall(s) in traditional marriage does this practice try to overcome or eliminate? What are the pitfalls of an open marriage?
# Open interval/Continuous image is not/Exercise

Show that the image of an open interval under a continuous function is not necessarily open.
# Open license

This Education resource is a "stub". Learn how you can help Wikiversity to develop it.
This Law resource is a "stub". Learn how you can help Wikiversity to develop it.
Open licensing refers to the use of copyright licenses which allow a greater degree of freedom than the default all rights reserved.
Examples of open licensing systems include:
* Creative Commons
* GFDL
Governments which have a default is open approach to public service information:
* New Zealand
* UK
* Australia
# Open management

This Philosophy resource is a "stub". Learn how you can help Wikiversity to develop it.
Open management is one of the suggested criteria for open academia, but the principles can apply in any organisation or task/group process.
# Software applications/Overview/Zenya Software Suite

Zenya provides an easy-to-use, integrated risk and quality management system and is used world-wide by more than 1 million professionals each day. The foundation for Zenya was laid more than 20 years ago by softwareproducer Infoland.
Keeping up with continuous change is a prerequisite for success in a dynamic world. As an organisation, as a team and as a professional. That’s where Zenya comes in: Zenya acts as a personal assistant for safety and quality management. From actions and tasks that are easily started or completed, to powerful dashboards that show people exactly where they stand, or where businesses can improve.
Zenya is a modular platform with tools benefiting professionals in a variety of segments.
* Zenya FLOW for incident management
* Zenya CHECK for audit management
* Zenya RISK for risk management
# Software applications/Overview/WhatsApp
## Index

* Applications
# Software applications/Overview/WordPress

* Category: Website builder
* Website: https://wordpress.org
* Wikipedia: WordPress
WordPress is a free and open-source content management system (CMS).
# Software applications/Overview/Weebly

* Category: Website builder
* Website: http://www.weebly.com
* Wikipedia: Weebly
Weebly is a web-hosting service featuring a drag-and-drop website builder.
# Software applications/Overview/Yik Yak

* Category: Learning
* Website: http://www.yikyakapp.com/
Yik Yak is like anonymous twitter. You can see messages from people in your area.

## Questions

* When will Yik Yak be available in the Netherlands?
# Software process management
## Resources

* Agile software development
* Software Process Improvement Models
* Configuration Management (SPM)
* Software Metrics and Measurement
* Process and People
# U. S. Government/Communication in the 1920s

* Change in communication
* Increased availability of telephones
* The telephone was invented in the late 1800's, but it wasn't until the 1920's when most homes had a telephone.
* Development of the radio
* By the end of the 1920s, most homes had a radio. In a time before the TV and the Internet, the radio was a fun source of news and entertainment.
* Development of the broadcasting industry
* Today, NBC broadcasts TV shows, but it started as the National Broadcasting Company and broadcasted radio programs.
* Development of Movies
* Al Jolson is pictured on the right in the first talking movie, "The Jazz Singer."
# U. S. Government/Criminal Law vs. Civil Law Notes

Courts resolve two kinds of legal conflicts, civil and criminal

## Criminal Case

In a criminal case, a court determines whether a person accused of breaking the law is guilty or not guilty of a misdemeanor or a felony.

### Criminal Procedure in felony cases

* Arrested - A person accused of a crime may be arrested if the police have probable cause.
* Jail or Bail - The accused may be committed to a jail or released on bail.
* Arraignment - At an arraignment, probable cause is reviewed, the defendant may be appointed an attorney, and a plea is entered: "Guilty or not Guilty".
* Trial - A court date is set and a trial is conducted
* Verdict - A guilty verdict (decision) may be appealed to the Court of Appeals or directly to the Supreme Court in certain cases.

## Civil Case

In a civil case, a court settles a disagreement between two parties to recover damages or receive compensation

### Procedure for civil cases

* Complaint - The plaintiff files a complaint to recover damages or receive compensation
* Trial - Case can be heard by judge or by jury
* Verdict - Cases can be appealed

## Juvenile Justice

For minors under 18.
* Judges have great latitude in handling juvenile cases
* Juveniles who commit serious crimes can be tried as adults
# U. S. Government/Industrialization/Captains of Industry
## Rockefeller

There is gold, and then there is black gold, or oil. In 1859, oil was struck in western Pennsylvania. This even gave an Ohioan, John Davidson Rockefeller, an idea. He decided that oil would become a big business and that his hometown of Cleveland was ideally situated to refine and distribute it to the nation. Rockefeller built a refinery there in 1862. Then, he put together the Standard Oil Trust. A trust is a group of companies that act as one. The Standard Oil came to control all phases of the oil industry, from extraction, to refining, and distribution. Standard Oil was the first of many "trusts" and monopolies formed in industries after the Civil War. Monopolies can be dangerous, because it can keep prices for its goods high without worrying about another business selling the same thing for less.

## Carnegie

American businessman, Andrew Carnegie, contributed to making Pittsburgh the nation's steel city. Carnegie saw the Bessemer process in use in England. Realizing that America's need for steel would soon soar, Carnegie built his first steel mill near Pittsburgh in 1873. It became the largest and most modern steel mill of its time. By 1901, 34 separated iron and steel plants lined Pittsburgh's rivers. Carnegie, however, often forced out small steel companies or consolidated (combined) the companies to create a large one. Eventually, Carnegie sold his company to J.P. Morgan for $225 million dollars.

## Ford

Henry Ford was born in 1863 on a farm next to a rural town west of Detroit, Michigan. He was the American founder of the Ford Motor Company and father of modern assembly lines used in mass production. His introduction of the Model T automobile revolutionized transportation and American industry. With the use of the assembly line, he produced an affordable car. In addition, unlike other "Captains of Industry", he paid his employees high wages so that they would be able to buy one of his cars. As owner of the Ford Company, he became one of the richest and best known people in the world.

## Vanderbilt

Vanderbilt was born to a poor family and quit school at the age of 11, to work for his father who was employed in boating. When he turned 16, he persuaded his mother to give him $100 loan for a boat to start his first business. He repaid the loan after the first year with an additional $1,000. In 1867, Vanderbilt decided that the wave of the future was in building a railroad empire. Vanderbilt bought the Central Railroad and merged it with the Hudson River Railroad. In addition, he spent millions on the improvement of the railroads lines by making them faster and safer. Upon his death, he was the richest man in the United States.