OpenLLM-France/wikimedia huggingface dataset:
# Muggles' Guide to Harry Potter/Magic/Goblet of Fire
## Overview

The Goblet of Fire is "a large, roughly-hewn wooden cup", "full to the brim with dancing blue-white flames", kept in an ornamental casket.

## Extended Description

Professor Dumbledore, in his introduction to the Triwizard Tournament, said that selection of the Champions to represent each school would be done by an "impartial judge". The Goblet of Fire is that judge; those who wish the chance to compete place their names in the Goblet, and at the start time (which is apparently Hallowe'en), the Goblet selects one person from each school by returning one name from each school.
In Harry Potter and the Goblet of Fire, four Champions are selected, apparently giving Hogwarts two shots at the championship. At the time, Professor Moody reminds everyone there that the Goblet is a "powerful magical artifact," far beyond the ability of a fourth-year student to alter, and suggests that the only way that could have happened would be if someone had hoodwinked the Goblet into believing that there were four schools competing, and then entering Harry's name as the sole competitor from this fictional fourth school. Karkaroff, the Headmaster of Durmstrang, demands that the names be re-submitted, but is told that the Goblet has extinguished itself and will not re-ignite until it is time for the next Tournament.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
* During the hiatus in the Triwizard Tournament, did the Goblet ignite every five years and wait, flaming patiently, for submissions?
* Where is the Goblet now?
# Muggles' Guide to Harry Potter/Magic/Grindylow
## Overview

The Grindylow is a pale green water demon. It is found throughout lakes in Britain and Ireland, including the lake at Hogwarts. It acts with aggression towards muggles and wizards alike, but Merpeople have been known to domesticate it. Grindylows have long fingers which will allow them to get a good grip, but which are easy to snap.

## Extended Description

The Grindylow appears first in Harry Potter and the Prisoner of Azkaban, where Professor Lupin brings it in for his Defence Against the Dark Arts classes. We see it only in passing there, being used only to illustrate Professor Lupin's willingness to communicate with the students in general, and Harry in particular. It reappears as part of the final exam for that course.
The Grindylow reappears in Harry Potter and the Goblet of Fire, as one of the obstacles that Harry must pass in order to complete the Second Challenge. It is in this section that we see that the Grindylow has been domesticated by merpeople; there are some tied up, like dogs, near some of the merpeople's houses.
The Grindylow is also mentioned in Harry Potter and the Deathly Hallows, in order to confirm that it is Harry he is talking to, Lupin asks Harry what was in Lupin's office the first time Harry was in there, and Harry replies, correctly, that it was a Grindylow in a tank.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Graphorn
## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Gurdyroot
## Overview

In appearance, the Gurdyroot looks like a large green onion.

## Extended Description

We first see the Gurdyroot when Luna Lovegood produces it from a pocket of her robes while searching for a message for Harry. She hands it to Ron, explaining that it is good for warding off Gulping Plimpies.
At Bill and Fleur's wedding, Xenophilius Lovegood says he has not yet given the happy couple his wedding gift. As he departs, Ron suggests that it might be a lifetime supply of Gurdyroots.
When Harry, Ron, and Hermione are visiting Xeno Lovegood after Christmas in Harry's seventh year, he prepares an infusion of Gurdyroots for them. The resulting tea is a very dark purplish red, like beetroot. Xeno seems to quite like the taste, but Harry finds it undrinkable.
No particular magical properties are ever attributed to Gurdyroot, except for its being able to ward off Gulping Plimpies.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Introduction to Paleoanthropology
## Table of Contents

* Introduction to Paleoanthropology
* Defining Paleoanthropology
* Origin of Paleoanthropology
* Importance of Bones
* Early Hominid Fossils
* Phylogeny and Chronology
* Early Hominid Behavior
* The Oldowan Period
* The Acheulean Period
* Hominids of the Acheulean
* Technology in the Acheulean
* Hominids of the Middle Paleolithic
* Technology of the Middle Paleolithic
* The Upper Paleolithic
* Suggested Supplemental Reading
* Dating Techniques
* Cultural Evolution
* Darwinian Thought
* Genetics
* Contemporary Primates
* Humans as Primates
* Origin of Language
* From Hunter-Gatherer to Food Producer
* Variation in Modern Human Populations
# Introduction to Open Educational Resources
## Contents

* Overview
* Open Courseware
* Copyright, Fair Use, and TEACH Act
* Tools to Identify and Select OER
* Accessibility and Universal Design
* OER Discipline-Specific Sources
* Use of Primary Sources
* Sources of Open Textbooks
* OER Development and Collaboration
* OER Delivery, Storage, and Organization
* Support for OER Use on Campus
* Future of OER Movement
# LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation-Based Properties
## Young’s Relation-Based Properties

1. Consider $X$,$Y\in \mathbb {R} ^{n\times m}$ and $Z\in \mathbb {S} ^{m}$. The matrix inequality given by
                               ${\begin{aligned}Z+XTY+YTX>0,\end{aligned}}$
    is satisfied if and only if there exist $Q\in \mathbb {S} ^{m}$, $P\in \mathbb {S} ^{n}$,$G1\in \mathbb {R} ^{n\times n}$,$G2\in \mathbb {R} ^{n\times m}$,$F\in \mathbb {R} ^{m\times n}$, and $H\in \mathbb {R} ^{m\times m}$, where $Q>0$
    and $P>0$, such that
                ${\begin{bmatrix}P&Y\\*&Q\end{bmatrix}}>0$ and ${\begin{bmatrix}Z+Q+X^{T}PX&F-X^{T}G_{2}&H-X^{T}G_{1}\\*&G_{1}+G_{1}^{T}-P&F^{T}+G_{2}-Y\\*&*&H^{T}+H-Q\end{bmatrix}}$
2. Consider $X$,$Y\in \mathbb {R} ^{n\times n}$ and $W\in \mathbb {S} ^{m}$, where $X$ is full rank and $W>0$. The matrix inequality given by
                               ${\begin{aligned}X^{T}-W>0,\end{aligned}}$
    is satisfied if there exist $\lambda \in \mathbb {R} _{>0}$ such that
                ${\begin{bmatrix}\lambda \mathbf {1} &\lambda \mathbf {1} &\mathbf {0} \\*&\mathbf {X} +\mathbf {X} ^{T}&\mathbf {W} ^{\frac {1}{2}}\\*&*&\lambda \mathbf {1} \end{bmatrix}}>0$

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (2.4.1 page 23)
# LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

## Matrix inequality

Consider $X,Y\in \mathbb {R} ^{n\times m}$ and $S\in \mathbb {S} ^{n\times n}$ , where $S$ >0, The matrix inequality given by
    ${\begin{aligned}\ X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\\\end{aligned}}$
which is named Young’s relation or Young’s inequality.

## Derivation

Young’s relation can be derived from a completion of the squares as follows.
    ${\begin{aligned}0\leq (X-SY)^{T}S^{-1}(X-SY)\\0\leq X^{T}S^{-1}X+Y^{T}SY-X^{T}Y-Y^{T}X\\X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\end{aligned}}$
which is named Young’s relation.

## Reformulation of Young’s Relation

Consider $X,Y\in \mathbb {R} ^{n\times m}$ and $S\in \mathbb {S} ^{n\times n}$ , where $S$ >0, The matrix inequality given by
    ${\begin{aligned}\ X^{T}Y+Y^{T}X\leq {\frac {1}{2}}(X+SY)^{T}S^{-1}(X+SY),\\\end{aligned}}$
is a reformulation of Young’s relation.

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (2.4.1 page 23)
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations

The information in this chapter is referenced from: "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization" by Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo, published in SIAM Review 2010, Volume 52, Issue 3, pages 471-501.

## Contents

      * Rank minimization problem: general form
      * Minimum-Order Linear System Realization via minimizing the rank of the Hankel matrix
      * Low-Rank Matrix Completion
      * Low-Dimensional Euclidean Embedding Problem
      * Nuclear Norm Minimization
      * Nuclear Norm Minimization
      * Low-Rank Parametrization
* ↑ "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization" (PDF). Retrieved 2019-11-08.
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Rank Matrix Completion

WIP
LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Rank Matrix Completion
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Minimum-Order Linear System Realization via minimizing the rank of the Hankel matrix

WIP
LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Minimum-Order Linear System Realization via minimizing the rank of the Hankel matrix
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Rank Parametrization

WIP
LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Rank Parametrization
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Nuclear Norm Minimization

WIP
LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Nuclear Norm Minimization
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Matrix and LMI Properties and Tools/Tangential Nevanlinna Pick
## Tangential Nevanlinna-Pick

The Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly $H_{\infty }$ robust and optimal control.
The problem is to try and find a function $H:C\to C^{pxq}$ which is analytic in $C_{+}$ and satisfies ‌ $H({\lambda }_{i})u_{i}=v_{i},$   $i=1,...,m$   with $||H||_{\infty }\leq 1$ $(1)$

## The System

$N_{(ij)}$ is a set of $pxq$ matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if $Im(H({\lambda }))\geq 0$ $({\lambda }\in {\pi }^{+})$.

## The Data

Given: ‌ Initial sequence of data points on real axis ${\lambda }_{1},...,{\lambda }_{m}$ with ${\lambda }_{i}\in C_{+}{\widehat {=}}(s|Re(s)>0)$, ‌ And two sequences of row vectors containing distinct target points $u_{1},....,u_{m}$ with $u_{i}\in C^{q}$, and $v_{1},...,v_{m}$ with $v_{i}\in C^{p},i=1,...,m$.

## The LMI: Tangential Nevanlinna- Pick

Problem (1) has a solution if and only if the following is true:

##### Nevanlinna-Pick Approach

$N_{ij}=$ ${\frac {u_{i}^{\ast }u_{j}-v_{i}^{\ast }v_{j}}{\ {\lambda }_{i}^{\ast }+{\lambda }_{j}}}$ ‌

##### Lyapunov Approach

N can also be found using the Lyapunov equation:
$A^{\ast }N+NA-(U^{\ast }U-V^{\ast }V)=0$
where $A=diag({\lambda }_{1},...,{\lambda }_{m}),U=[u_{1}...u_{m}],V=[v1...vm]$
The tangential Nevanlinna-Pick problem is then solved by confirming that $N\geq 0$.

## Conclusion:

If $N_{(ij)}$ is positive (semi)-definite, then there exists a norm-bounded analytic function, $H$ which satisfies $H({\lambda }_{i})u_{i}=v_{i},$  $i=1,...,m$   with $||H||_{\infty }\leq 1$

## Implementation

Implementation requires YALMIP and a linear solver such as sedumi. - MATLAB code for Tangential Nevanlinna-Pick Problem.

## Related LMIs

Nevalinna-Pick Interpolation with Scaling

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* Generalized Interpolation in $H_{\infty }$ by Donald Sarason.
* Tangential Nevanlinna-Pick Interpolation Problem With Boundary Nodes in the Nevanlinna Class And The Related Moment Problem by Yong Jian Hu and Xiu Ping Zhang.

## Return to Main Page:

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
# LMIs in Control/Matrix and LMI Properties and Tools/Variable Reduction Lemma
## Introduction

The variable reduction lemma allows the solution of algebraic Riccati inequality that involve a matrix of unknown dimension. This often arises when finding the controller that minimizes the H$_∞$ norm.

## The Data

In order to find the unknown matrix $M$ we need matrices $A$, $P$ & $Q$.

## The Optimization Problem

Given a symmetric matrix $A\in \mathbb {R} ^{n\times n}$ and two matrices $P$ & $Q$ of column dimension n, consider the problem of finding matrix $M$ of compatible dimensions such that
    ${\begin{aligned}\ A+P^{T}M^{T}Q+Q^{T}M^{T}P<0\\\end{aligned}}$
The above equation is solvable for some $M$ if and only if the following two conditions hold
    ${\begin{aligned}\ W_{P}^{T}AW_{P}<0\\\ W_{Q}^{T}AW_{Q}<0\\\end{aligned}}$
Where $W_{P}$ and $W_{Q}$ are matrices whose columns are bases for the null spaces of $P$ & $Q$, respectively.

## Implementation

This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

## Conclusion

Using this technique we can get the value of unknown matrix $M$.

## External Links

A list of references documenting and validating the LMI.
* https://web.mit.edu/braatzgroup/33 A tutorial on linear and bilinear matrix inequalities.pdf - A journal paper on the said LMI
* https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.4590040403 - Research paper on the said LMI and its proof

## Return to Main Page

https://en.wikibooks.org/wiki/LMIs_in_Control
# LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Dimensional Euclidean Embedding Problem

WIP
LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Low-Dimensional Euclidean Embedding Problem
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# Modern Photography/Printing

Printing is the process of placing a finished image on to paper or other media.

## Analog era

Traditionally, printing referred to the creation of a positive photographic print from a negative film or photographic plate, however today it generally refers to digital printing processes. In the late days of the analog era it was said of printing:

## Digital era

Today's digital printing processes take much of the drudgery and poisonous substance exposure out of printing photographs.
# Modern Photography/Restoration

Restoration is the process of attempting to wind-back the ravages of time on a photograph.
Types of damage that photos (positives) and/or film (negatives) may incur include:
* Dust
* Folding
* Scratching
* Chemical staining due to oil, solvents, impurities in development chemicals, physical proximity to other materials, water, condensation, oxidization, etc.
* Loss of contrast
* Tearing
* Sticking to other surfaces
Most of these types of damage can be repaired to some extent, usually by digitizing the photo or negative (if it's not already digitized) and then using image editing software such as the Darktable or the GIMP (both free and open source software), Lightroom or Photoshop (both commercial products from Adobe) or similar software packages.

## Strategies

Some of the strategies or general approaches that may be taken to restoration include the following.

### Cloning

Cloning is the act of duplicating the subject, color, texture or other visual properties from one part of an image to another. In restoration, this is usually used to mask out damaged areas such as dust, folding, staining or scratches.

### Obfuscation

Obfuscation is the act of reducing the apparent visual presence or impact of part of an image. This may be achieved, for instance, by reducing its contrast, lightening or darkening it, or changing its color to a more neutral tone.

### Contrast restoration

Faded images may often be significantly enhanced by restoring their original contrast, or even digitally enhancing contrast beyond what was probably originally there. This process is generally very easy to achieve with photo manipulation software, either through a dedicated image contrast function (generally preserving the existing color balance of the image) or through finer grained control over individual color channels such as red, green and blue.

### Cropping

Cropping or outright removing parts of an image is sometimes the fastest and cheapest way to restore an image to a higher apparent overall quality, and can be particularly useful if significant damage has occurred at the edges or corners of an image, for example through a bent or dog-eared corner. Unfortunately, it is not always possible since it relies upon no other significant elements within the image being similarly close to the cropped-away edge. For example, in a portrait it is generally unacceptable to cut away part of the subject's head.