OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/Algebraic Riccati Equation

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\\end{aligned}}$

## The Data

Following matrices are needed as Inputs:.
    $A,B,N$.

## The Optimization Problem

In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

## The LMI: Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.
       P>0, Q>0, R>0. The Algebraic Riccati inequality is given byA$^T$P+PA−(PB+N$^T$)R$^−$1(B$^T$P+N)+Q≥0 can be written using the Schur complement lemma as[A$^T$P+PA+QPB+N$^T$⋆R]≥0{\textstyle {\begin{aligned}P>0, \\Q>0, \\R>0.\\{\text{ The Algebraic Riccati inequality is given by}}\;\\A^{T}P+PA-(PB+N^{T})R^{-}1(B^{T}P+N)+Q&\geq 0\\\\{\text{ can be written using the Schur complement lemma as}}\;\\{\begin{bmatrix}A^{T}P+PA+Q&&PB+N^{T}\\\star &&R\\\end{bmatrix}}&\geq 0\end{aligned}}}

## Conclusion:

If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

## Implementation:

Matlab code for this LMI in the Github repository:
* REDIRECT- CODE

## External links

* -Optimal Solution to Matrix Riccati Equation
* https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory.- - A List of LMIs by Ryan Caverly and James Forbes
# LMIs in Control/pages/Algebraic Riccati Inequality

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccati Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\\end{aligned}}$

## The Data

Following matrices are needed as Inputs:.
    $A,B,N$.

## The Optimization Problem

In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

## The LMI: Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.
       P>0, Q>0, R>0. The Algebraic Riccati inequality is given byA$^T$P+PA−(PB+N$^T$)R$^−$1(B$^T$P+N)+Q≥0 can be written using the Schur complement lemma as[A$^T$P+PA+QPB+N$^T$⋆R]≥0{\textstyle {\begin{aligned}P>0, \\Q>0, \\R>0.\\{\text{ The Algebraic Riccati inequality is given by}}\;\\A^{T}P+PA-(PB+N^{T})R^{-}1(B^{T}P+N)+Q&\geq 0\\\\{\text{ can be written using the Schur complement lemma as}}\;\\{\begin{bmatrix}A^{T}P+PA+Q&&PB+N^{T}\\\star &&R\\\end{bmatrix}}&\geq 0\end{aligned}}}

## Conclusion:

If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

## Implementation:

Matlab code for this LMI in the Github repository:
* REDIRECT
# On 2D Inverse Problems/On the inverse problem of Calderon

The following inverse problem stated by Calderon has many potential practical applications and had received a lot of attention in the past decades, see [Uh]. This is the problem of recovering the conductivity of a body from its Dirichlet-to-Neumann operator.
Given a domain with positive measurable function on it, the Dirichlet-to-Neumann operator connects the Dirichlet and Neumann boundary values of $\gamma$-harmonic functions in $H^{1}(\Omega )$, defined on the domain. It is a pseudo-differential operator of order 1.
    $\Lambda :H^{1/2}(\partial \Omega )\rightarrow H^{-1/2}(\partial \Omega )$
One can only recover one number (an integral of reciprocal of conductivity) in 1D. The problem is overdetermined in dimensions higher than 2. It was settled in these cases @ [KV] & [SU].
The dimensions of the measurement parameter and the unknown one fit precisely in the case of 2D. It was recently proved that the Dirichlet-to-Neumann map uniquely determines the conductivity in a 2D simply connected bounded domain if the conductivity is in a weighted space including differentiable functions in [N], and if it's measurable and bounded from 0 and infinity in [AP].
    $0<1/c<\gamma <c<\infty .$
In the 2D case one can make sense of the operator for measurable conductivities (vs. differentiable), (see [AP]), using the Hilbert transform, defined later in the book, b/w the boundary values of harmonic functions and their conjugates.
In this book the problem is also stated for the planar electrical networks, that allows a different approach for solution of the inverse problem through discretization technique, that was implemented by Druskin et al. @ Schlumberger-Doll Research, see [BDK] and [BDMV].
# On 2D Inverse Problems/Overview

The inverse problems, that are the main objects of this book, have their roots in the study of the relationships between local and global properties of the embedded graphs and surfaces. The two-dimensional setting has an important structure, due to the duality b/w the harmonic conjugate functions and the connection to classical special matrices, that can be viewed as two-dimensional objects. Also, the dimension of the unknown parameter fits the dimension of the measurement data in several important instances of the inverse problems.
The solutions of the inverse problems are presented in a way bridging the discrete and continuous settings.
The techniques of solutions are motivated by the following commutative diagram:
The context of the inverse problems provides a unified point of view on the work of many great mathematicians, and even though there are references to many areas of mathematics in this book, it is practically self-contained, and is intended for a wide audience of readers interested in the subject. There are exercises, ranked by difficulty, through most of the chapters of the book, and it can be used as a textbook for an applied mathematics class.
# On 2D Inverse Problems/Pick-Nevanlinna interpolation

Certain questions about the layered inverse problems can be reduced to the Pick-Nevanlinna interpolation problem: given the values of a function at specific points of the domains D or C$^+$, find its analytic continuation to an automorphism of the domain.
More formally, if z$_1$, ..., z$_N$ and w$_1$, ..., w$_N$ are collections of points in the unit disc or the complex right half-plane, one seeks an analytic function f defined in the whole domain, such that
    $f:\mathbb {D} \to \mathbb {D} {\mbox{ or }}f:\mathbb {C} ^{+}\to \mathbb {C} ^{+}$,
and
    $f(z_{k})=w_{k},k=1,2,\dots ,N.$
The function f can be chosen a rational Stieltjes continued fraction or the Blaschke product, depending on the domain in the problem. The interpolating function exists, see [M], if and only if the matrices
    $\left({\frac {1-w_{k}{\overline {w_{l}}}}{1-z_{k}{\overline {z_{l}}}}}\right)_{k,l=1}^{N}{\mbox{ and }}\left({\frac {w_{k}+w_{l}}{z_{k}+z_{l}}}\right)_{k,l=1}^{N}$
are positive semi-definite, respectively. The interpolation function is unique if and only if the corresponding matrix is singular. If the matrix is not singular, then there're infinitely many interpolating continued fractions w/the number of levels larger than N.
Since the corresponding networks have equal Dirichle-to-Neumann operators, any pair of such networks can be transformed one to another by a finite sequence of Y-Δ moves. The intermidiate graphs do not have rotation symmetry, which provides an example of symmetry breaking.
Exercise (**). Using the solution for the Pick-Nevanlinna interpolation problem, find an algorithm for calculating the coefficients of the Stieltjes continued fraction from the interpolation data.
# On 2D Inverse Problems/Printed Version/About the book

for Nicole Delaittre
Preface
This book is about the inverse problems that take its roots in medical imaging such as electrical impedance tomography (EIT) and similar imaging methods from geophysics with applications to the oil & gas production industry. The study was motivated by the needs of non-destructive and non-intrusive methods for imaging of hidden objects such as internal human/animal organs or Earth's natural resources.
The inverse problems are formulated as the mathematical problems of recovering the coefficients of functional and differential systems of equations from data about their solutions. These problems are opposite in some sense to the forward problems of evaluating functions. The study of the inverse problems is well suited for computer simulations and many classical and current mathematical problems can be restated with ease as inverse problems.
The techniques of study and solutions of the inverse problems considered in this book allow one to "see inside" the objects using data about the sound waves and electro-magnetic fields observed at the boundary or outside the object.
Copyright © 2013 by David V. Ingerman
All rights reserved.
# On 2D Inverse Problems/Riemann mapping theorem

    Exercise (*). Proof that $K_{U}(\phi ,\theta )=|f'(\phi )|K_{D}(f(\phi ),f(\theta ))|f'(\theta )|$ off the diagonal.
    Exercise (**). Derive the formula for the kernel of the DN map for the unit disc D: $K_{D}(\phi ,\theta )={\frac {-1}{\pi (1-cos(\phi -\theta ))}}$ w/the kernel formula for the half-plane and the radial derivative of the Poisson kernel,solving Dirichlet boundary problem on the disc:
$K_{P}(z,\theta )={\frac {1-|z|^{2}}{2\pi |1-ze^{-i\theta }|^{2}}}.$
$\Lambda _{kl}={\begin{cases}{\frac {2n(n+1)}{3}},{\mbox{ }}k=l,\\{\frac {-1}{1-\cos {\frac {2\pi (k-l)}{2n+1}}}},{\mbox{ }}k\neq l,\end{cases}}$ where n is a natural number and k,l = 1,2, ... 2n+1.
    Exercise (**). Prove that the eigenvalues of the matrix are natural numbers(!) $2n,4n-2,6n-6,\ldots ,n(n+1)$ w/multiplicity two and 0 w/multiplicity one.
# On 2D Inverse Problems/On random processes

In this section the endpoints of edges of a considered graph can have an order, making it a directed graph. A graph with positive function/vector defined on its edges is called weighted graph.
Random walk of a particle on a graph G with discrete time is the following process:
* At moment t = 0 the particle occupies a vertex v of G.
* At moment t = n+1 the particle moves to a neighbor of its position at the moment t = n w/probability proportional to the weight of the edge connecting/pointing to the neighbor.
Choosing a subset of vertices of a graph as boundary, the harmonic measure of a subset S of the boundary is the function/vector on vertices of G that equals the probability that a particle, starting its random walk at a vertex p, occupies a boundary vertex in the set S before a boundary vertex not in S.
It follows from the definition that the harmonic measure at p of a single boundary node b equals to the sum over the finite paths through interior from p to b:
    $u_{b}(p)=\sum _{p{\xrightarrow[]{path}}b}(\prod _{e\in path}w(e)/\prod _{q\in (path-\{b\})}\sum _{q\rightarrow r}w(qr)),$
or
    $u_{b}(p)=\sum _{p{\xrightarrow[]{path}}b}\prod _{e=(qr)\in path}{\frac {w(e)}{\sum _{q\rightarrow r}w(qr)}}.$
Note, that an edge or a vertex may appear multiple times in a path.
The Brownian motion is a continuous/limiting analog of the random walk. It follows from the averaging property of the operator that the hitting probabilities of a particle under Brownian motion are described by harmonic functions, defined in the previous section. The harmonic functions are conformaly invariant.
# On 2D Inverse Problems/On inhomogeneous string of Krein

    ${\frac {1}{\rho (x)}}{\frac {\partial ^{2}f(x,\lambda )}{\partial x^{2}}}=\lambda f(x,\lambda ),$
where $\rho (x)={\frac {dm}{dx}}$ is the density of the string, possibly including atomic masses. One can express the coefficient in terms of the fundamental solution of the ODE:
    $H(\lambda )={\frac {f'(0,\lambda )}{f(0,\lambda )}},$
where, $f(l,\lambda )=0.$
    $\beta (\lambda )=\lambda H(-\lambda ^{2})$
is an analytic automorphism of the right half-plane $C^{+}$, that is positive on the real positive ray. The Herglotz theorem completely characterizes such functions by the following integral representation:
$\beta (\lambda )=\sigma _{\infty }\lambda +{\frac {\sigma _{0}}{\lambda }}+\int _{0}^{\infty }{\frac {\lambda (1+x^{2})d\sigma (x)}{\lambda ^{2}+x^{2}}},$
where,
$\sigma$ is positive measure of bounded variation on the closed positive ray $(0,\infty )$.
    Exercise(**). Use the theorem above, change of variables and the Fourier transform to characterize the set of Dirichlet-to-Neumann maps for a disc w/rotationally invariant conductivity.
# MIRC Scripting/Info/Manual of Style

This manual of style will be undergoing major work as the wikibook is belongs to is being constructed.

## Basic Wikibook Style

* The main format of the page hierarchy this book looks like this:
   * Title Page
      * Section
         * Chapter
            * Subpage (If any)
* Every chapter/subpage should contain sections. The main idea of a section should be a first-level header. A sub-topic should be a second-level header. An offshoot of the sub-topic should be a third-level. There should be no need to go beyond a third-level.
* Every content page MUST use the {{MIRC_Scripting/header}} template with the proper parameters and it must be the first item on the page. A content page is a page listed in, you guessed it, THE TABLE OF CONTENTS! This means do not use the header template on Images, Templates, pages in the section /Info/, or any other page that ISN'T a chapter of the book.
* To make sure you don't break Firefox browsers, use : in place of * when there's multiple bullets. e.g. Instead of **** use :::*, instead of ** use :*, and so on.
* For code examples, use a table, like so:
* <table><tr><td>
* <syntaxhighlight lang="langname">
* code goes here
* </syntaxhighlight>
* </td></tr></table>
* For code in the middle of a sentence or commands for a user to type, use <tt>
* For continuity's sake, most images should go on the right, because most of them are already over there, and it looks better.
* Consult the main author BEFORE making massive changes, or any changes to a template/image

## Development Stages Style

* Please don't forget |devstage= in the {{MIRC_Scripting/header}} template. Also, make sure that the dev stage on the bookshelf, total progress for the book header, and each ToC entry is up to date.

## Page Style

* All pages not images or templates should have [[Category:MIRC Scripting]] on them.

## Image Style

* All images need [[Category:MIRC Scripting Images]] on their page.

## Template Style

* All templates need a usage section. It should be inside <pre> tags and should explain the syntax and usage of the template.
* All templates also need [[Category:MIRC_Scripting_Templates]] on them.

## Author Page Style

* Nothing on the list of authors should be changed other than adding {{MIRC_Scripting/Author}} templates. The only other data besides an Author template should be the special thanks.
* If a field does not apply to you, leave it alone.

## To-Do Page Style

* Please sign things to do with the four tildes (~~~~).

## Use Common Sense

Even if something might go against the Manual of Style, this is not written in stone, and not everything is covered in here. As long as it looks alright and is helpful the book it will most likely be fine.
# MIRC Scripting

mIRC Scripting is a built-in, interpreted scripting language for the mIRC IRC client for Windows. It is an easy-to-use and flexible script for uses ranging from automating and simplifying IRC tasks to making mini-programs such as file servers and away systems. Some people even utilize scripts to completely skin and program the IRC interface. These are usually called "full scripts," while individual, smaller scripting projects are known as "remotes" or "addons," or "snippets" for really small scripts. This wikibook will attempt to explain all the aspects of mIRC Scripting, starting from the very basic parts of the language for beginners, and delving into the more advanced topic for experts.

## Table of Contents

If you're looking for something specific in this book, dive into the table of contents and find what you're looking for.
1. Starting From Scratch
   * Become Familiar with mIRC
   * Getting Started
   ** Types of scripts (alias, popups, and remotes)
   ** Raw and CTCP Events
   ** Identifiers and Variables
   ** Comments in Code
   ** Command Prefixes and Quiet Commands
   ** Control Codes
2. Learning the Ropes
   * Branching and Loops
   ** If-Then-Else
   ** While Loops
   ** Goto Loops
   ** Halt, Break, Continue, and Return
   * Timers
   * User Levels
   * Text Matching
   ** Wilcards
   ** Regular Expressions
   * File Handling
   ** Regular text files
   ** .INI Files
   * Hash Tables
   * Dialogs
   ** Non-Modal Dialogs
   ** Modal Dialogs
   ** Dialog Identifiers/Commands
   ** Dialog Events
   ** Dialog Studio
3. Advanced Subjects
   * Agents
   * Sockets
   * Custom Windows
   * Planned advanced topics: DLLs, COM Objects, and Error Checking and Handling.

## Wikibook information

* Manual of Style
* List of Authors
* To-Do Page
# MIRC Scripting/BeyondBasics/Text Matching

on TEXT
The on TEXT event triggers when you receive private and/or channel messages.
Format:
on <level>:TEXT:<matchtext>:<*><?><#[,#]>:<commands>
Example:
on 1:TEXT:*help*:#mirc,#irchelp:/msg $nick what is the problem?
The on ACTION and on NOTICE events use exactly the same format as on TEXT, and trigger on an action and on a notice event respectively.
The match text can be a wildcard string, where:
The match text can also be a regular expression. See the $ prefix section in Access Levels.
The location where this event occurs can be specified using:
Examples:
on 1:TEXT:hello*:#:/msg $chan Welcome to $chan $nick!
This listens on any channel for any line beginning with the word hello and welcomes the user who said it to the channel.
on 1:TEXT:*cookie*:#food:/describe $chan gives $nick a cookie :)
This listens on channel #food for any message containing the word cookie and gives the user who said it a cookie.
on 1:ACTION:moo:#:/msg $chan Aha, I see we have a cow among us.
This listens on any channel for an action that contains the word moo and responds accordingly.
on 1:NOTICE:*:?:/msg $nick I am AFK, back in a moment!
This listens for any private notice and responds with the message that you are away from the keyboard.
For more flexibility, you can also use Variables in place of both the matchtext and the channel parameters.
on 1:TEXT:%matchtext:%channel:/msg $nick You just said $1- on channel %channel
The value of %matchtext will be matched against whatever text the user sends, and the value of %channel will be matched against the channel to which the message was sent.
Note: You cannot test out these events by typing text to yourself. They can only be initiated by someone else saying something in a channel or in a private message.
# MIRC Scripting/Info/Authors
## List of Authors

These people contributed a lot to the writing of the Wikibook and have their contributions listed here.

## Special Thanks

These people didn't make major edits to the book, but still made valuable contributions to the writing process.
* Mike.lifeguard was a huge help to me as far as wiki markup, espeially Templates and categories, how to do things, and overall support.
* Darklama helped me out with my templates.
# Physics Study Guide/Authors

| Karl Wick | Adon Metcalfe | Brendan Abbott | Tristan Sabel | Fromund Hock | Martin Hoecker-Martinez |
# Muggles' Guide to Harry Potter/Places/Surrey Zoo

    "I suppose we could take him to the zoo, ... and leave him in the car...."
    — Petunia Dursley's reluctance to take Harry to the zoo.
Vernon Dursley and Petunia Dursley took their son Dudley Dursley to this zoo for his eleventh birthday on 23 June 1991, along with Harry.
Appearances
* Harry Potter and the Philosopher's Stone (First appearance)
* Harry Potter and the Chamber of Secrets (Mentioned only)
# Muggles' Guide to Harry Potter/Magic/Pogrebin
## 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/Portus
## Overview

A Portus spell is used to create a Portkey.

## Extended Description

A Portkey is a magical means of transportation between two specific points, potentially at a specific time. The spell Portus is used to create a portkey; the visible effect is that the object being charmed glows blue and vibrates briefly.
Professor Dumbledore creates a portkey on two occasions in Harry Potter and the Order of the Phoenix: when Mr. Weasley is injured, he creates a Portkey to carry the Weasley children and Harry to Number 12 Grimmauld Place so that they don't have to deal with Professor Umbridge; and after the battle at the Ministry, he creates one to carry Harry back to his office. In this latter case, Cornelius Fudge, the Minister for Magic, is present, and is scandalized that Dumbledore would dare to produce an unauthorized Portkey.

## Analysis

The Portus spell must be complex indeed, quite apart from the magic involved. We can safely assume that, like the Riddikulus spell and the Patronus charm, a significant part of the spell is actually derived from what you are thinking of at the time the spell is cast. There are a significant number of things that you must think of when casting Portus. Not only must you hold in your mind the destination that you desire, you must also decide whether the Portkey is to be a there-and-return key, in which case you must presumably also hold the return destination in mind; additionally you must set the time when the portkey is to activate, or whether it is to activate upon command or upon contact. It may be possible to make a personal Portkey as well, in which case you must know whose contact will activate it.

## 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/Prior Incantato
## Overview

The Prior Incantato charm is used to reveal the last spell cast by a wand. Once the spell cast has been determined, the Deletrius charm can be used to end the display. The effects are similar to the Priori Incantatem wand effect, but not entirely related.

## Extended Description

In Harry Potter and the Goblet of Fire, the House Elf Winky is found Stunned, holding a wand, shortly after the Dark Mark is cast. The Prior Incantato charm is used to determine that it was, in fact, the wand that was found with Winky that was used to cast the Dark Mark.
In Harry Potter and the Deathly Hallows, Hermione's wand is captured at Malfoy Manor. Harry assumes that, by means of the Prior Incantato charm, the Death Eaters there will be able to determine that Hermione had tried to repair Harry's wand, and had failed. Remote as this possibility may be, Harry decides that he must proceed as if Voldemort is aware that his wand has been destroyed.

## 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/Porlock
## Overview

Porlocks are magical beasts that guard horses. They are mistrustful of humans.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.