OpenLLM-France/wikimedia huggingface dataset:
# Magar/Akkha Script

The Akkha Script or AkkhaRika Script (also called Magar Akkha) is used to write the Magar language. It is a Brahmic type abugida.
# Magar/Animals

* Ant- Mhār🐜
* Cat- Suthū🐈
* Cow- Hyant🐄
* Dog- Chiū🐕
* Fish- Disyā🐟
* Goat- Rhā🐐
* HouseFly- Jhinghā
* Pig- Wāk 🐖
* Rat - Byū🐀
* Ox - Bārdā/Thor🐂
# Magar/Colours

""Colours Name""
* Black - Chicha/Chicklingcha⚫
* White - Bocha⚪
* Red - Gyācha🔴
* Green- Ficha🍏
* Yellow- Wórcha💛
# Magar/Conversation I

    Meeting [भेटघाट]
English- Good Morning[Greetings]/[Namaste]
    Magar- Jhorle [झोर्ले]
English- Hello, How are you?
    Magar- Nāko Sehema/Gepma na le.[नाको सेहेम/गेप्म न ले?]
English- I'm fine & u?
    Magar- Ngā Sehema/Gepma na le. Nākoke kunchà le? [ङा सेहेम/(गेप्म) न ले, नाको के कून्च ले?]
English- I'm Fine.
    Magar- Ngā Sehema(Jherma) na le. [ङा सेहेम(झेर्म) न‌ ले।]
English- I'm glad to see u.
    Magar- Nāko kathā dupnāng dhaling marhāngä. [ना कोकठ डूप्ना दुप्नाङ ढलीङ मर्हाङा]
English- Me too
    Magar- Ngāke ra dhaling marhngä. [ङाकेर ढलीङ मर्हाङा]
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Eigen Value Minimization

LMI for Minimizing Eigenvalue of a Matrix
Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

## The System

Assume that we have a matrix function of variables $x$:
${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$
where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

## The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

## The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ to minimize the following cost function:
${\begin{aligned}J(x)=\lambda _{\text{max}}(A(x))\end{aligned}}$
where $J(x)$ is the cost function and $\lambda _{\text{max}}(.)$ indicates the maximim eigenvalue of a matrix.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent
${\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}$
where $t$ is defined as the maximim eigenvalue of the matrix $A$.

## The LMI: LMI for eigenvalue minimization

This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)-tI\leq 0\end{aligned}}$

## Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem.
Moreover, we obtain the maximum eigenvalue, $t$, of the matrix $A(x)$.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

## Related LMIs

LMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix
LMI for Matrix Positivity

## External Links

* - LMI in Control Systems Analysis, Design and Applications
* State-space Representation of a System
* Eigenvalues and Eigenvectors of a Matrix

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Norm Minimization

LMI for Matrix Norm Minimization
This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an $H_{2}$ or an $H_{\infty }$ optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.

## The System

Assume that we have a matrix function of variables $x$:
${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$
where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

## The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

## The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ in order to minimize the following cost function:
${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$
where $J(x)$ is the cost function and $||.||_{2}$ indicates the norm of the matrix function $A$.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:
${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## The LMI: LMI for matrix norm minimization

This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
${\begin{aligned}&{\text{min}}\quad t&\\&{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem and we obtain $t$ that is the norm of matrix function $A(x)$.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

## Related LMIs

LMI for Matrix Norm Minimization
LMI for Generalized Eigenvalue Problem
LMI for Maximum Singular Value of a Complex Matrix
LMI for Matrix Positivity

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Minimizing Condition Number of Positive Definite Matrix

LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Minimizing Condition Number of Positive Definite Matrix

## The System:

A related problem is minimizing the condition number of a positive-definite matrix $M$ that depends affinely on the variable $x$, subject to the LMI constraint $F(x)$ > 0. This problem can be reformulated as the GEVP.

## The Optimization Problem:

The GEVP can be formulated as follows:
minimize $\gamma$
subject to $F(x)$ > 0;
$\mu$>0;
$\mu I$< $M(x)$ < $\gamma \mu I$.
We can reformulate this GEVP as an EVP as follows. Suppose,
$M(x)$= $M_{0}$ +$\sum _{n=1}^{m}$$x_{i}M_{i}$ , $F(x)$= $F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$

## The LMI:

Defining the new variables $\nu$=$1/\mu$ , ${\tilde {x}}$=$x/\mu$ we can express the previous formulation as the EVP with variables ${\tilde {x}},\nu$ and $\gamma$:
miminize$\gamma$
subject to $\nu F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$ >0; $I$ < $\nu M_{0}$+ $\sum _{n=1}^{m}$$x_{i}M_{i}$ < $\gamma I$

## Conclusion:

The LMI is feasible.

## References

* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Feasibility Problem

LMI for Feasibility Problem in Optimization
The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

## The System

Assume that we have two matrices as follows:
${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$
${\begin{aligned}B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$
which are matrix functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$.

## The Data

Suppose that the matrices ${\begin{aligned}A_{0},A_{1},...,A_{n}\quad \end{aligned}}$ and ${\begin{aligned}B_{0},B_{1},...,B_{n}\quad \end{aligned}}$ are given.

## The Optimization Problem

The optimization problem is to find variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that the following constraint is satisfied:
${\begin{aligned}A(x)<B(x)\end{aligned}}$

## The LMI: LMI for Feasibility Problem

This optimization problem can be converted to a standard LMI problem by adding a slack variable, $t$.
The mathematical description for this problem is to minimize $t$ in the following form of the LMI formulation:
${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)<B(x)+tI\\\end{aligned}}$

## Conclusion:

In this problem, $x$ and $t$ are decision variables of the LMI problem.
As a result, these variables are determined in the optimization problem such that the minimum value of $t$ is found while the inequality constraint is satisfied.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

## Related LMIs

LMI for Linear Programming

## External Links

* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Generalized eigenvalue problem

LMI for Generalized Eigenvalue Problem
Technically, the generalized eigenvalue problem considers two matrices, like $A$ and $B$, to find the generalized eigenvector, $x$, and eigenvalues, $\lambda$, that satisfies $Ax=\lambda Bx$. If the matrix $B$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

## The System

Assume that we have three matrice functions which are functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$ as follows:
$A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}$
$B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}$
$C(x)=C_{0}+C_{1}x_{1}+...+C_{n}x_{n}$
where are $A_{i}$, $B_{i}$, and $C_{i}$ ($i=1,2,...,n$) are the coefficient matrices.

## The Data

The $A(x)$, $B(x)$, and $C(x)$ are matrix functions of appropriate dimensions which are all linear in the variable $x$ and $A_{i}$, $B_{i}$, $C_{i}$ are given matrix coefficients.

## The Optimization Problem

The problem is to find ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that:
$A(x)<\lambda B(x)$, $B(x)>0$, and $C(x)<0$ are satisfied and $\lambda$ is a scalar variable.

## The LMI: LMI for Schur stabilization

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:
${\begin{aligned}&{\text{min}}\quad \lambda \\&{\text{s.t.}}\quad A(x)<\lambda B(x)\\&\quad \quad B(x)>0\\&\quad \quad C(x)<0\end{aligned}}$

## Conclusion:

The solution for this LMI problem is the values of variables $x$ such that the scalar parameter, $\lambda$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability

## Related LMIs

LMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix

## External Links

* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Generalized Eigenvalue Problem

LMI for Generalized Eigenvalue Problem
Technically, the generalized eigenvalue problem considers two matrices, like $A$ and $B$, to find the generalized eigenvector, $x$, and eigenvalues, $\lambda$, that satisfies $Ax=\lambda Bx$. If the matrix $B$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

## The System

Assume that we have three matrice functions which are functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$ as follows:
$A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}$
$B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}$
$C(x)=C_{0}+C_{1}x_{1}+...+C_{n}x_{n}$
where are $A_{i}$, $B_{i}$, and $C_{i}$ ($i=1,2,...,n$) are the coefficient matrices.

## The Data

The $A(x)$, $B(x)$, and $C(x)$ are matrix functions of appropriate dimensions which are all linear in the variable $x$ and $A_{i}$, $B_{i}$, $C_{i}$ are given matrix coefficients.

## The Optimization Problem

The problem is to find ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that:
$A(x)<\lambda B(x)$, $B(x)>0$, and $C(x)<0$ are satisfied and $\lambda$ is a scalar variable.

## The LMI: LMI for Schur stabilization

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:
${\begin{aligned}&{\text{min}}\quad \lambda \\&{\text{s.t.}}\quad A(x)<\lambda B(x)\\&\quad \quad B(x)>0\\&\quad \quad C(x)<0\end{aligned}}$

## Conclusion:

The solution for this LMI problem is the values of variables $x$ such that the scalar parameter, $\lambda$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability

## Related LMIs

LMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix

## External Links

* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Linear Programming

LMI for Linear Programming
Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

## The System

We define the objective function as:
$c_{1}x_{1}+c_{2}x_{2}+...c_{n}x_{n}$
and constraints of the problem as:
$a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}<b_{1}$
$a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}<b_{2}$
$a_{m1}x_{1}+a_{m2}x_{2}+...+a_{mn}x_{n}<b_{m}$

## The Data

Suppose that $c_{j}\in \mathbb {R} ^{n}$, $a_{ij}\in \mathbb {R} ^{n}$, and $b_{i}\in \mathbb {R} ^{m}$ are given parameters where $i=1,2,...,m$ and $j=1,2,...,n$. Moreover, $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}$ is an $n\times 1$ vector of positive variables.

## The Optimization Problem

The optimization problem is to minimize the objective function, $c^{\text{T}}x$ when the aforementioned linear constraints are satisfied.

## The LMI: LMI for linear programming

The mathematical description of the optimization problem can be readily written in the following LMI formulation:
${\begin{aligned}&{\text{min}}\quad c^{\text{T}}x&\\&{\text{s.t.}}\quad a_{i}^{\text{T}}x\leq b_{i}x\\\end{aligned}}$

## Conclusion:

Solving this problem results in the values of variables $x$ which minimize the objective function. It is also worthwhile to note that if $m\geq n$, the computational cost for solving this problem would be $mn^{2}$.
There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Linear-Programming

## Related LMIs

LMI for Feasibility Problem

## External Links

* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Generalized H2 Norm
## Generalized $H_{2}$ Norm

The $H_{2}$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $D$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $G$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

## The System

Consider a continuous-time, linear, time-invariant system $G:L_{2e}\to L_{2e}$ with state space realization $(A,B,C,0)$ where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, amd $A$ is Hurwitz. The generalized $H_{2}$ norm of $G$ is:
$\left\|G\right\|_{2,{\infty }}=\sup _{u\in L_{2},{u}\neq {\text{zero}}}$ ${\frac {\left\|Gu\right\|_{\infty }}{\ \left\|u\right\|_{2}}}$

## The Data

The transfer function $G$, and system matrices $A$, $B$, $C$ are known and $A$ is Hurwitz.

## The LMI: Generalized $H_{2}$ Norm LMIs

The inequality $\left\|G\right\|_{2,{\infty }}<\mu$ holds under the following conditions:
1. There exists $P\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:
       ${\begin{bmatrix}A^{T}P+PA&PB\\*&-\mu 1\end{bmatrix}}<0$.
       ${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.
2. There exists $Q\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $Q>0$ such that:
       ${\begin{bmatrix}QA^{T}+AQ&B\\*&-\mu 1\end{bmatrix}}<0$.
       ${\begin{bmatrix}Q&QC^{T}\\*&\mu 1\end{bmatrix}}>0$.
3. There exists $P\in \mathbb {S} ^{n},V\in \mathbb {R} ^{n\times n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:
       ${\begin{bmatrix}-(V+V^{T})&V^{T}A+P&V^{T}B&V^{T}\\*&-P&0&0\\*&**-\mu 1&0\\*&*&*&-P\end{bmatrix}}<0$.
       ${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.

## Conclusion:

The generalized $H_{2}$ norm of $G$ is the minimum value of $\mu \in \mathbb {R} _{>0}$ that satisfies the LMIs presented in this page.

## Implementation

This implementation requires Yalmip and Sedumi.
Generalized $H_{2}$ Norm - MATLAB code for Generalized $H_{2}$ Norm.

## Related LMIs

LMI for System H_{2} Norm

## External Links

* 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 Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

## Return to Main Page:

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Iterative Convex Overbounding

Iterative convex overbounding is a technique based on Young’s relation that is useful when solving an optimization problem with a BMI constraint.

## The System

Consider the matrices $Q=Q^{T}\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},R\in \mathbb {R} ^{m\times p},D\in \mathbb {R} ^{p\times q},S\in \mathbb {R} ^{q\times r}andC\in \mathbb {R} ^{r\times n}$, where S and R are design variables in the BMI given by
    ${\begin{aligned}\qquad Q+BRDSC+C^{T}S^{T}D^{T}R^{T}B^{T}<0\qquad (1)\\\end{aligned}}$

## The LMI:Iterative Convex Overbounding

Suppose that S$_0$ and R$_0$ are known to satisfy (1). The BMI of (1) is implied by the LMI
       ${\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&B(R-R0)U&C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\*&W^{\text{-1}}&0\\*&*&-W\\\end{bmatrix}}<0\qquad (2)$
where Φ(R,S) = B(RDS$_0$+R$_0$DS-R$_0$DS$_0$)C, W>0 is an arbitrary matrix, D=UV, and the matrices U and V$^T$ have full column rank. The LMI of (2) is equivalent to the BMI of (1) when R = R$_0$ and S = S$_0$, and is therefore non-conservative for values of R and S and are close to the previously known solutions R$_0$ and S$_0$.
Alternatively, the BMI of (1) is implied by the LMI
       ${\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\*&Z\\\end{bmatrix}}<0\qquad (3)$
where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and V$^T$ have full column rank. Again, the LMI of (4) is equivalent to the BMI of (2) when R = R$_0$ and S = S$_0$, and is therefore non-conservative for values of R and S and are close to the previously known solutions R$_0$ and S$_0$.

## Conclusion:

A benefit of convex overbounding compared to a linearization approach, is that in addition to ensuring conservatism or error is reduced in the neighborhood of R = R$_0$ and S = S$_0$, the LMIs of (2) and (3) imply (1). Iterative convex overbounding is particularly useful when used to solve an optimization problem with BMI constraints.

## 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.
* Young’s Relation
# Introduction

This Appendix contains the top 30 mistakes people ask when using Mac hardware and software. It's a good idea to read through this section before you use your Mac, as it will clarify some of the things you are likely to run into at first.

# The Mistakes
## Windows switcher mistakes

* Closing the Red Dot on the top left and thinking the program has quit like it might have in Windows.
* People using the control button on the keyboard like they would on a Windows operating system instead of Apple's alternative control button which is marked by the apple logo and is often referred to as the command button.

## Hardware Mistakes

* In Macintosh computers, the "Motherboard" is referred to as the "Logic Board"

## Software Mistakes

* Running programs directly from disk images without dragging them to the Applications folder. The disk image is just a virtual disk created by the .dmg, the program inside should be copied to your computer so you can eject the disk image and trash the .dmg file.
* The Dock only has links to files (with the exception of Trash), the actual Applications are located in your Applications folder

## Changes from Previous Versions

* Spotlight isn't just good for finding lost files, it is often a quicker way to open Applications and documents when compared to using Finder to find the item.
# Introduction

While we traditionally interact with our computers with a keyboard and mouse, Mac OS X supports two other kinds of input device: a microphone and a tablet.

# Speech
## Speech Recognition

Operating a computer with your voice may seem like something out of a futuristic sci-fi movie. But the future is now, and Mac OS X lets you control certain functions with your voice.

# Ink

"Ink" is the name of Apple's handwriting recognition technology, borrowed from the now-defunct Newton PDA. Now that it's been applied to Mac OS X, not only can you use your tablet to draw, but also to write.