text:
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{\Large EPI scaling} \\
Sean Bittner \\
September 17, 2020 \\
\end{center}

\section{Introduction}
A relative strength of EPI is its scalability in parameter dimension $|z|$.
This write-up compares the performance of EPI to SMC-ABC and SNPE as $|z|$ is increased.
The application is to condition arbitrary rank-2 RNNs on a regime of stable amplification.

\textbf{SMC-ABC}:
Traditional approaches to likelihood-free inference -- approximate Bayesian computation (ABC) methods -- randomly sample parameters $z$ until a suitable set is obtained.
State-of-the-art ABC methods leverage sequential monte-carlo (SMC) sampling techniques to obtain parameter sets more efficiently.
To obtain more parameter samples, SMC-ABC must be run from scratch again.
ABC methods do not confer log probabilities of samples.

\textbf{SNPE}: Like EPI, sequential neural posterior estimation (SNPE) uses deep learning to produce flexible posterior approximations.
Like traditional Bayesian inference methods, SNPE conditions directly on the statistics of data.
This differs from EPI, where posteriors are conditioned on emergent properties (moment constraints on the posterior predictive distribution).
Peculiarities of SNPE (density estimation approach, two deep networks) make scaling in $z$ prohibitive.

\textbf{Rank-2 RNN}: The model we use is a rank-2 RNN with N neurons:
\begin{equation*}
\begin{split}
U = \begin{bmatrix} u_1 & u_2 \end{bmatrix}, V = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \\ J = UV^\top + g\chi \\
u_1, u_2, v_1, v_2 \in \left[-1, 1 \right]^N,~~ g = 0.01 \\
\tau \dot{r} = -r + Jr
\end{split}
\end{equation*}
From \cite{bondanelli2020coding}, we know necessary and sufficient conditions for RNNs to exhibit stable amplification.  These two conditions are on the maximal real eigenvalue of $J$ ($\text{real}(\lambda_1)$) and the maximal eigenvalue of $J^s = \frac{J + J\top}{2} (\lambda^s_1)$:
\[\text{real}(\lambda_1) < 1, \lambda^s_1 > 1\]

In our analysis, we seek to condition rank-2 networks of increasing size on a regime of stable amplification.
Networks with $\text{real}(\lambda_1) = 0.5 \pm 0.5$ and $\lambda_1^s = 1.5 \pm 0.5$ will yield moderate (quantifiable) amplification.

\underline{EPI emergent property}: EPI can naturally condition on this emergent property.
\[\mathbb{E}\begin{bmatrix} \text{real}(\lambda_1) \\ \lambda^s_1 \\ (\text{real}(\lambda_1)-0.5)^2 \\ (\lambda^s_1 -1.5)^2  \end{bmatrix} = \begin{bmatrix} 0.5 \\ 1.5 \\ 0.25^2 \\ 0.25^2 \end{bmatrix}\]

\underline{SNPE observation}: SNPE cannot condition on the variance of observations across posterior.  Thus, we condition on an observation $x_0$ located at the mean of our desired emergent property.
\[x_0 = \begin{bmatrix} \text{real}(\lambda_1) \\ \lambda^s_1 \end{bmatrix} = \begin{bmatrix} 0.5 \\ 1.5 \end{bmatrix} \]

\underline{SMC-ABC criteria}: ABC methods define tolerance $\epsilon$ and distance for observed data $x_0$.\\
\begin{center}
$\epsilon = 0.5$, $l2$ distance, and  $x_0 = \begin{bmatrix} \text{real}(\lambda_1) \\ \lambda^s_1 \end{bmatrix} = \begin{bmatrix} 0.5 \\ 1.5 \end{bmatrix}$
\end{center}

\section{Results}
SMC-ABC was run with \href{https://pyabc.readthedocs.io/en/latest/index.html}{pyabc} package.  SNPE was run with \href{http://www.mackelab.org/delfi/}{delfi} package.  EPI was run with revision branch of \href{https://github.com/cunningham-lab/epi/tree/revision}{epi} package.  All methods use a uniform prior form -1 to 1 for each parameter.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.5]{figs/epi_both.png} \\
\end{center}
\caption{\small EPI scales with $z$ to high dimensions.
Convergence definitions:
EPI (blue) - satisfies all moment constraints,
SNPE (orange)- produces at least 2/$n_{\text{train}}$ parameter samples are in the bounds of emergent property (mean +- 0.5),
and SMC-ABC (red) - 100 particles with $\epsilon < 0.5$ are produced.}
\end{figure}

\textbf{Figure 1 points}:
\begin{enumerate}
\item EPI is capable of scaling to high dimensional parameter spaces.
\item SMC has many hyperparameters, of which pyABC selects automatically by running some initial diagnostics upon initialization.
In concurrence with the literacture, SMC-ABC fails to converge around 25-30 dimensions, since it's proposal samples never get close enough to the target statistics.
\item We searched over many SNPE hyperparameter choices: $n_{\text{train}} \in [2,000, 10,000, 100,000]$ is the number of simulations run per training epoch, and $n_{\text{mades}} \in [2, 3]$ is the number of masked autoregressive density estimators in the deep parameter distribution architecture.
The greater $n_{\text{train}}$, the longer each epoch will take, but the more likely SNPE may converge during that epoch.
Greater $n_{\text{mades}}$ increases the flexibility of the deep parameter distribution of SNPE, but slows optimization.
For the timing plot, we show the fastest among all of these choices, and for the convergence plot, we show the best convergence among all of these choices.
During optimization, we used $n_{\text{atom}}$=100 atomic proposals as is recommended.
\end{enumerate}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.75]{figs/StableAmplification_convergence.pdf}
\caption{\small Emergent property fidelity.  Top row: Posterior predictive distributions of EPI (blue), SNPE (orange), and SMC-ABC (red).
Gray star indicates emergent property mean, and gray dashed lines indicate two standard deviations corresponding to the variance constraint.
For $N <= 6$ where SMC-ABC converges, samples are not diverse (path degeneracies).
For $N >= 25$, SNPE does not produce a posterior approximation yielding parameters with simulations near $x_0$.
Bottom row: simulations of network parameters resulting from each method ($\tau=100ms$).  Each trace corresponds to simulation of one $z$.
Almost all EPI networks, few SNPE networks, and all SMC-ABC networks exhibit stable amplification.
There is little variation in SMC-ABC network simulations.}
\end{center}
\end{figure}

\textbf{Figure 2 points}:
\begin{enumerate}
\item EPI consistently produces the same distribution posterior predictive independent of the dimensionality.
\item SMC produces a limited variety of parameters due to the nature of its proposal generation algorithm, yet all parameters obtained produce stable amplification.
\item SNPE's posterior predictive distribution is not necessarily close to the conditioning point, and is very dependent on dimensionality.
\end{enumerate}

\bibliography{epi_scaling}
\bibliographystyle{unsrt}
\end{document}

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text:
% THIS IS AN EXAMPLE DOCUMENT FOR VLDB 2012
% based on ACM SIGPROC-SP.TEX VERSION 2.7
% Modified by  Gerald Weber <gerald@cs.auckland.ac.nz>
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\title{Question Answering over a Probabilistic Knowledge Base}

% possible, but not really needed or used for PVLDB:
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%\titlenote{A full version of this paper is available as\textit{Author's Guide to Preparing ACM SIG Proceedings Using \LaTeX$2_\epsilon$\ and BibTeX} at \texttt{www.acm.org/eaddress.htm}}}

% ****************** AUTHORS **************************************

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\author{\alignauthor Christan Grant, Kun Li, Yang Chen, Daisy Zhe Wang\\
       \affaddr{University of Florida}\\
       \affaddr{Computer \& Information Science \& Engineering Department}\\
       \affaddr{Gainesville, Florida}\\
       \email{\{cgrant,kli,yang,daisyw\} @ cise.ufl.edu}
}

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\input{content/overview}

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\section{Acknowledgments}
This work was partially supported by DARPA under FA8750-12-2-0348-2
(DEFT/CUBISM) and Christan Grant was supported by a NSF Graduate Research
Fellowship, Grant DGE-0802270.\\

\small
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text:
\documentclass[main.tex]{subfiles}
\begin{document}

\marginpar{Tuesday\\ 2021-6-29, \\ compiled \\ \today}

Anile and others wrote complicated books on relativistic hydrodynamics.
With differential forms, in the Carter formalism the proofs become trivial differential geometry identities.

Today is the last formal full lecture.
Next, there will be about two weeks for which there will be more hands-on sessions.

There will be two exercise slots, on Tuesday and Thursday.

\section{Hyperbolic free-evolution schemes}

We have discussed earlier the fact that the constraints are propagated along the dynamics.

We want a schema to provide the free evolution of \emph{constraint-satisfying} initial data \(\Sigma_0 \).
We solve the ``time-dependent'' 3+1 EFE, and monitor the constraints as a check for numerical errors, since theoretically they should always hold.

Other approaches are possible, such as a fully constrained formulation,
which is not really used --- it maximizes the number of hyperbolic equations.

Another idea is to do constrained formulations exploiting symmetry.

The key issue for free-evolution schemas is the well-posedness of the Initial Value Boundary Problem.

We will discuss only the 3+1 Cauchy approach, for which \(\Sigma \) is always spacelike.
It is not the only options: we could do \emph{characteristic evolution} with null foliations, or hyperboloidal evolution with spacelike foliations reaching null infinity.

In the strong-field region these lightcones yield singularities, which is why they are not often used; however there are waveform extracting techniques which are based on this kind of thing.

Let us define hyperbolicity and hyperbolic equations.
The prototype hyperbolic equations is
%
\begin{align}
\partial_{tt} \phi - c^2 \triangle \phi = 0
\,,
\end{align}
%
where \(\phi \) is some sort of scalar field. An IBVP for this is imposes constraints in the form \(\phi (t= 0, x) = I(x)\) for \(x \in \Omega \) and \(\phi (t, x) = b(t, x)\) for \(x \in \partial \Omega \).

This can be made first-order in time by introducing an auxilliary field \(\Pi \):
%
\begin{align}
\begin{cases}
    \partial_{t} \phi &= \Pi   \\
    \partial_{t} \Pi &= c^2 \triangle \phi
\,.
\end{cases}
\end{align}

There also is a fully first order (in time and space) reduction:
%
\begin{align}
\begin{cases}
    \partial_{t} \phi &= \Pi  \\
    \partial_{t} \chi_i &= c \partial_{i} \Pi
    \partial_{t} \Pi &= c \partial_{i} \chi^{i}
    \chi_i &= c \partial_{t} \phi
\,.
\end{cases}
\end{align}

This can be written in matrix form in terms  of the vector \(u = (\phi , \chi _i, \Pi )\):
%
\begin{align}
\partial_{t} u + A^{i} (u) \partial_{i} u = s(u)
\,,
\end{align}
%
which is the general form of a hyperbolic equation.
If \(A\) is a constant, then we can diagonalize the system: \(LAR = R^{-1} A R = \Lambda \), where \(\Lambda \) is diagonal and \(L\), \(R \) are the left and right eigenvalue matrices.
We can also change variables accordingly: \(w = R u\), which are called \emph{characteristic variables}.
After doing this, we are left with a system of equations of the form
%
\begin{align}
\partial_{t} w + \lambda \partial_{x} w = 0
\,,
\end{align}
%
which is simply a translation of the initial profile, according to the eigenvalues of the matrix \(\Lambda \).

Therefore, the spectrum of \(A^{i}\) plays an important role in determining the solution and well-posedness.

Another way to see it is as follows: if we consider a Fourier mode
%
\begin{align}
u = \widetilde{u}_k e^{ik (\vec{x} \cdot \hat{r} - vt)}
\,.
\end{align}

Here \(v = \omega / k\).
We can plug this in:
%
\begin{align}
- i k v \widetilde{u}_k + ik A^{j} u_j \widetilde{u}_k &= s  \\
 - v \widetilde{u}_k + A^{j} u_j \widetilde{u}_k &= \frac{s}{ik}
\,,
\end{align}
%
which in the high-frequency limit becomes
%
\begin{align}
\qty(A^{j} u_j) \widetilde{u}_k &= v \widetilde{u}_k
\,,
\end{align}
%
an eigenvalue system for the \emph{principal symbol } \(P(\hat{n}) = A^{j} u_j\) matrix.
Iff there is a complete set of eigenvectors and eigenvalues, then we are lookiing at a \emph{strongly hyperbolic PDE}.
If so, the IVP is \emph{well-posed}!

What does well-posedness mean?
A characterization is as follows: there exists a norm \(\norm{\cdot}\) such that \(\norm{u(t, x)} \leq k e^{\alpha t} \norm{u(0, x)}\), where \(K\) and \(\alpha \) are constants independent of initial deta.

What this means is: there are no Fourier modes that are growing in time.

An example of a (non-hyperbolic) non-well-posed system is the inverse heat equation: \(\partial_{t} u = - \partial_{xx} u\).
If we take \(u = \widetilde{u}_k e^{ikx + \sigma t}\), we get
%
\begin{align}
\sigma e^{ikx + \sigma t} \widetilde{u}_k - (ik)^2 e^{ikx - \sigma t} \widetilde{u}_k
\,,
\end{align}
%
so \(\sigma = k^2\), and therefore the solution is in the form \(e^{k^2 t + ik x}\).
The solution grows in a way which depends on the initial data: no universal constant can bound it.

Another example is the regular heat equation: \(\partial_{t} u = \partial_{xx} u \). Here the Fourier modes satisfy \(\sigma = - k^2\), therefore the system is well-posed.

Another example is the inverse wave equation: \(\partial_{tt} u = - \partial_{xx} u\). We can prove that this is also ill-posed.

The definition of hyperbolicity can be stated in several ways: in term of the principal symbol \(P(n^i) = A^{i} n_i\) a PDE system is called
\begin{enumerate}
    \item weakly hyperbolic iff \(P\) has real eigenvalues but not a complete set of them;
    \item strongly hyperbolic iff \(P\) has a complete set of real eigenvalues;
    \item symmetry hyperbolic iff the symmetrizer \(H\) is independent of \(n^i\), which is equivalent to saying that all the \(A^{i}\) matrices are symmetric.
\end{enumerate}

The symmetrizer is defined as
%
\begin{align}
H = \qty(R^{-1})^\dag R^{-1}
\,.
\end{align}

The three conditions imply each other in order, with weak hyperbolicity \(\implies\) strong hyperbolicity \(\implies\) symmetry hyperbolicity.
Going back to the definition of well-posedness: strong hyperbolicity implies that the system admits a well-posed IVP.

A symmetry hyperbolic system is also well-posed, and it is typically easier to work with.

How about GR? Our goal is to get a free evolution scheme which is at least strongly hyperbolic.

A way to do so is using \textbf{Generalized Hyperbolic Gauge} (GHG).
The EFE in vacuo are
%
\begin{align}
0 = R_{ab} = \underbrace{- \frac{1}{2} g^{cd} \partial_{c} \partial_{d} g_{ab}}_{\text{principal part}} + \nabla_{(a} \Gamma_{b)} + g^{cd} g^{ef}
\qty( \partial_{e} g_{ca} \partial_{f} g_{bd} - \Gamma \Gamma )
\,,
\end{align}
%
where the highest-order derivatives are found in the \emph{pprincipal part}, which is the one we must consider when assessing hyperbolicity.
Here we also defined \(\Gamma_{a} g^{bc} \Gamma_{abc}\).

The term \(\square_g g_{ab}\) is good (manifestly hyperbolic), the term \(\nabla_{(a} \Gamma_{b)}\) is bad. The idea behind harmonic gauge is to take \(\Gamma_a \equiv 0\).
The EFE in harmonic gauge are therefore manifestly symmetry hyperbolic.
If can be easily shown that these coordinates are allowed, since we just need to set \(0 = \square x^{\mu }\).

This can be generalized: \(\Gamma _a\) can be set to a prescribed function \(H_a\), which can be determined \emph{a priori} or as something satisfying certain hyperbolic equations.
If this is done carefully, it allows us to preserve the symmetric hyperbolicity.
The reason to do so is in order to be more flexible with the gauge driver.

If we define the function \(Z_a = \Gamma _a - H_a\), it is possible to prove that the GHG is equivalent to
%
\begin{align}
0 = \overline{R}_{ab} = R_{ab} - \nabla_{(a } Z_{b)}
\,,
\end{align}
%
under the algebraic constraint \(0 = Z_a\).
This is just the Z_4 system for a particular choice of \(Z_a\).
There is a 2005 paper discussing this.
GHG was used for
\begin{enumerate}
    \item the BBH breakthrough by Pretorius (GHG + constraint damping terms);
    \item NR codes based on spectral methods (since the ``patches'' with different Fourier bases must be matched to each other).
\end{enumerate}

What about ADMY? Is it hyperbolic?
This is way more complicated, but it can be proven under the conditions that
\begin{enumerate}
    \item \(\beta^{i}\) is prescribed;
    \item \(\alpha \) belongs to the Bona-Masso family or the densitized lapse \(\overline{\alpha} = \alpha / \sqrt{\gamma }\) is prescribed;
    \item \(C_i \equiv 0\) is satisfied along the evolution.
\end{enumerate}

If we drop the third condition, then ADMY is weakly hyperbolic, and no well-posedness result can apply.

How can we then use ADMY? We can change those equations in order to get a strongly hyperbolic system.
Consider the ADMY second order equations:
%
\begin{align}
\partial_{tt} \gamma_{ij} \simeq - 2 \alpha  R_{ij} + \dots \approx \underbrace{\triangle \gamma_{ij}}_{\text{good term}} + \underbrace{\gamma_{ik} \partial_{j} \partial_{l} \gamma^{kl} + \gamma_{jk} \partial_{i} \partial_{l} \gamma^{kl}}_{\text{bad term}}  + \dots
\,,
\end{align}
%
so, without setting the bad term to zero directly we can define the auxilliary variable \(f^{k} = \partial_{l} \gamma^{kl}\), the ``coordinate divergence of \(\gamma \)'' and find an evolution equation for \(f^{k}\) which guarantees hyperbolicity.
What could be this equation?
Observe that its time derivative reads
%
\begin{align}
\partial_{t} f^{k} = \partial_{t} \partial_{l} \gamma^{kl} = \partial_{l} \partial_{t} \gamma^{kl} \sim \partial_{l} k^{kl}  \sim C^{k}
\,,
\end{align}
%
so this looks like the momentum constraint!

Modern evolution schemas are based on this, but using \emph{conformal ADMY}.
The conformal Ricci reads
%
\begin{align}
\widetilde{R}_{ij} \simeq - \frac{1}{2} \widetilde{\gamma}^{kl} \qty(
D_k D_l \widetilde{\gamma}_{ij} + \widetilde{\gamma}_{ik} D_j D_k \widetilde{\gamma}^{kl} + \widetilde{\gamma}_{jk} D_i D_k \widetilde{\gamma}^{kl}
)  + \dots
\,,
\end{align}
%
so like before we define \(\widetilde{\Gamma}^{i} = - D_j \widetilde{\gamma}^{ij}\), \emph{auxiliary conformal variables}.
We use the momentum constraint in order to find \(\partial_{t} \widetilde{\Gamma}^{i} = \dots\), an additional equation to add to the ADMY in conformal variables to obtain the BSSN (BSSNOK).

The Z4c formulation is the same as BSSN with an additional equation voor the variable \(\theta = Z^{a} n_a\): \(\partial_{t} \theta \sim C^{0}\), which is convenient because it adds a constraint damping term to BSSN.

\end{document}

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\subsection{Programming Languages}
\begin{itemize}
\item Expert: C/C++, CMake, CUDA
\item Proficient: C\#, Python, Java, Lua, \textsc{Matlab}, Bash
\end{itemize}

\subsection{C++ Libraries and Projects}
STL, Boost, LLVM/Clang, Qt5, OpenGL, DirectX 12, Ogre3D, OpenMP, Intel TBB, MPI, Protobuf

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text:
\SetAPI{J-C}
\section{Paging Query}
\label{feature:PagingQuery}
\ClearAPI
\TODO

\begin{lstlisting}[style=Java,caption={Usage example for the \type{Paging Query} (Java)}]
IPagingQuery<IMyEntity> pagingQuery = qb.buildPaging(whereClause);
PagingRequest pagingRequest = pagingQuery.createRequest(page, size);
IPagingResponse<IMyEntity> pagingResponse = pagingQuery.retrieve(pagingRequest);
List<IMyEntity> result = pagingResponse.getResult();
\end{lstlisting}

The example above retrieves page number \type{page} with the maximum number of elements specified by \type{size}.
\begin{itemize}
\item To retrieve the first 150 elements, set page to 0 and size to 150.
\item If you have a paging grid displaying 30 elements, size would be 30, and page would be the number of the current page (starting with 0)
\end{itemize}

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text:
\section*{Introduction} \label{sec:intro}

\subsection*{Statement of the problem} \label{subsec:intro-problem}

\subsection*{Yearly milestones} \label{subsec:intro-milestones}

\subsection*{Conceptual solution to solve the problem} \label{subsec:intro-solution}

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text:
\chapter{Introduction}

\section{<section here>}

\subsection{<sub section here>}

\blindtext

\subsubsection{<sub sub section here>}

\blindtext

%some text\cite{citation-1-name-here}, some more text
%even more text\footnote{<footnote here>}, and even more.

\section{Adding Content}

\subsection{Images}

\begin{figure}[ht]
    \begin{center}
        \includegraphics[width=0.50\textwidth]{./vit-logo.png}
        \caption{Sample Image.}
        \label{fig:sample-image}
    \end{center}
\end{figure}

\noindent To view the image code please refer to the `sample.tex' file.

\subsection{Tables}

\begin{table}[ht]
    \caption{Sample Table}
    \centering
    \begin{tabular}{ | c | c | c | }
        \hline
        cell1 & cell2 & cell3 \\
        \hline
        cell1 & cell5 & cell6 \\
        \hline
        cell7 & cell8 & cell9 \\
        \hline
    \end{tabular}
    \label{tab:sample-table}
\end{table}

\noindent To view the table code please refer to the `sample.tex' file.

\subsection{References, Citations, and Footnotes}

You can refer to an image as Fig. \ref{fig:sample-image} and to a table as Table. \ref{tab:sample-table}.

\noindent You can cite items from the `references.bib' file as \cite{10.5120/ijca2020920393}.

\noindent You can also add a footnote \footnote{Sample Footnote}.

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text:
\chapter{MySql}
\label{chp:mysql}

Simply run the command;
\begin{lstlisting}
sudo apt-get install mysql-server
\end{lstlisting}

And remember the password and username specified during installation.

Most programs that require the use of MySql will create and manage their own databases.
\vspace{2cm}
\textbf{IMPORTANT} Do not run MySql under the same user and group as apache, as an attack on one, could leave the other vulnerable.

\url{http://www.cyberciti.biz/faq/mysql-command-to-show-list-of-databases-on-server/}

\url{http://www.bios.niu.edu/johns/bioinform/mysql_commands.htm}

\subsection{MySQL basic commands}

Here are a list of basic commands for MySQL. Note the semicolon at the end of each command.

\begin{table}[htb]
\begin{tabular}{cc}
\hline
Command & Description\\
\hline
SHOW DATABASES; & Lists all databases in MySQL\\
USE [NAME]; & Switches to specified database\\
quit; & Quit from MySQL\\
\hline
\end{tabular}
\caption{Table of Basic commands for MySQL}
\label{tab:mysqlComm}
\end{table}

\subsection{Reset mysql root password}
\label{sec:mysqlresetpass}

Sometime we forget passwords, so it is important we know how to change them, here is a six part solution to change the password of the mysql root user.

\subsubsection{Stop MySQL}

\begin{lstlisting}
sudo service mysql stop
\end{lstlisting}

\subsubsection{Start in safe mode}

\begin{lstlisting}
sudo mysql_safe --skip-grant-tables &
\end{lstlisting}

\subsubsection{Then change database}

\begin{lstlisting}
use mysql;
\end{lstlisting}

\subsubsection{Change Password}

Change the password using the following, and putting a new password instead of newpasswordhere below.

\begin{lstlisting}
update user set password=PASSWORD ("newpasswordhere") where User = 'root';
\end{lstlisting}

\subsubsection{flush privileges \& quit}

\begin{lstlisting}
flush privileges;
quit;
\end{lstlisting}

\subsubsection{Restart MySQL}

\begin{lstlisting}
sudo service mysql restart
\end{lstlisting}

\subsection{Backup and Restore MySQL databases}
\label{ssec:mySQLBackup}

\url{http://webcheatsheet.com/sql/mysql_backup_restore.php}

In this section we will see how to backup and restore the mysql databases.

\subsubsection{Backing up}

Before logging in to mysql, run the following commands:

\begin{lstlisting}
mysqldump -u [USERNAME] -p [DATABASE NAME] > [BACKUPFILE.sql]
\end{lstlisting}

Don't put the square brackets in the command, and replace where necessary, you will be asked for your password after running the command.

You can backup multiple databases at once by separating them with a space.

This command will lock out users when it is running, unless you have a very large database, you shouldn't need to worry.

\subsubsection{Restoring a Backup}

There are two ways to restore, one to add a backup database into a new install (effectively creating a new database with the old data), and one to overwrite an existing database.

To simply add the backup database to mysql where it didn't exist before run the following:

\begin{lstlisting}
mysql -u [USERNAME] -p [DATABASE NAME] < [BACKUPFILE.sql]
\end{lstlisting}

However if the database does exist but you wish to overwrite the data, simply type:

\begin{lstlisting}
mysqlimport -u [USERNAME] -p [DATABASE NAME] [BACKUPFILE.sql]
\end{lstlisting}


text:
\title{Tensorboard}

\subsection{Tensorboard}

TensorBoard provides a suite of visualization tools to make it easier
to understand, debug, and optimize Edward programs. You can use it
``to visualize your TensorFlow graph, plot quantitative metrics about
the execution of your graph, and show additional data like images that
pass through it''
(\href{https://www.tensorflow.org/get_started/summaries_and_tensorboard}
{tensorflow.org}).

A Jupyter notebook version of this tutorial is available
\href{http://nbviewer.jupyter.org/github/blei-lab/edward/blob/master/notebooks/tensorboard.ipynb}{here}.

\includegraphics[width=750px]{/images/tensorboard-scalars.png}

To use TensorBoard, we first need to specify a directory for storing
logs during inference. For example, if manually controlling inference,
call

\begin{lstlisting}[language=Python]
inference.initialize(logdir='log')
\end{lstlisting}

If you're using the catch-all \texttt{inference.run()}, include
\texttt{logdir} as an argument. As inference runs, files are
outputted to \texttt{log/} within the working directory. In
commandline, we run TensorBoard and point to that directory.

\begin{lstlisting}[language=JSON]
tensorboard --logdir=log/
\end{lstlisting}

The command will provide a web address to access TensorBoard. By
default, it is \url{http://localhost:6006}. If working correctly, you
should see something like the above picture.

You're set up!

Additional steps need to be taken in order to clean up TensorBoard's
naming. Specifically, we might configure names for random variables
and tensors in the computational graph. To provide a concrete example,
we extend the
\href{http://edwardlib.org/tutorials/supervised-regression}
{supervised learning tutorial},
where the task is to infer hidden structure from labeled examples
$\{(x_n, y_n)\}$.

\subsubsection{Data}

Simulate training and test sets of $40$ data points. They comprise of
pairs of inputs $\mathbf{x}_n\in\mathbb{R}^{5}$ and outputs
$y_n\in\mathbb{R}$. They have a linear dependence with normally
distributed noise.

\begin{lstlisting}[language=Python]
def build_toy_dataset(N, w):
  D = len(w)
  x = np.random.normal(0.0, 2.0, size=(N, D))
  y = np.dot(x, w) + np.random.normal(0.0, 0.01, size=N)
  return x, y

ed.set_seed(42)

N = 40  # number of data points
D = 5  # number of features

w_true = np.random.randn(D) * 0.5
X_train, y_train = build_toy_dataset(N, w_true)
X_test, y_test = build_toy_dataset(N, w_true)
\end{lstlisting}

\subsubsection{Model}

Posit the model as Bayesian linear regression \citep{murphy2012machine}.
For a set of $N$ data points $(\mathbf{X},\mathbf{y})=\{(\mathbf{x}_n, y_n)\}$,
the model posits the following distributions:

\begin{align*}
  p(\mathbf{w})
  &=
  \text{Normal}(\mathbf{w} \mid \mathbf{0}, \sigma_w^2\mathbf{I}),
  \\[1.5ex]
  p(b)
  &=
  \text{Normal}(b \mid 0, \sigma_b^2),
  \\
  p(\mathbf{y} \mid \mathbf{w}, b, \mathbf{X})
  &=
  \prod_{n=1}^N
  \text{Normal}(y_n \mid \mathbf{x}_n^\top\mathbf{w} + b, \sigma_y^2).
\end{align*}

The latent variables are the linear model's weights $\mathbf{w}$ and
intercept $b$, also known as the bias.
Assume $\sigma_w^2,\sigma_b^2$ are known prior variances and $\sigma_y^2$ is a
known likelihood variance. The mean of the likelihood is given by a
linear transformation of the inputs $\mathbf{x}_n$.

Let's build the model in Edward, fixing $\sigma_w,\sigma_b,\sigma_y=1$.

\begin{lstlisting}[language=Python]
with tf.name_scope("model"):
  X = tf.placeholder(tf.float32, [N, D], name="X")
  w = Normal(loc=tf.zeros(D, name="weights/loc"),
             scale=tf.ones(D, name="weights/loc"),
             name="weights")
  b = Normal(loc=tf.zeros(1, name="bias/loc"),
             scale=tf.ones(1, name="bias/scale"),
             name="bias")
  y = Normal(loc=ed.dot(X, w) + b,
             scale=tf.ones(N, name="y/scale"),
             name="y")
\end{lstlisting}

Here, we define a placeholder \texttt{X}. During inference, we pass in
the value for this placeholder according to batches of data.  We also
use a name scope. This adds the scope's name as a prefix
(\texttt{"model/"}) to all tensors in the \texttt{with} context.
Similarly, we name the parameters in each random variable under a
grouped naming system.

\subsubsection{Inference}

We now turn to inferring the posterior using variational inference.
Define the variational model to be a fully factorized normal across
the weights. We add another scope to group naming in the variational
family.

\begin{lstlisting}[language=Python]
with tf.name_scope("posterior"):
  qw = Normal(loc=tf.Variable(tf.random_normal([D]), name="qw/loc"),
              scale=tf.nn.softplus(tf.Variable(tf.random_normal([D]), name="qw/unconstrained_scale")),
              name="qw")
  qb = Normal(loc=tf.Variable(tf.random_normal([1]), name="qb/loc"),
              scale=tf.nn.softplus(tf.Variable(tf.random_normal([1]), name="qb/unconstrained_scale")),
              name="qb")
\end{lstlisting}

Run variational inference with the Kullback-Leibler divergence.
We use $5$ latent variable samples for computing
black box stochastic gradients in the algorithm.
(For more details, see the
\href{/tutorials/klqp}{$\text{KL}(q\|p)$ tutorial}.)

\begin{lstlisting}[language=Python]
inference = ed.KLqp({w: qw, b: qb}, data={X: X_train, y: y_train})
inference.run(n_samples=5, n_iter=250, logdir='log/n_samples_5')
\end{lstlisting}

\begin{lstlisting}
250/250 [100%] ██████████████████████████████ Elapsed: 5s | Loss: 50.865
\end{lstlisting}

Optionally, we might include an \texttt{"inference"} name scope.
If it is absent, the charts are partitioned naturally
and not automatically grouped under the monolithic \texttt{"inference"}.
If it is added, the TensorBoard graph is slightly more organized.

\subsubsection{Criticism}

We can use TensorBoard to explore learning and diagnose any problems.
After running TensorBoard with the command above, we can navigate the
tabs.

Below we assume the above code is run twice with different
configurations
of the \texttt{n_samples} hyperparameter.
We specified the log directory to be \texttt{log/n_samples_*}.
By default, Edward also includes a timestamped subdirectory so that
multiple runs of the same experiment have properly organized logs for
TensorBoard. You can turn it off by specifying
\texttt{log_timestamp=False} during inference.

\textbf{TensorBoard Scalars.}

\includegraphics[width=750px]{/images/tensorboard-scalars.png}

Scalars provides scalar-valued information across iterations of the
algorithm, wall time, and relative wall time. In Edward, the tab
includes the value of scalar TensorFlow variables in the model or
approximating family.

With variational inference, we also include information such as the
loss function and its decomposition into individual terms. This
particular example shows that \texttt{n_samples=1} tends to have higher
variance than \texttt{n_samples=5} but still converges to the same solution.

\textbf{TensorBoard Distributions.}

\includegraphics[width=750px]{/images/tensorboard-distributions.png}

Distributions display the distribution of each non-scalar TensorFlow
variable in the model and approximating family across iterations.

\textbf{TensorBoard Histograms.}

\includegraphics[width=750px]{/images/tensorboard-histograms.png}

Histograms displays the same information as Distributions but as a 3-D
histogram changing aross iteration.

\textbf{TensorBoard Graphs.}

\includegraphics[width=750px]{/images/tensorboard-graphs-0.png} \\
\includegraphics[width=750px]{/images/tensorboard-graphs-1.png}

Graphs displays the computational graph underlying the model,
approximating family, and inference. Boxes denote tensors grouped
under the same name scope. Cleaning up names in the graph makes it
easy to better understand and optimize your code.

\subsubsection{Acknowledgments}

We thank Sean Kruzel for writing the initial version of this
tutorial.

A TensorFlow tutorial to TensorBoard can be found
\href{https://www.tensorflow.org/get_started/summaries_and_tensorboard}
{here}.

\subsubsection{References}\label{references}

max_stars_repo_name: NunoEdgarGFlowHub/edward

max_issues_repo_name: NunoEdgarGFlowHub/edward

max_forks_repo_name: NunoEdgarGFlowHub/edward


text:
% -- Encoding UTF-8 without BOM
% -- XeLaTeX => PDF (BIBER)

\documentclass[]{cv-style}          % Add 'print' as an option into the square bracket to remove colours from this template for printing.
                                    % Add 'espanol' as an option into the square bracket to change the date format of the Last Updated Text

\sethyphenation[variant=british]{english}{} % Add words between the {} to avoid them to be cut

\begin{document}

\header{Jincy}{ Johny}           % Your name

%----------------------------------------------------------------------------------------
%	SIDEBAR SECTION  -- In the aside, each new line forces a line break
%----------------------------------------------------------------------------------------

\begin{aside}
%
\section{Contact}
336,GH-IV, NITK
 Mangalore, India
~
+91 8547512639
~
jincyjohny94@gmail.com
~
%
\section{Personal Traits}
 Problem solving skills
Ability to take the initiative
     Ability to work well under pressure
     communication skills
     ~
%
\section{Softwares}
Wireshark,Nessus
\LaTeX{},MySQL
%
\section{Operating
   System}
   Ubuntu
    Kali Linux(Debian)
Windows
%
\section{Languages}
 English
 Malayalam
 Hindi
%
\end{aside}

%----------------------------------------------------------------------------------------
%	EDUCATION SECTION
%----------------------------------------------------------------------------------------
\section{CAREER OBJECTIVE}
To work with satisfaction, based on knowledge in a company which consider human values
%----------------------------------------------------------------------------------------
%	EDUCATION SECTION
%----------------------------------------------------------------------------------------
 \section{EDUCATION}

\begin{entrylist}
%------------------------------------------------
\entry
{2016--Present}
     { National Institute of Technology,Surathkal}
{}
{\jobtitle{M.TECH}

{Computer Science and Information Security}

\begin{itemize}
\item \textbf{CPI-8.39/10}

\end{itemize}}

{\vspace{-0.3cm}}
\end{entrylist}
%------------------------------------------------
\begin{entrylist}
\entry
{2011-2015}
{Government Engineering College,Palakkad}
{}
{\jobtitle{B.TECH}

{Computer Science and Engineering}

\begin{itemize}
\item \textbf{CPI-8.63/10}

\end{itemize}}

\end{entrylist}
%------------------------------------------------
\begin{entrylist}
\entry
{2009--2011}
{BHSS Kalady, Kerala}
{}
{\jobtitle{Higher Secondary}
\begin{itemize}
\item \textbf{Percentage Secured-90.4}
\end{itemize}}
%------------------------------------------------

\end{entrylist}
%------------------------------------------------

\begin{entrylist}
\entry
{2008--2009}
{St.Joseph's GHS Karukutty}
{}
{\jobtitle{High School}
\begin{itemize}
\item \textbf{Percentage Secured-90}
\end{itemize}}
%------------------------------------------------

\end{entrylist}

\section{PROJECTS}

\begin{entrylist}
%------------------------------------------------
\entry
{Feb-2015}
{Buffer-less Routing in Network-on-chip }
{}
{\vspace{-0.3cm}}

\end{entrylist}
%
\begin{entrylist}
%------------------------------------------------
\entry
{Nov-2016}
{Enhanced SHA-192 Algorithm}
{}
{\vspace{-0.3cm}}
\end{entrylist}
\begin{entrylist}
%------------------------------------------------
\entry
{Dec-2016}
{Anti-phishing Scheme }
{}
{\vspace{-0.3cm}}
%
\end{entrylist}
%
\begin{entrylist}
%------------------------------------------------
\entry
{May-2017}
{Detection of network anomaly}
{}
{\vspace{-0.3cm}}

%------------------------------------------------
\end{entrylist}
%
\begin{entrylist}
%------------------------------------------------
\entry
{May--2017}
{Design of Modified Fuzzy Median Filter For Noise Reduction in Images }
{}
{\vspace{-0.3cm}}

\end{entrylist}
%
%----------------------------------------------------------------------------------------
%	SKILLS SECTION
%----------------------------------------------------------------------------------------

\section{PROGRAMMING SKILLS}

  \vspace{-0.2cm}
  \begin{entrylist}
%------------------------------------------------
\entry
 {\textbf{Languages}}
{}
{}
{\begin{tabular}{p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth}}
 C & Python &  JavaScript &  Matlab  \\
\end{tabular}}
  {\vspace{-0.3cm}}
%------------------------------------------------
\end{entrylist}
%
\begin{entrylist}
%------------------------------------------------
\entry
 {\textbf{Familar}}
{}
{}
{\begin{tabular}{p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth}}
 Java & C++ &  HTML\\
\end{tabular}}
%------------------------------------------------
\end{entrylist}
%
%----------------------------------------------------------------------------------------
%	OTHER QUALIFICATIONS SECTION
%----------------------------------------------------------------------------------------
\section{INTERESTS}
\vspace{-0.3cm}
  {}
%------------------------------------------------

%------------------------------------------------
\begin{entrylist}
%------------------------------------------------
\entry
 {\textbf{Professional}}
{}
  {}
  {Data Structures and Algorithm, Database, Computer Networks\\ Information Security }
%------------------------------------------------
\end{entrylist}
%
\begin{entrylist}
%------------------------------------------------
\entry
 {\textbf{Personal}}
{}
  {}
  {Music, Cycling, Badminton, Photography }
%--------------

\end{entrylist}
%
\section{LINKS}
\begin{itemize}
\item https://github.com/jinu12345
\item https://www.facebook.com/jincy.johny1
\item https://www.linkedin.com/in/jincy-johny-848537147
\end{itemize}

%

%----------------------------------------------------------------------------------------
%	Training
%----------------------------------------------------------------------------------------

%----------------------------------------------------------------------------------------

%----------------------------------------------------------------------------------------
%	Training
%----------------------------------------------------------------------------------------

%----------------------------------------------------------------------------------------

\end{document}


text:
%!TEX root=cargo.tex
\section{Getting Started}

\subsection{Obtaining a Copy}

There are two ways to obtain a copy.
\begin{itemize}
    \item Add release package to GitHub.

    \item Clone the master \code{libcargo} git repository to your local computer:
\begin{verbatim}
git clone https://github.com/jamjpan/Cargo.git
\end{verbatim}

The source repository is roughly 78 MB. It is large because it includes problem
instances and road networks. The problem instances together are 40.2 MB, and
the road networks are 34.1 MB.
\end{itemize}

\subsection{Installing}

The \code{libcargo} library is a \emph{static} library; it does not need to be
installed. But, it does need to be built in order to be usable. The release
package already contains the pre-built library, but if you would like to build
the library yourself, or if the pre-built version does not work on your
machine, follow the instructions below to build from source.

\subsection{Building from Source}

Make sure your machine satisfies the prerequisites (Section~\ref{2.
Prerequisites}).  Then, \code{cd} into the \code{Cargo} directory and type
\code{make}. The command will create two directories, \code{build} and
\code{lib}. the first directory stores temporary build objects, and the
second stores the library with filename \code{libcargo.a}.

\subsection{Usage}

Copy the \code{include} directory and \code{libcargo.a} into your own project's root directory. Then
use \code{\#include "libcargo.h"} in your code to access the library. Use
\code{using namespace cargo;} at the top of your code to save prepending the
\code{cargo} namespace to library objects and functions. When building your
project, link \code{libcargo} and all its dependencies. For example, using \code{g++}:
\begin{verbatim}
g++ -std=c++11 -O3 -c -Iinclude -o my_project.o my_project.cpp
g++ my_project.o -lcargo -pthread -lrt -lmetis -ldl
\end{verbatim}
The first \code{g++} command compiles \code{my\_project.cpp} into the
\code{my\_project.o} binary. The \code{-I} flag instructs \code{g++} where to
look for includes; this should match wherever you placed \code{libcargo}'s \code{include}
directory.

max_stars_repo_name: jamjpan/libezrs

max_issues_repo_name: jamjpan/libezrs

max_forks_repo_name: jamjpan/libezrs


text:
\hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward}{}\section{numpp\+:\+:differentiation\+:\+:automatic\+:\+:forward$<$ T $>$ Class Template Reference}
\label{classnumpp_1_1differentiation_1_1automatic_1_1forward}\index{numpp\+::differentiation\+::automatic\+::forward$<$ T $>$@{numpp\+::differentiation\+::automatic\+::forward$<$ T $>$}}

{\ttfamily \#include $<$structure.\+hpp$>$}

\subsection*{Public Types}
\begin{DoxyCompactItemize}
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_ad360786eabc5ef8ba98e13431dc43f9e}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_ad360786eabc5ef8ba98e13431dc43f9e}}
using {\bfseries value\+\_\+type} = T
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a5cf8dbc6f2eaea64516093d742238cef}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a5cf8dbc6f2eaea64516093d742238cef}}
using {\bfseries size\+\_\+type} = std\+::size\+\_\+t
\end{DoxyCompactItemize}
\subsection*{Public Member Functions}
\begin{DoxyCompactItemize}
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_ae322ba3bb0f58f80595bf463907b8312}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_ae322ba3bb0f58f80595bf463907b8312}}
constexpr {\bfseries forward} (T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_a64a2339eab5772de4da0483fd5c4f895}{value}, T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_abcaca75164fca6c9d423d6b11b5517e9}{derivative}=1)
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_aee3aa201c624dcba4d203b667eebdc42}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_aee3aa201c624dcba4d203b667eebdc42}}
constexpr {\bfseries forward} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&)=default
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a6977f29fb9a93d34ccf17997a762da79}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a6977f29fb9a93d34ccf17997a762da79}}
constexpr {\bfseries forward} (\hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&\&)=default
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_ae986202a11b02333dce05cead206cc4f}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_ae986202a11b02333dce05cead206cc4f}}
constexpr \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \& {\bfseries operator=} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&)=default
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a082c8f6cb5faada6017dad74cc64370a}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a082c8f6cb5faada6017dad74cc64370a}}
constexpr \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \& {\bfseries operator=} (\hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&\&)=default
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a5600c62c6caba359305c2f8facbe06a0}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a5600c62c6caba359305c2f8facbe06a0}}
constexpr auto {\bfseries operator+} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&other)
\item
{\footnotesize template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ }\\constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_ac97b21e3e666a20f4f731a65faafae79}{operator+} (U other)
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a4f7666eff1d116c6b7c3d9aefe19d69c}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a4f7666eff1d116c6b7c3d9aefe19d69c}}
constexpr auto {\bfseries operator-\/} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&other)
\item
{\footnotesize template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ }\\constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_afe3627daed4fc827b0e31886ddfd5dd9}{operator-\/} (U other)
\item
{\footnotesize template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ }\\constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_a006d157f743938e11eb54668c41da769}{operator$\ast$} (U other)
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a021e057d791fb9786c0a5cd0f1c35fbe}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a021e057d791fb9786c0a5cd0f1c35fbe}}
constexpr auto {\bfseries operator$\ast$} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&other)
\item
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a1e7b473c400fcf510d8d24712821e72e}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a1e7b473c400fcf510d8d24712821e72e}}
constexpr auto {\bfseries operator/} (const \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{forward} \&other)
\item
{\footnotesize template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ }\\constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_a630cf420f7160392da3f4c575b2d8e66}{operator/} (U other)
\end{DoxyCompactItemize}
\subsection*{Public Attributes}
\begin{DoxyCompactItemize}
\item
const T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_a64a2339eab5772de4da0483fd5c4f895}{value}
\item
const T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward_abcaca75164fca6c9d423d6b11b5517e9}{derivative}
\end{DoxyCompactItemize}

\subsection{Detailed Description}
\subsubsection*{template$<$typename T$>$\newline
class numpp\+::differentiation\+::automatic\+::forward$<$ T $>$}

\begin{DoxyTemplParams}{Template Parameters}
{\em T} & arithmetic type contained in forward class \\
\hline
\end{DoxyTemplParams}
\begin{DoxyWarning}{Warning}
You have to specialize std\+::is\+\_\+arithmetic if you want to use type other than built-\/ins.
\end{DoxyWarning}
{\bfseries  value data member contains function value, derivative data member contains certain derivative }

\begin{DoxyCode}
\textcolor{preprocessor}{#include"numpp/differentiation/automatic/forward.h"}
\end{DoxyCode}

\subsection{Member Function Documentation}
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a006d157f743938e11eb54668c41da769}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a006d157f743938e11eb54668c41da769}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!operator$\ast$@{operator$\ast$}}
\index{operator$\ast$@{operator$\ast$}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{operator$\ast$()}{operator*()}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ \\
constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::operator$\ast$ (\begin{DoxyParamCaption}\item[{U}]{other }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}

operator$\ast$ for built-\/in arithmetic\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_ac97b21e3e666a20f4f731a65faafae79}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_ac97b21e3e666a20f4f731a65faafae79}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!operator+@{operator+}}
\index{operator+@{operator+}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{operator+()}{operator+()}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ \\
constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::operator+ (\begin{DoxyParamCaption}\item[{U}]{other }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}

operator+ for built-\/in arithmetic\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_afe3627daed4fc827b0e31886ddfd5dd9}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_afe3627daed4fc827b0e31886ddfd5dd9}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!operator-\/@{operator-\/}}
\index{operator-\/@{operator-\/}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{operator-\/()}{operator-()}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ \\
constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::operator-\/ (\begin{DoxyParamCaption}\item[{U}]{other }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}

operator-\/ for built-\/in arithmetic\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a630cf420f7160392da3f4c575b2d8e66}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a630cf420f7160392da3f4c575b2d8e66}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!operator/@{operator/}}
\index{operator/@{operator/}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{operator/()}{operator/()}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
template$<$typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+arithmetic$<$\+U$>$\+::value$>$$>$ \\
constexpr auto \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::operator/ (\begin{DoxyParamCaption}\item[{U}]{other }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}

operator/ for built-\/in arithmetic

\subsection{Member Data Documentation}
\mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_abcaca75164fca6c9d423d6b11b5517e9}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_abcaca75164fca6c9d423d6b11b5517e9}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!derivative@{derivative}}
\index{derivative@{derivative}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{derivative}{derivative}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
const T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::derivative}

Contains calculated derivative of an expression \mbox{\Hypertarget{classnumpp_1_1differentiation_1_1automatic_1_1forward_a64a2339eab5772de4da0483fd5c4f895}\label{classnumpp_1_1differentiation_1_1automatic_1_1forward_a64a2339eab5772de4da0483fd5c4f895}}
\index{numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}!value@{value}}
\index{value@{value}!numpp\+::differentiation\+::automatic\+::forward@{numpp\+::differentiation\+::automatic\+::forward}}
\subsubsection{\texorpdfstring{value}{value}}
{\footnotesize\ttfamily template$<$typename T$>$ \\
const T \hyperlink{classnumpp_1_1differentiation_1_1automatic_1_1forward}{numpp\+::differentiation\+::automatic\+::forward}$<$ T $>$\+::value}

Contains calculated value of an expression

The documentation for this class was generated from the following file\+:\begin{DoxyCompactItemize}
\item
differentiation/automatic/forward/structure.\+hpp\end{DoxyCompactItemize}

max_stars_repo_name: szymonmaszke/numpp

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text:
%-----------------------------------------------------------Header Starts--------------------------------------

\documentclass[main.tex]{subfiles}

%-----------------------------------------------------------Header Ends--------------------------------------

\graphicspath{ {./image_intro/} }

\begin{document}

\chapter{Introduction}

\pagenumbering{arabic}

\thispagestyle{empty}

\pagestyle{fancy}
\fancyhf{}

\rhead{Page \thepage}
\lhead{\chaptername \ \thechapter}
%\cfoot{Page \thepage}

\section{Background of the research}
Modern technology stands on the shoulder of semiconductor materials for their immensely useful adaptabilities of physical properties specially the electrical conductivity with an applied electrical field. Despite of having superior conductivity, metals have been outcast by the uniqueness of semiconductor materials in controlling the flow of current. For example, in order to perform a logic function, the ability to distinguish between one and zero is necessary. Semiconductor materials have the capability to transform from insulating to conducting with the help of very small amount of voltage known as gate voltage. When the gate voltage is above a certain level, current flows through a small channel between source and drain; making the material conductive. Again, this flow of current is interrupted when gate voltage is below the level. This directional flow of current in semiconductor is basically driven by very dilute concentration of dopants into a host semiconductor (i.e. 10$^{13}$ to 10$^{18}$ dopant atoms/cm$^{3}$ in Si as host material which contains 5$ \times $10$^{22}$ atoms/cm$^{3}$). By inducing such excess or shortage of electrons via doping, variable electrical resistance can be achieved which makes semiconductor materials useful for switching, amplification or energy conversion applications. This class of semiconductor is known as Transistor and they led the most important paradigm shift in technology by miniturization of devices which in turns reduces the energy utilization by thousand folds. The placement of these transistors on the integrated circuit (IC) is important because the more miniaturized dense IC could be made, the less heat loss would happen which eventually will raise the speed of information process. In 1965, Gordon Moore, one of the cofounder of Intel, predicted that the number of transistors placed on an IC will double in every two years. Companies like Intel, Samsung and NVidia are now fabricating transistors less than 20 nm in size at mass scale and NVidia’s new graphics card GF100 is an example of how dense miniaturized a device can become which contains more than 3 Billion transistors. But, this miniaturization to speed up the information process has now reached its saturation limit since it is almost at the edge of atomic scale. On the other side, we are producing more than 2.5 quintillion bytes of data everyday according to Forbes. With days passing by, we are becoming fonder of online based social and financial activities. The energy required for processing this large amount of data is causing global warming at an increased alarming rate. As, the miniaturization of device can no longer provide benefits to us in energy conservation via faster communication, we now must have to think otherwise. \\

In electronics, only the charge of the electron is utilized while it’s another degree of freedom “Spin” remains less remembered. If the spin states can be incorporated with electronic charge to store and process information, the communication will increase by many folds which has been a fascination for decades. Spintronics, an acronym for SPIN TRansport electrONICS, is one of such promising areas which brought us the novel concept of utilizing both the charge and spin of the electrons. Apart from the formerly known classical states of electron's spinning i.e. rotating clockwise or anticlockwise, there are infinite number of spin states that an electron can possess according to quantum mechanics. If these spin states could be controlled like the current flow in transistor, we would have more options beside the conventional switching technique i.e. current off(0) or on(1). That being said, the electrons carry the spin state must be non-volatile as well as spin polarized and the ferromagnetism needs to be capable of gating.  For example, Kreutz \textit{et al.} showed that an organic monolayer, deposited on surface, can provide electrons to compensate the hole carriers of GaMnAs which restrain the ferromagnetism (\(T_C < 170 K\)) \cite{kreutz2003modification}. The control over ferromagnetism by changing carrier concentration like Kreutz's research is what we want to achieve at room temperature. \\

Initially, spin injection attempts were limited to the heterostructure comprising a metallic ferromagnet on semiconductor device \cite{Datta1990,DasSarma2000}. But due to serious deficiency of spin injection at metal-semiconductor interface, this idea was replaced by a quest for the ferromagnetism in semiconductor. The concept of ferromagnetic semiconductor was initiated by introducing dilute concentration of magnetic dopants (Mn$^{2+}$) into a semiconductor material, \(Ga_{(1-x)}Mn_xAs \ (x=0.015–0.071)\) \cite{matsukura1998transport}. The ferromagnetism observed in such non-magnetic semiconductors are supposed to be carrier mediated. The excess electrons provided by the dopants delocalize near the fermi level. For certain cases, these delocalized impurity bands become spin polarized and tend to show half-metallic behavior. The materials showed such property were started to be termed as “Dilute Magnetic Semiconductor (DMS)”. In an ideal DMS, the charge carriers are spin polarized and behave like half-metallic which enable the charge carriers to be transported by inducing very small applied electrical or magnetic field. By showing high promises of carrier mediated spin-charge transport, DMS materials have raised a lot of enthusiasm in scientific community in the recent years to extend the established CMOS (Complementary Metal Oxide Semiconductor) based electronics towards spintronics devices. \\

\section{Dilute Magnetic Semiconductors (DMS)}

Dilute magnetic semiconductors are actually non – magnetic materials containing very small amounts of magnetic ions as dopants. Due to these magnetic dopants, the non – magnetic materials show dilute ferromagnetism. But the basic difference between DMS and other ferromagnetic materials is DMS’s half metallic behavior which alone controls the spin-charge transport in these materials. Hence, it is noteworthy to revisit the concept of spin polarization and half metallic behavior before going further about DMS. \\

Spin polarization can be expressed as the ratio of the density of states of up-spin and down-spin electrons at fermi level. The degree of polarization is \( P=\frac{N_{up}-N_{down}}{N_{up}+N_{down}} \) where up-spin ($N_{up}$) and down-spin ($N_{down}$) electrons denote the electrons polarized in opposite directions. In paramagnetic materials, the degree of polarization is 0 and in conventional ferromagnetic materials the value is between 0 and 1. Only for the cases of half-metal, the degree of polarization is 1. The schematic density of states of an ideal half metal is given in Fig 1(g). In an ideal half-metal, for specific polarized electrons (either $N_{up}$ or $N_{down}$) will have density of states in fermi level which does not have any gap between conduction and valence band. For other set of polarized electrons, there will be no density of states in fermi level just like an intrinsic semiconductor. This means that all electrons at fermi level in half-metal are completely polarized and for the presence of gapless density of states at fermi level, the polarized electrons will behave like free electrons of metal. For this reason, these materials are called as half-metal. The concept of half-metal was first predicted by Groot \textit{et al.} during the electronic band structure calculation of NiMnSb and PtMnSb \cite{groot1983}. Coey \textit{et al.} explained the half metallic behavior in Cr$O_2$ and predicted that the half-metallicity in semiconductors can be classified in two types as showed in Fig 1 (h) $\&$ (i) \cite{coey2002half}.

\begin{figure}[!htb]
\centering
	\begin{subfigure}[h]{0.32\textwidth}
		\centering
		\includegraphics[width=\linewidth]{dms1}
  		\caption{Magnetic material}
   		\label{fig:sub-first}
	\end{subfigure}
	%\hfill
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dms2}
  		\caption{Non-magnetic material}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dms3}
  		\caption{Dilute magnetic semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_intrinsic_semiconductor}
  		\caption{Intrinsic semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_ntype_semiconductor}
  		\caption{n type semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_ptype_semiconductor}
  		\caption{p type semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_half_metal}
  		\caption{Ideal half metal}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_half_metallic1}
  		\caption{Type-I half metallic semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
	\begin{subfigure}[h]{0.32\textwidth}
  		\centering
  		\includegraphics[width=\linewidth]{dos_half_metallic2}
  		\caption{Type-II half metallic semiconductor}
  		\label{fig:sub-second}
	\end{subfigure}
\caption{(a-c) Difference between ferromagnetic material, non-ferromagnetic material and dilute magnetic semiconductor. (d-i) Density of states of different types of semiconductor, ideal half metal and half-metallic semiconductors}
\label{fig:TEM_PT}

\end{figure}
\FloatBarrier

The bands at fermi level are originated from the delocalized d or f orbital electrons in both types of half-metallic semiconductors. From Fig 1 (h) $\&$ (i), it can be clearly understood that the bands in type I material are comprised of less than half filled d or f orbital electrons while in type II, the bands are comprised of more than half-filled d or f orbital electrons. Although the density of states of half-metallic semiconductor deviate from the gapless states of half-metals, the spin polarized delocalized electrons can be transported by hopping from one site to another \cite{coey2002half}. \\

In dilute magnetic semiconductor, the magnetic dopants have excess electrons which delocalize at fermi level. It is noteworthy to mention that the dilute magnetic semiconductors should have non-zero spin polarization to achieve half-metallicity. If the half-metallic behavior can be made stable at room temperature, only then the DMS materials will be able to bring a paradigm shift in technology by integrated in logic devices, spin polarized light emitting diodes, non – volatile memory storages, spin field effect transistors etc.

\section{Current challenges in DMS research}

The earliest discovery of dilute magnetic semiconductors were reported for Mn doped II-IV and III-V alloys. In 1988, Furdyna \textit{et al.} studied the physical properties of $Cd_{(1-x)}Mn_{x}Se$ and $Hg_{(1-x)}Mn_{x}Te$ but the curie temperature was around 40 K \cite{furdyna1988diluted}. In 1989, Munekata \textit{et al.} observed the properties of dilute magnetic semiconductor in $Ga_{(1-x)}Mn_{x}As$ while the curie temperature was around 100 K \cite{munekata1989diluted}. Since then half metallic behavior in these materials specially in $Ga_{(1-x)}Mn_{x}As$ was deliberately studied but the curie temperature could not be raised above 170 K. The quest for a room temperature DMS got a new dimension when Matsumoto \textit{et al.} reported extrinsic ferromagnetism in $Ti_{(1-x)}Co_{x}O_{2}$ which has very high curie temperature (600 K) \cite{matsumoto2001room}. To this day, dilute ferromagnetism at room temperature has been observed in non-magnetic oxides like ZnO, $In_{2}O_{3}$, $CeO_{2}$, $SnO_{2}$, $CuO_{2}$ etc \cite{yi2010ferromagnetism, ueda2001magnetic, ogale2003high, philip2004high}. But the concept of making DMS by doping of magnetic impurities was changed when room temperature ferromagnetism was observed in undoped non-magnetic oxides. Besides, magnetic clusters and secondary phases have been found to contribute to extrinsic ferromagnetism in these oxides \cite{sundaresan2006ferromagnetism, hong2006room, venkatesan2004thin}. Therefore, creation of any spintronics device based on these oxide based DMS materials remains halted due to the true understanding of the origin of ferromagnetism in these materials.

\section{Objective of this research}

The aim of this research is focused on the dilute ferromagnetism in $Ti_{(1-x)}Sm_{x}O_{2}$ (0 $\leq x \leq$ 0.2) nanoparticles. Among the wide bandgap semiconductors, Ti$O_{2}$ is the most extensively studied material for its superior chemical stability and high curie temperature. The primary reason for $Sm^{3+}$ substitution in Ti$O_{2}$ is to create oxygen vacancy. The electronic configuration of Sm and Ti are [Xe] $4f^{6}$ $6s^{2}$ and [Ar] $3d^{2}$ $4s^{2}$ respectively. But the preference of $Sm^{3+}$ ($Sm^{3+}$=[Xe] $4f^{5}$) substitution stems from the fact that it has odd number of excess electrons when it substitutes $Ti^{4+}$ in Ti$O_{2}$. The number of excess electrons is important because they are supposed to be delocalized and induced non-zero spin polarization at fermi level. The delocalized electrons tend to pair themselves by crystal field splitting for energy minimization. For example, $Fe^{2+}$ ([Ar] $3d^{6}$) has four unpaired d oribital electrons but after crystal field splitting $Fe^{2+}$ tends to have 3 pairs of d electrons which in turns yield near to zero spin polarization. On another case, $Fe^{3+}$ ([Ar] $3d^{5}$) five unpaired electrons in valence band. Because having odd number of electrons, it is impossible to nullify the net spin polarization of $Fe^{3+}$ by electron's ownselves pairing mechanism. However, in this research $Sm^{3+}$ was chosen for its odd numbers of f orbital electrons in valence band which are supposed to exhibit strong spin-orbit interaction. \\

Another reason of $Sm^{3+}$ substitution in Ti$O_{2}$ is its larger ionic radius ($Sm^{3+}$=1.08 $\AA$ and $Ti^{4+}$=0.68 $\AA$). One of the main goals of this research is to investigate what happens beyond the solid solubility limit of $Sm^{3+}$ substitution. As it is previously stated that there are discrepancies on the origin of ferromagnetism in oxide DMS materials which mostly revolve around the nature of substitution. Hence it is crucial to investigate the structural purity beyond solid solubility limit of substitution such as the formation $\&$ growth of the interface of any 2nd phase and their effect on optical and magnetic properties. So, the aim of this research can be summarized as:

\begin{itemize}
  \item To study the effect of $Sm^{3+}$ substitution on the structural, optical and magnetic properties of Ti$O_{2}$
  \item To investigate the nature of formation and growth of any second phase beyond solid solubility limit of substitution
\end{itemize}

\section{Thesis overview}

Chapter 2 starts with a brief review on recent progress in dilute magnetic semiconductors and the experimental techniques to investigate the spin polarization in these materials. It will also discuss several theoretical models on the origin of ferromagnetism in DMS. Finally there will be a brief review on recent theoretical and experimental research on Ti$O_{2}$. Chapter 3 describes the experimental methodology and introduces the detailed synthesis process followed by the structural, morphology, optical and magnetic characterization techniques. Chapter 4 discusses the results obtained from experiments and how they support an elucidation of this research which is concisely given in Chapter 5.

\thispagestyle{fancy}

\end{document}

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text:
\subsection{Gauss-Markov theorem}

Mean of errors zero
+ If the model should only have errors on upside or downside for some reason, OLS will not provide this.

Homoscedastic (all have the same variance)
+ The results aren’t biased, but variances etc are

Errors are uncorrelated
+ (this would mean you should add lagged variables etc)

show bias from each GM violation

OLS is BUE under normally distributed errors

OLS is BLUE for non-normally distribed errors

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text:
\chapter{Introduction}
\begin{flushleft}
Emotion is, perhaps, the most critical attribute of living beings that is critical to detect and generate artificially. Its detection always remains a classical well-explored problem.Emotionally-aware systems
would have a rich context from which to make appropriate
decisions about how to interact with the user or adapt their
system response. There exist many approaches for determining human emotions based on facial expression analysis \cite{ioannou}, thermal imaging of faces \cite{kong}, gesture and pose tracking \cite{gunes}, voice intonation \cite{kwon}, etc.We conducted a field study where we collected
participants’ keystrokes, mouse usages and their emotional states via a survey system by inducing emotion through multimedia components.From this data, we extracted important features,
and created classifiers for 10 emotional states. Our top
results include classification of 3-level emotion(positive emotion, negative emotion and neutral emotion), 4-class dominant emotion(amusement, surprise, anger, disgust), dominant user classification etc.
\end{flushleft}

max_stars_repo_path: Thesis Paper/Latex Files of Paper/graduate-thesis/Chapters/Chapter1.tex

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text:
\section{Well ordering and induction}

\label{well-ordering-section}

We begin this section with some important notation. Summation notation\index{summation notation}, written $\sum_{i=1}^{j} i$, represents a sum. Here, $i$ is called the index of the sum, and we add iterations until $i=j$. For example,
\[
\sum_{i=1}^{j} i = 1 + 2 + \ldots + j
\]
Another example:
\[
a_{11} + a_{12} + a_{13}  = \sum_{i=1}^{3} a_{1i}
\]

The following notation is a specific use of summation notation.

\begin{notation}{Summation notation}{summation}
Let $a_{ij}$ be real numbers, and suppose  $1\leq i\leq r$ while $
1\leq j\leq s$. These numbers can be listed in a rectangular array as given by
\begin{equation*}
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1s} \\
a_{21} & a_{22} & \cdots & a_{2s} \\
\vdots & \vdots &  & \vdots \\
a_{r1} & a_{r2} & \cdots & a_{rs}
\end{array}
\end{equation*}

Then $\sum_{j=1}^{s}\sum_{i=1}^{r} a_{ij}$ means to first sum the numbers
in each column (using $i$ as the index) and then to add the sums which result (using $j$ as the index). Similarly,
$\sum_{i=1}^{r}\sum_{j=1}^{s} a_{ij}$ means to sum the vectors in
each row (using $j$ as the index) and then to add the sums which result (using $i$ as the index).
\end{notation}

Notice that since addition is commutative, $\sum_{j=1}^{s}\sum_{i=1}^{r} a_{ij} = \sum_{i=1}^{r}\sum_{j=1}^{s} a_{ij}$.

We now consider the main concept of this section. Mathematical induction and well ordering are two extremely important
principles in math. They are often used to prove significant things which
would be hard to prove otherwise.

\begin{definition}{Well ordered}{well-ordered}
 A set is well ordered if every nonempty subset $S$, contains a
smallest element $z$ having the property that $z\leq x$ for all $x\in S$.
\end{definition}

In particular, the set of natural numbers defined as
\begin{equation*}
\N = \set{1,2,\ldots}
\end{equation*}
is well ordered\index{well ordered}\index{mathematical induction}.

Consider the following proposition.

\begin{proposition}{Well ordered sets}{well-ordered}
Any set of integers larger than a given number is well ordered.
\end{proposition}

This proposition claims that if a set has a lower bound which is a real number,
then this set is well ordered.

Further, this proposition implies the principle of mathematical induction\index{mathematical induction}. The symbol $\Z$ denotes the set of all
integers. Note that if $a$ is an integer, then there are no integers between
$a$ and $a+1$.

\begin{theorem}{Mathematical induction}{math-induction}
 A set $S\subseteq \Z$, having
the property that $a\in S$ and $n+1\in S$ whenever $n\in S$, contains all
integers $x\in \Z$ such that $x\geq a$.
\end{theorem}

\begin{proof}
Let $T$ consist of all integers larger than or equal to $a$
which are not in $S$. The theorem will be proved if $T=\emptyset$. If
$T\neq \emptyset $ then by the well ordering principle, there would have to
exist a smallest element of $T$, denoted as $b$. It must be the case that
$b>a$ since by definition, $a\notin T$. Thus $b\geq a+1$, and so $b-1\geq a$
and $b-1\notin S$ because if $b-1\in $ $S$, then $b-1+1=b\in S$ by the
assumed property of $S$. Therefore, $b-1\in T$ which contradicts the choice
of $b$ as the smallest element of $T$. ($b-1$ is smaller.) Since a
contradiction is obtained by assuming $T\neq \emptyset$, it must be the
case that $T=\emptyset $ and this says that every integer at least as large
as $a$ is also in $S$.
\end{proof}

Mathematical induction is a very useful device for proving theorems about
the integers. The procedure is as follows.

\begin{procedure}{Proof by mathematical induction}{math-induction}
Suppose $S_{n}$ is a statement which is a function of the number $n$, for $n=1,2,\ldots$, and
we wish to show that $S_{n}$ is true for all $n \geq 1$. To do so using mathematical induction\index{mathematical induction}, use the
following steps.
\begin{enumerate}
\item \textbf{Base Case:}\index{base case} Show $S_{1}$ is true.

\item Assume $S_{n}$ is true for some $n$, which is the \textbf{induction hypothesis}\index{induction hypothesis}. Then, using this assumption,
show that $S_{n+1}$ is true.
\end{enumerate}

Proving these two steps shows that $S_{n}$ is true for all $n = 1,2,\ldots$.
\end{procedure}

We can use this procedure to solve the following examples.

\begin{example}{Proving by induction}{proof-by-induction}
Prove by induction that $\sum_{k=1}^{n}k^{2}=\displaystyle
\frac{n(n+1) (2n+1) }{6}$.
\end{example}

\begin{solution}
By Procedure~\ref{proc:math-induction}, we first need to show that this statement is true for $n=1$.
When $n=1$, the statement says that
\begin{eqnarray*}
\sum_{k=1}^{1}k^{2}&=&\frac{1(1+1) (2(1)+1) }{6}\\
&=&\frac{6}{6} \\
&=& 1
\end{eqnarray*}
The sum on the left-hand side also equals $1$, so this equation is true for $n=1$.

Now suppose this formula is valid for some $n\geq
1$ where $n$ is an integer. Hence, the following equation is true.
\begin{equation}
\sum_{k=1}^{n}k^{2}=
\frac{n(n+1) (2n+1) }{6}
\label{induction1}
\end{equation}
We want to show that this is true for $n+1$.

Suppose we add $(n+1)^2$ to both sides of equation {\eqref{induction1}}.
\begin{eqnarray*}
\sum_{k=1}^{n+1}k^{2}&=&\sum_{k=1}^{n}k^{2}+(n+1) ^{2} \\
&=&\frac{n(n+1) (2n+1) }{6}+(n+1) ^{2}
\end{eqnarray*}
The step going from the first to the second line is based on the assumption
that the formula is true for $n$.
Now simplify the expression in the second line,
\begin{equation*}
\frac{n(n+1) (2n+1) }{6}+(n+1) ^{2}
\end{equation*}
This equals
\begin{equation*}
(n+1) (\frac{n(2n+1) }{6}+(n+1)
)
\end{equation*}
and
\begin{equation*}
\frac{n(2n+1) }{6}+(n+1) =\frac{6(n+1)
+2n^{2}+n}{6}=\frac{\allowbreak (n+2) (2n+3) }{6}
\end{equation*}
Therefore,
\begin{equation*}
\sum_{k=1}^{n+1}k^{2}=\frac{\allowbreak (n+1) (n+2)
(2n+3) }{6}=\frac{\allowbreak (n+1) ((
n+1) +1) (2(n+1) +1) }{6}
\end{equation*}
showing the formula holds for $n+1$ whenever it holds for $n$. This proves
the formula by mathematical induction. In other words, this formula is true for all $n = 1, 2, \ldots$.
\end{solution}

Consider another example.

\begin{example}{Proving an inequality by induction}{inequality-proof-by-induction}
Show that for all $n\in \N$, $\displaystyle \frac{1}{2}\cdot
\displaystyle \frac{3}{4}\cdot\ldots\cdot\displaystyle \frac{2n-1}{2n}<\displaystyle
\frac{1}{\sqrt{2n+1}}$.
\end{example}

\begin{solution}
Again we will use the procedure given in Procedure~\ref{proc:math-induction} to prove that this statement is true for all $n$.
Suppose $n=1$. Then the statement says
\begin{equation*}
\frac{1}{2}< \frac{1}{\sqrt{3}}
\end{equation*}
 which is true.

Suppose then that
the inequality holds for $n$. In other words,
\begin{equation*}
 \frac{1}{2}\cdot \frac{3}{4}\cdot\ldots\cdot \frac{2n-1}{2n} < \frac{1}{\sqrt{2n+1}}
\end{equation*}
is true.

Now multiply both sides of this inequality by $\frac{2n+1}{2n+2}$.
This yields
\begin{equation*}
\frac{1}{2}\cdot \frac{3}{4}\cdot\ldots\cdot \frac{2n-1}{2n}\cdot \frac{2n+1}{2n+2}<
\frac{1}{\sqrt{2n+1}}\frac{2n+1}{2n+2}=\frac{\sqrt{2n+1}}{2n+2}
\end{equation*}
The theorem will be proved if this last expression is less than $
\displaystyle\frac{1}{\sqrt{2n+3}}$. This happens if and only if
\begin{equation*}
\paren{\frac{1}{\sqrt{2n+3}}} ^{2}=\frac{1}{2n+3}>\frac{2n+1}{(
2n+2) ^{2}}
\end{equation*}
which occurs if and only if $(2n+2) ^{2}>(2n+3)
(2n+1) $ and this is clearly true which may be seen from
expanding both sides. This proves the inequality.
\end{solution}

Let's review the process just used. If $S$ is the set of integers at least as
large as $1$ for which the formula holds, the first step was to show $1\in S$
and then that whenever $n\in S$, it follows $n+1\in S$. Therefore, by the
principle of mathematical induction, $S$ contains $[1,\infty )\cap \Z
$, all positive integers. In doing an inductive proof of this sort, the set $
S$ is normally not mentioned. One just verifies the steps above.

max_stars_repo_name: selinger/linear-algebra

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text:
\documentclass[a4paper]{article}
\usepackage[a4paper, left=25mm, right=25mm, top=8mm, bottom=8mm, landscape]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{indentfirst}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}

\newcommand{\R}{\mathbb{R}}

\title{CAQM: Convexity Analysis of Quadratic Maps}
\date{}
\author{Anatoly Dymarsky, Elena Gryazina, Sergei Volodin, Boris Polyak}

\begin{document}
\maketitle

\section{Additional functions in {\rm library/}}

\renewcommand{\arraystretch}{1.8}
\hspace{-3em}
\begin{tabular}{|p{35mm}|c|p{50mm}|c|p{30mm}|c|}
	\hline
	\bf Name & \bf Input & \bf Call & \bf Description & \bf Return value & \bf Exception\\\hline
	\bf Random map & {Dimensions n, m} & {\tt get\_random\_f(n, m, is\_complex)} & Generates random map $f$ & {\tt [A, b]} & None\\\hline
	\bf Value at x & The point $x\in X$ & {\tt quadratic\_map(A, b, x)} & Calculates $f(x)$ & $y=f(x)$ & None\\\hline
	\bf Product $c\cdot A$ & Normal vector $c$ & {\tt get\_Ac(A, c)} & Calculates $c\cdot A$ & $A_c=c\cdot A$ & None\\\hline
	\bf Get $H_c$ & $c,y\in\R^m$ & {\tt get\_H\_c(A, b, c, y)} & Calculates $H_c=\left(\begin{array}{cc}A_c & b_c\\b_c' &-(c,y) \end{array}\right)$ & $H_c$ & None\\\hline
	\bf Minimize $z(c)$ & $c,c_+$, step $\beta$ & {\tt minimize\_z\_c(A, b, c, c\_plus, beta\_initial, max\_step)} & Calculates $\inf\limits_{c\in C_-}z(c)$ & {\tt [z, c\_array, z\_array]} & If failed\\\hline
	\bf  $\R^n$ projection & & {\tt project(A, b, c, x\_0, delta\_c, normal, search\_area\_size)} & Projects $c+\Delta c$ to $C_-$ & {\tt [c\_new, lambda]} & If failed\\\hline
	\bf $\mathbb{C}^n$ projection & & {\tt project\_descent(A, b, c, normal\_1, normal\_2)} & Projects $c$ to $C_-$ & {\tt [c\_new, distance]} & If failed\\\hline
	\bf Gradient $\frac{\partial z}{\partial c}$ & Normal $c$ & {\tt get\_dz\_dc(A, b, c)} & Calculates $Q$, $\nabla z(c)$, normal vectors $n_1$, $n_2$ & {\tt [Q, Q\_inv, k, v, lambda\_min, z, dz\_dc, normal\_re, normal\_im, drho\_dc]} & None\\\hline
	\bf Change of basis & $c_+$ & {\tt change\_basis(A, b, c\_plus)} & Finds $\begin{cases}
	x = S(x'+x_0)\\
	y = y' + y_0
	\end{cases}
	$ s.t.
	$\begin{cases}
	c_+\cdot A_0=I\\
	c_+\cdot b_0=0
	\end{cases}$ & {\tt [A\_new, b\_new, x0, y0]} & None\\\hline
	\bf Boundary point $\partial F_c$ & Normal $c$ & {\tt boundary\_point(A, b, c)} & Calculates pre-image $x\colon f(x)\in\partial F_c$ & {\tt [x]} & If $c\cdot A<0$\\\hline
\end{tabular}

\newpage

\section{Numerical parameters in {\rm library/get\_config.m}}

\hspace{-7em}
\renewcommand{\arraystretch}{2.2}
\begin{tabular}{|p{47mm}|p{27mm}|p{34mm}|p{130mm}|c|}
	\hline
	\bf Parameter name & \bf Target function & \bf Purpose & \bf Description & \bf Default value\\\hline
	{\tt Q\_inv\_eps} & {\tt get\_dz\_dc} & Pseudoinverse & Tolerance for {\tt pinv} function for $Q$ matrix & $10^{-5}$ \\\hline
	{\tt c\_plus\_min\_lambda} & {\tt is\_c\_plus} & $\lambda > 0$ check & Minimal eigenvalue which is still considered $> 0$ & $10^{-4}$ \\\hline
	{\tt c\_minus\_lambda\_min} & {\tt is\_nonconvex} & $\lambda > 0$ check & Maximal eigenvalue which is still considered to be $0$ & $10^{-7}$ \\\hline
	{\tt c\_minus\_h\_rank} & {\tt is\_nonconvex} & Rank check & Tolerance for {\tt rank} function in homogeneous case & $10^{-7}$ \\\hline
	{\tt c\_minus\_ortho} & {\tt is\_nonconvex} & Orthogonality criteria & Maximal value of scalar product of $|x_0^T (c \cdot b)|$ s.t. they are still considered orthogonal & $10^{-7}$ \\\hline
	{\tt c\_minus\_f1f2rank} & {\tt is\_nonconvex} & Rank check & Tolerance for {\tt rank} function for $f_1 \nparallel f_2$ check & $10^{-7}$ \\\hline
	{\tt descent\_min\_norm} & {\tt minimize\_z\_c} & Stopping criteria & Minimal norm of a vector (gradient, norm) & $10^{-3}$ \\\hline
	{\tt descent\_max\_z} & {\tt minimize\_z\_c} & Stopping criteria & Maximal value of $z$ & $10^{9}$ \\\hline
	{\tt descent\_rank\_Q} & {\tt minimize\_z\_c} & Stopping criteria & Tolerance for {\tt rank(Q)} check & $10^{-5}$ \\\hline
	{\tt descent\_min\_beta} & {\tt minimize\_z\_c} & Stopping criteria & Minimal step in direction of the gradient & $10^{-15}$ \\\hline
	{\tt descent\_theta} & {\tt minimize\_z\_c} & Step reduction & Multiplicative factor to reduce beta if projection failed & $0.5$ \\\hline
	{\tt descent\_max\_cos} & {\tt minimize\_z\_c} & Stopping criteria & Maximal cos between grad and n s.t. they are still considered to be not parallel & $1-10^{-4}$ \\\hline
	{\tt project\_descent\_max\_dist} & {\tt project\_descent} & Stopping criteria & Maximal tolerable distance to $C_-$ as value of constraint & $10^{-10}$ \\\hline
	{\tt project\_descent\_min\_alpha} & {\tt project\_descent} & Stopping criteria & Minimal step in the direction of the gradient & $10^{-20}$ \\\hline
	{\tt project\_descent\_theta} & {\tt project\_descent} & Step reduction & Multiplicative factor to reduce alpha & $0.4$ \\\hline
	{\tt project\_bisection\_min\_rl} & {\tt project} & Stopping criteria & Minimal value of $|r - l|$ & $10^{-8}$ \\\hline
	{\tt project\_bisection\_min\_value} & {\tt project} & Stopping criteria & Minimal value of distance to $C_-$ s.t. considered inside this set & $10^{-9}$ \\\hline
	{\tt project\_bisection\_max\_value} & {\tt project} & Precondition & Maximal value of the initial distance to $C_-$ s.t. projection is not attempted & $10^{-3}$ \\\hline
\end{tabular}

\end{document}

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