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LaTeX Skill

This skill governs the production of LaTeX output for a researcher in computational and applied mathematics whose work spans numerical optimisation, deep learning theory, electrical impedance tomography (EIT), regularisation theory, data-driven inversion, numerical analysis, PINNs, and related areas. All output must satisfy the mathematical standards of journals such as SIAM Journal on Scientific Computing, Inverse Problems, Mathematics of Computation, and Journal of Computational Physics.

Step 1. Classify the Request

Before writing anything, determine which of the three output modes applies. Check for BEAMER MODE first, since it has the strongest surface signals.

Beamer Mode ← CHECK THIS FIRST

Produce a complete Beamer presentation (.tex file with \documentclass{beamer}) when the request contains any of:

• The words "slides", "presentation", "beamer", "slide deck", "talk", "seminar", "conference talk", or "present my work on".
• A request to prepare a talk from a manuscript, paper draft, or set of results.
• A request for a "title slide", "table of contents slide", or named academic sections such as "introduction slide", "convergence results slide", "numerical experiments slide".

Proceed to Step 2B for the beamer preamble and Step 3B for beamer content rules. Do NOT use the article preamble (Step 2) or article document structure (Step 4) for Beamer Mode output.

Content sourcing for Beamer Mode:

• If the user supplies a complete manuscript or paper: extract content from it faithfully to populate each slide section.
• If the user supplies only a title, abstract, or partial notes: generate a fully structured template with \todo{} placeholders in every slide that lacks content.
• Never fabricate theorems, lemmas, or numerical results that were not provided.

Document Mode

Produce a complete, standalone .tex file (preamble through \end{document}) using \documentclass{article} when the request involves:

• A theorem, lemma, proposition, corollary, definition, or remark, with or without proof.
• A multi-step derivation or mathematical argument.
• A convergence analysis or complexity result.
• An algorithm presented alongside its theoretical justification.
• An introduction, related-work survey, or structured academic section.
• A self-contained report, technical note, or preprint draft.

Save the output as a .tex file and present it to the user for download.

Snippet Mode

Produce raw LaTeX body content only (no \documentclass, no preamble, no \begin{document}) when the request involves:

• A single equation or aligned system of equations.
• An algorithm or pseudocode block in isolation.
• A numerical results table.
• A TikZ figure or pgfplots graph.
• Any fragment intended to be inserted into the user's own template.

Deliver snippet output as a labelled code block in the chat, not as a file, unless the user explicitly requests a file.

When in doubt, ask: "Should I produce a complete standalone document, a snippet, or a Beamer presentation?"

Step 2. Apply the Standard Preamble (Document Mode Only)

Use the preamble below verbatim. Do not improvise or abbreviate it.

\documentclass[11pt,a4paper]{article}

% ---------------------------------------------------------------
% Core mathematics packages
% ---------------------------------------------------------------
\usepackage{amsmath, amssymb, amsthm, mathtools}
\usepackage{bm}            % bold math symbols
\usepackage{dsfont}        % indicator function \mathds{1}

% ---------------------------------------------------------------
% Algorithms
% ---------------------------------------------------------------
\usepackage[ruled,vlined,linesnumbered]{algorithm2e}

% ---------------------------------------------------------------
% Graphics, figures, and plots
% ---------------------------------------------------------------
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{subfigure}     % side-by-side figures
\usepackage{adjustbox}     % fallback rescaling for wide TikZ diagrams

% ---------------------------------------------------------------
% Tables
% ---------------------------------------------------------------
\usepackage{booktabs}
\usepackage{tabularx}
\usepackage{siunitx}       % decimal-aligned columns, SI units

% ---------------------------------------------------------------
% Cross-referencing and citations
% ---------------------------------------------------------------
\usepackage[numbers,sort&compress]{natbib}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage[capitalise,noabbrev]{cleveref}

% ---------------------------------------------------------------
% Typography
% ---------------------------------------------------------------
\usepackage{microtype}
\usepackage{parskip}
\setlength{\parindent}{0pt}
\setlength{\parskip}{6pt}

% ---------------------------------------------------------------
% Page layout
% ---------------------------------------------------------------
\usepackage[margin=2.5cm]{geometry}

% ---------------------------------------------------------------
% Miscellaneous
% ---------------------------------------------------------------
\usepackage{xcolor}
\usepackage{todonotes}     % \todo{} markers for placeholders

% ---------------------------------------------------------------
% Theorem environments (amsthm)
% ---------------------------------------------------------------
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% ---------------------------------------------------------------
% Notation macros (computational and applied mathematics)
% ---------------------------------------------------------------
% Norms and inner products
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\normF}[1]{\left\lVert #1 \right\rVert_{\mathrm{F}}}
\newcommand{\ip}[2]{\left\langle #1,\, #2 \right\rangle}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}

% Calculus and optimisation
\newcommand{\grad}{\nabla}
\newcommand{\Hess}{\nabla^2}
\newcommand{\Lapl}{\Delta}
\newcommand{\divergence}{\operatorname{div}}
\newcommand{\curl}{\operatorname{curl}}

% Function spaces
\newcommand{\Lp}[1]{L^{#1}}
\newcommand{\Sob}[2]{H^{#1}(#2)}
\newcommand{\SobZ}[2]{H^{#1}_0(#2)}

% Operators and matrices
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bJ}{\mathbf{J}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bI}{\mathbf{I}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}

% Probability and statistics
\newcommand{\E}{\mathbb{E}}
\newcommand{\Prob}{\mathbb{P}}
\newcommand{\Var}{\operatorname{Var}}
\newcommand{\Cov}{\operatorname{Cov}}
\newcommand{\indicator}{\mathbf{1}}

% Asymptotic notation
\newcommand{\bigO}[1]{\mathcal{O}\!\left(#1\right)}
\newcommand{\smallO}[1]{o\!\left(#1\right)}

% Common domains
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}

% Regularisation (EIT / inverse problems)
\newcommand{\Reg}{\mathcal{R}}
\newcommand{\Forward}{\mathcal{F}}
\newcommand{\Tikhonov}[3]{\norm{#1 - #2}^2 + #3\,\Reg(#2)}

% Iterates and step sizes
\newcommand{\xk}{x_k}
\newcommand{\alphak}{\alpha_k}
\newcommand{\etak}{\eta_k}

Step 2B. Beamer Preamble (Beamer Mode Only)

Use the preamble below for ALL Beamer presentations. Do NOT mix with the article preamble in Step 2. The two preambles are mutually exclusive.

Theme selection: The default theme is Madrid. If the user specifies a different standard theme (e.g., Berlin, AnnArbor, Warsaw, Copenhagen, Frankfurt, Singapore, Boadilla, CambridgeUS), substitute it in \usetheme{}. Never invent custom themes; use only named Beamer built-in themes.

Color theme: Default is default (matching the chosen outer theme's palette). If the user requests a specific color theme (e.g., dolphin, beaver, crane, orchid, rose, seagull, seahorse, whale, wolverine), apply it with \usecolortheme{}.

\documentclass[aspectratio=169,10pt]{beamer}
% aspectratio=169 gives 16:9 widescreen; use 43 for 4:3 if user requests it.

% ---------------------------------------------------------------
% Theme — change \usetheme{} to any standard Beamer theme name
% ---------------------------------------------------------------
\usetheme{Madrid}
% \usecolortheme{dolphin}   % Uncomment and change to apply a colour theme

% ---------------------------------------------------------------
% Core mathematics
% ---------------------------------------------------------------
\usepackage{amsmath, amssymb, amsthm, mathtools}
\usepackage{bm}

% ---------------------------------------------------------------
% Algorithms — use ONE of the two options below; comment out the other
% ---------------------------------------------------------------
% OPTION A: algorithm2e (recommended for pseudocode with line numbers)
\usepackage[ruled,vlined,linesnumbered]{algorithm2e}

% OPTION B: algorithmicx + algpseudocode (uncomment if preferred)
% \usepackage{algorithmic}
% \usepackage{algorithmicx}
% \usepackage{algpseudocode}

% ---------------------------------------------------------------
% Graphics and plots
% ---------------------------------------------------------------
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{subfigure}

% ---------------------------------------------------------------
% Tables
% ---------------------------------------------------------------
\usepackage{booktabs}
\usepackage{tabularx}

% ---------------------------------------------------------------
% Theorem-like blocks — configurable style
% ---------------------------------------------------------------
% Beamer provides: theorem, lemma, corollary, proof, definition,
% example, block, alertblock, exampleblock — all built-in.
%
% For coloured framed boxes (tcolorbox style), uncomment below:
% \usepackage{tcolorbox}
% \tcbuselibrary{theorems,skins}
% Then define custom tcolorbox environments as needed (see Step 3B).

% ---------------------------------------------------------------
% Footnote citation control
% ---------------------------------------------------------------
\usepackage{perpage}   % resets footnote counter on each slide
\MakePerPage{footnote}

% ---------------------------------------------------------------
% Miscellaneous
% ---------------------------------------------------------------
\usepackage{xcolor}
\usepackage{multicol}  % for two-column slides

% ---------------------------------------------------------------
% Notation macros (shared with article mode for consistency)
% ---------------------------------------------------------------
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\ip}[2]{\left\langle #1,\, #2 \right\rangle}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\grad}{\nabla}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\bigO}[1]{\mathcal{O}\!\left(#1\right)}
\newcommand{\xk}{x_k}
\newcommand{\alphak}{\alpha_k}
\newcommand{\etak}{\eta_k}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bJ}{\mathbf{J}}

% ---------------------------------------------------------------
% Presentation metadata — fill in before \begin{document}
% ---------------------------------------------------------------
\title[Short Title]{Full Title of the Presentation}
\subtitle{Subtitle or Paper Title (if applicable)}
\author[H.~Mohammad]{Hassan Mohammad}
\institute[BUK]{%
  Numerical Optimisation Research Group\\
  Department of Mathematical Sciences\\
  Faculty of Physical Sciences\\
  Bayero University, Kano, Nigeria
}
\date{\today}

Step 3B. Beamer Content Standards (Beamer Mode Only)

3B.1 Section Structure — Flexible Defaults

The eight sections below are the default scaffold. The user may omit any section or reorder them. When a section is omitted, remove its \section{} declaration and all corresponding frames. Never leave an empty \section{} block.

If the user provides only a title or partial content, generate all eight sections with \todo{} placeholder text inside each frame body. If the user provides a full manuscript, populate each section from the manuscript, preserving mathematical notation exactly.

3B.2 Footnote Citations

Beamer Mode uses \footnotemark / \footnotetext{} pairs exclusively. There is NO reference section at the end of the presentation; every cited source appears as a footnote on the slide where it is first cited.

Citation pattern — use verbatim:

% Within slide body text, place the mark:
...as shown by La Cruz et al.\footnotemark{}...

% Immediately before \end{frame}, place the text:
\footnotetext{W.~La Cruz, J.~Mart\'{\i}nez, and M.~Raydan,
  ``Spectral residual method without gradient information for solving large-scale
  nonlinear systems of equations,''
  \textit{Math.\ Comp.}, vol.~75, no.~255, pp.~1429--1448, 2006.}

Rules for footnote citations:

1. Every \footnotemark must have a matching \footnotetext within the same frame.
2. Use \MakePerPage{footnote} (already in the preamble) so the counter resets per slide.
3. If more than two references appear on one slide, consider splitting the slide or using a smaller font for the \footnotetext entries: {\tiny \footnotetext{...}}.
4. Format: Author(s), ``Title,'' \textit{Journal/Proceedings}, vol., no., pp., Year. For books: Author(s), \textit{Title}, Publisher, Year.
5. Never list a reference in a \footnotetext that does not have a corresponding \footnotemark on the same slide.
6. If the user provides BibTeX keys without full details, supply the correct bibliographic entry from knowledge of the standard literature. If genuinely ambiguous, insert: \footnotetext{\todo{Fill in full bibliographic details.}}

3B.3 Theorem-like Blocks

Default (Beamer built-in environments): Use \begin{theorem}, \begin{lemma}, \begin{corollary}, \begin{definition}, \begin{proof} directly inside frames. Beamer styles these automatically with coloured headers matching the chosen theme.

\begin{frame}{Convergence Result}
  \begin{theorem}[Global Convergence]\label{thm:global}
    Let Assumptions~1 and~2 hold. If $\{x_k\}$ is the sequence generated by
    Algorithm~1, then
    \[
      \lim_{k \to \infty} \|F(x_k)\| = 0.
    \]
  \end{theorem}
  \footnotetext{\todo{Citation if theorem is from a paper.}}
\end{frame}

Optional tcolorbox style (activate in preamble): If the user requests coloured framed boxes resembling the sample slides, uncomment the tcolorbox lines in the preamble and add definitions such as:

\usepackage{tcolorbox}
\tcbuselibrary{theorems,skins}
\newtcbtheorem[number within=section]{thm}{Theorem}{%
  colback=blue!5, colframe=blue!40!black,
  fonttitle=\bfseries}{thm}
\newtcbtheorem[number within=section]{lem}{Lemma}{%
  colback=green!5, colframe=green!40!black,
  fonttitle=\bfseries}{lem}

Do NOT load tcolorbox by default; only include it when the user explicitly requests coloured framed boxes.

3B.4 Algorithm Pseudocode on Slides

Use algorithm2e (already loaded) inside a frame with [fragile] option, because verbatim-like environments require it.

\begin{frame}[fragile]{Algorithm: Name of Algorithm}
  \begin{algorithm}[H]
  \caption{AlgorithmName}\label{alg:main}
  \KwIn{Initial point $x_0 \in \R^n$, tolerance $\varepsilon > 0$}
  \KwOut{Approximate solution $x^*$}
  Set $k \leftarrow 0$\;
  \While{$\|F(x_k)\| > \varepsilon$}{
    Compute search direction $d_k$\;
    Find steplength $\alpha_k$ via line search\;
    $x_{k+1} \leftarrow x_k + \alpha_k d_k$\;
    $k \leftarrow k + 1$\;
  }
  \Return{$x_k$}\;
  \end{algorithm}
\end{frame}

If using algorithmicx instead, the frame must also be [fragile]:

\begin{frame}[fragile]{Algorithm: Name}
  \begin{algorithmic}[1]
    \Require $x_0$, $\varepsilon > 0$
    \Ensure $x^*$
    \For{$k = 0, 1, 2, \ldots$}
      \State Compute $d_k$
      \State $x_{k+1} \leftarrow x_k + \alpha_k d_k$
    \EndFor
  \end{algorithmic}
\end{frame}

3B.5 Performance Profile and Numerical Results Slides

For numerical experiments, use a two-column layout to show profiles and tables side by side:

\begin{frame}{Numerical Results — Performance Profiles}
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \begin{figure}
        \includegraphics[width=\linewidth]{fig_iterations.pdf}
        \caption{Number of iterations}
      \end{figure}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{figure}
        \includegraphics[width=\linewidth]{fig_fevals.pdf}
        \caption{Function evaluations}
      \end{figure}
    \end{column}
  \end{columns}
  \vspace{0.3em}
  {\small Dolan--Mor\'{e} performance profiles; higher is better.}
\end{frame}

For tables of numerical results:

\begin{frame}{Numerical Results — Comparison Table}
  \begin{table}
    \centering
    \small
    \begin{tabular}{lrrr}
      \toprule
      Method & Iter. & F-Evals & CPU (s) \\
      \midrule
      Method A & 42 & 89 & \textbf{0.31} \\
      Method B & \textbf{38} & \textbf{76} & 0.45 \\
      \bottomrule
    \end{tabular}
    \caption{Comparison on test problems ($n = 1000$).}
  \end{table}
\end{frame}

3B.6 Frame Construction Rules

1. Every frame must have a non-empty \frametitle{} argument (or use the {Title} short form of \begin{frame}{Title}).
2. Never overload a single frame; limit each slide to one main idea, result, or algorithm step. If content overflows, split into multiple frames with the same section and a subtitle distinguishing them (e.g., "Proof — Part I", "Proof — Part II").
3. Use \pause sparingly; prefer complete slides for printed handouts.
4. Use \alert{} to highlight a single key term or result per frame, not multiple items.
5. Use \begin{itemize} / \begin{enumerate} with no more than five items per frame. Sub-items are allowed but limit nesting to two levels.
6. Equations on slides must be display-style whenever they span more than a short inline fragment. Prefer \[ ... \] over inline $ ... $ for anything non-trivial.
7. Every frame that cites a source must have at least one \footnotemark and its matching \footnotetext (see Section 3B.2).

3B.7 Table of Contents Slide

Use \tableofcontents with [hideallsubsections] to show only section-level entries. For a long talk, use [currentsection] at the start of each section to re-show the TOC with the current section highlighted.

% Main TOC slide (after title)
\begin{frame}{Outline}
  \tableofcontents[hideallsubsections]
\end{frame}

% Optional: section-entry TOC slide at the start of each section
\AtBeginSection[]{
  \begin{frame}{Outline}
    \tableofcontents[currentsection, hideallsubsections]
  \end{frame}
}

Step 3. Mathematical Content Standards

3.1 Theorem-like Environments

Use amsthm environments throughout. Apply consistent label prefixes: thm:, lem:, prop:, cor:, def:, ass:, rem:, ex:, eq:, alg:, fig:, tab:.

Every numbered environment must carry a \label{}. Every \ref{} and \eqref{} must match an existing label. Use \cref{} (from cleveref) in preference to \ref{} wherever the environment type should appear in text. Use \qed or rely on amsthm's automatic QED symbol at the end of proofs. If assumptions are referenced repeatedly, define them as numbered assumption environments.

Example with full proof scaffold:

\begin{assumption}\label{ass:smoothness}
  The function $f \colon \R^n \to \R$ is $L$-smooth: there exists $L > 0$ such that
  \begin{equation}\label{eq:smoothness}
    \norm{\grad f(x) - \grad f(y)} \leq L\norm{x - y}
    \quad \text{for all } x, y \in \R^n.
  \end{equation}
\end{assumption}

\begin{theorem}[Convergence of gradient descent]\label{thm:gd-convergence}
  Suppose \cref{ass:smoothness} holds and $f$ is bounded below. Let $\{x_k\}_{k \geq 0}$
  be the iterates of gradient descent with constant step size $\alpha \in (0, 1/L]$.
  Then
  \begin{equation}\label{eq:gd-rate}
    \min_{0 \leq k \leq K-1} \norm{\grad f(x_k)}^2
    \leq \frac{2\bigl(f(x_0) - f^*\bigr)}{\alpha K},
  \end{equation}
  where $f^* \coloneqq \inf_{x \in \R^n} f(x)$.
\end{theorem}

\begin{proof}
  By $L$-smoothness (\cref{ass:smoothness}), the descent lemma gives
  \[
    f(x_{k+1}) \leq f(x_k) - \frac{1}{2L}\norm{\grad f(x_k)}^2.
  \]
  Summing from $k = 0$ to $K-1$ and dividing by $K$ yields \cref{eq:gd-rate}.
\end{proof}

If the user provides a theorem statement without a proof, insert a scaffold:

\begin{proof}
  \todo{Proof to be completed. Suggested outline: (i) establish the descent lemma;
  (ii) telescope; (iii) divide by $K$.}
\end{proof}

Do not fabricate a proof.

3.2 Equations and Alignment

Use \coloneqq (from mathtools) for definitions. Use \text{where}, \text{for all}, and similar inside display math. Place punctuation inside displayed equations when the surrounding sentence requires it. Never use \text{O} for asymptotic notation; always use \bigO{\cdot} as defined in the preamble.

3.3 Notation Conventions

Be consistent within every document. The conventions below apply unless the user specifies otherwise.

For EIT and inverse problems, adopt the following when not overridden by the user:

• Conductivity distribution: $\sigma \in L^\infty(\Omega)$ with $\sigma \geq \sigma_{\min} > 0$.
• Dirichlet-to-Neumann map: $\Lambda_\sigma \colon H^{1/2}(\partial\Omega) \to H^{-1/2}(\partial\Omega)$.
• Measurement data: $\mathbf{V} \in \R^{m \times n_e}$.
• Tikhonov functional: $\Tikhonov{\Forward(\sigma)}{\mathbf{V}}{\lambda}$ per the macro.

3.4 Algorithm Pseudocode

Use the algorithm2e package (loaded with ruled, vlined, linesnumbered). Number lines whenever the proof or analysis refers to specific steps. Add inline comments with \tcp{}.

\begin{algorithm}[H]
\caption{Iteratively Regularised Gauss--Newton (IRGN)}\label{alg:irgn}
\KwIn{Initial guess $\sigma^{(0)}$; data $\mathbf{V}$;
      regularisation parameters $\{\lambda_k\}$; tolerance $\varepsilon > 0$}
\KwOut{Approximate solution $\sigma^{(K)}$}
Set $k \leftarrow 0$\;
\While{$\norm{\sigma^{(k+1)} - \sigma^{(k)}} > \varepsilon$}{
  Compute Jacobian $\bJ_k \leftarrow \Forward'(\sigma^{(k)})$\;
  \tcp{Solve the linearised regularised subproblem}
  $\sigma^{(k+1)} \leftarrow \arg\min_{\sigma}
    \bigl\|\bJ_k(\sigma - \sigma^{(k)}) - (\mathbf{V} - \Forward(\sigma^{(k)}))\bigr\|^2
    + \lambda_k \Reg(\sigma)$\;
  $k \leftarrow k + 1$\;
}
\Return{$\sigma^{(k)}$}\;
\end{algorithm}

3.5 Tables of Numerical Results

Rules (apply without exception):

1. Use booktabs (\toprule, \midrule, \bottomrule). Never use vertical rules in the body.
2. Bold the best entry in each column with \textbf{}.
3. Report uncertainties as $\mu \pm \sigma$ using \pm.
4. Use siunitx with the S column type for decimal alignment when precision matters.
5. Never allow a table to exceed \linewidth. Choose the construction method by column count:
   • Narrow (<=4 cols): plain tabular
   • Wide (5-7 cols): tabularx with \linewidth
   • Very wide (8+ cols): \resizebox{\linewidth}{!}{...}

Prefer tabularx over \resizebox wherever possible — rescaling reduces font size relative to surrounding text.

See Section 3.5 examples below:

% Narrow table
\begin{table}[ht]
\centering
\caption{Relative reconstruction error. Bold indicates the lowest error.}\label{tab:relerr}
\begin{tabular}{lccc}
\toprule
Method & $\delta = 0.01$ & $\delta = 0.05$ & $\delta = 0.10$ \\
\midrule
Tikhonov ($\ell^2$) & $0.142 \pm 0.008$ & $0.231 \pm 0.011$ & $0.318 \pm 0.014$ \\
TV regularisation   & $\mathbf{0.103 \pm 0.006}$ & $\mathbf{0.187 \pm 0.009}$ & $0.274 \pm 0.013$ \\
\bottomrule
\end{tabular}
\end{table}

% Wide table
\begin{table}[ht]
\centering
\caption{Performance metrics across five noise levels.}\label{tab:perf}
\begin{tabularx}{\linewidth}{l *{5}{>{\centering\arraybackslash}X}}
\toprule
Method & $\delta_1$ & $\delta_2$ & $\delta_3$ & $\delta_4$ & $\delta_5$ \\
\midrule
Method A & val & val & val & val & val \\
\bottomrule
\end{tabularx}
\end{table}

3.6 TikZ Figures and pgfplots Graphs

Every figure must include: axis labels, a legend when multiple series are plotted, a \caption, and a \label. For convergence plots, use ymode=log.

Width policy (mandatory): Every tikzpicture and pgfplots axis must declare an explicit width using a relative length. Never use absolute cm or pt values.

% Standard convergence plot
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{semilogyaxis}[
    xlabel={Iteration $k$},
    ylabel={$f(x_k) - f^*$},
    legend pos=north east,
    grid=major,
    width=0.75\textwidth,
    height=0.5\textwidth
]
\addplot[blue, thick] coordinates { ... };
\addlegendentry{Method A}
\addplot[red, dashed, thick] coordinates { ... };
\addlegendentry{Method B}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Convergence comparison. Vertical axis in logarithmic scale.}\label{fig:convergence}
\end{figure}

% Wide diagram fallback
\begin{figure}[ht]
\centering
\adjustbox{max width=\textwidth}{%
  \begin{tikzpicture}
    % wide architecture diagram
  \end{tikzpicture}%
}
\caption{Network architecture.}\label{fig:arch}
\end{figure}

Step 4. Document Structure

4.1 Theorem or Proof Document

\section{Problem Setup}        % domain, objective, algorithm
\section{Assumptions}          % numbered assumption environments
\section{Main Result}          % preliminary lemmas, then main theorem with proof
\section{Discussion}           % implications, special cases, connections (optional)

4.2 Convergence Analysis

\section{Problem Formulation}       % objective, domain, algorithm statement
\section{Assumptions}               % numbered assumption environments
\section{Main Convergence Theorem}  % theorem + proof
\section{Complexity Analysis}       % oracle and iteration complexity (when applicable)
\section{Numerical Experiments}     % tables and figures

4.3 Derivation Document

\section{Setup}       % define all objects, notation, spaces
\section{Derivation}  % step-by-step with align environments
\section{Result}      % \boxed{} final expression

Highlight a key final result with:

\begin{equation}\label{eq:main}
\boxed{ x_{k+1} = x_k - \alphak \grad f(x_k) }
\end{equation}

4.4 Literature Review or Introduction

Structure:

1. Motivate the problem with context and practical relevance.
2. Survey related work using \citet{} and \citep{}.
3. Identify the gap or limitation addressed by the present work.
4. State contributions precisely (numbered or itemised list).
5. Outline the remainder of the document.

Example:

Early work by \citet{calderon1980} established the theoretical basis for EIT.
Regularisation strategies were studied by \citet{engl1996} and \citet{vogel2002}.
Data-driven approaches have recently gained attention~\citep{hamilton2018, adler2017}.

Step 5. Reference Section (Document Mode)

5.0 Choose a Bibliography Workflow

Ask the user which workflow they use before writing the reference section. If no preference is stated, default to Option A. The options are architecturally incompatible: do not mix commands or packages from both in the same document.

Option A — Self-Contained (thebibliography)

Place immediately before \end{document}. Set the argument to the widest label expected (e.g., {99}). Do not include \bibliographystyle{} or \bibliography{}.

\bibitem Format by Source Type

% Journal article
\bibitem{citekey}
A.~Surname and B.~Surname,
``Title,''
\textit{Journal Name},
vol.~X, no.~Y, pp.~NNN--NNN, Year.

% Conference paper
\bibitem{citekey}
A.~Surname and B.~Surname,
``Title,''
in \textit{Proceedings of the Conference (ACRONYM)},
City, Country, Year, pp.~NNN--NNN.

% Book
\bibitem{citekey}
A.~Surname,
\textit{Title of the Book},
Publisher, City, Year.

% PhD / MSc thesis
\bibitem{citekey}
A.~Surname,
``Title of the thesis,''
Ph.D.\ dissertation, Department, University, City, Country, Year.

% Technical report
\bibitem{citekey}
A.~Surname,
``Title,''
Tech.\ Rep.\ TR-XXXX, Institution, Year.

% arXiv preprint
\bibitem{citekey}
A.~Surname and B.~Surname,
``Title,''
\textit{arXiv preprint} arXiv:XXXX.XXXXX, Year.

If the user cites a key without providing bibliographic details, populate from knowledge of the standard literature. For well-known works, supply complete and accurate entries. If genuinely ambiguous, insert:

\bibitem{citekey}
\todo{Fill in full bibliographic details for \texttt{citekey}.}

Do not omit the \bibitem; a missing entry causes a compilation error.

Option B1 — BibTeX with natbib

Replace the natbib preamble line with:

\usepackage[numbers,sort&compress]{natbib}

End-of-document block:

\bibliographystyle{plainnat}   % or: unsrtnat, ieeetr, siam, plain
\bibliography{refs}

Compilation: pdflatex → bibtex → pdflatex × 2.

Common .bst files: plainnat (author-year), plain (numbered), ieeetr (IEEE), siam (SIAM), amsplain (AMS), unsrt (citation order).

Option B2 — BibLaTeX with Biber

Remove \usepackage{natbib} entirely. Replace with:

\usepackage[
  backend=biber,
  style=authoryear,
  sorting=nyt,
  maxbibnames=99,
  giveninits=true,
  doi=false,
  url=false,
  eprint=true
]{biblatex}
\addbibresource{refs.bib}

End-of-document: \printbibliography. Do not use \bibliographystyle{} or \bibliography{}.

Compilation: pdflatex (or lualatex) → biber → pdflatex × 2.

Citation commands: \parencite{} (parenthetical), \textcite{} (textual), \citeyear{} (year only), \citeauthor{} (author only).

Sample .bib entry:

@article{nesterov1983,
  author  = {Nesterov, Yurii},
  title   = {A method for solving the convex programming problem with convergence
             rate {$\mathcal{O}(1/k^2)$}},
  journal = {Doklady Akademii Nauk SSSR},
  volume  = {269},
  number  = {3},
  pages   = {543--547},
  year    = {1983}
}

If the user cites a key without a .bib entry, supply the correct entry from knowledge of the standard literature. If genuinely ambiguous, insert:

@misc{citekey,
  note = {TODO: Fill in full bibliographic details.}
}

Step 6. Writing Quality Standards

6.1 Prohibited Characters and Typographic Flags

The following must never appear in any output:

6.2 Sentence and Paragraph Construction

Write in complete, declarative sentences. Every sentence carries one identifiable claim or step. Paragraph breaks signal a shift in focus.

Prefer active constructions: "We derive an upper bound" over "An upper bound is derived." Reserve the passive for results independent of the authors: "The problem is ill-posed in the sense of Hadamard."

Avoid vague intensifiers: "very", "quite", "rather", "highly", "extremely". Quantify where possible: "the condition number grows as $\bigO{h^{-2}}$".

6.3 Mathematical Prose

Every displayed equation referred to subsequently must be labelled and introduced by a complete grammatical sentence. Treat the equation as part of the sentence with appropriate punctuation.

Define every symbol before or at its first use. When citing a result, state precisely which part of the cited work is being used (e.g., \citet[Theorem~2.1]{engl1996}). Avoid vague attributions such as "as shown in [3]".

Place quantitative conditions in numbered assumption environments rather than burying them inside theorem statements, whenever those conditions are reusable across multiple results.

6.4 Consistency Checks

Before outputting any document, verify:

1. Every symbol introduced in the preamble is used at least once in the body; remove unused macros.
2. The same physical quantity uses the same symbol throughout; no silent switching.
3. All theorem environments are closed; all proofs end with \end{proof}.
4. Every \begin{} has a matching \end{}.
5. No conflicting packages (e.g., amsmath not loaded twice; algorithm2e and algorithmic not both loaded).

Step 7. Pre-Output Quality Checklist

Run through all items below before producing the final output.

Compilability

• Every \begin{} has a matching \end{}.
• No undefined control sequences.
• All required packages loaded in the preamble.
• No package conflicts.

Label Consistency

• Every numbered environment has a \label{}.
• Every \ref{}, \eqref{}, \cref{} resolves to an existing label.
• No label defined more than once.

Notation Consistency

• Same symbol used for the same object throughout.
• Step sizes, regularisation parameters, and iterates follow Section 3.3 conventions.
• All macros used in the body are defined in the preamble.

Mathematical Correctness

• Inequalities point in the correct direction.
• Convergence rates and complexity bounds are dimensionally consistent.
• Cited results are used correctly and not misrepresented.
• Proof steps follow logically from stated assumptions.

Bibliography Completeness

• Option A: Every \cite{} key has a fully populated \bibitem{}; no orphan entries.
• Option B1: Every cited key exists in the .bib file; \bibliographystyle{} and \bibliography{} present; biblatex not loaded.
• Option B2: Every cited key in the .bib file; \printbibliography present; no \bibliographystyle{} or \bibliography{}; natbib not loaded.

Margin Safety

• No plain tabular with more than 4 columns unless wrapped in tabularx or \resizebox.
• Every pgfplots axis uses a relative width; no absolute cm or pt values.
• Every free-standing wide tikzpicture wrapped in \adjustbox{max width=\textwidth}.
• Side-by-side subfigures use width=\linewidth inside their subfigure environment.

Writing Quality

• No em dashes, en dashes (outside LaTeX ranges), or prose ellipses.
• No contractions, exclamation marks, or rhetorical questions.
• No prohibited transitional intensifiers.
• Every symbol defined before or at first use.
• Every displayed equation introduced by a complete grammatical sentence with correct punctuation.

Step 8. Handling Ambiguous or Incomplete Requests

Missing proof

If the user requests "write up this theorem" without providing a proof, produce the theorem statement and a \begin{proof}...\end{proof} scaffold with \todo{} markers. Do not construct a proof that was not provided.

Partial notation

If the user provides incomplete notation or an unfinished derivation, ask exactly one clarifying question before proceeding. Do not guess at notation.

Mixed content

If the request combines document and snippet content, produce a complete document and include all elements within it.

User-provided notation

If the user provides existing mathematical content, preserve their notation and mathematical choices exactly. Normalise only formatting and typographic conventions; do not alter the mathematics or rename symbols.

Step 9. Domain-Specific Reference Entries

The following \bibitem entries cover foundational works in EIT, regularisation theory, inverse problems, numerical analysis, deep learning, PINNs, and numerical optimisation. Insert directly when the corresponding key is cited.

% --- Inverse problems and EIT ---

\bibitem{calderon1980}
A.~P.~Calder\'{o}n,
``On an inverse boundary value problem,''
in \textit{Seminar on Numerical Analysis and its Applications to Continuum Physics},
Rio de Janeiro, Brazil, 1980, pp.~65--73.

\bibitem{cheney1990}
M.~Cheney, D.~Isaacson, J.~C.~Newell, S.~Simske, and J.~Goble,
``NOSER: An algorithm for solving the inverse conductivity problem,''
\textit{International Journal of Imaging Systems and Technology},
vol.~2, no.~2, pp.~66--75, 1990.

\bibitem{engl1996}
H.~W.~Engl, M.~Hanke, and A.~Neubauer,
\textit{Regularization of Inverse Problems},
Kluwer Academic Publishers, Dordrecht, 1996.

\bibitem{vogel2002}
C.~R.~Vogel,
\textit{Computational Methods for Inverse Problems},
SIAM, Philadelphia, PA, 2002.

\bibitem{kaltenbacher2008}
B.~Kaltenbacher, A.~Neubauer, and O.~Scherzer,
\textit{Iterative Regularization Methods for Nonlinear Ill-Posed Problems},
de Gruyter, Berlin, 2008.

\bibitem{borcea2002}
L.~Borcea,
``Electrical impedance tomography,''
\textit{Inverse Problems},
vol.~18, no.~6, pp.~R99--R136, 2002.

% --- Regularisation and optimisation ---

\bibitem{tikhonov1943}
A.~N.~Tikhonov,
``On the stability of inverse problems,''
\textit{Doklady Akademii Nauk SSSR},
vol.~39, no.~5, pp.~195--198, 1943.

\bibitem{rudin1992}
L.~I.~Rudin, S.~Osher, and E.~Fatemi,
``Nonlinear total variation based noise removal algorithms,''
\textit{Physica D: Nonlinear Phenomena},
vol.~60, no.~1--4, pp.~259--268, 1992.

\bibitem{nesterov1983}
Y.~Nesterov,
``A method for solving the convex programming problem with convergence rate
$\mathcal{O}(1/k^2)$,''
\textit{Doklady Akademii Nauk SSSR},
vol.~269, no.~3, pp.~543--547, 1983.

\bibitem{nocedal2006}
J.~Nocedal and S.~J.~Wright,
\textit{Numerical Optimization},
2nd~ed.,
Springer, New York, NY, 2006.

\bibitem{boyd2004}
S.~Boyd and L.~Vandenberghe,
\textit{Convex Optimization},
Cambridge University Press, Cambridge, 2004.

% --- Deep learning for inverse problems ---

\bibitem{adler2017}
J.~Adler and O.~\"{O}ktem,
``Solving ill-posed inverse problems using iterative deep neural networks,''
\textit{Inverse Problems},
vol.~33, no.~12, p.~124007, 2017.

\bibitem{hamilton2018}
S.~J.~Hamilton and A.~Hauptmann,
``Deep d-bar: Real-time electrical impedance tomography imaging with deep
neural networks,''
\textit{IEEE Transactions on Medical Imaging},
vol.~37, no.~10, pp.~2367--2377, 2018.

\bibitem{fan2020}
Y.~Fan and L.~Ying,
``Solving traveltime tomography with deep learning,''
\textit{Research in the Mathematical Sciences},
vol.~7, no.~3, p.~17, 2020.

\bibitem{kingmaba2015}
D.~P.~Kingma and J.~Ba,
``Adam: A method for stochastic optimization,''
in \textit{Proceedings of the 3rd International Conference on Learning
Representations (ICLR)},
San Diego, CA, USA, 2015.

\bibitem{goodfellow2016}
I.~Goodfellow, Y.~Bengio, and A.~Courville,
\textit{Deep Learning},
MIT Press, Cambridge, MA, 2016.

% --- PINNs and scientific machine learning ---

\bibitem{raissi2019}
M.~Raissi, P.~Perdikaris, and G.~E.~Karniadakis,
``Physics-informed neural networks: A deep learning framework for solving forward
and inverse problems involving nonlinear partial differential equations,''
\textit{Journal of Computational Physics},
vol.~378, pp.~686--707, 2019.

\bibitem{lagaris1998}
I.~E.~Lagaris, A.~Likas, and D.~I.~Fotiadis,
``Artificial neural networks for solving ordinary and partial differential equations,''
\textit{IEEE Transactions on Neural Networks},
vol.~9, no.~5, pp.~987--1000, 1998.

% --- Numerical analysis ---

\bibitem{golub2013}
G.~H.~Golub and C.~F.~Van~Loan,
\textit{Matrix Computations},
4th~ed.,
Johns Hopkins University Press, Baltimore, MD, 2013.

\bibitem{trefethen1997}
L.~N.~Trefethen and D.~Bau,
\textit{Numerical Linear Algebra},
SIAM, Philadelphia, PA, 1997.

\bibitem{brenner2008}
S.~C.~Brenner and L.~R.~Scott,
\textit{The Mathematical Theory of Finite Element Methods},
3rd~ed.,
Springer, New York, NY, 2008.

Step 10. Complete Document Template

Replace all placeholder text with actual content.

\documentclass[11pt,a4paper]{article}

% [Insert full preamble from Step 2 here]

\title{Title of the Document}
\author{Author Name\\
  Department, Institution, City, Country\\
  \texttt{email@example.com}}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
  A concise summary (150--250 words) of the problem, main results, methods used, and
  significance. Written in the third person; past tense for results, present tense for
  conclusions.
\end{abstract}

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

\section{Problem Formulation}\label{sec:problem}

\section{Assumptions}\label{sec:assumptions}

\begin{assumption}\label{ass:main}
  [State the assumption precisely.]
\end{assumption}

\section{Main Results}\label{sec:results}

\begin{theorem}[Descriptive title]\label{thm:main}
  Under \cref{ass:main}, \ldots
\end{theorem}

\begin{proof}
  [Proof.]
\end{proof}

\section{Numerical Experiments}\label{sec:experiments}

\section{Conclusion}\label{sec:conclusion}

% --- CHOOSE ONE ending below; delete the other two ---

% OPTION A: Self-contained
\begin{thebibliography}{99}
% [Populate \bibitem entries from Step 5 Option A and Step 9.]
\end{thebibliography}

% OPTION B1: BibTeX + natbib
% \bibliographystyle{plainnat}
% \bibliography{refs}

% OPTION B2: BibLaTeX + Biber
% \printbibliography

\end{document}

Step 10B. Complete Beamer Presentation Template (Beamer Mode Only)

This is the canonical full-file scaffold for a Beamer presentation. Populate every section from the user's manuscript or insert \todo{} placeholders for missing content. Omit or reorder any section the user does not need (see Section 3B.1 for the section table). Compile with pdflatex (two passes to resolve cross-references).

\documentclass[aspectratio=169,10pt]{beamer}

% ---------------------------------------------------------------
% Theme — substitute any standard Beamer theme name
% ---------------------------------------------------------------
\usetheme{Madrid}
% \usecolortheme{dolphin}

% ---------------------------------------------------------------
% Core mathematics
% ---------------------------------------------------------------
\usepackage{amsmath, amssymb, amsthm, mathtools}
\usepackage{bm}

% ---------------------------------------------------------------
% Algorithms
% ---------------------------------------------------------------
\usepackage[ruled,vlined,linesnumbered]{algorithm2e}

% ---------------------------------------------------------------
% Graphics
% ---------------------------------------------------------------
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{subfigure}

% ---------------------------------------------------------------
% Tables
% ---------------------------------------------------------------
\usepackage{booktabs}
\usepackage{tabularx}

% ---------------------------------------------------------------
% Footnote citation control
% ---------------------------------------------------------------
\usepackage{perpage}
\MakePerPage{footnote}

% ---------------------------------------------------------------
% Miscellaneous
% ---------------------------------------------------------------
\usepackage{xcolor}
\usepackage{multicol}

% ---------------------------------------------------------------
% Notation macros
% ---------------------------------------------------------------
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\ip}[2]{\left\langle #1,\, #2 \right\rangle}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\grad}{\nabla}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\bigO}[1]{\mathcal{O}\!\left(#1\right)}
\newcommand{\xk}{x_k}
\newcommand{\alphak}{\alpha_k}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bJ}{\mathbf{J}}

% ---------------------------------------------------------------
% Presentation metadata
% ---------------------------------------------------------------
\title[Short Title]{Full Title of the Presentation}
\subtitle{Subtitle or Paper Title (if applicable)}
\author[H.~Mohammad]{Hassan Mohammad}
\institute[BUK]{%
  Numerical Optimisation Research Group\\
  Department of Mathematical Sciences\\
  Faculty of Physical Sciences\\
  Bayero University, Kano, Nigeria
}
\date{\today}

% ---------------------------------------------------------------
% Optional: re-show TOC at each section start
% ---------------------------------------------------------------
% \AtBeginSection[]{
%   \begin{frame}{Outline}
%     \tableofcontents[currentsection, hideallsubsections]
%   \end{frame}
% }

\begin{document}

% =============================================================
% FRAME 1 — Title Page
% =============================================================
\begin{frame}
  \titlepage
\end{frame}

% =============================================================
% FRAME 2 — Table of Contents
% =============================================================
\begin{frame}{Outline}
  \tableofcontents[hideallsubsections]
\end{frame}

% =============================================================
% SECTION 1: Introduction
% =============================================================
\section{Introduction}

\begin{frame}{Introduction}
  \begin{itemize}
    \item \todo{State the problem and its practical or theoretical motivation.}
    \item \todo{Describe the specific problem considered in this work.}
    \item \todo{Outline the main contributions.}
  \end{itemize}
\end{frame}

\begin{frame}{Problem Formulation}
  We consider the problem of finding $x^* \in \R^n$ such that
  \[
    F(x^*) = 0,
  \]
  where $F \colon \R^n \to \R^n$ \todo{[describe properties of $F$]}.

  The iterative scheme has the form
  \[
    x_{k+1} = x_k + \alphak d_k, \quad k = 0, 1, 2, \ldots,
  \]
  where $\alphak > 0$ is the steplength and $d_k$ is the search direction.
\end{frame}

% =============================================================
% SECTION 2: Related Work / Literature Review / Motivation
% =============================================================
\section{Related Work}

\begin{frame}{Related Work}
  \begin{itemize}
    \item \todo{Author(s) (Year)\footnotemark{} — brief description of contribution.}
    \item \todo{Author(s) (Year)\footnotemark{} — brief description.}
    \item \todo{Author(s) (Year)\footnotemark{} — brief description.}
  \end{itemize}
  \footnotetext{\todo{Full citation 1.}}
  \footnotetext{\todo{Full citation 2.}}
  \footnotetext{\todo{Full citation 3.}}
\end{frame}

\begin{frame}{Motivation and Gap}
  \begin{itemize}
    \item \todo{Identify the limitation in existing methods that this work addresses.}
    \item \todo{State why the proposed approach is needed.}
  \end{itemize}
\end{frame}

% =============================================================
% SECTION 3: Methodology / Algorithm
% =============================================================
\section{Methodology}

\begin{frame}{Assumptions}
  \begin{block}{Assumption 1}
    \todo{State Assumption 1 precisely (e.g., Lipschitz continuity of $F$).}
  \end{block}
  \vspace{0.5em}
  \begin{block}{Assumption 2}
    \todo{State Assumption 2 precisely (e.g., boundedness of the Jacobian).}
  \end{block}
\end{frame}

\begin{frame}{Search Direction}
  The search direction $d_k$ is defined by
  \[
    d_k = \begin{cases}
      \todo{[initial direction]}, & k = 0, \\
      \todo{[recursive formula]}, & k \geq 1,
    \end{cases}
  \]
  where \todo{define all parameters}.
\end{frame}

\begin{frame}[fragile]{Algorithm}
  \begin{algorithm}[H]
  \caption{\todo{AlgorithmName}}\label{alg:main}
  \KwIn{\todo{Inputs: initial point, tolerances, parameters}}
  \KwOut{\todo{Output}}
  Set $k \leftarrow 0$\;
  \While{$\norm{F(x_k)} > \varepsilon$}{
    Compute search direction $d_k$\;
    Find steplength $\alphak$ via line search\;
    $x_{k+1} \leftarrow x_k + \alphak d_k$\;
    $k \leftarrow k + 1$\;
  }
  \Return{$x_k$}\;
  \end{algorithm}
\end{frame}

% =============================================================
% SECTION 4: Convergence Analysis
% =============================================================
\section{Convergence Analysis}

\begin{frame}{Preliminary Lemmas}
  \begin{lemma}\label{lem:prelim1}
    \todo{State Lemma 1.}
  \end{lemma}
  \vspace{0.5em}
  \begin{lemma}\label{lem:prelim2}
    \todo{State Lemma 2.}
  \end{lemma}
\end{frame}

\begin{frame}{Main Convergence Theorem}
  \begin{theorem}[Global Convergence]\label{thm:global}
    \todo{Under Assumptions 1 and 2, if $\{x_k\}$ is generated by the algorithm, then
    state the convergence result.}
  \end{theorem}
  \vspace{0.5em}
  \begin{proof}[Proof sketch]
    \todo{Outline the main steps of the proof.}
  \end{proof}
\end{frame}

\begin{frame}{Rate of Convergence}
  \begin{theorem}[R-linear Convergence]\label{thm:rate}
    \todo{State the rate-of-convergence theorem. For example:
    there exist $C > 0$ and $\mu \in (0,1)$ such that
    $\norm{x_k - x^*} \leq C\mu^k$.}
  \end{theorem}
\end{frame}

% =============================================================
% SECTION 5: Numerical Experiments
% =============================================================
\section{Numerical Experiments}

\begin{frame}{Experimental Setup}
  \begin{itemize}
    \item \todo{Describe the test problems, dimension, and parameter settings.}
    \item \todo{List the competing methods.}
    \item \todo{State the stopping criterion and hardware/software details.}
  \end{itemize}
\end{frame}

\begin{frame}{Performance Profiles}
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \begin{figure}
        \includegraphics[width=\linewidth]{\todo{fig\_iterations}}
        \caption{Number of iterations}
      \end{figure}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{figure}
        \includegraphics[width=\linewidth]{\todo{fig\_fevals}}
        \caption{Function evaluations}
      \end{figure}
    \end{column}
  \end{columns}
  {\small Dolan--Mor\'{e} performance profiles;\footnotemark{} higher curve = better.}
  \footnotetext{E.~D.~Dolan and J.~J.~Mor\'{e},
    ``Benchmarking optimization software with performance profiles,''
    \textit{Math.\ Program.}, vol.~91, no.~2, pp.~201--213, 2002.}
\end{frame}

\begin{frame}{Results Summary}
  \begin{table}
    \centering
    \small
    \begin{tabular}{lrrr}
      \toprule
      Method & Iter. & F-Evals & CPU (s) \\
      \midrule
      \todo{Method A} & \todo{--} & \todo{--} & \todo{--} \\
      \todo{Method B} & \todo{--} & \todo{--} & \todo{--} \\
      \bottomrule
    \end{tabular}
    \caption{\todo{Caption describing the table.}}
  \end{table}
\end{frame}

% =============================================================
% SECTION 6: Conclusion / Further Research
% =============================================================
\section{Conclusion}

\begin{frame}{Concluding Remarks}
  \begin{itemize}
    \item \todo{Summarise what was presented.}
    \item \todo{State the main theoretical guarantee established.}
    \item \todo{Comment on practical performance.}
  \end{itemize}
\end{frame}

\begin{frame}{Future Research Questions}
  \begin{enumerate}
    \item \todo{Open question 1.}
    \item \todo{Open question 2.}
    \item \todo{Open question 3.}
  \end{enumerate}
\end{frame}

% =============================================================
% CLOSING FRAME
% =============================================================
\begin{frame}[plain]
  \begin{center}
    {\Large \textbf{Thank You}}\\[1em]
    \todo{Author name}\\
    \texttt{\todo{email@institution.edu}}
  \end{center}
\end{frame}

\end{document}

Step 10C. Beamer Pre-Output Quality Checklist (Beamer Mode Only)

Run through all items below before delivering the final Beamer .tex file.

Compilability

• \documentclass{beamer} is present; NOT \documentclass{article}.
• The perpage package is loaded and \MakePerPage{footnote} is called.
• Every frame with algorithm2e or verbatim content uses [fragile].
• No geometry, cleveref, or natbib packages are loaded (incompatible with beamer).

Frame Structure

• Every \begin{frame} has a matching \end{frame}.
• Every frame has a non-empty title (\begin{frame}{Title} or \frametitle{}).
• No frame contains more than five itemize entries at the top level.
• No frame body exceeds approximately 10 lines of display content (split if necessary).

Footnote Citation Consistency

• Every \footnotemark on a slide has a matching \footnotetext within the SAME frame.
• No \footnotetext is present without a corresponding \footnotemark on the same slide.
• No bibliography section (\begin{thebibliography}, \printbibliography) is present.

Mathematical Correctness

• All macros used in the body are defined in the preamble.
• Same notation used throughout; no silent symbol reassignment between frames.
• Every theorem, lemma, and definition that comes from the user's manuscript is reproduced faithfully; no content is fabricated.

Placeholder Hygiene

• All \todo{} markers represent genuinely missing information (not leftover from copy-paste). If the user provided the content, it must appear — not a \todo{}.

Reference Files

• references/preamble.md — Legacy preamble reference. The canonical preamble is now embedded in Step 2. Consult only for additional macro variants not covered in Step 2.