\begin
Opdag hvorfor 18 millioner mennesker verden rundt stoler på Overleaf med deres arbejde.
% --- LaTeX Presentation Template - S. Venkatraman ---
% --- Set document class ---
% Remove "handout" when presenting to include pauses
\documentclass[dvipsnames, handout]{beamer}
\usetheme{default}
% Make content that is hidden by pauses "transparent"
\setbeamercovered{transparent}
% --- Slide layout settings ---
% Set line spacing
\renewcommand{\baselinestretch}{1.15}
% Set left and right text margins
\setbeamersize{text margin left=12mm, text margin right=12mm}
% Add slide numbers in bottom right corner
\setbeamertemplate{footline}[frame number]
% Remove navigation symbols
\setbeamertemplate{navigation symbols}{}
% Allow local line spacing changes
\usepackage{setspace}
% Change itemized list bullets to circles
\setbeamertemplate{itemize item}{$\bullet$}
\setbeamertemplate{itemize subitem}{$\circ$}
% --- Color and font settings ---
\usepackage{xcolor}
% Slide title background color
\definecolor{background}{HTML}{ede6d8}
% Slide title text color
\definecolor{titleText}{HTML}{B40404}
% Other possible color schemes
% - Light green/dark green -
%\definecolor{background}{HTML}{e4ede4}
%\definecolor{titleText}{HTML}{2e592f}
% - Light blue/dark blue -
%\definecolor{background}{HTML}{d5d9e8}
%\definecolor{titleText}{HTML}{2d375e}
% - Beige/dark blue -
%\definecolor{background}{HTML}{e8e2d5}
%\definecolor{titleText}{HTML}{2d3375}
% Set colors
\setbeamercolor{frametitle}{bg=background, fg=titleText}
\setbeamercolor{subtitle}{fg=titleText}
% Set font sizes for frame title and subtitle
\setbeamerfont{frametitle}{size=\fontsize{15}{16}}
\setbeamerfont{framesubtitle}{size=\small}
% --- Math/Statistics commands ---
% Add a reference number to a single line of a multi-line equation
% Usage: "\numberthis\label{labelNameHere}" in an align or gather environment
\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
% Shortcut for bold text in math mode, e.g. $\b{X}$
\let\b\mathbf
% Shortcut for bold Greek letters, e.g. $\bg{\beta}$
\let\bg\boldsymbol
% Shortcut for calligraphic script, e.g. %\mc{M}$
\let\mc\mathcal
% \mathscr{(letter here)} is sometimes used to denote vector spaces
\usepackage[mathscr]{euscript}
% Convergence: right arrow with optional text on top
% E.g. $\converge[p]$ for converges in probability
\newcommand{\converge}[1][]{\xrightarrow{#1}}
% Weak convergence: harpoon symbol with optional text on top
% E.g. $\wconverge[n\to\infty]$
\newcommand{\wconverge}[1][]{\stackrel{#1}{\rightharpoonup}}
% Equality: equals sign with optional text on top
% E.g. $X \equals[d] Y$ for equality in distribution
\newcommand{\equals}[1][]{\stackrel{\smash{#1}}{=}}
% Normal distribution: arguments are the mean and variance
% E.g. $\normal{\mu}{\sigma}$
\newcommand{\normal}[2]{\mathcal{N}\left(#1,#2\right)}
% Uniform distribution: arguments are the left and right endpoints
% E.g. $\unif{0}{1}$
\newcommand{\unif}[2]{\text{Uniform}(#1,#2)}
% Independent and identically distributed random variables
% E.g. $ X_1,...,X_n \iid \normal{0}{1}$
\newcommand{\iid}{\stackrel{\smash{\text{iid}}}{\sim}}
% Sequences (this shortcut is mostly to reduce finger strain for small hands)
% E.g. to write $\{A_n\}_{n\geq 1}$, do $\bk{A_n}{n\geq 1}$
\newcommand{\bk}[2]{\{#1\}_{#2}}
% Math mode symbols for common sets and spaces. Example usage: $\R$
\newcommand{\R}{\mathbb{R}} % Real numbers
\newcommand{\C}{\mathbb{C}} % Complex numbers
\newcommand{\Q}{\mathbb{Q}} % Rational numbers
\newcommand{\Z}{\mathbb{Z}} % Integers
\newcommand{\N}{\mathbb{N}} % Natural numbers
\newcommand{\F}{\mathcal{F}} % Calligraphic F for a sigma algebra
\newcommand{\El}{\mathcal{L}} % Calligraphic L, e.g. for L^p spaces
% Math mode symbols for probability
\newcommand{\pr}{\mathbb{P}} % Probability measure
\newcommand{\E}{\mathbb{E}} % Expectation, e.g. $\E(X)$
\newcommand{\var}{\text{Var}} % Variance, e.g. $\var(X)$
\newcommand{\cov}{\text{Cov}} % Covariance, e.g. $\cov(X,Y)$
\newcommand{\corr}{\text{Corr}} % Correlation, e.g. $\corr(X,Y)$
\newcommand{\B}{\mathcal{B}} % Borel sigma-algebra
% Other miscellaneous symbols
\newcommand{\tth}{\text{th}} % Non-italicized 'th', e.g. $n^\tth$
\newcommand{\Oh}{\mathcal{O}} % Big-O notation, e.g. $\O(n)$
\newcommand{\1}{\mathds{1}} % Indicator function, e.g. $\1_A$
% Additional commands for math mode
\DeclareMathOperator*{\argmax}{argmax} % Argmax, e.g. $\argmax_{x\in[0,1]} f(x)$
\DeclareMathOperator*{\argmin}{argmin} % Argmin, e.g. $\argmin_{x\in[0,1]} f(x)$
\DeclareMathOperator*{\spann}{Span} % Span, e.g. $\spann\{X_1,...,X_n\}$
\DeclareMathOperator*{\bias}{Bias} % Bias, e.g. $\bias(\hat\theta)$
\DeclareMathOperator*{\ran}{ran} % Range of an operator, e.g. $\ran(T)
\DeclareMathOperator*{\dv}{d\!} % Non-italicized 'with respect to', e.g. $\int f(x) \dv x$
\DeclareMathOperator*{\diag}{diag} % Diagonal of a matrix, e.g. $\diag(M)$
\DeclareMathOperator*{\trace}{trace} % Trace of a matrix, e.g. $\trace(M)$
\DeclareMathOperator*{\supp}{supp} % Support of a function, e.g., $\supp(f)$
% --- Presentation begins here ---
\begin{document}
% --- Title slide ---
\title{\color{titleText}Presentation title here}
\subtitle{\color{Blue}Subtitle here (e.g., name of conference)}
\author{Sara Venkatraman\vspace{-.3cm}}
\institute{Cornell University, Statistics and Data Science}
\date{}
\begin{frame}
\titlepage
\vspace{-1.2cm}
\begin{center}
\includegraphics[width=2.3cm]{Cornell}\bigskip
{\begin{spacing}{1.2}\scriptsize
Joint work with FirstName LastName, FirstName LastName
\end{spacing}}
\end{center}
\end{frame}
% --- Main content ---
% Example slide: use \pause to sequentially unveil content
\begin{frame}{Mercer's theorem}
\framesubtitle{Eigendecomposition of a kernel}
A continuous function $K:([0,1]\times[0,1])\to\R$ is a \textit{kernel} if it is: \vspace{-0.4em}
\pause
\begin{itemize}
\item Symmetric: $K(a,b)=K(b,a)$\pause
\item Positive semi-definite: $$\sum_{i=1}^n\sum_{j=1}^nK(x_i,x_j)c_ic_j\geq 0$$ for all $x_1,...,x_n\in[0,1]$ and $c_1,...,c_n\in\R$
\end{itemize}
\pause
Now define an integral operator on $\El^2[0,1]$ using $K$:
\begin{align*}
(T_Kf)(x) = \int_0^1K(x,s)f(s)\dv s
\end{align*}
\end{frame}
% Example slide 2: Image
\begin{frame}{Example image}
\begin{figure}[H]
\centering\includegraphics[width=7cm]{ExampleImage}
\end{figure}
\end{frame}
% --- Bibliography slide ---
\begin{frame}{References}
\begin{thebibliography}{10}
\beamertemplatearticlebibitems
{\small
\bibitem{Paper1}
Paper 1 Title.
\newblock Paper 1 Authors. \\
{\it Journal Name} Edition, Year.
\bibitem{Paper 2}
Paper 2 Title.
\newblock Paper 2 Authors.\\
arXiv:1234.56789.
}
\end{thebibliography}
\end{frame}
% --- Thank you slide ---
\begin{frame}
\begin{center}
{\large\color{titleText} Thank you for listening!}
\vspace{1cm}
Sara Venkatraman \\[1em]
skv24@cornell.edu \\
https://sara-venkatraman.github.io
\end{center}
\end{frame}
% --- Presentation ends here ---
\end{document}