第 9、10 和 11 张幻灯片的顺序错误。第 9 张应该在 11 张之后。页脚中的页码也是错误的。
请帮我
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beamer Presentation
% LaTeX Template
% Version 2.0 (March 8, 2022)
%
% This template originates from:
% https://www.LaTeXTemplates.com
%
% Author:
% Vel ([email protected])
%
% License:
% CC BY-NC-SA 4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%----------------------------------------------------------------------------------------
% PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------
\documentclass[
11pt, % Set the default font size, options include: 8pt, 9pt, 10pt, 11pt, 12pt, 14pt, 17pt, 20pt
%t, % Uncomment to vertically align all slide content to the top of the slide, rather than the default centered
%aspectratio=169, % Uncomment to set the aspect ratio to a 16:9 ratio which matches the aspect ratio of 1080p and 4K screens and projectors
]{beamer}
\graphicspath{{Images/}{./}} % Specifies where to look for included images (trailing slash required)
\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule for better rules in tables
% \usepackage{media9}
\usepackage{graphicx}
\usepackage{dsfont}
\usepackage{float}
\usepackage{algorithmicx}
\usepackage{algorithm}% http://ctan.org/pkg/algorithms
\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx
%----------------------------------------------------------------------------------------
% SELECT LAYOUT THEME
%----------------------------------------------------------------------------------------
% Beamer comes with a number of default layout themes which change the colors and layouts of slides. Below is a list of all themes available, uncomment each in turn to see what they look like.
%\usetheme{default}
%\usetheme{AnnArbor}
%\usetheme{Antibes}
%\usetheme{Bergen}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Boadilla}
%\usetheme{CambridgeUS}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{Dresden}
%\usetheme{Frankfurt}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}
%\usetheme{Pittsburgh}
%\usetheme{Rochester}
%\usetheme{Singapore}
%\usetheme{Szeged}
%\usetheme{Warsaw}
%----------------------------------------------------------------------------------------
% SELECT COLOR THEME
%----------------------------------------------------------------------------------------
% Beamer comes with a number of color themes that can be applied to any layout theme to change its colors. Uncomment each of these in turn to see how they change the colors of your selected layout theme.
%\usecolortheme{albatross}
%\usecolortheme{beaver}
%\usecolortheme{beetle}
%\usecolortheme{crane}
%\usecolortheme{dolphin}
%\usecolortheme{dove}
%\usecolortheme{fly}
%\usecolortheme{lily}
%\usecolortheme{monarca}
%\usecolortheme{seagull}
%\usecolortheme{seahorse}
%\usecolortheme{spruce}
%\usecolortheme{whale}
%\usecolortheme{wolverine}
%----------------------------------------------------------------------------------------
% SELECT FONT THEME & FONTS
%----------------------------------------------------------------------------------------
% Beamer comes with several font themes to easily change the fonts used in various parts of the presentation. Review the comments beside each one to decide if you would like to use it. Note that additional options can be specified for several of these font themes, consult the beamer documentation for more information.
\usefonttheme[onlymath]{serif} % Typeset using the default sans serif font
%\usefonttheme{serif} % Typeset using the default serif font (make sure a sans font isn't being set as the default font if you use this option!)
%\usefonttheme{structurebold} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in bold
%\usefonttheme{structureitalicserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in italic serif
%\usefonttheme{structuresmallcapsserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in small caps serif
%------------------------------------------------
%\usepackage{mathptmx} % Use the Times font for serif text
\usepackage{palatino} % Use the Palatino font for serif text
%\usepackage{helvet} % Use the Helvetica font for sans serif text
\usepackage[default]{opensans} % Use the Open Sans font for sans serif text
%\usepackage[default]{FiraSans} % Use the Fira Sans font for sans serif text
%\usepackage[default]{lato} % Use the Lato font for sans serif text
%----------------------------------------------------------------------------------------
% SELECT INNER THEME
%----------------------------------------------------------------------------------------
% Inner themes change the styling of internal slide elements, for example: bullet points, blocks, bibliography entries, title pages, theorems, etc. Uncomment each theme in turn to see what changes it makes to your presentation.
%\useinnertheme{default}
\useinnertheme{circles}
%\useinnertheme{rectangles}
%\useinnertheme{rounded}
%\useinnertheme{inmargin}
%----------------------------------------------------------------------------------------
% SELECT OUTER THEME
%----------------------------------------------------------------------------------------
% Outer themes change the overall layout of slides, such as: header and footer lines, sidebars and slide titles. Uncomment each theme in turn to see what changes it makes to your presentation.
%\useoutertheme{default}
%\useoutertheme{infolines}
%\useoutertheme{miniframes}
%\useoutertheme{smoothbars}
%\useoutertheme{sidebar}
%\useoutertheme{split}
%\useoutertheme{shadow}
%\useoutertheme{tree}
%\useoutertheme{smoothtree}
%\setbeamertemplate{footline} % Uncomment this line to remove the footer line in all slides
%\setbeamertemplate{footline}[page number] % Uncomment this line to replace the footer line in all slides with a simple slide count
%\setbeamertemplate{navigation symbols}{} % Uncomment this line to remove the navigation symbols from the bottom of all slides
%----------------------------------------------------------------------------------------
% PRESENTATION INFORMATION
%----------------------------------------------------------------------------------------
\title[Mathematical Epidemiology Models]{Neural Network Solvers in Epidemiology Models} % The short title in the optional parameter appears at the bottom of every slide, the full title in the main parameter is only on the title page
\subtitle{A Prediction Model for Covid-19 Cases in Turkey} % Presentation subtitle, remove this command if a subtitle isn't required
\author[Günel \and Ahmad]{Korhan Günel \inst{1} \and Muhammad Jalil Ahmad \inst{2}}
% Affiliations
\institute[ADÜ \and UMBC]{\inst{1} Aydin Adnan Menderes University, Turkey\and \inst{2} University of Maryland, Baltimore Country, USA} % Presenter name(s), the optional parameter can contain a shortened version to appear on the bottom of every slide, while the main parameter will appear on the title slide
%\institute[]{University of Maryland, Baltimore County \\ \smallskip \textit{[email protected]}} % Your institution, the optional parameter can be used for the institution shorthand and will appear on the bottom of every slide after author names, while the required parameter is used on the title slide and can include your email address or additional information on separate lines
\date[November 11–13, 2022]{The 2022 International BEER Symposium \\ November 11–13, 2022} % Presentation date or conference/meeting name, the optional parameter can contain a shortened version to appear on the bottom of every slide, while the required parameter value is output to the title slide
%----------------------------------------------------------------------------------------
\begin{document}
%----------------------------------------------------------------------------------------
% TITLE SLIDE
%----------------------------------------------------------------------------------------
\begin{frame}
\titlepage % Output the title slide, automatically created using the text entered in the PRESENTATION INFORMATION block above
\end{frame}
%----------------------------------------------------------------------------------------
% TABLE OF CONTENTS SLIDE
%----------------------------------------------------------------------------------------
% The table of contents outputs the sections and subsections that appear in your presentation, specified with the standard \section and \subsection commands. You may either display all sections and subsections on one slide with \tableofcontents, or display each section at a time on subsequent slides with \tableofcontents[pausesections]. The latter is useful if you want to step through each section and mention what you will discuss.
%\begin{frame}
% \frametitle{Presentation Overview} % Slide title, remove this %command for no title
% \tableofcontents % Output the table of contents (all sections on %one slide)
%\tableofcontents[pausesections] % Output the table of contents %(break sections up across separate slides)
%\end{frame}
%----------------------------------------------------------------------------------------
% PRESENTATION BODY SLIDES
%----------------------------------------------------------------------------------------
\subsection{Epidemiology Models}
\begin{frame}
\frametitle{Susceptible-Infected ($SI$) Model}
\begin{itemize}
\item $S(t)$ represents number of people susceptible to disease at time $t$.
\item $I(t)$ represents number of people already infected at time $t$.
\item Total number of people in the population is given by $$N=S(t)+I(t)$$
\end{itemize}
\begin{center}
\includegraphics[scale=0.75]{SI model}
\end{center}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected ($SI$) Model}
\begin{itemize}
\item Then the $SI$ model can be represented by the following system of ODEs:
$$\dfrac{dS(t)}{dt} = -\dfrac{\beta S(t) I(t)}{N} $$
$$\dfrac{dI(t)}{dt} = \dfrac{\beta S(t) I(t)}{N} $$
with initial values, $S(0)=s_0>0$ and $I(0)=i_0>0$\\
\item $\beta$ represents the rate at which the disease is spreading.
\item Analytic solution of this system is give by:
$$I(t) = \dfrac{i_0}{i_0+(1-i_0)e^{-\beta t}}
$$ $$S(t)=N-I(t)$$
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
\begin{itemize}
\item In this model we will consider the case in which infected people become susceptible again after recovering from the disease.
\begin{center}
\includegraphics[scale=0.75]{SIS model}
\end{center}
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
\begin{itemize}
\item $SIS$ model is given by:
$$\dfrac{dS(t)}{dt} = -\beta S(t) I(t)+\gamma I(t) $$
$$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
where $\gamma$ represents the rate at which infected people recover and become susceptible again.
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
\begin{itemize}
\item Analytic solution of this system is given by:
$$I(t) = \dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$
$$S(t)=N-\dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$
where $\alpha=\beta N-\gamma$ and $C=\dfrac{\alpha-i_0 \beta}{\alpha i_0}$
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected-Recovered ($SIR$) Model}
\begin{itemize}
\item $SIR$ model is given by:
$$\dfrac{dS(t)}{dt} = -\beta S(t) I(t) $$
$$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
$$\dfrac{dR(t)}{dt} = \gamma I(t) $$
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Susceptible-Infected-Susceptible ($SIR$) Model}
\begin{itemize}
\item Analytic solution of this system is given by:
$$S(u)=s_0 u$$
$$I(u)= \dfrac{\gamma}{\beta} \log u - \alpha u - \dfrac{C_1}{\beta}$$
$$R(u)=-\dfrac{\gamma}{\beta} \log u$$
where $C_1$ is an integration constant and $$t-t_0 = \int_{u_0}^u \dfrac{d\epsilon}{\epsilon (C_1-\gamma \log \epsilon + \alpha \beta u)}$$
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{algorithm}
\caption{MADS Algorithm}\label{alg:mABC}
\small
Given $f:\mathds{R}^n \rightarrow \mathds{R}$ ve $x^0 \in \mathds{R}^n$ is the starting point
\begin{algorithmic}[1]
\State Initialisation
\State $\Delta^0 \in (0,\infty)$ \Comment{ initial frame size parameter }
\State $D = GZ $\Comment{ positive spanning matrix }
\State $\tau \in (0,1), \: \tau \in \mathbb{Q} $\Comment{ mesh size adjustment parameter }
\State $\epsilon_{stop} \in [0, \infty) $\Comment{ stopping tolerance }
\State $k \gets 0$\Comment{ iteration counter }
\State Parameter Update
\State set the mesh size parameter to $\delta^k = min\lbrace \Delta^k , {(\Delta^k)}^2\rbrace $
\State Search
\State if $f(t) < f(x^k)$ for some $t$ in a finite subset $S^k$ of $M^k$ set $x^{k+1} \gets t$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$ and go to 14 otherwise go to 11
\algstore{bkbreak}
\end{algorithmic}
\end{algorithm}
%------------------------------------------------
\begin{algorithm}[h]
\begin{algorithmic}
\algrestore{bkbreak}
\State Poll
\State select a positive spanning set $\mathds{D}^k_\Delta$ such that $ P^k = \lbrace x^k + \delta^k d : d \in \mathds{D}^k_\Delta \rbrace $ is a subset of the frame $F^k$ of some extent $\Delta^k$.\\ if $f(t) < f(x^k)$ for some $t \in P^k$ set $x^{k+1} \gets t$ ve $\delta^{k+1} \gets \tau^{-1} \delta^k$ otherwise set $x^{k+1} \gets x^k$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$.
\State Termination
\State If $\Delta^{k+1} \geq \epsilon_{stop}$ increment $k \gets k+1$ and go to 9 otherwise stop.
\end{algorithmic}
\end{algorithm}
%------------------------------------------------
\begin{frame}
\frametitle{2-D Example for working of MADS Algorithm}
\begin{center}
\includegraphics[scale=0.5]{Frame 1}
\end{center}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{2-D Example for working of MADS Algorithm}
\begin{center}
\includegraphics[scale=0.5]{Frame 2}
\end{center}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Artificial Neural Networks}
\begin{itemize}
\item Single Layers Neural Nets
\begin{itemize}
\item For linear functions
\end{itemize}
\item Multi-Layers Neural Nets
\begin{itemize}
\item For non-linear functions
\end{itemize}
\end{itemize}
\begin{center}
\begin{figure}
\includegraphics[scale=0.5]{NeuralNet}
\caption{Topology of Neural Network}
\end{figure}
\end{center}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}
$$S_t=s_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_S}) $$
$$T_t=i_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_I}) $$
\begin{itemize}
\item $h>0$ is step size
\item $t_j=t_0+j \cdot h $ for $j=0,1, \ldots ,n$ gives us discretization of the interval $[t_0, t_n]$
\item $\overrightarrow{p_S}=\overrightarrow{p_S}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ and $\overrightarrow{p_I}=\overrightarrow{p_I}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ are unknown parameters of the neural network
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}
\begin{itemize}
\item Points $t_j$ are used to train neural network
\item The output of the NN is given by $$Net(t_j, s_j, i_j, \overrightarrow{p_S})= \sum_{i=1}^{m_1} \alpha_{S i} \ \sigma(z_{1i}) $$
\item $\sigma(z) = \dfrac{1}{1+e^{-z}} $ is the activation function
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}
$$\dfrac{\partial S_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_S})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_S})}{\partial t} $$
$$\dfrac{\partial I_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_I})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_I})}{\partial t} $$
$$\dfrac{dS(t)}{dt}=f(t, S(t),I(t)) $$
$$\dfrac{dI(t)}{dt}=g(t, S(t),I(t)) $$
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}
Cost function for $SI$ model is given by:
$$E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i))\right]^2 + \left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i))\right]^2\right\rbrace $$
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Cost function for $SIR$}
Similarly cost function for $SIR$ model is given by:
Similarly cost function for $SIR$ model is given by:
\begin{multline*} E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i),R(t_i))\right]^2 + \right.\\
\left.\left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i),R(t_i))\right]^2 + \left[ \dfrac{\partial R(t_i)}{\partial t_i}-h(t_i, S(t_i),I(t_i),R(t_i))\right]^2\right\rbrace
\end{multline*}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Error Calculation}
Absolute error for the models is given by:
$$E(S)=|s_j - S_t(t_j)| $$
$$E(I)=|i_j - I_t(t_j)| $$
$$E(R)=|r_j - R_t(t_j)| $$
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Training Neural Network}
\begin{itemize}
\item Data announced during November 27, 2020 and May 28, 2021 by Health Ministry of Turkey is used.
\item $S$ represents number of reported cases, $I$ is number of confirmed cases and $R$ is number of people recovered in one day.
\item There are 7 layers in the NN.
\item Number of neurons in each layer is 10, 6, 6, 4, 4, 3, 3 respectively.
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Results for $SI$ Model}
\begin{figure}
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_1} % first figure itself
\caption{Results For Training Set}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_2} % second figure itself
\caption{Results For Test Set}
\end{minipage}
\end{figure}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Results for $SIS$ Model}
\begin{figure}
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_1} % first figure itself
\caption{Results For Training Set}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_2} % second figure itself
\caption{Results For Test Set}
\end{minipage}
\end{figure}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Results for $SIR$ Model}
\begin{figure}
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_1} % first figure itself
\caption{Results For Training Set}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{SI_2} % second figure itself
\caption{Results For Test Set}
\end{minipage}
\end{figure}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Conclusions}
\begin{itemize}
\item Results show $3-5 \% $ error for the training set. The reasons are:
\begin{itemize}
\item Inconsistency in the data.
\item The coefficients $\beta$ and $\gamma$ obtained using the linear regression of real data are constant during the study.
\item In MADS, the initial positions of the basis vectors have a significant effect on reaching the solution.
\end{itemize}
\item If this was some other problem then these results would have been considered good but when we talk about COVID-19 which effects all humanity, this error should be minimized further.
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Future Work}
\begin{itemize}
\item Techniques like data approximation are advised to overcome the inconsistency in data.
\item The coefficients $\beta$ and $\gamma$ should be defined as a function of time.
\item Algorithms like Particle Swarm Optimization (PSO) should be tested where the initial value doesnt play a big role as it does in MADS.
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame} % Use [allowframebreaks] to allow automatic splitting across slides if the content is too long
\frametitle{References}
\begin{figure}
\includegraphics[scale=0.6]{References}
\end{figure}
\end{frame}
%----------------------------------------------------------------------------------------
\begin{frame} % The optional argument 'plain' hides the headline and footline
\begin{center}
{\Huge Thank You}
\bigskip\bigskip % Vertical whitespace
{\LARGE Questions? Comments?}
\end{center}
\end{frame}
%----------------------------------------------------------------------------------------
\end{document}
答案1
将的两部分放在algorithmic同一个框架中，避免使用begin{algorithm}[h!]会生成浮动内容的元素，因为在 beamer 环境中这是完全不需要的。
\documentclass[11pt]{beamer}
\usepackage{graphicx}
\usepackage{dsfont}
%\usepackage{float} % do not use <<<<<<<<
%\usepackage{algorithmicx}
%\usepackage{algorithm}% http://ctan.org/pkg/algorithms
\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx
\usetheme{Madrid}
\usefonttheme[onlymath]{serif} % Typeset using the default sans serif font
\usepackage{palatino} % Use the Palatino font for serif text
\usepackage[default]{opensans} % Use the Open Sans font for sans serif text
\useinnertheme{circles}
\begin{document}
\begin{frame}
\frametitle{Susceptible-Infected-Susceptible ($SIR$) Model}
\begin{itemize}
\item Analytic solution of this system is given by:
$$S(u)=s_0 u$$
$$I(u)= \dfrac{\gamma}{\beta} \log u - \alpha u - \dfrac{C_1}{\beta}$$
$$R(u)=-\dfrac{\gamma}{\beta} \log u$$
where $C_1$ is an integration constant and $$t-t_0 = \int_{u_0}^u \dfrac{d\epsilon}{\epsilon (C_1-\gamma \log \epsilon + \alpha \beta u)}$$
\end{itemize}
\end{frame}
%------------------------------------------------
\begin{frame} % added <<<<<<<<<<<<<<<
\frametitle{MADS Algorithm 1/2}% added <<<<<<<<<<<<<<<
% \begin{algorithm}[h!]
% \caption{MADS Algorithm}
\label{alg:mABC}
\small
Given $f:\mathds{R}^n \rightarrow \mathds{R}$ ve $x^0 \in \mathds{R}^n$ is the starting point
\begin{algorithmic}[1]
\State Initialisation
\State $\Delta^0 \in (0,\infty)$ \Comment{ initial frame size parameter }
\State $D = GZ $\Comment{ positive spanning matrix }
\State $\tau \in (0,1), \: \tau \in \mathbb{Q} $\Comment{ mesh size adjustment parameter }
\State $\epsilon_{stop} \in [0, \infty) $\Comment{ stopping tolerance }
\State $k \gets 0$\Comment{ iteration counter }
\State Parameter Update
\State set the mesh size parameter to $\delta^k = min\lbrace \Delta^k , {(\Delta^k)}^2\rbrace $
\State Search
\State if $f(t) < f(x^k)$ for some $t$ in a finite subset $S^k$ of $M^k$ set $x^{k+1} \gets t$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$ and go to 14 otherwise go to 11
\algstore{bkbreak}
\end{algorithmic}
% \end{algorithm}
\end{frame} % added <<<<<<<<<<<<<<<
%------------------------------------------------
\begin{frame}% added <<<<<<<<<<<<<<<
\frametitle{MADS Algorithm 2/2} % added <<<<<<<<<<<<<<<
% \begin{algorithm}[h!]
\begin{algorithmic}
\algrestore{bkbreak}
\State Poll
\State select a positive spanning set $\mathds{D}^k_\Delta$ such that $ P^k = \lbrace x^k + \delta^k d : d \in \mathds{D}^k_\Delta \rbrace $ is a subset of the frame $F^k$ of some extent $\Delta^k$.\\ if $f(t) < f(x^k)$ for some $t \in P^k$ set $x^{k+1} \gets t$ ve $\delta^{k+1} \gets \tau^{-1} \delta^k$ otherwise set $x^{k+1} \gets x^k$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$.
\State Termination
\State If $\Delta^{k+1} \geq \epsilon_{stop}$ increment $k \gets k+1$ and go to 9 otherwise stop.
\end{algorithmic}
% \end{algorithm}
\end{frame}% added <<<<<<<<<<<<<<<
%------------------------------------------------
\begin{frame}
\frametitle{2-D Example for working of MADS Algorithm}
\begin{center}
\includegraphics[scale=0.5]{example-image}
\end{center}
\end{frame}
%------------------------------------------------
\end{document}