clt_presentation2.tex

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\title{Dissertation Title}
\author{Dissertation Author}
%\committee{Advisor}{Committee Member}{Committee Member}{Committee Member}{Committee Member}
\date{January 1, 1970}

%\input{abstract.tex}
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%\dedication{To Wawa}

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\newacronym{unc}{UNC}{The University of North Carolina at Chapel Hill}
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\linespread{1.0}
\begin{document}
	
	\begin{frame}[plain]
		\maketitle
	\end{frame}
	
	\begin{frame}{About this Beamer project}
		\begin{itemize}
			\item This template is available for both Japanese and English.
		\end{itemize}
		\note{
			Please write presenter's note here.
		}
	\end{frame}

	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%
	%
	%
	%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	% Chapter 0
	% Everything, disorganized	
	\begin{frame}{Chapter 0}
	\end{frame}
	
	\begin{frame}{Neural Networks}
	\end{frame}
	\begin{frame}{Fully-Connected Architecture}
	\end{frame}
	\begin{frame}{Optimization and Backpropagation}
	\end{frame}

	\begin{frame}{Activation Functions}
	\end{frame}

	\begin{frame}{Overview: PINN and Use Cases}
	\end{frame}

	\begin{frame}{PINNs Intro}
	\end{frame}

	\begin{frame}{PINNs Review of Applications}
	\end{frame}

	\begin{frame}{NN and FEM Review}
	\end{frame}

	\begin{frame}{PINN: Convergence Review}
	\end{frame}

%	\begin{frame}{FOSLS-PINN Motivation}
%	\end{frame}
%	\begin{frame}{LSFEM Review Elasticity}
%	\end{frame}
	\begin{frame}{Stokes/Linear Elasticity}
	\end{frame}
	\begin{frame}{Pure Traction}
	\end{frame}
	\begin{frame}{Pure Displacement}
	\end{frame}
	\begin{frame}{Mixed Formulations for Linear Elasticity}
	\end{frame}
	\begin{frame}{Mixed Formulations for Hyperelasticity}
	\end{frame}
%	LSFEM Review Fluid Mechanics
%	Navier-Stokes (Incompressible Newtonian Fluids)
%	VVP
%	VVG
%	VVT
%	(Variable Viscosity e.g. Carreau) Navier-Stokes (Incompressible Newtonian Fluids)
%	VPS
%	PINN: Linear Elasticity Plane Displacement
%	Linear elasticity: Mixed Formulation
%	(Spring 2022)
%	PINN Application: Navier-Stokes Idealized Vessels
%	Newtonian
%	Carreau
%	Singly and Doubly Stenotic
%	FOSLS-PINN: Linear Elasticity Plane Stress
%	Linear elasticity: Mixed Formulation
%	Linear elasticity: VVG Formulation
%	PINN and FOSLS-PINN: Channel Flow 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Sampled Points
%	PINN and FOSLS-PINN: Bercovier-Engelman 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Steady Cylinder 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Block Stenosis 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Lid Driven Cavity 2D
%	Importance Sampling
%	Signed Distance Function Weighting (Corner Singularities)
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Annular Ring 2D
%	Regular
%	Hard BCs
%	Residual Gradient
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	Variational PINN (VPINN)
%	Poisson, Laplace, or Linear Elasticity
%	VPINN
%	FOSLS VPINN
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 0
%	% Everything, disorganized	
%%	\begin{frame}{Chapter 0}
%%	\end{frame}
%	\begin{frame}{
%%			\include{}
%		}
%	\end{frame}
%\include{slide-UseCasePINNs.tex}

	\begin{frame}{PINN Applications, Use Cases}
%	\include{slide-UseCasePINNs.tex}
		%\section{Use Cases of PINNs vs Classical Numerical Methods}
		Current AI methods will be slower than the traditional solvers for training of a single case/geometry. 
		\begin{enumerate}
			\item Multiple (parameterized) cases where there are several different configurations in the analysis space
			\item Parametric Models
			\item Surrogate Models 
			\item Design Optimization
			\item Inverse problems (data is given but determine coefficients of PDEs)
			\item Cases where measured data is available but the Physics is too complicated or not fully understood
			\item Data assimilation cases where there is measured or simulated data for some variables but not the entire field (e.g. digital twins, medical imaging, full waveform inversion etc.)
			\item Training AI model on compute intensive part of the solver and call that from the solver during the analysis to improve the speed (constitutive models, turbulence models or radiation viewfactors etc.)
			\item Runge Kutta simulations (extreme order)
			\item Point clouds can sometimes present higher quality solutions than mesh based simulations (since without convergence studies, mesh based solutions can yield questionable results).
		\end{enumerate}
	\end{frame}
%		different model configurations (subject to physical constraints) based on a multitude of parameters.
%Compare autlod: Appearance-Driven Automatic 3D Model Simplification

	\begin{frame}{
			\include{slide-NeuralNetworks.tex}
		}
	\end{frame}

%	\begin{frame}{
%			\include{slide-PINNs.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{slide-Methods.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{slide-LSFEM.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS1D-Formulations.tex}
%	\end{frame}
%	%\begin{frame}{
%			\include{./notes/FOSLS/sec-EllipticFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS2D-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVP-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVG-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVT-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-Results.tex}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityStokesFOSLS-Formulations.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureDisplacement.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureTraction.tex}
%	\end{frame}
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%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-Mixed.tex}
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%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Formulations.tex}
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%			\include{./notes/FOSLS/sec-HyperElasticityFOSLS-Formulations-Mixed.tex}
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%			\include{./notes/FOSLS/slide-NavierStokesFOSLS-Formulations.tex}
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%			\include{./notes/FOSLS/slide-NavierStokesFOSLS-Results.tex}
%		}
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%			\include{./notes/FOSLS/slide-SteadyNavierStokes2D-FOSLS-Results.tex}
%		}
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%			\include{./notes/FOSLS/sec-HagenPoiseuilleChannel2D.tex}
%		}
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%			\include{./notes/FOSLS/sec-CouetteChannel2D.tex}
%		}
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%	
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%			\include{./notes/FOSLS/sec-KovasznayRectangle2D.tex}
%		}
%	\end{frame}
%	
%%	\begin{frame}{
%			\include{./notes/FOSLS/sec-BercovierSquare2D.tex}
%%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-RuasDisk2D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen2D.tex}
%		}
%	\end{frame}
%	
%%	\begin{frame}{
%			\include{./notes/FOSLS/sec-CylinderChannel2D.tex}
%%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-BlockStenosisChannel2D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-AnnularRing2D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-SteadyNavierStokes3D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen3D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-ABCFlow3D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokes2D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen2DT.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokes3D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-Beltrami3DT.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen3DT.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-Cylinder2DT.tex}
%		}
%	\end{frame}
%	
%	
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-LDC2D.tex}
%		}
%	\end{frame}
%
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Cylinder2D.tex}
%		}
%	\end{frame}
%
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Block2D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-AnnularRing.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-AnnularRing1.tex}
%		}
%	\end{frame}
%	
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Cylinder2D_inverse.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Block2D_inverse.tex}
%		}
%	\end{frame}
%
%%	\input{./slide-Chapter0.tex}
%	%\input{./notes/FOSLS/SECTION_ANNULUARRING_PRESSUREMONITORS_STEADY_2D}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 1
%	% Overview of Neural Networks and Supervised (Learning) NN Approximation
%	\begin{frame}{Chapter 1}
%		}
%	\end{frame}
%
%	%\input{chapter1.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 2
%	% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application? ROM and surrogate models, 
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 3
%	% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application?
%	\begin{frame}{Chapter 3}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 2D (Unconstrained?) PDEs
%	% Deep FOSLS vs PINNs comparison.  
%	% Poisson-like Equations
%	% Poisson
%	% Diffusion
%	% Helmholtz
%	%
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%laplace
%	%\input{poisson2d.tex}
%	%\input{surfacepoisson2d.tex}
%	%%\input{diffusion2d.tex}
%	%%\input{diffusionreaction2d.tex}
%	%singularlyperturbed
%	%\input{advectiondiffusionreaction2d.tex}
%	%\input{helmholtz2d.tex}
%	%\input{chapter2.tex}
%	
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Introduction: FEM and LSFEM methods
%	% Linear and Hyperelastic Elasticity
%	% PINNs vs Deep FOSLS (Deep Energy Method?)comparison
%	% thesisLeast-squares mixed finite elements for geometricallySteegerDiss.pdf
%	\begin{frame}{Finite Element Methods and Least-Squares FEM in Solid and Fluid Mechanics}
%		}
%	\end{frame}
%
%	%%\input{Galerkin and mixed Galerkin finite elements.tex}
%	%\input{GalerkinandMixedGalerkinFiniteElements.tex}
%	%%\input{Least-squares mixed finite elements.tex}
%	%\input{LeastSquaresMixedFiniteElements.tex}
%	%%\input{Outline}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 2D Static Solid Mechanics 
%	% Linear and Hyperelastic Elasticity
%	% PINNs vs Deep FOSLS (Deep Energy Method?)comparison
%	% thesisLeast-squares mixed finite elements for geometricallySteegerDiss.pdf
%	\begin{frame}{Continuum mechanical background}
%		}
%	\end{frame}
%	%\input{KinematicsAndDeformationMeasures.tex}
%	%\input{StressQuantities.tex}
%	%\input{Hyperelasticmaterials.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%	\begin{frame}{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\input{LinearElasticityPureTractionFormulation.tex}
%	%\input{LinearElasticityPureDisplacementFormulation.tex}
%	%\input{LinearElasticityTractionDisplacementFormulation.tex}
%	%\input{hyperelasticfosls.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%	\begin{frame}{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\input{VPFormulation.tex}
%	%\input{VVFormulation.tex}
%	%\input{VVPFormulation.tex}
%	%\input{VPSFormulation.tex}
%	%\input{VVGFormulation.tex}
%	%\input{VVTFormulation.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% PINN Approximations to Navier-Stokes: 
%	% 2D Steady Newtonian Fluid Mechanics
%	\begin{frame}{PINN Approximations to Navier-Stokes: Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\input{chapter2.tex}
%	\subimport{notes/}{windkessel_overview.tex}
%	%%\input{./notes/windkessel_overview.tex}
%	%%\input{../notes/FOSLS/BLOCKSTENOSISSTEADY2D.tex}
%	%%\input{/notes/FOSLS/BLOCKSTENOSISSTEADY2D.tex}
%	%%\input{../notes/FOSLS/BLOCKSTENOSIS_STEADY_2D.tex}
%	%%\input{../notes/FOSLS/SECTION_ANNULARRING_STEADY_2D.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 2D Steady Non-Newtonian Fluid Mechanics, Special Methods, Applications
%%	\begin{frame}{Chapter 2}
%%		}
%%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Time-Dependent FOSLS Method, 1D/2D Advection-Diffusion-Reaction, Burger's, KdV, NL Schrodinger
%%	\begin{frame}{Chapter 2}
%%		}
%%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% PIGAN FOSLS vs Primitive FOSLS: Pressure Reconstruction, Darcy, and Elasticity
%%	\begin{frame}{Chapter 2}
%%		}
%%	\end{frame}
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Bayesian PINNs vs Bayesian FOSLS PINNs for UQ, noise distributions
%%	\begin{frame}{Chapter 2}
%%		}
%%	\end{frame}
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Operator Learning (DeepONet, MIONet, and Fourcastnet) 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%%	\begin{frame}{Chapter 2}
%%		}
%%	\end{frame}
%	%\input{chapter2.tex}
%	{\tiny }
%	
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%% BIBLIOGRAPHY AND OTHER LISTS
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%%	\backmatter
%	
%	% Bibliography
%	\bibliography{dissertation}
%	\addtocontents{toc}{\protect\vspace*{\baselineskip}}
%	
%	% Add bibliography to contents page
%	%\addcontentsline{toc}{chapter}{Bibliography} %'Bibliography' into toc
%	
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%% APPENDICES
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	\appendix
%	\begin{frame}{}
%	\section{}
%			}
%	\end{frame}

	
\end{document}