diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.log b/Lectures_my/MC_2016/Lecture1/mchrzasz.log
index dd7011c..ba9cccd 100644
--- a/Lectures_my/MC_2016/Lecture1/mchrzasz.log
+++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.log
@@ -1,4 +1,4 @@
-This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1)  23 FEB 2016 16:59
+This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1)  25 FEB 2016 09:55
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf
index 68ca850..72b9786 100644
--- a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf
+++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf
Binary files differ
diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex
index 2c7916b..8aa468c 100644
--- a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex
+++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex
@@ -418,7 +418,7 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\begin{frame}\frametitle{Let's test the $\sqrt{N}$, $\rm \color{RubineRed}{Lecture1/Euler\_number}$  }
+\begin{frame}\frametitle{Let's test the $\sqrt{N}$}
 
 \only<1>
 {
@@ -589,7 +589,7 @@
 \begin{exampleblock}{}
 A sum of large number random variables has always a Gaussian distributions, independent of what kind of PDF was used to generate the sample. The only requirement is that the PDFs have a finite expected value.	
 \end{exampleblock}
-\ARROW E(1.3) Using ROOT draw the Gauss distribution:
+\ARROW E(1.4) Using ROOT draw the Gauss distribution:
 \begin{equation}
 \rho(x;\mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi} } e^{ \frac{-(x-\mu)^2}{2\sigma^2}} \nonumber
 \end{equation}
@@ -623,7 +623,7 @@
 E(x_i)=1/2   \Rightarrow E(R_n)=n/2 \\
 V(x_i)=1/12    \Rightarrow V(R_n)=n/12
 \end{align*}
-\ARROW E(1.4) Calculate the above.\\
+\ARROW E(1.5) Calculate the above.\\
 \ARROW From above we get:
 \begin{equation}
 \dfrac{R_n-n/12}{\sqrt{n/12}} \xrightarrow{N\to \infty} N(0,1).\nonumber
@@ -631,7 +631,7 @@
 aka we get a Gaussian distribution.
 \ARROW For $n=12$ $\Rightarrow~(R_{12}-6)$ ''practical'' Gauss generator.\\
 \ARROW Attention: This kind of generators are not good in reproducing the tails of the Gauss!\\
-\ARROW E(1.5) Construct the Gauss generator based on the math above in ROOT.
+\ARROW E(1.6) Construct the Gauss generator based on the math above in ROOT.
 
 
 \end{frame}
@@ -844,7 +844,7 @@
 \end{frame}
 
 
-\begin{frame}\frametitle{Buffon needle, $\rm \color{RubineRed}{Lecture1/Heads\_tails}$  }
+\begin{frame}\frametitle{Buffon needle}
 \begin{small}
 
 
diff --git a/Lectures_my/MC_2016/Lecture2/Makefile b/Lectures_my/MC_2016/Lecture2/Makefile
deleted file mode 100644
index e4cf8e8..0000000
--- a/Lectures_my/MC_2016/Lecture2/Makefile
+++ /dev/null
@@ -1,9 +0,0 @@
-all: mchrzasz.pdf
-
-mchrzasz.pdf: mchrzasz.tex
-	xelatex mchrzasz.tex
-#	bibtex mchrzasz
-	xelatex mchrzasz.tex
-#	xelatex mchrzasz.tex
-clean:
-	rm *.aux *.bbl *.blg *.log *.out *.pdf *.sh *.xml *.snm *.toc *.nav 
diff --git a/Lectures_my/MC_2016/Lecture2/beamercolorthemeFlip.sty b/Lectures_my/MC_2016/Lecture2/beamercolorthemeFlip.sty
deleted file mode 100755
index d6b9c50..0000000
--- a/Lectures_my/MC_2016/Lecture2/beamercolorthemeFlip.sty
+++ /dev/null
@@ -1,179 +0,0 @@
-% Color style file for the BEAMER FLIP THEME
-% Copyright 2012 by Flip Tanedo
-% This file may be distributed and/or modified
-% 	1. under the LaTeX Project Public License and/or
-% 	2. under the GNU Public License.
-
-
-
-\mode<presentation>
-\newif\ifbeamer@Flip@dark
-\beamer@Flip@darkfalse
-\DeclareOptionBeamer{dark}{\beamer@Flip@darktrue}
-\ProcessOptionsBeamer
-
-
-% These are a few colors that I like.
-
-\definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
-\definecolor{darkcharcoal}{RGB}{25,25,25}		% Darker gray
-\definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
-\definecolor{ash}{RGB}{100,100,100}			% medium gray
-\definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
-\definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
-\definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
-\definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
-\definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
-\definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
-
-
-% Define some "invariant colors" so that a color that doen't work with a
-% light/dark background will automatically be substituted with one that does.
-% These will be rewritten in beamercolorthemeFlipDark
-
-\ifbeamer@Flip@dark
-	\definecolor{FlipSand}{RGB}{204,204,102}		% paleale, Only for dark BG
-	\definecolor{FlipGreen}{RGB}{0,255,0}			% Neon-like green, only for dark BG
-\else
-	\definecolor{FlipSand}{RGB}{140,110,10}			% lager, Only for light BG
-	\definecolor{FlipGreen}{RGB}{51,153,0}			% turtlegreen, works for both
-\fi
-
-%% Future revision: It would be nice to define these in terms of the palette.
-
-
-\ifbeamer@Flip@dark
-	\setbeamercolor{normal text}{fg=white, bg=black}		
-	\setbeamercolor{structure}{fg=gray, bg=black}		% fg controls bullets
-	\setbeamercolor*{footerlike}{fg=white,bg=black}		% Controls the footerline
-\else
-	% Default to black on white
-	\setbeamercolor*{footerlike}{fg=white,bg=charcoal}	% for charcoal footer
-	\setbeamercolor{structure}{fg=charcoal, bg=white}	% fg controls bullets
-	% \setbeamercolor{normal text}{fg=white, bg=black}	
-\fi
-
-
-
-
-
-% These defined colors can be used in \textcolor{colorname}{text in that color}.
-
-% Colors for Keynote background: DARK
-\definecolor{keynotebottom}{rgb}{0.32,0.3,0.38}
-\definecolor{keynotemiddle}{rgb}{0.08,0.08,0.16}
-\definecolor{keynotetop}{rgb}{0,0,0}
-%%% Example of usage:
-%% \setbeamertemplate{background canvas}[vertical shading]
-%% [bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
-
-
-
-
-% ALERT AND COMMENT COLORS (just copies of the above colors)
-% (I don't know how ot do this more elegantly)
-\ifbeamer@Flip@dark
-	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
-	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
-	\definecolor{alert}{RGB}{204,204,102}		% paleale		
-	\definecolor{charcoal}{RGB}{51,51,51}		% charcoal
-	\definecolor{comment}{RGB}{80,80,80}		% light charcoal
-	\definecolor{Comment}{RGB}{100,100,100}		% ash
-	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
-\else
-%	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
-	\definecolor{ALERT}{RGB}{28,134,238}		% DodgerBlue2
-%	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
-	\definecolor{Alert}{RGB}{255,127,36}		% chocolate1
-%	\definecolor{alert}{RGB}{140,110,10}		% lager
-	\definecolor{alert}{RGB}{0,205,102} 		% SpringGreen3
-	\definecolor{charcoal}{RGB}{80,80,80}		% light charcoal
-	\definecolor{comment}{RGB}{51,51,51}		% charcoal
-	\definecolor{Comment}{RGB}{100,100,100}		% ash
-	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
-\fi
-
-
-% \definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
-% \definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
-% \definecolor{ash}{RGB}{100,100,100}			% medium gray
-% \definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
-% \definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
-% \definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
-% \definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
-% \definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
-% \definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
-
-
-% A few useful colors
-\setbeamercolor{alerted text}{fg=alert}
-
-%% I would like ot use these colors, but there seems to be a bug in Beamer
-%% where the footnote color infects the normal text color
-%% See:  http://tex.stackexchange.com/questions/8264/beamer-changing-the-footnote-color-also-changes-the-normal-text-color/8268#8268
-
-% \setbeamercolor{footnote}{parent=normal text, fg=FlipSand}
-% \setbeamercolor{footnote mark}{parent=normal text, fg=FlipSand}
-
-% Blocks
-\setbeamercolor{block title}{fg=gray!30!white,bg=charcoal}
-\setbeamercolor{block body}{parent=normal text,bg=gray, fg=white}
-
-\setbeamercolor{block title example}{fg=turtlegreen!30!white,bg=turtlegreen!40!black}
-\setbeamercolor{block body example}{parent=normal text,bg=turtlegreen, fg=white}
-% 
-\setbeamercolor{block title alerted}{fg=crimsonred!30!white,bg=crimsonred!40!black}
-\setbeamercolor{block body alerted}{parent=normal text,bg=crimsonred, fg=white}
-
-
-
-% These "setbeamercolor" commands are more standardized. 
-
-
-%% From the Beamer User Guide:
-% Outer themes base the color of navigational elements and, possibly, also of other elements, on the 
-% four palette colors. The “primary” palette should be used for the most important navigational elements, 
-% which are usually the ones that change most often and hence require the most attention by the audience. 
-% The “secondary” and “tertiary” are less important, the “quaternary” one is least important. 
-% By default, the palette colors do not have a background and the foreground ranges from structure.fg to black. 
-% For the sidebar, there is an extra set of palette colors, see palette sidebar primary.
-
-\setbeamercolor*{palette primary}{fg=crimsonred,bg=charcoal}
-\setbeamercolor*{palette secondary}{fg=paleale,bg=lager}
-\setbeamercolor*{palette tertiary}{fg=turtlegreen,bg=jeans}
-\setbeamercolor*{palette quaternary}{fg=paleblue,bg=regal}
-
-% \setbeamercolor*{palette primary}{fg=crimsonred,bg=white}
-% \setbeamercolor*{palette secondary}{fg=charcoal,bg=white}
-% \setbeamercolor*{palette tertiary}{fg=paleblue,bg=white}
-% \setbeamercolor*{palette quaternary}{fg=turtlegreen,bg=white}
-
-
-
-%% These are some structures that I define for use in the outer theme.
-
-\setbeamercolor*{crimsontown}{fg=paleblue,bg=crimsonred}
-\setbeamercolor*{bluetown}{fg=white,bg=DodgerBlue2}
-%\setbeamercolor*{titlelike}{fg=crimsonred}
-\setbeamercolor*{titlelike}{fg=DodgerBlue2}
-\setbeamercolor*{topbarlike}{fg=white, bg=charcoal}
-
-
-
-
-
-%% Some old definitions that I might want to get back to
-%
-% \setbeamercolor*{frametitle}{bg=white,fg=charcoal}
-% \setbeamercolor*{part title}{bg=white,fg=charcoal}
-% \setbeamercolor*{item}{fg=charcoal, bg=white}
-% 
-% \setbeamercolor*{separation line}{}
-% \setbeamercolor*{fine separation line}{}
-
-
-
-
-
-\mode
-<all>
diff --git a/Lectures_my/MC_2016/Lecture2/beamerinnerthemeFlip.sty b/Lectures_my/MC_2016/Lecture2/beamerinnerthemeFlip.sty
deleted file mode 100755
index 9910f4d..0000000
--- a/Lectures_my/MC_2016/Lecture2/beamerinnerthemeFlip.sty
+++ /dev/null
@@ -1,95 +0,0 @@
-% Inner style file for the BEAMER FLIP THEME
-% Copyright 2010 by Flip Tanedo
-% This file may be distributed and/or modified
-% 	1. under the LaTeX Project Public License and/or
-% 	2. under the GNU Public License.
-%
-% Based on: beamerinnerthemeFlip by Marco Barisione
-
-\mode<presentation>
-
-% % Use alternative title page style.
-% \DeclareOptionBeamer{alternativetitlepage}[true]{\def\beamer@Flip@alternativetitlepage{#1}}
-
-% Logo to use in the alternative title page.
-% \def\beamer@Flip@titlepagelogo{}
-% \DeclareOptionBeamer{titlepagelogo}{\def\beamer@Flip@titlepagelogo{#1}}
-
-% Bullet shape.
-\DeclareOptionBeamer{bullet}{\def\beamer@Flip@bullet{#1}}
-\DeclareOptionBeamer{shadow}[true]{\def\beamer@themerounded@shadow{#1}}
-\ExecuteOptionsBeamer{shadow=false}
-
-
-% \ExecuteOptionsBeamer{alternativetitlepage=false,bullet=square}
-\ProcessOptionsBeamer
-
-
-
-% Margins.
-\newlength{\beamer@Flip@normalmargin}
-\setlength{\beamer@Flip@normalmargin}{.06\paperwidth}
-\setbeamersize{text margin left=\beamer@Flip@normalmargin}
-\setbeamersize{text margin right=\beamer@Flip@normalmargin}
-\setlength\leftmargini{.6\beamer@Flip@normalmargin}
-\setlength\leftmarginii{.6\beamer@Flip@normalmargin}
-\setlength\leftmarginiii{.6\beamer@Flip@normalmargin}
-
-% Normal title page.
-\defbeamertemplate*{title page normal}{Flip theme}[1][]
-{
-  \vbox{}
-  \vfill
-  \begin{centering}
-    \begin{beamercolorbox}[wd=\paperwidth,sep=8pt,center,#1]{title page header}
-      \usebeamerfont{title}\inserttitle\par%
-      \ifx\insertsubtitle\@empty%
-      \else%
-        \vskip0.25em%
-        {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
-      \fi%     
-    \end{beamercolorbox}%
-    \vskip1em\par
-    \begin{beamercolorbox}[sep=8pt,center,#1]{author}
-      \usebeamerfont{author}\insertauthor
-    \end{beamercolorbox}
-    \begin{beamercolorbox}[sep=8pt,center,#1]{institute}
-      \usebeamerfont{institute}\insertinstitute
-    \end{beamercolorbox}
-    \begin{beamercolorbox}[sep=8pt,center,#1]{date}
-      \usebeamerfont{date}\insertdate
-    \end{beamercolorbox}\vskip0.5em
-    {\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
-  \end{centering}
-  \vfill
-}
-
-
-% Items.
-\defbeamertemplate{itemize item}{squarealt}%
-{\tiny\raise.5ex\hbox{\donotcoloroutermaths$\blacksquare$}}
-\defbeamertemplate{itemize subitem}{squarealt}%
-{\tiny\raise.4ex\hbox{\donotcoloroutermaths$\square$}}
-\defbeamertemplate{itemize subsubitem}{squarealt}%
-{\tiny\raise.3ex\hbox{\donotcoloroutermaths$\blacksquare$}}
-% 
-\defbeamertemplate{itemize item}{circlealt}%
-{\small\raise.2ex\hbox{\donotcoloroutermaths$\bullet$}}
-\defbeamertemplate{itemize subitem}{circlealt}%
-{\small\raise.1ex\hbox{\donotcoloroutermaths$\circ$}}
-\defbeamertemplate{itemize subsubitem}{circlealt}%
-{\scriptsize\raise.1ex\hbox{\donotcoloroutermaths$\bullet$}}
-\setbeamertemplate{items}[circlealt]
-% 
-\def\circletext{circle}
-\ifx\beamer@Flip@bullet\circletext
-  \setbeamertemplate{items}[circlealt]
-\else
-  \setbeamertemplate{items}[squarealt]
-\fi
-
-\setbeamertemplate{blocks}[rounded][shadow=\beamer@themerounded@shadow]
-
-
-\mode<all>
-
diff --git a/Lectures_my/MC_2016/Lecture2/beamerouterthemeFlip.sty b/Lectures_my/MC_2016/Lecture2/beamerouterthemeFlip.sty
deleted file mode 100755
index d12f6d6..0000000
--- a/Lectures_my/MC_2016/Lecture2/beamerouterthemeFlip.sty
+++ /dev/null
@@ -1,189 +0,0 @@
-% Outer style file for the BEAMER FLIP THEME 212
-% Copyright 2012 by Flip Tanedo
-% This file may be distributed and/or modified
-% 	1. under the LaTeX Project Public License and/or
-% 	2. under the GNU Public License.
-%
-% Based on: beamerouterthemesplit by Till Tantau
-
-
-\mode<presentation>
-
-% Watermark
-\DeclareOptionBeamer{watermark}{\def\beamer@Flip@watermark{#1}}
-\ExecuteOptionsBeamer{watermark=} % Default value is empty
-
-
-\newif\ifbeamer@Flip@bigpagenumber
-\beamer@Flip@bigpagenumberfalse
-\DeclareOptionBeamer{bigpagenumber}{\beamer@Flip@bigpagenumbertrue}
-
-
-\def\beamer@Flip@truetext{true} % Kludge for true/false so I can use ifx
-
-\DeclareOptionBeamer{topline}[true]{\def\beamer@Flip@topline{#1}}
-\ExecuteOptionsBeamer{topline=false} % Default
-
-
-
-% \ExecuteOptionsBeamer{topline=} % Default value is empty
-
-
-\ProcessOptionsBeamer
-
-
-\usepackage{tikz}	% For fancy decorations
-
-
-% I should fix these up to be more robust against color changes
-\setbeamercolor{section in head/foot}{parent=footerlike}
-\setbeamercolor{subsection in head/foot}{parent=footerlike}
-\setbeamercolor{author in head/foot}{parent=footerlike}
-\setbeamercolor{title in head/foot}{parent=footerlike}
-
-
-% \setbeamercolor{author in head/foot}{parent=section in head/foot}
-% \setbeamercolor{title in head/foot}{parent=subsection in head/foot}
-
-%\setbeamercolor{topbar}{parent=crimsontown}
-\setbeamercolor{topbar}{parent=bluetown}
-
-
-
-
-\usesectionheadtemplate
-  {\hfill\insertsectionhead}
-  {\hfill\color{fg!50!bg}\insertsectionhead}
-
-
-%%%%%%%%%%%%%%%%%%
-% The footer bar %
-%%%%%%%%%%%%%%%%%%
-
-\defbeamertemplate*{footline}{Flip theme}
-{%
-  \leavevmode% 
-% OBSERVATION: the "%" symbols inside hbox are all very important here.
-% The \hbox enviroment will insert spaces whenever there's whitespace
-% Adding a % at the END of each line ensures that any hard returns aren't
-% interpreted as white space. This allows the color boxes to be flush against
-% one another. If a faint white line appears one could probably do something like
-% hspace{-1px} and add an additional pixel-or-two wide beamercolorbox.	
-
-\hbox{%
-\begin{beamercolorbox}[wd=.4\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm, rightskip=.3cm plus1fil]{author in head/foot}%
-	\usebeamerfont{author in head/foot}\insertshortauthor  \end{beamercolorbox}%
-%
-\begin{beamercolorbox}[wd=.6\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}%
-    \usebeamerfont{title in head/foot}{\textit \insertshorttitle}%
-    \hskip2ex plus1fill%
-		% \insertpagenumber\,/\,\insertpresentationendpage %% Want FRAME, not SLIDE number
-		\insertframenumber/\inserttotalframenumber
- \end{beamercolorbox}%
-}%
-
-% This is a complete kludge for the faint white line which appears
-% between the beamer color boxes above. (It doesnt appear in presentation mode
-% but sometimes looks funny when the pdfs are viewed in a window)
-\begin{tikzpicture}[overlay]
-	\usebeamercolor[bg]{author in head/foot}
-	\draw[thick] (0,3.625ex) -- (\paperwidth,3.625ex);
-	\draw (.4\paperwidth,0) -- (.4\paperwidth,3.625ex);
-	\draw[thick] (0,0) -- (\paperwidth,0);
-\end{tikzpicture}
-
-
-
-\ifbeamer@Flip@bigpagenumber
-	\begin{tikzpicture}[overlay]
-	\usebeamercolor[bg]{author in head/foot}
-	\draw[fill] (\paperwidth-5ex,3.625ex) circle (6ex);
-	\usebeamercolor[fg]{author in head/foot}
-	\draw[fill] (\paperwidth-5ex,3.625ex) circle (5ex);
-	\usebeamercolor[bg]{author in head/foot}
-	\node at (\paperwidth-5ex,4ex) {\small$^\text{\insertframenumber}/_\text{\inserttotalframenumber}$};
-	\end{tikzpicture}
-\fi
-}
-
-
-
-\defbeamertemplate*{frametitle}{Flip theme}[1][left]
-{
-% \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
-\@tempdima=\textwidth%
-\advance\@tempdima by\beamer@leftmargin%
-\advance\@tempdima by\beamer@rightmargin%
-
-\vbox{}\vskip-3.5ex%
-%
-\begin{beamercolorbox}[sep=0.3cm,#1,wd=\the\@tempdima]{titlelike}
-	\usebeamerfont{frametitle}%
-	\vbox{}\vskip-1ex%
-	\if@tempswa\else\csname beamer@fte#1\endcsname\fi%
-	% \strut{\bf\insertframetitle}\strut\par% 					%% Include the \bf
-	\strut{\insertframetitle}\strut\par%
-	{%
-  		\ifx\insertframesubtitle\@empty%
-  		\else
-			{\usebeamerfont{framesubtitle}
-			 \usebeamercolor[fg]{framesubtitle}
-			\insertframesubtitle\strut\par}%
-  		\fi
- 	}%
-	%
-  	\vskip-1ex%
-  	\if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
-\end{beamercolorbox}%
-%%
-
-% \def\beamer@fteright{\vskip0.35cm\advance\leftskip by 1.7cm\advance\rightskip by1.7cm}
-}
-
-
-
-
-% Sidebar right is important: anything placed here will be
-% visible *behind* the main text. This is where to put any
-% interesting watermarks.
-%
-% 
-\defbeamertemplate*{sidebar right}{Flip theme}
-{
-% \llap{\includegraphics[width=50px]{footdecoration}}
-% \llap{\includegraphics[width=\paperwidth,height=\paperheight]{upperleft_watermark_transp}}
-%
-% THIS WOULD BE A NICE WAY TO PUT THE PARTICLE BUBBLE CHAMBER BG
-% ... make a png of the background with TRANSPARENT BG
-% ... then put the image here (use \hfill to get it on the left)
-% Now one can FURTHER apply a slight gradient in the BG
-%
-% 
-% % Top Line 
-\ifx\beamer@Flip@topline\beamer@Flip@truetext
-	\begin{tikzpicture}[overlay]
-	\usebeamercolor[fg]{titlelike}	% alternately, topbarlike would be a good beamercolor
-%	\usebeamercolor[fg]{topbarlike}	% alternately, topbarlike would be a good beamercolor
-	\draw[fill] (-\paperwidth,0) rectangle (0,-3pt);
-	\end{tikzpicture}
-\fi
-%
-% 
-  \vskip.2\beamer@leftmargin%
-   %\llap{\insertlogo\hskip.5\beamer@leftmargin}%
-   \llap{\insertlogo\hskip.08\beamer@leftmargin}%
-    \vfill%
-  \if \beamer@Flip@watermark\@empty
-	% This is kind of a kludge since if I put anything in here, I automatically get errors
-	% Somehow LaTeX wants to read the \else part!
-	\else%
-	\llap{\includegraphics[width=\paperwidth,height=\paperheight]{\beamer@Flip@watermark}}
-  \fi
-}
-
-
-
-\mode<all>
-\endinput
-
-
diff --git a/Lectures_my/MC_2016/Lecture2/beamerouterthemedecolines.sty b/Lectures_my/MC_2016/Lecture2/beamerouterthemedecolines.sty
deleted file mode 100755
index 669e493..0000000
--- a/Lectures_my/MC_2016/Lecture2/beamerouterthemedecolines.sty
+++ /dev/null
@@ -1,168 +0,0 @@
-% Copyright 2007 by Marco Barisione
-%
-% This file may be distributed and/or modified
-%
-% 1. under the LaTeX Project Public License and/or
-% 2. under the GNU Public License.
-
-\mode<presentation>
-
-% String used between the current page and the total page count.
-\def\beamer@decolines@pageofpages{/}
-\DeclareOptionBeamer{pageofpages}{\def\beamer@decolines@pageofpages{#1}}
-
-% Show a line below the frame title.
-\DeclareOptionBeamer{titleline}[true]{\def\beamer@decolines@titleline{#1}}
-
-% Image used for the watermark.
-\def\beamer@decolines@watermarkorig{}
-\DeclareOptionBeamer{watermark}{\def\beamer@decolines@watermarkorig{#1}}
-
-% Height of the watermark.
-\def\beamer@decolines@watermarkheight{100px}
-\DeclareOptionBeamer{watermarkheight}{\def\beamer@decolines@watermarkheight{#1}}
-
-% The original image height is watermarkheightmult * watermarkheight.
-\def\beamer@decolines@watermarkheightmult{1}
-\DeclareOptionBeamer{watermarkheightmult}{\def\beamer@decolines@watermarkheightmult{#1}}
-
-\ExecuteOptionsBeamer{titleline=false}
-\ProcessOptionsBeamer
-
-% Enable/disable the watermark.
-\def\watermarkon{%
-  \def\beamer@decolines@watermark{\beamer@decolines@watermarkorig}%
-}
-\def\watermarkoff{\def\beamer@decolines@watermark{}}
-
-% Initially enable the watermark.
-\watermarkon
-
-% Colors.
-\setbeamercolor*{lineup}{parent=palette primary}
-\setbeamercolor*{linemid}{parent=palette secondary}
-\setbeamercolor*{linebottom}{parent=palette tertiary}
-\setbeamercolor*{page header}{parent=titlelike}
-
-% Lengths
-\newlength{\headerheight}
-\setlength{\headerheight}{.045\paperheight}
-\newlength{\beamer@decolines@lineup}
-\setlength{\beamer@decolines@lineup}{.025\paperheight}
-\newlength{\beamer@decolines@linemid}
-\setlength{\beamer@decolines@linemid}{.015\paperheight}
-\newlength{\beamer@decolines@linebottom}
-\setlength{\beamer@decolines@linebottom}{.01\paperheight}
-
-% The height of the watermark part below the 3 bottom lines.
-\newlength{\beamer@decolines@watermarkheightbottom}
-\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@lineup}
-\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linemid}
-\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linebottom}
-
-% The height of the watermark part over the 3 bottom lines before shrinking.
-\newlength{\beamer@decolines@watermarkheightupperorig}
-\setlength{\beamer@decolines@watermarkheightupperorig}{\beamer@decolines@watermarkheight}
-\addtolength{\beamer@decolines@watermarkheightupperorig}{-\beamer@decolines@watermarkheightbottom}
-\multiply\beamer@decolines@watermarkheightupperorig by \beamer@decolines@watermarkheightmult
-
-% % Footer.
-% \defbeamertemplate*{footline}{decolines theme}
-% {
-%   \leavevmode%
-%   % Page number.
-%   \hbox{%
-%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=0ex,dp=0ex,center]{}%
-%     \usebeamerfont{palette primary}\insertframenumber{} \beamer@decolines@pageofpages{} \inserttotalframenumber%
-%   \end{beamercolorbox}%
-%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=0ex,dp=0ex]{}%
-%   \end{beamercolorbox}%
-%   } %
-%   % First line.
-%   \hbox{%
-%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{}%
-%   \end{beamercolorbox}%
-%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{lineup}%
-%   \end{beamercolorbox}%
-%   } %
-%   % Second line.
-%   \hbox{%
-%   \begin{beamercolorbox}[wd=\paperwidth,ht=\beamer@decolines@linemid,dp=0pt]{linemid}%
-%   \end{beamercolorbox}%
-%   } %
-%   % Third line.
-%   \hbox{%
-%   \begin{beamercolorbox}[wd=.1\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{}%
-%   \end{beamercolorbox}%
-%   \begin{beamercolorbox}[wd=.9\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{linebottom}%
-%   \end{beamercolorbox}%
-%   }%
-%   % This seems to fix some alignment problems with the watermark. It has to be
-%   % always applied if you do not want to see the footer moving up and down when
-%   % moving from a page with watermark to a page without or vice versa.
-%   \vskip-.5px%
-%   % Watermark.
-%   \if\beamer@decolines@watermark\@empty\else%
-%     \vskip-\beamer@decolines@watermarkheightbottom%
-%     \llap{\includegraphics[height=\beamer@decolines@watermarkheightbottom,clip=true,%
-%       trim=0pt 0pt 0pt \beamer@decolines@watermarkheightupperorig]{\beamer@decolines@watermark}\hskip-\paperwidth}%
-%   \fi%
-% }
-
-\defbeamertemplate*{headline}{decolines theme}
-{
-  \leavevmode%
-  \hbox{%
-  \begin{beamercolorbox}[wd=\paperwidth,ht=\headerheight,dp=0pt]{page header}%
-  \end{beamercolorbox}%
-  } %
-  \vskip0pt%
-}
-
-\defbeamertemplate*{frametitle}{decolines theme}[1][left]
-{
-  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
-  \@tempdima=\textwidth%
-  \advance\@tempdima by\beamer@leftmargin%
-  \advance\@tempdima by\beamer@rightmargin%
-  \vbox{}\vskip-.5\beamer@leftmargin%
-  \begin{beamercolorbox}[sep=\beamer@leftmargin,#1,wd=\the\@tempdima]{}
-    \usebeamerfont{frametitle}\usebeamercolor[bg]{framesubtitle}%
-    \vbox{}\vskip0ex%
-    \if@tempswa\else\csname beamer@fte#1\endcsname\fi%
-    \strut\insertframetitle\strut\par%
-    {%
-      \ifx\insertframesubtitle\@empty%
-      \else%
-      {\usebeamerfont{framesubtitle}\usebeamercolor[bg]{framesubtitle}\insertframesubtitle\strut\par}%
-      \fi
-    }%
-    \vskip-1ex%
-    \if@tempswa\else\vskip-\beamer@leftmargin\fi
-  \end{beamercolorbox}%
-  \def\beamer@decolines@truetext{true}%
-  \ifx\beamer@decolines@titleline\beamer@decolines@truetext%
-    \vskip-.5\beamer@leftmargin%
-    \begin{beamercolorbox}[wd=\textwidth,ht=.1ex,dp=0ex]{linemid}%
-    \end{beamercolorbox}%
-  \fi
-}
-
-% Frame title continuations, default
-\defbeamertemplate*{frametitle continuation}{decolines theme}{(\insertcontinuationcount)}
-
-
-
-\defbeamertemplate*{sidebar right}{decolines theme}
-{
-  \vskip.1\beamer@leftmargin%
-  \llap{\insertlogo\hskip.5\beamer@leftmargin}%
-  \vfill%
-  \if\beamer@decolines@watermark\@empty\else%
-  \llap{\includegraphics[height=\beamer@decolines@watermarkheight]{\beamer@decolines@watermark}}%
-  \vskip-\beamer@decolines@watermarkheightbottom%
-  \fi
-}
-
-\mode
-<all>
diff --git a/Lectures_my/MC_2016/Lecture2/beamerthemeFlip.sty b/Lectures_my/MC_2016/Lecture2/beamerthemeFlip.sty
deleted file mode 100755
index a615ff5..0000000
--- a/Lectures_my/MC_2016/Lecture2/beamerthemeFlip.sty
+++ /dev/null
@@ -1,59 +0,0 @@
-% Main style file for the BEAMER FLIP THEME 2012
-% Copyright 2012 by Flip Tanedo
-% This file may be distributed and/or modified
-% 	1. under the LaTeX Project Public License and/or
-% 	2. under the GNU Public License.
-%
-% Comments: This is still a work in progress.
-% A good way to present: http://www.cs.hmc.edu/~oneill/freesoftware/pdftokeynote.html
-
-% \usepackage{beamerthemesplit}
-\mode<presentation>
-
-
-%% These are the options that are fed in through the driver file
-%% Some of them get passed on to the other theme files
-\DeclareOptionBeamer{bullet}{\PassOptionsToPackage{bullet=#1}{beamerinnerthemeFlip}}
-\DeclareOptionBeamer{bigpagenumber}{\PassOptionsToPackage{bigpagenumber}{beamerouterthemeFlip}}
-\DeclareOptionBeamer{topline}[true]{\PassOptionsToPackage{topline=#1}{beamerouterthemeFlip}}
-\DeclareOptionBeamer{shadow}[false]{\PassOptionsToPackage{shadow=#1}{beamerinnerthemeFlip}}
-
-\DeclareOptionBeamer{watermark}{\PassOptionsToPackage{watermark=#1}{beamerouterthemeFlip}}
-
-\ProcessOptionsBeamer
-
-\useinnertheme{Flip}	% Calls beamerinnerthemeFlip.sty
-\useoutertheme{Flip}	% Calls beamerouterthemeFlip.sty
-\usecolortheme{Flip}	% Calls beamercolorthemeFlip.sty
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%
-% Background  Gradient %
-%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%% Usually I would leave this on
-%% However, there seems to be problems with XeLaTeX
-%% http://tex.stackexchange.com/questions/29497/xelatex-preventing-beamer-from-using-different-backgrounds
-\setbeamertemplate{background canvas}[vertical shading][bottom=blue!.1, middle=white, top=white]
-
-%% HOWEVER: it seems to make it impossible to change this later on in the document
-
-
-%% For a dark background: 
-% \setbeamertemplate{background canvas}[vertical shading][bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
-
-
-%% An alternate way: \beamertemplateshadingbackground{blue!.1}{red!2}
-%% In general: you want the gradient to be subtle!
-
-
-%%%%%%%%%%%%%%%%%%%%%%%
-% Navigation  symbols %
-%%%%%%%%%%%%%%%%%%%%%%%
-
-\setbeamertemplate{navigation symbols}{} 	% Turns off PDF navigation symbols
-
-\mode<all>
-\endinput
\ No newline at end of file
diff --git a/Lectures_my/MC_2016/Lecture2/images/Focal_stability.png b/Lectures_my/MC_2016/Lecture2/images/Focal_stability.png
deleted file mode 100644
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diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz+mp.sh b/Lectures_my/MC_2016/Lecture2/mchrzasz+mp.sh
deleted file mode 100644
index 1a24852..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz+mp.sh
+++ /dev/null
@@ -1 +0,0 @@
-#!/bin/sh
diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz-blx.bib b/Lectures_my/MC_2016/Lecture2/mchrzasz-blx.bib
deleted file mode 100644
index 9e405b3..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz-blx.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@Comment{$ biblatex control file $}
-@Comment{$ biblatex version 2.3 $}
-Do not modify this file!
-
-This is an auxiliary file used by the 'biblatex' package.
-This file may safely be deleted. It will be recreated as
-required.
-
-@Control{biblatex-control,
-  options = {2.3:0:0:1:0:0:1:1:0:0:0:0:1:1:3:1:79:+},
-}
diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz.aux b/Lectures_my/MC_2016/Lecture2/mchrzasz.aux
deleted file mode 100644
index fae0953..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz.aux
+++ /dev/null
@@ -1,202 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
-\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
-\global\let\oldcontentsline\contentsline
-\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
-\global\let\oldnewlabel\newlabel
-\gdef\newlabel#1#2{\newlabelxx{#1}#2}
-\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
-\AtEndDocument{\ifx\hyper@anchor\@undefined
-\let\contentsline\oldcontentsline
-\let\newlabel\oldnewlabel
-\fi}
-\fi}
-\global\let\hyper@last\relax 
-\gdef\HyperFirstAtBeginDocument#1{#1}
-\providecommand*\HyPL@Entry[1]{}
-\bibstyle{biblatex}
-\bibdata{mchrzasz-blx,bib}
-\citation{biblatex-control}
-\HyPL@Entry{0<</P(1)>>}
-\providecommand \oddpage@label [2]{}
-\@writefile{toc}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
-\@writefile{lof}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
-\@writefile{lot}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
-\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\beamer@endinputifotherversion {3.24pt}}
-\@writefile{nav}{\defcounter {refsection}{0}\relax }\@writefile{nav}{\beamer@endinputifotherversion {3.24pt}}
-\select@language{english}
-\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\select@language{english}}
-\@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\select@language{english}}
-\@writefile{lot}{\defcounter {refsection}{0}\relax }\@writefile{lot}{\select@language{english}}
-\@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}}
-\@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}}
-\pgfsyspdfmark {pgfid3}{0}{0}
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-\HyPL@Entry{1<</P(2)>>}
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deleted file mode 100644
index e69de29..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz.out
+++ /dev/null
diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture2/mchrzasz.pdf
deleted file mode 100644
index be0d5f9..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz.pdf
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deleted file mode 100644
index 3368f59..0000000
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diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz.snm b/Lectures_my/MC_2016/Lecture2/mchrzasz.snm
deleted file mode 100644
index e69de29..0000000
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz.snm
+++ /dev/null
diff --git a/Lectures_my/MC_2016/Lecture2/mchrzasz.tex b/Lectures_my/MC_2016/Lecture2/mchrzasz.tex
index 1d8e744..d0c78f3 100644
--- a/Lectures_my/MC_2016/Lecture2/mchrzasz.tex
+++ b/Lectures_my/MC_2016/Lecture2/mchrzasz.tex
@@ -2,9 +2,7 @@
 
 \usepackage[english]{babel} 
 \usepackage{polski}         
-\usepackage[skins,theorems]{tcolorbox}
-\tcbset{highlight math style={enhanced,
-  colframe=red,colback=white,arc=0pt,boxrule=1pt}}
+
 
 \usetheme[
 	bullet=circle,		% Other option: square
@@ -20,11 +18,11 @@
 \usepackage[lf]{berenis}
 \usepackage[LY1]{fontenc}
 \usepackage[utf8]{inputenc}
-
+\usepackage{cases}
+\usepackage{mathtools}
 \usepackage{emerald}
 \usefonttheme{professionalfonts}
-\usepackage[no-math]{fontspec}	
-\usepackage{listings}
+\usepackage[no-math]{fontspec}		
 \defaultfontfeatures{Mapping=tex-text}	% This seems to be important for mapping glyphs properly
 
 \setmainfont{Gillius ADF}			% Beamer ignores "main font" in favor of sans font
@@ -138,9 +136,9 @@
 
 % SOME COMMANDS THAT I FIND HANDY
 % \renewcommand{\tilde}{\widetilde} % dinky tildes look silly, dosn't work with fontspec
-%\newcommand{\comment}[1]{\textcolor{comment}{\footnotesize{#1}\normalsize}} % comment mild
-%\newcommand{\Comment}[1]{\textcolor{Comment}{\footnotesize{#1}\normalsize}} % comment bold
-%\newcommand{\COMMENT}[1]{\textcolor{COMMENT}{\footnotesize{#1}\normalsize}} % comment crazy bold
+\newcommand{\comment}[1]{\textcolor{comment}{\footnotesize{#1}\normalsize}} % comment mild
+\newcommand{\Comment}[1]{\textcolor{Comment}{\footnotesize{#1}\normalsize}} % comment bold
+\newcommand{\COMMENT}[1]{\textcolor{COMMENT}{\footnotesize{#1}\normalsize}} % comment crazy bold
 \newcommand{\Alert}[1]{\textcolor{Alert}{#1}} % louder alert
 \newcommand{\ALERT}[1]{\textcolor{ALERT}{#1}} % loudest alert
 %% "\alert" is already a beamer pre-defined
@@ -201,12 +199,13 @@
 \def\e{{\rm e}}
 \def\Br{{\rm Br}}
 \def\fixme{{\color{red} FIXME!}}
-\def\mc{{\color{Magenta}{MC}}}
+\def\mc{{\color{Magenta}{MC}}\xspace}
 \def\pdf{{\rm p.d.f.}}
+\def\ARROW{{\color{JungleGreen}{$\Rrightarrow$}}\xspace}
 
 \author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
 \institute{UZH}
-\title[Random number generators and application]{Random number generators and application}
+\title[Introduction to \\Monte Carlo methods]{Introduction to \\Monte Carlo methods}
 \date{\fixme}
 
 
@@ -220,7 +219,7 @@
 \begin{center}
 	\begin{columns}
 		\begin{column}{0.9\textwidth}
-			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Random number generators and application}
+			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Introduction to \\Monte Carlo methods}
 		\end{column}
 		\begin{column}{0.2\textwidth}
 		  %\includegraphics[width=\textwidth]{SHiP-2}
@@ -242,519 +241,638 @@
 \vspace{1em}
 %		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
 \vspace{0.5em}
-	\textcolor{normal text.fg!50!Comment}{Experimental Methods in Particle Physics, \\ 25 November, 2015}
+	\textcolor{normal text.fg!50!Comment}{Monte Carlo methods, \\ 3 March, 2016}
 \end{center}
 \end{frame}
 }
 
 
-\begin{frame}\frametitle{Random and pseudorandom numbers}
-  \begin{exampleblock}{John von Neumann:}                                                                                                                                                                                                                                                                                                
-''Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.''
+
+
+\begin{frame}\frametitle{Monte Carlo and integration}
+\begin{footnotesize}
+$\hookrightarrow$ {\color{BrickRed}{\textbf{All MC calculations are equivalent to preforming an integration.}}}\\
+$\rightrightarrows$ Assumptions: $r_i$ random numbers from $\mathcal{U}(0,1)$. The MC result:
+\begin{align*}
+F=F(r_1,r_2,...r_n)
+\end{align*}
+is unbias estimator of an integral:
+\begin{align*}
+I=\int_0^1...\int_0^1 F(x_1,x_2,...,x_n)dx_1,dx_2...,dx_n
+\end{align*}
+aka the expected value of the $I$ integral is:
+\begin{align*}
+E(F)=I.
+\end{align*}
+    \begin{exampleblock}{}                                                                                                                                                                                                                                                                                                
+$\Rrightarrow$ This mathematical identity is the most useful property of the MC methods. It is a link between mathematical analysis and statistic world. Now we can use the best of the both world!
+        \end{exampleblock} 
+If we want to calculate the integral in different range then $(0,1)$ we just scale the the previous result:
+\begin{align*}
+\dfrac{1}{N}\sum_{i=1}^N f(x_i) \xrightarrow{N\to \infty} E(f)=\dfrac{1}{b-a}\int_a^b f(x)dx
+\end{align*}
+
+
+\end{footnotesize}
+\end{frame}
+
+
+\begin{frame}\frametitle{Uncertainty from Monte Carlo methods}
+\begin{footnotesize}
+$\Rrightarrow$ In practice we do not have $N\to \infty$ so we will never know the exact result of an integral :(\\
+$\longmapsto$ Let's use the {\color{BrickRed}{statistical}} world and estimate the uncertainty of an integral in this case :)\\
+$\rightarrowtail$ A variance of a MC integral:
+\begin{align*}
+V(\hat{I}) = \dfrac{1}{n} \Big\lbrace E(f^2) - E^2(f) \Big\rbrace = \dfrac{1}{n} \Big\lbrace \dfrac{1}{b-a} \int_a^b f^2(x)dx - I^2 \Big\rbrace
+\end{align*}
+    \begin{alertblock}{}                                                                                                                                                                                                                                                                                                
+$\looparrowright$ To calculate $V(\hat{I})$ one needs to know the value of $I$!
+\end{alertblock}
+$\Rrightarrow$ In practice $V(\hat{I})$ is calculated via estimator:
+\begin{columns}
+\column{2in}
+\begin{align*}
+\hat{V}(\hat{I})=\dfrac{1}{n}\hat{V}(f),
+\end{align*}
+\column{3in}
+\begin{align*}
+\hat{V}(f) = \dfrac{1}{n-1}\sum_{i=1}^n  \Big[ f(x_i)-\dfrac{1}{n} \sum_{i=1}^nf(x_i)\Big]^2.
+\end{align*}
+\end{columns}
+
+
+$\Rrightarrow$ MC estimator of standard deviation: $\hat{\sigma}=\sqrt{\hat{V}(\hat{I})}$
+
+
+\end{footnotesize}
+\end{frame}
+
+
+\begin{frame}\frametitle{Buffon needle - $\pi$ number calculus}
+\begin{footnotesize}
+
+$\Rrightarrow$ Buffon needle (Buffon 1777, Laplace 1886):
+We are throwing a needle (of length $l$) on to a surface covered with parallel lines width distance $L$. If a thrown needle touches a line we count a hit, else miss. Knowing the number of hits and misses one can calculate the $\pi$ number.
+\vspace{0.3cm}
+\begin{columns}
+\column{0.1in}
+{~}
+\column{2in}
+{\color{ForestGreen}{Experiment:}}
+\column{2.8in}
+{\color{Cerulean}{Theory:}}
+
+\end{columns}
+
+
+\begin{columns}
+\column{0.1in}
+{~}
+\column{2in}
+
+\includegraphics[width=0.9\textwidth]{images/buffon.png}\\
+$n$ -  number of hits\\
+$N$ number of hits and misses,\\
+aka number of tries.
+
+\column{2.8in}
+$\Rightarrow$ x - angle between needle and horizontal line, $x \in \mathcal{U}(0,\pi)$.
+$\Rightarrow$ the probability density function (\pdf) for x:
+\begin{align*}
+\rho(x)=\dfrac{1}{\pi}
+\end{align*}
+$\Rightarrow$  $p(x)$ probability to hit a line for a given x value:
+\begin{align*}
+p(x)=\dfrac{l}{L}\vert \cos x \vert
+\end{align*}
+$\Rightarrow$ Total hit probability:
+\begin{align*}
+P = E[p(x)]=\int_0^{\pi}p(x)\rho(x)dx=\dfrac{2l}{\pi L}
+\end{align*}
+
+\end{columns}
+Now one can calculate $\hat{P}$ from MC : $\hat{P}=\dfrac{n}{N} \xrightarrow{N\to \infty} P= \dfrac{2l}{\pi L} \Rightarrow \hat{\pi}=\dfrac{2Nl}{nL}$
+
+
+\end{footnotesize}
+\end{frame}
+
+
+
+
+
+
+
+
+
+
+
+\begin{frame}\frametitle{Buffon needle - Simplest Carlo method}
+\begin{footnotesize}
+{\color{MidnightBlue}{Monte Carlo type ''hit or miss''}}\\
+Let's use the summery of $p(x)$ function and stake $0<x<\frac{\pi}{2}$.\\
+$\Rightarrow$ Algorithm:\\
+\begin{columns}
+\column{0.1in}
+{~}
+\column{3.2in}
+
+
+
+Generate 2 dim. distribution:
+\begin{align*}
+(x,y): \mathcal{U}(0,\dfrac{\pi}{2})\times \mathcal{U}(0,1) {\rm{~and~}}
+\end{align*}
+\begin{align*}
+y 
+\begin{cases}
+  \leq p(x): & \text{hit},\\
+> p(x):             & \text{miss}.
+\end{cases}
+\end{align*}
+
+\column{2.5in}
+\includegraphics[width=0.75\textwidth]{images/result.png}
+
+
+
+\end{columns}
+Let's define weight function: $w(x,y)=\Theta(p(x)-y)$, \\
+where $\Theta(x)$ is the step function.\\
+$\rightarrowtail$ \pdf : $\varrho(x,y)=\rho(x)g(y)=\frac{2}{\pi} \cdot 1$\\
+$\Rightarrow$ Integrated probability:
+\begin{align*}
+P=E(w)=\int w(x,y) \varrho(x,y)dx dy = \dfrac{2l}{\pi L} \xleftarrow{N\to \infty}\hat{P}=\frac{1}{N} \sum_{i=1}^N w(x_i,y_i)= \dfrac{n}{N}
+\end{align*}
+Standard deviation for $\hat{P}$: $\hat{\sigma}=\dfrac{1}{\sqrt{N-1}}\sqrt{\dfrac{n}{N}\Big(1-\dfrac{n}{N}\Big)} $
+
+\end{footnotesize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Heads or tails \mc}
+
+    \begin{exampleblock}{}                                                                                                                                                                                                                                                                                                
+$\Rrightarrow$ \mc estimator of an integral that is based on counting the numbers of positive trials compared to the failed ones is called ''hit or miss''
+        \end{exampleblock}
+
+\ARROW The probability is described by the Bernoulli distribution:
+\begin{equation}
+\mathcal{P}(n) = \binom{n}{k} P^n (1-P)^{N-n}, \nonumber
+\end{equation}
+where $P$ is the probability of success, N is the number of trials and n is the number of successes.\\
+\ARROW The following are true:
+\begin{equation}
+E(n)=NP, \nonumber
+\end{equation}
+\begin{equation}
+V(n)=NP(1-P), \nonumber
+\end{equation}
+\ARROW Translating this into probability basis:
+\begin{equation}
+E(\hat{P})=P,{~}{~}V(\hat{P})\dfrac{P(1-P)}{N}. \nonumber
+\end{equation}
+\ARROW E2.1 prove the above.
+
+
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Buffon needle}
+\begin{small}
+
+
+$\Rrightarrow$ Lets make this toy experiment and calculate the $\pi$ number.\\
+$\hookrightarrow$ We can simulate the central position $(y)$ of an needle between $(-L, L)$ from $\mathcal{U}(-L, L)$. 
+
+  \begin{exampleblock}{Symmetry:}                                                                                                                                                                                                                                                                                                
+Please note the symmetry of the problem, if the position of the needle would be $>L$ then we can shift the needle by any number of $L$'s.
         \end{exampleblock} 
 
-$\color{PineGreen}\Rrightarrow$ Random number: a given value that is taken by a random variable $ \twoheadrightarrow$ by definition cannot be predicted.\\
-%$\color{PineGreen}\Rrightarrow$ Sequence of random numbers $\twoheadrightarrow$ 
-$\color{PineGreen}\Rrightarrow$ Sources of truly random numbers:
-\begin{itemize}
-\item Mechanical
-\item Physical
-\end{itemize}
-$\color{PineGreen}\Rrightarrow$ Disadvantages of physical generators:
-\begin{itemize}
-\item To slow for typical applications, especially the mechanical ones!
-\item Not stable; small changes in boundary conditions might lead to completely different results!
-\end{itemize}
-
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Random numbers - history remark}
-$\color{PineGreen}\Rrightarrow$ In the past there were books with random numbers:
-\begin{center}
-\includegraphics[width=0.55\textwidth]{images/million-random-digits-open.jpg}
-  \end{center}
-$\color{PineGreen}\Rrightarrow$ It's obvious that they didn't become very popular ;)\\
-$\color{PineGreen}\Rrightarrow$ This methods are comming back!\\
-$\color{PineGreen}\twoheadrightarrow$ Storage device are getting more cheap and bigger (CD, DVD).\\
-$\color{PineGreen}\twoheadrightarrow$ 1995: G. Marsaglia, $650\rm MB$ of random numbers, ''White and Black Noise''.
-
-
-\end{frame}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Pseudorandom numbers}
-$\color{PineGreen}\Rrightarrow$ Pseudorandom numbers are numbers that are generated accordingly to strict mathematical formula. \\
-$\color{PineGreen}\looparrowright$ Strictly speaking they are non random numbers, how ever they have all the statistical properties of random numbers.\\
-$\color{PineGreen}\looparrowright$  Discussing those properties is a wide topic so let's just say that without knowing the formula they are generated by one cannot say if those numbers are random or not.\\
-$\color{PineGreen}\Rrightarrow$  Mathematical methods of producing pseudorandom numbers:
-\begin{itemize}
-\item Good statistical properties of generated numbers.
-\item Easy to use and fast!
-\item Reproducible!
-\end{itemize}
-$\color{PineGreen}\Rrightarrow$ Since mathematical pseudorandom genrators are dominantly:  pseudorandom $\rightarrowtail$ random.
-
-
-\end{frame}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Middle square generator; von Neumann}
-$\color{PineGreen}\Rrightarrow$  The first mathematical generator (middle square) was proposed by von Neumann (1964).\\{~}\\
-$\color{PineGreen}\looparrowright$ Formula: $
-\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
-  boxrule=0pt,frame hidden]{ X_n= \lfloor X_{n-1}^2\cdot 10 ^{-m} \rfloor - \lfloor X_{n-1}^2\cdot 10^{-3m} \rfloor \cdot 10^{2m}}
-$
-
-$\color{PineGreen}\looparrowright$ where $X_0$ is a constant (seed), $\lfloor\cdot\rfloor$ is the cut-off of a number to integer.\\
-$\color{PineGreen}\Rrightarrow$  Example:\\
-{~}{~}Let's put $m=2$ and $X_0=2045$:\\
-\begin{columns}
-\column{0.1\textwidth}
-{~}
-\column{0.4\textwidth}
-$\color{PineGreen}\looparrowright$ $X_0^2=\underbrace{04}_{\rm rej}1820\underbrace{25}_{\rm rej}$
-\column{0.3\textwidth}
- $\Rightarrow X_1=1820$
-\end{columns}
-\begin{columns}
-\column{0.1\textwidth}
-{~}
-\column{0.4\textwidth}
-$\color{PineGreen}\looparrowright$ $X_1^2=\underbrace{03}_{\rm rej}3124\underbrace{00}_{\rm rej}$
-\column{0.3\textwidth}
- $\Rightarrow X_1=3124$
-\end{columns}
-$\color{PineGreen}\looparrowright$ Simple generator but unfortunately quite bad generator. Firstly the sequences are very short and strongly dependent on the $X_0$ number.
-
-
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Linear generators $\rm \color{RubineRed}{Lecture2/Linear\_gen1}$}
-$\color{PineGreen}\Rrightarrow$  This was a first generator written and it's a good example how to not write generators.\\
-$\color{PineGreen}\Rrightarrow$ It's highly non stable!
-\includegraphics[width=0.8\textwidth]{images/shit.png}
-
-
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Linear generators}
-
-$\color{PineGreen}\Rrightarrow$ General equation:
-\begin{equation}
-\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
-  boxrule=0pt,frame hidden]{X_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m    ,} \nonumber
-\end{equation}
-$\color{PineGreen}\looparrowright$ where $a_i,c,m$ are parameters of a generator(integer numbers).\\
-$\color{PineGreen}\looparrowright$  Generator initialization $\color{PineGreen} \rightleftarrows$ setting those parameters.\\
-$\color{PineGreen}\Rrightarrow$ Very old generators. (often used in Pascal, or first C versions):
-\begin{columns}
-\column{0.1\textwidth}
-{~}
-\column{0.8\textwidth}
-$k=1:~X_n=(aX_{n-1} +c )~{\rm mod} m,$
-$c=
-\begin{cases}
-=0, {\rm multiplicative geneator}\\
-\neq 0, {\rm mix geneator}
-\end{cases}
-$
-\column{0.1\textwidth}
-{~}
-
-
-\end{columns}
-$\color{PineGreen}\Rrightarrow$ The period can be achieved by tuning the seed parameters:\\
-$P_{{\rm max}}=
-\begin{cases}
-2^{L-2};~{\rm for~} m=2^L\\
-m-1;~{\rm for~} m= {\rm prime~number}
-\end{cases}
-$
-\end{frame}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Shift register generator}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ General equation:
-\begin{equation}
-\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
-  boxrule=0pt,frame hidden]{b_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~2    ,} \nonumber
-\end{equation}
-where $a_i \subset(\lbrace0,1\rbrace )$\\
-$\color{PineGreen}\Rrightarrow$ Super fast and easy to implement due to: $(a+b) ~{\rm mod}~2 = a~{\rm xor}~b$
-\begin{center}
-\begin{tabular}{||c|c|c||}
-\hline \hline a & b & a xor b\\ \hline
-0 & 0 & 0\\
-1 & 0 & 1\\
-0 & 1 & 1\\
-1 & 1 & 0\\ \hline \hline
-\end{tabular}
-\end{center}
-$\color{PineGreen}\Rrightarrow$ Maximal period is $2^k-1$.\\
-$\color{PineGreen}\Rrightarrow$ Example (Tausworths generator):\\
-$a_p=a_q=1$, other $a_i=0$ and $p>q$. Then: $b_n=b_{n-p}~{\rm xor}~b_{n-q}$
-
-$\color{PineGreen}\Rrightarrow$ How to get numbers from bits (for example):\\
-$U_i = \sum_{j=1}^L 2^{-j} b_{is+j},~s<L$.
-
+$\hookrightarrow$ New we simulate the angle $(\phi)$ with a flat distribution from $(0,\pi)$. 
+$\hookrightarrow$ The maximum and minimum $y$ position of the needle are:
+\begin{align*}
+y_{\max}=y+\vert \cos \phi \vert l\\
+y_{\min}=y-\vert \cos \phi \vert l
+\end{align*}
+$\hookrightarrow$ Now we check if the needle touches any of the lines: $y=L$, $y=0$ or $y=-L$. If yes we count the events.
 \end{small}
-\end{frame}
+\begin{center}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Fibonacci generator}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ In 1202 Fibonacci with Leonardo in Piza:
-\begin{equation}
-f_n=f_{n-2}+f_{n-1},~n\geqslant 2 \nonumber
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ Based on this first generator was created (Taussky and Todd, 1956):
-\begin{equation}
-X_n=(X_{n-2}+X_{n-1})~{\rm mod}~m,~n\geqslant 2 \nonumber
-\end{equation}
-This generator isn't so good in terms of statistics tests.\\
-$\color{PineGreen}\Rrightarrow$ Generalization:
-\begin{equation}
-X_n=(X_{n-r} \odot X_{n-s})~{\rm mod}~m,~n \geqslant r ,~s \geqslant 1 \nonumber
-\end{equation}\begin{center}
 
-\begin{tabular}{||c|c|c||}
-\hline \hline $\bigodot$ & $P_{max}$ & Stat. properties \\ \hline
-$+,-$ & $(2^r-1)2^{L-1}$ & good \\
-$x$ & $(2^r-1)2^{L-13}$ & very good \\
-$xor$ & $(2^r-1)$ & poor \\ \hline \hline 
+\begin{tiny}
+
+
+\begin{tabular}{r c c c }
+$N$ & $\hat{\pi}$ & $\hat{\pi} - \pi$ & $\sigma(\hat{\pi})$ \\
+10000 & $3.12317$ & $-0.01842$ & $0.03047$\\
+100000 & $3.14707$ & $0.00547$ & $0.00979$\\
+1000000 & $3.13682$ & $-0.00477$ & $0.00307$\\
+10000000 & $3.14096$ & $-0.00063$ & $0.00097$\\
+
+
 \end{tabular}
 
-
+\end{tiny}
 \end{center}
+\begin{footnotesize}
 
-\end{small}
+\ARROW E2.2 Write the program that calculates the $\pi$ number using the Buffon needle.
+\end{footnotesize}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\begin{frame}\frametitle{Crude Monte Carlo method of integration}
+\begin{footnotesize}
+$\Rrightarrow$ {\color{MidnightBlue}{Crude Monte Carlo method of integration is based on Law of Large Numbers: }}\\
+\begin{align*}
+\dfrac{1}{N} \sum_{i=1}^N f(x_i) \xrightarrow{N\to \infty} \dfrac{1}{b-a}\int_a^b f(x)dx =E(f)
+\end{align*}
+$\Rrightarrow$ The standard deviation can be calculated:
+\begin{align*}
+\sigma = \dfrac{1}{\sqrt{N}} \sqrt{\Big[ E(f^2) -E^2(f)\Big] }
+\end{align*}
+
+$\Rrightarrow$ From LNT we have:
+\begin{align*}
+P= \int w(x) \rho(x) dx = \int_0^{\pi/2} (\frac{l}{L} \cos x ) \frac{2}{\pi} dx= \dfrac{2l}{\pi L}  \xrightarrow{N\to \infty} \frac{1}{N}\sum_{i=1}^N w(x_i)
+\end{align*}
+$\Rrightarrow$ Important comparison between ''Hit and mishit'' and Crude \mc~methods. One can analytically calculate:
+
+\begin{align*}
+\hat{\sigma}^{{\rm{Crude}}} < \hat{\sigma}^{{\rm{Hit~and~mishit}}}
+\end{align*}
+
+
+$\Rrightarrow$ Crude \mc~is \textbf{always} better then ''Hit and mishit'' method. We will prove this on an example (can be proven analytically as well).
+
+
+\end{footnotesize}
 \end{frame}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Multiply with carry, generator}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ We start from:
-\begin{equation}
-b_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m, \nonumber
-\end{equation}
-where $a_1,..,a_k \in \mathbb{N}$ are constant parameters.\\
-$\color{PineGreen}\Rrightarrow$  The c parameters is calculated foe each step:
-\begin{equation}
-c=\lfloor (a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)/m\rfloor, \nonumber 
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ Initialization: $a_1,..,a_k,c$.\\
-$\color{PineGreen}\Rrightarrow$ Advantages:
-\begin{itemize}
-\item Fast and easy to implement.
-\item Large period.
-\item Good statistical properties.
-\item First proposed by Marsaglia.
-\end{itemize}
 
-\end{small}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Subtract with borrow, generator}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$  Created again by Marsaglia (1991):
+\begin{frame}\frametitle{Crude vs ''hit or miss''}
+\begin{footnotesize}
+\ARROW The Crude \mc is never worse then the "hit or miss" method.\\
+\ARROW Prove: Let's assume we calculate an integral:
 \begin{equation}
-X_n=(X_{n-r} \circleddash X_{n-s})~{\rm mod}~m,~r,s  \nonumber \in \mathbb{N},
+I=\int_0^1f(x)dx,~{\rm{and~}}0\leq f(x)\leq 1 {~}	\forall x \in (0,1) \nonumber
 \end{equation}
-where :
+\ARROW  The variation for the ''hit-or-miss''(HM) method: $V(\hat{I}_{HM})=\dfrac{1}{N}(I-I^2)$
+\ARROW  The variation for the crude method:  $V(\hat{I}_{HM})=\dfrac{1}{N}[\int_0^1 f^2(x)dx-I^2]$
+\ARROW Now the difference:
+\begin{align*}
+V(\hat{I}_{HM}) - V(\hat{I}_{HM})= \dfrac{1}{N}[I-\int_0^1 f^2(x)dx]= \dfrac{1}{N}\int_0^1 f(x)[1-f(x)]dx\geq 0~{\rm{q.e.d}}
+\end{align*}
+\ARROW E2.3 Calculate the following integrals with uncertainties using ''hit or miss'' and crude methods:
 \begin{equation}
-x \circleddash y = \begin{cases}
-x-y-c +m,~c=1,~{\rm{ when~x-y-c<0}}\\
-x-y-c,~c=0,~{\rm{ when~x-y-c\geq0}} \nonumber 
-\end{cases}
+\int_0^1 \dfrac{1}{\sqrt{2\pi}} e^{-\dfrac{x^2}{2}} dxd \nonumber
+\end{equation}
+\begin{equation}
+\int_{x^2+y^2\leq 1} \dfrac{1}{4}\sqrt{1-(x^2+y^2)} dx dy\nonumber
 \end{equation}
 
 
-$\color{PineGreen}\Rrightarrow$ Initialization: $X_1,...,X_{n-r}$ and $c=0$.\\
-$\color{PineGreen}\Rrightarrow$ Fast and easy :)\\
-$\color{PineGreen}\Rrightarrow$ Fails some of the basic statistics tests.
-\end{small}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Non linear generators}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ The natural solutions to problems of linear generators are the non-linear generators (second part of 1980s).\\
-$\color{PineGreen}\Rrightarrow$ Eichenauera i Lehna (1986):
-\begin{equation}
-X_n= (a X_{n−1}^{-1} + b)~{\rm mod}~m, \nonumber 
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ Eichenauera-Hermanna (1993)
-\begin{equation}
-X_n= [a(n+n_0)+b]^{-1}~{\rm mod}~m, \nonumber 
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ L. Blum, M. Blum, Shub (1986):
-\begin{equation}
-X_n= X_{n-1}^2~{\rm mod}~m, \nonumber 
-\end{equation}
-$\color{PineGreen}\rightarrowtail$ Very popular in cryptography.\\
-$\color{PineGreen}\Rrightarrow$ Pros and cons:
-\begin{itemize}
-\item {\color{PineGreen}{They all pass all statistical tests.}}
-\item {\color{red}{Much slower then linear generators.}}
-\end{itemize}
-\end{small}
+\end{footnotesize}
 \end{frame}
 
 
 
+\begin{frame}\frametitle{Generalization to multi-dimension case, Crude method}
+\begin{footnotesize}
 
+\ARROW Let $x=(x_1, x_2,..., x_n)$- vector in the n-dim vector space $\mathcal{R}^n$.\\
+$\Omega \subset \mathcal{R}^n$ - some subspace in the n-dim space.\\
+$V\equiv(\Omega)$ - volume of the $\Omega$ subspaces.
+\begin{equation}
+I=\int_{\Omega} f(x) dx = V \int_{\Omega} f(x) dx/V= V \int_{\Omega} f(x) dp(x)  \equiv V J = V E(f), \nonumber
+\end{equation}
+where the \mc estimator:
+\begin{equation}
+\hat{J}=\dfrac{1}{N}\sum_{i=1}^Nf(x^{(i)}),{~}x\in\mathcal{U}(\Omega) \nonumber
+\end{equation}
+\ARROW The standard deviation:
+\begin{equation}
+\hat{\sigma}(\hat{J})\dfrac{1}{\sqrt{N(N-1)}}\sqrt{\sum_{i=1}^N f^2(x^{(i)})-\dfrac{1}{N}[\sum_{i=1}^{N}  f(x^{(i)})]^2 } \nonumber
+\end{equation}
+\ARROW In the end we get:
+\begin{equation}
+\hat{I}=V\hat{J},{~}{~}\hat{\sigma}(\hat{I})=V\sigma(\hat{J}) \nonumber
+\end{equation}
 
-
-
-
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%5
-%% Stat tests$
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{\texttt{RANLUX} generator}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ All described generators are based on some mathematical algorithms and recursion. The typical scheme is of constructing a MC generator:
-\begin{itemize}
-\item Think of a formula that takes some initial values.
-\item Generate large number of random numbers and put them through statistical tests.
-\item If the test are positive we accept the the generator.
-\end{itemize}
-$\color{PineGreen}\Rrightarrow$ Now let's think: why the hell numbers obtained that way are showing some random number properties?\pause ~There is no science behind it, it's pure luck!\\
-$\color{PineGreen}\Rrightarrow$ M.Luscher (1993) \href{http://arxiv.org/pdf/hep-lat/9309020v1.pdf}{hep-lat/9309020}\\
-$\color{PineGreen}\Rrightarrow$ Generator RANLUX based on Kolomogorow entropy and Lyapunov exponent. { \color{PineGreen} Effectively we are building inside the generator the chaos theory}.\\
-$\color{PineGreen}\Rrightarrow$  RANLUX and Mersenne Twister (TRandom1, TRandom3) are the 2 most powerful generators in the world that passed every known statistical test.
-
-\end{small}
+\end{footnotesize}
 \end{frame}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Chaos theory in a nut shell}
-\begin{small}
-\begin{columns}
-\column{0.1in}
-{~}
-\column{3in}
-$\color{PineGreen}\Rrightarrow$ We know that the solution of classical systems is described by trajectory in phase spaces. Now the problem with this picture starts to be when arround one point in this phase space we are getting more and more trajectories that are drifting a part later on.\\
-$\color{PineGreen}\Rrightarrow$ The Lyapunov exponent tells us how a two solutions drift apart with time:
-\begin{equation}
-\vert \delta X(t) \vert \approx  e^{\lambda t} \vert \delta X_0 \vert \nonumber
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ Kolomogorow entropy:
-\begin{equation}
-h_K = \int_P \lambda d \mu  \nonumber
-\end{equation}
-\column{2in}
 
-\includegraphics[width=0.9\textwidth]{images/Focal_stability.png}\\
-\includegraphics[width=0.9\textwidth]{images/LogisticMap_BifurcationDiagram.png}
 
-\end{columns}
-\end{small}
+\begin{frame}\frametitle{Generalization to multi-dimension case, ''Hit-or-miss''}
+\begin{footnotesize}
+\ARROW Let $x=(x_1, x_2,..., x_n)$- vector in the n-dim vector space $\mathcal{R}^n$.\\
+$\Omega \subset \mathcal{R}^n$ - some subspace in the n-dim space.\\
+$V\equiv(\Omega)$ - volume of the $\Omega$ subspaces.
+\begin{equation}
+I=\int_{\Omega}   = \int_{\Omega} dx \int_{0}{f_{max}} dy \Theta(f(x)-y)=Vf_{max} \int_{\Omega} \dfrac{dx}{V} \int_{0}^{f_{max}} \dfrac{dy}{f_{max}} \Theta(f(x)-y) \nonumber
+\end{equation}
+where $(x,y)\in \mathcal{U}(\Omega \times [0,f_{max}]$.
+\ARROW Now we define $K$:
+\begin{equation}
+K=\int_{\Omega} \dfrac{dx}{V}\int_{0}^{f_{max}} \dfrac{dy}{f_{max}} \Theta(f(x)-y)=E(\Theta) \nonumber
+\end{equation}
+\ARROW We generator: $(x,y) \in \mathcal{U}(\Omega \times [0,f_{max}])$ and check:
+\begin{equation}
+y = \begin{cases}
+                        \leq f(x) \text{hit, weight=1} \\
+                        > f(x) \text{hit, weight=0} \\
+                    \end{cases} \nonumber
+\end{equation}
+\ARROW In the end:
+\begin{equation}
+\hat{K}=\dfrac{n}{N},{~}{~}{~}\hat{\sigma}(\hat{K})=\dfrac{1}{\sqrt{N-1}}\sqrt{\hat{K}(1-\hat{K})} \nonumber
+\end{equation}
+\begin{equation}
+\hat{I}=f_{max}V\hat{K},{~}{~}{~}\hat{\sigma}(\hat{I})=f_{max}V \hat{\sigma}(\hat{K}) \nonumber
+\end{equation}
+\end{footnotesize}
 \end{frame}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{HEP simulation}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ There is some ambiguity what particle physicist call MC. Normally those are mathematical theorises but when we say MC we usually mean MC simulation of a physics process.
-$\color{PineGreen}\Rrightarrow$ There are plenty of things that need to be simulated:
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Crude MC vs ”Hit and mishit”}
+
+$\Rrightarrow$ We can repeat a toy MC studies as we did in the Euler needle case.\\
+$\hookrightarrow$ In this example we want to calculate $\int_0^{\pi/2} \cos x dx$
+
 \only<1>{
-\includegraphics[width=0.95\textwidth]{images/gen1.png}
+\begin{center}
+\includegraphics[angle=-90,width=0.8\textwidth]{images/results_0.pdf}
+\end{center}
 }
 \only<2>{
-\includegraphics[width=0.95\textwidth]{images/gen2.png}
+\begin{center}
+\includegraphics[angle=-90,width=0.8\textwidth]{images/results_1.pdf}
+\end{center}
 }
 \only<3>{
-\includegraphics[width=0.95\textwidth]{images/gen3.png}
-}
-\only<4>{
-\includegraphics[width=0.95\textwidth]{images/gen4.png}
-}
-\only<5>{
-\includegraphics[width=0.95\textwidth]{images/gen5.png}
-}
-\only<6>{
-\includegraphics[width=0.95\textwidth]{images/gen6.png}
-}
-\end{small}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Detector simulation}
+\begin{center}
 
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ Things do not get simpler on the detector side simulation.\\
-$\color{PineGreen}\Rrightarrow$ Lots of effects need to be taken into account:
+\includegraphics[angle=-90,width=0.4\textwidth]{images/results_fit_0.pdf}
+\includegraphics[angle=-90,width=0.4\textwidth]{images/results_fit_1.pdf}\\
+\end{center}
+$\Rrightarrow$ One clearly sees that both methods follow $1/\sqrt{N}$ dependence and that the Crude MC is always better then the ''Hit and mishit''.\\
+$\Rrightarrow$ Please note that for the ''Hit and mishit'' we are suing 2 times more random numbers than for the Crude method so in terms of timing the Crude MC is also much faster.
+
+}
+
+
+
+\end{frame}
+
+\begin{frame}\frametitle{Classical methods of variance reduction}
+\begin{footnotesize}
+
+$\Rrightarrow$ In Monte Carlo methods the statistical uncertainty is defined as:
+\begin{align*}
+\sigma = \dfrac{1}{\sqrt{N}}\sqrt{V(f)}
+\end{align*}
+$\Rrightarrow$ Obvious conclusion:
+\begin{itemize}
+\item To reduce the uncertainty one needs to increase $N$.\\
+$\rightrightarrows$ Slow convergence. In order to reduce the error by factor of 10 one needs to simulate factor of 100 more points!
+\end{itemize}
+$\Rrightarrow$ How ever the other handle ($V(f)$) can be changed! $\longrightarrow$ Lot's of theoretical effort goes into reducing this factor.\\
+$\Rrightarrow$ We will discuss {\color{Mahogany}{four}} classical methods of variance reduction:
+\begin{enumerate}
+\item Stratified sampling.
+\item Importance sampling.
+\item Control variates.
+\item Antithetic variates.
+\end{enumerate}
+
+
+\end{footnotesize}
+\end{frame}
+
+
+
+
+\begin{frame}\frametitle{Stratified sampling}
+\begin{footnotesize}
+$\Rrightarrow$ The most intuitive method of variance reduction. The idea behind it is to divide the function in different ranges and to use the Riemann integral property:
+\begin{align*}
+I = \int_0^1 f(u) du = \int_0^a f(u)du + \int_a^1 f(u) du,~ 0<a<1.
+\end{align*}
+
+$\Rrightarrow$ The reason for this method is that in smaller ranges the integration function is more flat. And it's trivial to see that the more flatter you get the smaller uncertainty. 
+$\rightrightarrows$ A constant function would have zero uncertainty!
+
+    \begin{exampleblock}{General schematic:}                                                                                                                                                                                                                                                                                                
+Let's take our integration domain and divide it in smaller domains. In the $j^{th}$ domain with the volume $w_j$ we simulate $n_j$ points from uniform distribution. We sum the function values in each of the simulated points for each of the domain. Finally we sum them with weights proportional to $w_i$ and anti-proportional to $n_i$. 
+        \end{exampleblock} 
+
+\end{footnotesize}
+\end{frame}
+
+\begin{frame}\frametitle{Stratified sampling - mathematical details}
+\begin{footnotesize}
+Let's define our integrals and domains:
+\begin{align*}
+I=\int_{\Omega} f(x) dx,{~}{~} \Omega=\bigcup_{i=1}^k w_i
+\end{align*}
+The integral over $j^{th}$ domain:
+\begin{align*}
+I_j=\int_{w_j} f(x) dx,{~}{~} \Rightarrow I = \sum_{j=1}^k I_i
+\end{align*}
+$\rightrightarrows$ $p_j$ uniform distribution in the $w_j$ domain: $dp_j=\frac{dx}{w_j}$.\\
+$\rightrightarrows$ The integral is calculated based on crude \mc~method. The estimator is equal:
+\begin{align*}
+\hat{I}_j = \frac{w_j}{n_j} \sum_{i=1}^{n_j}f(x_j^i)
+\end{align*}
+Now the total integral is just a sum:
+\begin{align*}
+\hat{I} = \sum_{j=1}^k \hat{I}_j = \sum_{j=1}^k \frac{w_j}{n_j} \sum_{i=1}^{n_j} f(x_j^{(i)}), 
+\end{align*}
 \begin{columns}
 \column{0.2in}
-{~}
 \column{2in}
-$\color{PineGreen}\rightarrowtail$ Bremsstrahlung\\
-$\color{PineGreen}\rightarrowtail$ Interactions with different detector materials\\
-$\color{PineGreen}\rightarrowtail$ Particle identification\\
-$\color{PineGreen}\rightarrowtail$ Showers\\
+Variance:
+$V(\hat{I})=\sum_{j=1}^k \dfrac{w_j^2}{n_j}V_j(f)$,
 \column{3in}
-\includegraphics[width=0.95\textwidth]{{images/lhcb2_h-640x408}.jpg}
+and it's estimator:
+$\hat{V} (\hat{I}) = \sum_{j=1}^k \dfrac{w_j^2}{n_j} \hat{V}_j(f)$
 \end{columns}
-$\color{PineGreen}\Rrightarrow$ Example of generators:\\
-$\color{PineGreen}\rightarrowtail$ FLUKA\\
-$\color{PineGreen}\rightarrowtail$ Geant
 
-\end{small}
+\end{footnotesize}
 \end{frame}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Method of Moments}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ Now real cool things!\\
-$\color{PineGreen}\Rrightarrow$ Let's consider we want to study a rare decay: $\PB^{\pm} \to \PK^{\pm} \Pmu \Pmu$.  The decay is described by the following PDF:
-\begin{equation}
-\dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =\dfrac{3}{4}(1-F_H)(1-\cos^2 \theta_l)+F_H/2 + A_{FB}\cos \theta_l \nonumber
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ PDF by construction is normalized: $\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =1$
+\begin{frame}\frametitle{Importance sampling}
+\begin{footnotesize}
+$\Rrightarrow$ If the function is changing rapidly in its domain one needs to use a more elegant method: make the function more stable.\\
+$\rightrightarrows$ The solution is from first course of mathematical analysis: change the integration variable :)
+\begin{align*}
+f(x)dx \longrightarrow \frac{f(x)}{g(x)}dG(x),~{\rm{where}}~g(x)=\frac{dG(x)}{dx}
+\end{align*}
+    \begin{exampleblock}{Schematic:}                                                                                                                                                                                                                                                                                                
+\begin{itemize}
+\item Generate the distribution from $G(x)$ instead of $\mathcal{U}$.
+\item For each generate point calculate the weight: $w(x)=\frac{f(x)}{g(x)}$.
+\item We calculate the expected value $\hat{E}(w)$ and its variance $\hat{V}_G(w)$ for the whole sample.
+\end{itemize}
+        \end{exampleblock} 
+
+
+
+\begin{itemize}
+
+\item If $g(x)$ is choose correctly the resulting variance can be much smaller.
+\item There are some mathematical requirements:
+\begin{itemize}
+\item $g(x)$ needs to be non-negative and analytically integrable on its domain.
+\item $G(x)$ invertible or there should be a direct generator of $g$ distribution.
+\end{itemize}
+\end{itemize}
+
+
+\end{footnotesize}
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}\frametitle{Importance sampling - Example}
+\begin{footnotesize}
+
+$\Rrightarrow$ Let's take our good old $\pi$ determination example.
+\vspace{0.3cm}
 \begin{columns}
 \column{0.1in}
-{~}
-\column{2.2in}
+\column{3in}
+$\Rrightarrow$ Let's take here for simplicity: $L=l$. 
 \begin{itemize}
-\item Normally we do a likelihood fit and we are done.
-\item There is a second way!
+\item Let's take a trivial linear weight function: $g(x)=\frac{4}{\pi}(1-\frac{2}{\pi}x)$
+\item It's invertible analytically: $G(x)=\frac{4}{\pi}x(1-\frac{x}{\pi})$
+\item The weight function:
+\begin{align*}
+w(x)=\frac{p(x)}{g(x)}=\frac{\pi}{4}\frac{\cos x}{1-2x/ \pi}
+\end{align*}
 \end{itemize}
-\column{2.8in}
-\includegraphics[width=0.95\textwidth]{images/Kmumu_LL.png}
+
+\column{2in}
+\includegraphics[width=0.95\textwidth]{images/result_weight.png}
 \end{columns}
-
-\end{small}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Method of Moments}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ Let's calculate the integrals:
-\begin{equation}
-\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} \nonumber
-\end{equation}
-\begin{equation}
-\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15} \nonumber
-\end{equation}
-$\color{PineGreen}\Rrightarrow$ So we can get our parameters that we searched for by doing a integration. So now what?\pause \\
-$\color{PineGreen}\Rrightarrow$ Well nature is the best random number generator so let's take the data and treat and calculate the integral estimates:
-\begin{equation}
-\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} = \dfrac{1}{N} \sum_{i=1}^N  \cos \theta_{l,i} \nonumber 
-\end{equation}
-\begin{equation}
-\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15}= \dfrac{1}{N} \sum_{i=1}^N  \cos^2 \theta_{l,i} \nonumber
-\end{equation}
-
-\end{small}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Method of Moments}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ So what did we do?
 \begin{itemize}
-\item We have just estimated a parameters of interests without using any fit!!
+\item Now the new standard deviation is smaller:
 \end{itemize}
-$\color{PineGreen}\Rrightarrow$ Pros and cones of method of moments:
+\begin{align*}
+\sigma_{\pi}^{{\rm{IS}}} \simeq \frac{0.41}{\sqrt{N}} < \sigma_{\pi}\simeq \frac{1.52}{\sqrt{N}}
+\end{align*}
 \begin{itemize}
-\item {\color{PineGreen}{Are very immune to bias.}}
-\item {\color{PineGreen}{Do not suffer from boundary problems.}}
-\item {\color{PineGreen}{Require less statistic to work then likelihood fit.}}
-\item {\color{PineGreen}{They always have a Gaussian error}}.
-\item {\color{Red}{Estimator has a larger uncertainty.}}
-
-
+\item Importance sampling has advantages:
+\begin{itemize}
+\item Big improvements of variance reduction.
+\item The only method that can cope with singularities.
+\end{itemize}
 \end{itemize}
 
-\end{small}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Method of Moments, uncertainty estimator}
-\begin{small}
-\begin{columns}
-\column{0.1in}
-{~}
-\column{2.5in}
-$\color{PineGreen}\Rrightarrow$ It can be proven that Method of Moments estimator converges slower then the maximum likelihood fit.
-\column{2.5in}
-\includegraphics[width=0.95\textwidth]{images/S7.png}
-\end{columns}
 
-\begin{columns}
-\column{0.1in}
-{~}
-\column{2.5in}
-\includegraphics[width=0.95\textwidth]{images/S8F_950.png}
-\column{2.5in}
-\includegraphics[width=0.95\textwidth]{images/S8.png}
-\end{columns}
-
-\end{small}
+\end{footnotesize}
 \end{frame}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Other application of MC - testing your analysis}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$  Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows:
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+
+
+
+
+
+\begin{frame}\frametitle{Control variates}
+\begin{footnotesize}
+$\Rrightarrow$ Control variates uses an other nice property of Riemann integral:
+\begin{align*}
+\int f(x) dx = \int [f(x)-g(x)]dx+ \int g(x)dx
+\end{align*}
 \begin{itemize}
-\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure.
-\item Simulate an assembly of simulation events for different values of parameters you want to measure.
-\item Do the analysis on this pseudo data.
-\item See if you are getting back what you have simulated.
+\item $g(x)$ needs to be analytically integrable.
+\item The uncertainty comes only from the integral: $\int [f(x)-g(x)]dx$. 
+\item Obviously: $V(f\to g) \xrightarrow{f\to g} 0$
+\end{itemize}
+$\Rrightarrow$ Advantages:\\
+\begin{itemize}
+\item Quite stable, immune to the singularities.
+\item $g(x)$ doesn't need to be invertible analytically.
+\end{itemize}
+$\Rrightarrow$ Disadvantage:\\
+\begin{itemize}
+\item Useful only if you know $\int g(x)dx$
 \end{itemize}
 
-\end{small}
+\end{footnotesize}
 \end{frame}
 
-\begin{frame}\frametitle{Testing your analysis,  $\rm \color{RubineRed}{Lecture2/Test\_met}$}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$  Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows:
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}\frametitle{Antithetic variates}
+\begin{footnotesize}
+$\Rrightarrow$ In \mc~methods usually one uses the independent random variables. The Antithetic variates method on purpose uses a set of correlated variables (negative correlation is the important property):
 \begin{itemize}
-\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure.
-\item Simulate an assembly of simulation events for different values of parameters you want to measure.
-\item Do the analysis on this pseudo data.
-\item See if you are getting back what you have simulated.
+\item Let $f$ and $f\prime$ will be functions of x on the same domain.
+\item The variance: $V(f+f\prime)=V(f)+V(f\prime)+2 Cov(f,f\prime)$.
+\item If $Cov(f,f\prime)<0$ then you can reduce the variance.
 \end{itemize}
-\includegraphics[width=0.4\textwidth]{{images/mean_estimator}.png}
-\end{small}
+$\Rrightarrow$ Advantages:
+\begin{itemize}
+\item If you can pick up $f$ and $f\prime$ so that they have negative correlation one can significantly reduce the variance!
+\end{itemize}
+
+$\Rrightarrow$ Disadvantages:
+\begin{itemize}
+\item There are no general methods to produce such a negative correlations.
+\item Hard to generalize this for multidimensional case.
+\item You can't generate events from $f(x)$ with this method.
+\end{itemize}
+
+
+
+\end{footnotesize}
 \end{frame}
 
 
+
 \begin{frame}\frametitle{Wrap up}
-\begin{small}
-$\color{PineGreen}\Rrightarrow$ Things to remember:
+\begin{footnotesize}
+$\Rrightarrow$ To sum up:
 \begin{itemize}
-\item Computer cannot produce random numbers, only pseudorandom numbers.
-\item We use pseudorandon numbers as random numbers if they are statistically acting the same as random numbers.
-\item Linear generators are not commonly used nowadays.
-\item State of the art generators are the ones based on Kolomogorows theorem.
-\item MC methods used to simulate physics process, detector response and validating the estimators. 
+\item We discussed basic mathematical properties of \mc~methods.
+\item We shown that besides the stochastic nature of \mc~ they can be used to determine totally non stochastic quantities.
+\item We demonstrated there is a perfect isomorphism between \mc~method and integration.
+\item We learned how co calculate integrals and estimate the uncertainties.
+\item Finally we discussed several classical methods of variance reduction.
 \end{itemize}
-\end{small}
+
+
+\end{footnotesize}
 \end{frame}
 
 
 
+
+
+
+
 \backupbegin   
 
-\begin{frame}\frametitle{Backup}
-
-
-\end{frame}
 
 \backupend			
 
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-% Tikz Feynman Diagrams
-% by Flip Tanedo
-% 4 January 2011, work in progress
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%% TIKZ - for drawing Feynman diagrams %%%%
-%%% ... use with pdflatex   			%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\usepackage{tikz}
-\usetikzlibrary{arrows,shapes}
-\usetikzlibrary{trees}
-\usetikzlibrary{matrix,arrows} 				% For commutative diagram
-											% http://www.felixl.de/commu.pdf
-\usetikzlibrary{positioning}				% For "above of=" commands
-\usetikzlibrary{calc,through}				% For coordinates
-\usetikzlibrary{decorations.pathreplacing}  % For curly braces
-\usepackage[tikz]{bclogo} 					% For cute logo boxes
-\usepackage{pgffor}							% For repeating patterns
-
-\usetikzlibrary{decorations.pathmorphing}	% For Feynman Diagrams
-\usetikzlibrary{decorations.markings}
-\tikzset{
-	% >=stealth', %% Different kind of arrows
-    vector/.style={decorate, decoration={snake}, draw},
-    fermion/.style={postaction={decorate},
-        decoration={markings,mark=at position .55 with {\arrow{>}}}},
-    fermionbar/.style={draw, postaction={decorate},
-        decoration={markings,mark=at position .55 with {\arrow{<}}}},
-    fermionnoarrow/.style={},
-    gluon/.style={decorate,
-        decoration={coil,amplitude=4pt, segment length=5pt}},
-    scalar/.style={dashed, postaction={decorate},
-        decoration={markings,mark=at position .55 with {\arrow{>}}}},
-    scalarbar/.style={dashed, postaction={decorate},
-        decoration={markings,mark=at position .55 with {\arrow{<}}}},
-    scalarnoarrow/.style={dashed,draw},
-%
-%% 	Special vectors (when you need to fine-tune wiggles)
-	provector/.style={decorate, decoration={snake,amplitude=2.5pt}, draw},
-	antivector/.style={decorate, decoration={snake,amplitude=-2.5pt}, draw},
-}
-
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diff --git a/Lectures_my/MC_2016/Lecture3/mchrzasz.nav b/Lectures_my/MC_2016/Lecture3/mchrzasz.nav
index 0f2bda2..74b2e11 100644
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diff --git a/Lectures_my/MC_2016/Lecture3/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture3/mchrzasz.pdf
index 2ef368f..be0d5f9 100644
--- a/Lectures_my/MC_2016/Lecture3/mchrzasz.pdf
+++ b/Lectures_my/MC_2016/Lecture3/mchrzasz.pdf
Binary files differ
diff --git a/Lectures_my/MC_2016/Lecture3/mchrzasz.tex b/Lectures_my/MC_2016/Lecture3/mchrzasz.tex
index d936aca..1d8e744 100644
--- a/Lectures_my/MC_2016/Lecture3/mchrzasz.tex
+++ b/Lectures_my/MC_2016/Lecture3/mchrzasz.tex
@@ -220,7 +220,7 @@
 \begin{center}
 	\begin{columns}
 		\begin{column}{0.9\textwidth}
-			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Applications of Monte Carlo methods}
+			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Random number generators and application}
 		\end{column}
 		\begin{column}{0.2\textwidth}
 		  %\includegraphics[width=\textwidth]{SHiP-2}
@@ -242,411 +242,271 @@
 \vspace{1em}
 %		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
 \vspace{0.5em}
-	\textcolor{normal text.fg!50!Comment}{Experimental Methods in Particle Physics, \\ 26 November, 2015}
+	\textcolor{normal text.fg!50!Comment}{Experimental Methods in Particle Physics, \\ 25 November, 2015}
 \end{center}
 \end{frame}
 }
 
 
+\begin{frame}\frametitle{Random and pseudorandom numbers}
+  \begin{exampleblock}{John von Neumann:}                                                                                                                                                                                                                                                                                                
+''Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.''
+        \end{exampleblock} 
 
-
-
-\begin{frame}\frametitle{Markov Chain MC}
+$\color{PineGreen}\Rrightarrow$ Random number: a given value that is taken by a random variable $ \twoheadrightarrow$ by definition cannot be predicted.\\
+%$\color{PineGreen}\Rrightarrow$ Sequence of random numbers $\twoheadrightarrow$ 
+$\color{PineGreen}\Rrightarrow$ Sources of truly random numbers:
 \begin{itemize}
-\item Consider a finite possible states: $S_1$, $S_2$, ...
-\item And the time steps of time, labelled as $1$, $2$, ...
-\item At time $t$ the state is denoted $X_t$.
-\item The conditional probability is defined as:
+\item Mechanical
+\item Physical
 \end{itemize}
-\begin{equation}
-P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) \nonumber
-\end{equation}
+$\color{PineGreen}\Rrightarrow$ Disadvantages of physical generators:
 \begin{itemize}
-\item The Markov chain is then if the probability depends only on previous step.
-\end{itemize}
-\begin{equation}
-P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) =  P(X_t=S_j \vert X_{t-1}=S_{j-1} )\nonumber
-\end{equation}
-\begin{itemize}
-\item For this reason this reason MCMC is also knows as drunk sailor walk.
-\item Very powerful method. Used to solve linear eq. systems, invert matrix, solve differential equations, etc.
-\end{itemize}
-\end{frame}
-
-
-
-\begin{frame}\frametitle{Linear Equations}
-\begin{itemize}
-\item Lets say we have a linear equation system:
-\end{itemize}
-\begin{equation}
-\begin{array}{lcl} X & = & pY + (1-p) A \\ Y & = & qX + (1-q)B \end{array} \nonumber
-\end{equation}
-\begin{itemize}
-\item We know $A,B,p,q$; $X$ and $Y$ are meant to be determined.
-\item Algorithm:
-\begin{enumerate}
-\item We choose first element of the first equation with probability $p$ and second with probability $1-p$.
-\item We we choose the second one, the outcome of this MCMC is $W=A$.
-\item If we choose the first we go to second equation and choose the first element with probability $q$ and the second with $1-q$.
-\item We we choose the second one, the outcome of this MCMC is $W=B$.
-\item If we choose the first we go to the first equation back again.
-\item We repeat the procedure.
-\end{enumerate}
-\item We can estimate the solution of this system:
-\end{itemize}
-\begin{equation}
-\hat{X} = \dfrac{1}{N}\sum_{i=1} W_i{~}{~}{~}{~}{~} \hat{\sigma_X}=\dfrac{1}{\sqrt{N-1}}\sqrt{\dfrac{1}{N} \sum_{i=1}^N W_i^2-\hat{X}^2} \nonumber
-\end{equation}
-
-\end{frame}
-
-\begin{frame}\frametitle{Neumann-Ulam method}
-\begin{itemize}
-\item Let's try apply the basic MCMC method to solve a simple linear equation system:
-\end{itemize}
-\begin{equation}
-A \overrightarrow{x} = \overrightarrow{b} \nonumber
-\end{equation}
-\begin{itemize}
-\item The above system can be (always, see linear algebra lecture) translated into system:
-\end{itemize}
-\begin{equation}
-\overrightarrow{x} = \overrightarrow{a} + H \overrightarrow{x} \nonumber
-\end{equation}
-\begin{itemize}
-\item For this method we assume that the norm of the matrix is:
-\end{itemize}
-\begin{equation}
-\Vert H \Vert =  \underset{1 \leq i \leq n}{max} \sum_{j=1}^n \vert h_{ij} \vert <1 \nonumber \end{equation}
-\begin{itemize}
-\item Which we can write in a form:
-\end{itemize}
-\begin{equation}
-(1 -H)\overrightarrow{x}=\overrightarrow{a} \nonumber
-\end{equation}
-\end{frame}
-
-
-
-
-\begin{frame}\frametitle{Neumann-Ulam method}
-\begin{itemize}
-\item The solution would be then:
-\end{itemize}
-\begin{equation}
-\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} \nonumber
-\end{equation}
-\begin{itemize}
-\item We can Taylor expend this:
-\end{itemize}
-\begin{equation}
-\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} = \overrightarrow{a} + H \overrightarrow{a} + H^2 \overrightarrow{a} + H^3 \overrightarrow{a} +.... \nonumber
-\end{equation}
-\begin{itemize}
-\item For the $i$-th component of the $\overrightarrow{x}$ vector:
-\end{itemize}
-\begin{equation}
-x_0^i= a_i + \sum_{j=1}^n h_{ij} a_{j_1} + \sum_{j_1=1}^n \sum_{j_2=1}^n  h_{ij_1} h_{ij_2}  a_{j_2} +  \sum_{j_1=1}^n \sum_{j_2=1}^n \sum_{j_3=1}^n  h_{ij_1} h_{ij_2} h_{ij_3} a_{j_3} + ...\nonumber
-\end{equation}
-\begin{itemize}
-\item One can construct probabilistic behaviour of a system that follows the path of equation above.
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Neumann-Ulam method}
-\begin{itemize}
-\item To do so we add to our matrix an additional column of the matrix:
-\end{itemize}
-\begin{equation}
-h_{i,0} = 1-\sum_{j=1}^n h_{ij} > 0 \nonumber
-\end{equation}
-\begin{itemize}
-\item The system has states: $\lbrace 0,1,2...,n\rbrace$
-\item State at $t$ time is denoted as $i_t$.
-\item We make a random walk accordingly to to the following rules:
-\begin{itemize}
-\item At the begging of the walk ($t=0$) we are at $i_0$. 
-\item In the $t$ moment we are in the $i_t$ position then in $t+1$ time stamp we move to state $i_{t+1}$ with the probability $h_{i_t i_{t+1}}$. 
-\item We stop walking if we are in state $0$. 
-\end{itemize}
-\item The path $X(\gamma) = (i_0, i_1, i_2, ..., i_k, 0)$ is called trajectory.
-\item It can be proven that $x_i^0 =E \lbrace X (\gamma) \vert i_0=j \rbrace$.
-\end{itemize}
-
-
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Neumann-Ulam method, $\rm \color{RubineRed}{Lecture3/Markov}$}
-\begin{itemize}
-\item For example lets try to solve this equation system:
-\end{itemize}
-\begin{equation}
-\overrightarrow{x} = 
-\left(\begin{array}{c}
- 1.5  \\
--1.0\\
-0.7  \end{array} \right) 
-+
-\left(\begin{array}{ccc}
-0.2 &  0.3 & 0.1  \\
-0.4 &  0.3 & 0.2 \\
-0.3 &  0.1 & 0.1  \end{array} \right) \overrightarrow{x}
- \nonumber
-\end{equation}
-\begin{itemize}
-\item The solution is $\overrightarrow{x}_0 = (2.154303, 0.237389, 1.522255)$.
-\end{itemize}
-\begin{columns}
-
-\column{0.1in}
-
-\column{2.5in}
-\begin{itemize}
-\item The propability matrix $h_{ij}$ has the shape:
-\end{itemize}
-\begin{tabular}{|c|cccc|}
-\hline
-$i/j$ &  1 & 2 & 3 & 4  \\ \hline
-1 & 0.2 & 0.3 & 0.1 & 0.4 \\
-2 & 0.4 & 0.3 & 0.2 & 0.1 \\
-3 & 0.3 & 0.1 & 0.1 & 0.5 \\ \hline
-\end{tabular}
-
-\column{2.5in}
-\begin{itemize}
-\item An example solution:
-\end{itemize}
-\includegraphics[width=0.95\textwidth]{images/mark.png}
-
-\end{columns}
-
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Neumann-Ulam dual method}
-\begin{itemize}
-\item The problem with Neumann-Ulam method is that you need to repeat it for each of the coordinates of the $\overrightarrow{x}_0$ vector.
-\item The dual method calculates the whole $\overrightarrow{x}_0$ vector.
-\item The algorithm:
-\begin{itemize}
-\item On the indexes: $\lbrace0,1,...,n\rbrace$ we set a probability distribution:\\ $q_1, q_2,..., q_n$, $q_i>0$ and $\sum_i=1^n q_i=1$. 
-\item The starting point we select from $q_i$ distribution.
-\item If in $t$ time we are in $i_t$ state then with probability $p(i_{t+1} \vert i_t) = h_{i_{t+1},i_{t}}$ in $t+1$ we will be in state $i_1$. For  $i_{t+1}=0$ we define the probability: $h_{0,i_{t}}=1-\sum_{j=1}^n h_{j,i_{t}}$. Here we also assume that $h_{j,i_{t}} > 0$.
-\item NOTE: there the matrix is transposed compared to previous method: $H^{T}$.
-\item Again we end our walk when we are at state $0$. 
-\item For the trajectory: $\gamma = (i_0, i_1,...,i_k, 0)$, we assign the vector:
-\end{itemize}
-\begin{equation}
-\overrightarrow{Y}(\gamma) = \dfrac{a_{i_0}}{ q_{i_{0}} p(0 \vert i_k)   }  \widehat{e}_{i_{k}} \in \mathcal{R}^n \nonumber
-\end{equation}
-\item The solution will be : $\overrightarrow{x}^0 = \dfrac{1}{N} \sum \overrightarrow{Y}(\gamma)$
+\item To slow for typical applications, especially the mechanical ones!
+\item Not stable; small changes in boundary conditions might lead to completely different results!
 \end{itemize}
 
 
 \end{frame}
 
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}\frametitle{Neumann-Ulam dual method, $\rm \color{RubineRed}{Lecture3/Markov2}$}
-\begin{itemize}
-\item Let's try to solve the equation system:
-\end{itemize}
-\begin{equation}
-\overrightarrow{x} = 
-\left(\begin{array}{c}
- 1.5  \\
--1.0\\
-0.7  \end{array} \right) 
-+
-\left(\begin{array}{ccc}
-0.2 &  0.3 & 0.1  \\
-0.4 &  0.3 & 0.2 \\
-0.1 &  0.1 & 0.1  \end{array} \right) \overrightarrow{x}
- \nonumber
-\end{equation}
-\begin{itemize}
-\item The solution is: $\overrightarrow{x}_0 = (2.0, 0.0, 1.0)$.
-\item Let's put the initial probability as constant:
-\end{itemize}
-\begin{equation}
-q_1=q_2=q_3=\dfrac{1}{3}  \nonumber
-\end{equation}
-\begin{columns}
-
-\column{0.1in}
-
-\column{2.5in}
-\begin{itemize}
-\item The propability matrix $h_{ij}$ has the shape:
-\end{itemize}
-\begin{tabular}{|c|cccc|}
-\hline
-$i/j$ &  1 & 2 & 3 & 4  \\ \hline
-1 & 0.2 & 0.4 & 0.1 & 0.3 \\
-2 & 0.3 & 0.3 & 0.1 & 0.3 \\
-3 & 0.1 & 0.2 & 0.1 & 0.6 \\ \hline
-\end{tabular}
-
-\column{2.5in}
-\begin{itemize}
-\item An example solution:
-\end{itemize}
-\includegraphics[width=0.95\textwidth]{images/mark2.png}
-
-\end{columns}
-\end{frame}
-
-
-
-
-\begin{frame}\frametitle{Look elsewhere effect, $\rm \color{RubineRed}{Lecture3/LEE}$}
-\begin{itemize}
-\item Look elsewhere effect addresses the following problem:
-\begin{itemize}
-\item Imagine you observed a $3\sigma$ deviation in one of the observable that you measured. 
-\item Before you get excited one needs to understand if given the fact that you had so many measurements this might happen!
-\end{itemize}
-\item Example: Let's say we have measured 50 observables. What is the probability to observed 1 that is $3\sigma$ away from theory prediction?
-\item Let's simulate 50 Gaussian distribution centred at 0 and width of 1. We count how simulations where at least one of the 50 numbers have the absolute value $>3$.
-\item More complicated example: what if you observed 3 in a row $2\sigma$ fluctuations among 50 measurements?
-\item This kind of studies are the best solvable by MC simulations.
-\end{itemize}
-
-
-\end{frame}
-
-
-
-
-
-
-\begin{frame}\frametitle{Travelling Salesman Problem}
-\begin{itemize}
-\item Salesman starting from his base has to visit $n-1$ other locations and return to base headquarters. The problem is to find the shortest way.
-\item For large $n$ the problem can't be solver by brutal force as the complexity of the problem is $(n-1)!$
-\item There exist simplified numerical solutions assuming factorizations. Unfortunately even those require anonymous computing power.
-\item Can MC help? YES :)
-\item The minimum distance $l$ has to depend on 2 factors: $P$ the area of the city the Salesman is travelling and the density of places he wants to visit: $\dfrac{n}{P}$
-\item Form this we can assume:
-\end{itemize}
-\begin{equation}
-l \sim P^a (\dfrac{n}{P})^b=P^{a-b}n^b. \nonumber
-\end{equation}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-
-
-
-
-
-\begin{frame}\frametitle{Traveling Salesman Problem}
-\begin{itemize}
-\item From dimension analysis:
-\end{itemize}
-\begin{equation}
-a-b=\dfrac{1}{2}. \nonumber
-\end{equation}
-\begin{itemize}
-\item To get $l$ we need square root of area.
-\item From this it's obvious:
-\end{itemize}
-\begin{equation}
-l \sim P^a (\dfrac{n}{P})^b=P^{0.5}n^{a-0.5}. \nonumber
-\end{equation}
-\begin{itemize}
-\item Now we can multiply the area by alpha factor that keeps the density constant then:
-\end{itemize}
-\begin{equation}
-l \sim \alpha^0.5 \alpha6{a-0.5} = \alpha^a \nonumber
-\end{equation}
-\begin{itemize}
-\item In this case the distance between the clients will not change, but the number of clients will increase by $\alpha$ so:
-\end{itemize}
-\begin{equation}
-l \sim \alpha \nonumber
-\end{equation}
-\begin{itemize}
-\item In the end we get: $a=1$
-\end{itemize}
-\end{frame}
-
-
-
-\begin{frame}\frametitle{Traveling Salesman Problem}
-\begin{itemize}
-\item In total:
-\end{itemize}
-\begin{equation}
-l \sim k (nP)^{0.5}\nonumber
-\end{equation}
-\begin{itemize}
-\item Of course the k depends on the shape of the area and locations of client. However for large $n$ the k starts loosing the dependency. It's an asymptotically free estimator.
-\item To use the above formula we need to somehow calculate k.
-\item How to estimate this? Well make a TOY MC: take a square put uniformly $n$ points. Then we can calculate $l$. Then it's trivial:
-\end{itemize}
-\begin{equation}
-k= l(nP)^{-0.5} \nonumber
-\end{equation}
-\end{frame}
-
-
-\begin{frame}\frametitle{Traveling Salesman Problem}
-\begin{itemize}
-\item This kind of MC experiment might require large CPU power and time. The adventage is that once we solve the problem we can use the obtained k for other cases (it's universal constant!).
-\item It turns out that:
-\end{itemize}
-\begin{equation}
-k \sim \dfrac{3}{4} \nonumber
-\end{equation}
-\begin{itemize}
-\item Ok, but in this case we can calculate $l$ but not the actual shortest way! Why the hell we did this exercise?!
-\item Turns out that for most of the problems we are looking for the solution that is close to smallest $l$ not the exact minimum.
-\end{itemize}
-\end{frame}
-
-\begin{frame}\frametitle{War Games}
-
-\begin{itemize}
-\item S. Andersoon 1966 simulated for Swedish government how would a tank battle look like.
-\item Each of the sides has  15 tanks. that they allocate on the battle field. 
-\item The battle is done in time steps.
-\item Each tank has 5 states:
-\begin{itemize}
-\item OK
-\item Tank can only shoot
-\item Tank can only move
-\item Tank is destroyed
-\item Temporary states
-\end{itemize}
-\item This models made possible to check different fighting strategies.
-
-\end{itemize}
-\end{frame}
-
-
-
-\begin{frame}
-
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Random numbers - history remark}
+$\color{PineGreen}\Rrightarrow$ In the past there were books with random numbers:
 \begin{center}
-\begin{Huge}
- Q \& A
-\end{Huge}
+\includegraphics[width=0.55\textwidth]{images/million-random-digits-open.jpg}
+  \end{center}
+$\color{PineGreen}\Rrightarrow$ It's obvious that they didn't become very popular ;)\\
+$\color{PineGreen}\Rrightarrow$ This methods are comming back!\\
+$\color{PineGreen}\twoheadrightarrow$ Storage device are getting more cheap and bigger (CD, DVD).\\
+$\color{PineGreen}\twoheadrightarrow$ 1995: G. Marsaglia, $650\rm MB$ of random numbers, ''White and Black Noise''.
+
+
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Pseudorandom numbers}
+$\color{PineGreen}\Rrightarrow$ Pseudorandom numbers are numbers that are generated accordingly to strict mathematical formula. \\
+$\color{PineGreen}\looparrowright$ Strictly speaking they are non random numbers, how ever they have all the statistical properties of random numbers.\\
+$\color{PineGreen}\looparrowright$  Discussing those properties is a wide topic so let's just say that without knowing the formula they are generated by one cannot say if those numbers are random or not.\\
+$\color{PineGreen}\Rrightarrow$  Mathematical methods of producing pseudorandom numbers:
+\begin{itemize}
+\item Good statistical properties of generated numbers.
+\item Easy to use and fast!
+\item Reproducible!
+\end{itemize}
+$\color{PineGreen}\Rrightarrow$ Since mathematical pseudorandom genrators are dominantly:  pseudorandom $\rightarrowtail$ random.
+
+
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Middle square generator; von Neumann}
+$\color{PineGreen}\Rrightarrow$  The first mathematical generator (middle square) was proposed by von Neumann (1964).\\{~}\\
+$\color{PineGreen}\looparrowright$ Formula: $
+\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
+  boxrule=0pt,frame hidden]{ X_n= \lfloor X_{n-1}^2\cdot 10 ^{-m} \rfloor - \lfloor X_{n-1}^2\cdot 10^{-3m} \rfloor \cdot 10^{2m}}
+$
+
+$\color{PineGreen}\looparrowright$ where $X_0$ is a constant (seed), $\lfloor\cdot\rfloor$ is the cut-off of a number to integer.\\
+$\color{PineGreen}\Rrightarrow$  Example:\\
+{~}{~}Let's put $m=2$ and $X_0=2045$:\\
+\begin{columns}
+\column{0.1\textwidth}
+{~}
+\column{0.4\textwidth}
+$\color{PineGreen}\looparrowright$ $X_0^2=\underbrace{04}_{\rm rej}1820\underbrace{25}_{\rm rej}$
+\column{0.3\textwidth}
+ $\Rightarrow X_1=1820$
+\end{columns}
+\begin{columns}
+\column{0.1\textwidth}
+{~}
+\column{0.4\textwidth}
+$\color{PineGreen}\looparrowright$ $X_1^2=\underbrace{03}_{\rm rej}3124\underbrace{00}_{\rm rej}$
+\column{0.3\textwidth}
+ $\Rightarrow X_1=3124$
+\end{columns}
+$\color{PineGreen}\looparrowright$ Simple generator but unfortunately quite bad generator. Firstly the sequences are very short and strongly dependent on the $X_0$ number.
+
+
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Linear generators $\rm \color{RubineRed}{Lecture2/Linear\_gen1}$}
+$\color{PineGreen}\Rrightarrow$  This was a first generator written and it's a good example how to not write generators.\\
+$\color{PineGreen}\Rrightarrow$ It's highly non stable!
+\includegraphics[width=0.8\textwidth]{images/shit.png}
+
+
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Linear generators}
+
+$\color{PineGreen}\Rrightarrow$ General equation:
+\begin{equation}
+\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
+  boxrule=0pt,frame hidden]{X_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m    ,} \nonumber
+\end{equation}
+$\color{PineGreen}\looparrowright$ where $a_i,c,m$ are parameters of a generator(integer numbers).\\
+$\color{PineGreen}\looparrowright$  Generator initialization $\color{PineGreen} \rightleftarrows$ setting those parameters.\\
+$\color{PineGreen}\Rrightarrow$ Very old generators. (often used in Pascal, or first C versions):
+\begin{columns}
+\column{0.1\textwidth}
+{~}
+\column{0.8\textwidth}
+$k=1:~X_n=(aX_{n-1} +c )~{\rm mod} m,$
+$c=
+\begin{cases}
+=0, {\rm multiplicative geneator}\\
+\neq 0, {\rm mix geneator}
+\end{cases}
+$
+\column{0.1\textwidth}
+{~}
+
+
+\end{columns}
+$\color{PineGreen}\Rrightarrow$ The period can be achieved by tuning the seed parameters:\\
+$P_{{\rm max}}=
+\begin{cases}
+2^{L-2};~{\rm for~} m=2^L\\
+m-1;~{\rm for~} m= {\rm prime~number}
+\end{cases}
+$
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Shift register generator}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ General equation:
+\begin{equation}
+\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
+  boxrule=0pt,frame hidden]{b_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~2    ,} \nonumber
+\end{equation}
+where $a_i \subset(\lbrace0,1\rbrace )$\\
+$\color{PineGreen}\Rrightarrow$ Super fast and easy to implement due to: $(a+b) ~{\rm mod}~2 = a~{\rm xor}~b$
+\begin{center}
+\begin{tabular}{||c|c|c||}
+\hline \hline a & b & a xor b\\ \hline
+0 & 0 & 0\\
+1 & 0 & 1\\
+0 & 1 & 1\\
+1 & 1 & 0\\ \hline \hline
+\end{tabular}
+\end{center}
+$\color{PineGreen}\Rrightarrow$ Maximal period is $2^k-1$.\\
+$\color{PineGreen}\Rrightarrow$ Example (Tausworths generator):\\
+$a_p=a_q=1$, other $a_i=0$ and $p>q$. Then: $b_n=b_{n-p}~{\rm xor}~b_{n-q}$
+
+$\color{PineGreen}\Rrightarrow$ How to get numbers from bits (for example):\\
+$U_i = \sum_{j=1}^L 2^{-j} b_{is+j},~s<L$.
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Fibonacci generator}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ In 1202 Fibonacci with Leonardo in Piza:
+\begin{equation}
+f_n=f_{n-2}+f_{n-1},~n\geqslant 2 \nonumber
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ Based on this first generator was created (Taussky and Todd, 1956):
+\begin{equation}
+X_n=(X_{n-2}+X_{n-1})~{\rm mod}~m,~n\geqslant 2 \nonumber
+\end{equation}
+This generator isn't so good in terms of statistics tests.\\
+$\color{PineGreen}\Rrightarrow$ Generalization:
+\begin{equation}
+X_n=(X_{n-r} \odot X_{n-s})~{\rm mod}~m,~n \geqslant r ,~s \geqslant 1 \nonumber
+\end{equation}\begin{center}
+
+\begin{tabular}{||c|c|c||}
+\hline \hline $\bigodot$ & $P_{max}$ & Stat. properties \\ \hline
+$+,-$ & $(2^r-1)2^{L-1}$ & good \\
+$x$ & $(2^r-1)2^{L-13}$ & very good \\
+$xor$ & $(2^r-1)$ & poor \\ \hline \hline 
+\end{tabular}
+
+
 \end{center}
 
+\end{small}
+\end{frame}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Multiply with carry, generator}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ We start from:
+\begin{equation}
+b_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m, \nonumber
+\end{equation}
+where $a_1,..,a_k \in \mathbb{N}$ are constant parameters.\\
+$\color{PineGreen}\Rrightarrow$  The c parameters is calculated foe each step:
+\begin{equation}
+c=\lfloor (a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)/m\rfloor, \nonumber 
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ Initialization: $a_1,..,a_k,c$.\\
+$\color{PineGreen}\Rrightarrow$ Advantages:
+\begin{itemize}
+\item Fast and easy to implement.
+\item Large period.
+\item Good statistical properties.
+\item First proposed by Marsaglia.
+\end{itemize}
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Subtract with borrow, generator}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$  Created again by Marsaglia (1991):
+\begin{equation}
+X_n=(X_{n-r} \circleddash X_{n-s})~{\rm mod}~m,~r,s  \nonumber \in \mathbb{N},
+\end{equation}
+where :
+\begin{equation}
+x \circleddash y = \begin{cases}
+x-y-c +m,~c=1,~{\rm{ when~x-y-c<0}}\\
+x-y-c,~c=0,~{\rm{ when~x-y-c\geq0}} \nonumber 
+\end{cases}
+\end{equation}
+
+
+$\color{PineGreen}\Rrightarrow$ Initialization: $X_1,...,X_{n-r}$ and $c=0$.\\
+$\color{PineGreen}\Rrightarrow$ Fast and easy :)\\
+$\color{PineGreen}\Rrightarrow$ Fails some of the basic statistics tests.
+\end{small}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Non linear generators}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ The natural solutions to problems of linear generators are the non-linear generators (second part of 1980s).\\
+$\color{PineGreen}\Rrightarrow$ Eichenauera i Lehna (1986):
+\begin{equation}
+X_n= (a X_{n−1}^{-1} + b)~{\rm mod}~m, \nonumber 
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ Eichenauera-Hermanna (1993)
+\begin{equation}
+X_n= [a(n+n_0)+b]^{-1}~{\rm mod}~m, \nonumber 
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ L. Blum, M. Blum, Shub (1986):
+\begin{equation}
+X_n= X_{n-1}^2~{\rm mod}~m, \nonumber 
+\end{equation}
+$\color{PineGreen}\rightarrowtail$ Very popular in cryptography.\\
+$\color{PineGreen}\Rrightarrow$ Pros and cons:
+\begin{itemize}
+\item {\color{PineGreen}{They all pass all statistical tests.}}
+\item {\color{red}{Much slower then linear generators.}}
+\end{itemize}
+\end{small}
 \end{frame}
 
 
@@ -656,6 +516,239 @@
 
 
 
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%5
+%% Stat tests$
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{\texttt{RANLUX} generator}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ All described generators are based on some mathematical algorithms and recursion. The typical scheme is of constructing a MC generator:
+\begin{itemize}
+\item Think of a formula that takes some initial values.
+\item Generate large number of random numbers and put them through statistical tests.
+\item If the test are positive we accept the the generator.
+\end{itemize}
+$\color{PineGreen}\Rrightarrow$ Now let's think: why the hell numbers obtained that way are showing some random number properties?\pause ~There is no science behind it, it's pure luck!\\
+$\color{PineGreen}\Rrightarrow$ M.Luscher (1993) \href{http://arxiv.org/pdf/hep-lat/9309020v1.pdf}{hep-lat/9309020}\\
+$\color{PineGreen}\Rrightarrow$ Generator RANLUX based on Kolomogorow entropy and Lyapunov exponent. { \color{PineGreen} Effectively we are building inside the generator the chaos theory}.\\
+$\color{PineGreen}\Rrightarrow$  RANLUX and Mersenne Twister (TRandom1, TRandom3) are the 2 most powerful generators in the world that passed every known statistical test.
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Chaos theory in a nut shell}
+\begin{small}
+\begin{columns}
+\column{0.1in}
+{~}
+\column{3in}
+$\color{PineGreen}\Rrightarrow$ We know that the solution of classical systems is described by trajectory in phase spaces. Now the problem with this picture starts to be when arround one point in this phase space we are getting more and more trajectories that are drifting a part later on.\\
+$\color{PineGreen}\Rrightarrow$ The Lyapunov exponent tells us how a two solutions drift apart with time:
+\begin{equation}
+\vert \delta X(t) \vert \approx  e^{\lambda t} \vert \delta X_0 \vert \nonumber
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ Kolomogorow entropy:
+\begin{equation}
+h_K = \int_P \lambda d \mu  \nonumber
+\end{equation}
+\column{2in}
+
+\includegraphics[width=0.9\textwidth]{images/Focal_stability.png}\\
+\includegraphics[width=0.9\textwidth]{images/LogisticMap_BifurcationDiagram.png}
+
+\end{columns}
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{HEP simulation}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ There is some ambiguity what particle physicist call MC. Normally those are mathematical theorises but when we say MC we usually mean MC simulation of a physics process.
+$\color{PineGreen}\Rrightarrow$ There are plenty of things that need to be simulated:
+\only<1>{
+\includegraphics[width=0.95\textwidth]{images/gen1.png}
+}
+\only<2>{
+\includegraphics[width=0.95\textwidth]{images/gen2.png}
+}
+\only<3>{
+\includegraphics[width=0.95\textwidth]{images/gen3.png}
+}
+\only<4>{
+\includegraphics[width=0.95\textwidth]{images/gen4.png}
+}
+\only<5>{
+\includegraphics[width=0.95\textwidth]{images/gen5.png}
+}
+\only<6>{
+\includegraphics[width=0.95\textwidth]{images/gen6.png}
+}
+\end{small}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Detector simulation}
+
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ Things do not get simpler on the detector side simulation.\\
+$\color{PineGreen}\Rrightarrow$ Lots of effects need to be taken into account:
+\begin{columns}
+\column{0.2in}
+{~}
+\column{2in}
+$\color{PineGreen}\rightarrowtail$ Bremsstrahlung\\
+$\color{PineGreen}\rightarrowtail$ Interactions with different detector materials\\
+$\color{PineGreen}\rightarrowtail$ Particle identification\\
+$\color{PineGreen}\rightarrowtail$ Showers\\
+\column{3in}
+\includegraphics[width=0.95\textwidth]{{images/lhcb2_h-640x408}.jpg}
+\end{columns}
+$\color{PineGreen}\Rrightarrow$ Example of generators:\\
+$\color{PineGreen}\rightarrowtail$ FLUKA\\
+$\color{PineGreen}\rightarrowtail$ Geant
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Method of Moments}
+
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ Now real cool things!\\
+$\color{PineGreen}\Rrightarrow$ Let's consider we want to study a rare decay: $\PB^{\pm} \to \PK^{\pm} \Pmu \Pmu$.  The decay is described by the following PDF:
+\begin{equation}
+\dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =\dfrac{3}{4}(1-F_H)(1-\cos^2 \theta_l)+F_H/2 + A_{FB}\cos \theta_l \nonumber
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ PDF by construction is normalized: $\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =1$
+\begin{columns}
+\column{0.1in}
+{~}
+\column{2.2in}
+\begin{itemize}
+\item Normally we do a likelihood fit and we are done.
+\item There is a second way!
+\end{itemize}
+\column{2.8in}
+\includegraphics[width=0.95\textwidth]{images/Kmumu_LL.png}
+\end{columns}
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Method of Moments}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ Let's calculate the integrals:
+\begin{equation}
+\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} \nonumber
+\end{equation}
+\begin{equation}
+\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15} \nonumber
+\end{equation}
+$\color{PineGreen}\Rrightarrow$ So we can get our parameters that we searched for by doing a integration. So now what?\pause \\
+$\color{PineGreen}\Rrightarrow$ Well nature is the best random number generator so let's take the data and treat and calculate the integral estimates:
+\begin{equation}
+\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} = \dfrac{1}{N} \sum_{i=1}^N  \cos \theta_{l,i} \nonumber 
+\end{equation}
+\begin{equation}
+\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15}= \dfrac{1}{N} \sum_{i=1}^N  \cos^2 \theta_{l,i} \nonumber
+\end{equation}
+
+\end{small}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Method of Moments}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ So what did we do?
+\begin{itemize}
+\item We have just estimated a parameters of interests without using any fit!!
+\end{itemize}
+$\color{PineGreen}\Rrightarrow$ Pros and cones of method of moments:
+\begin{itemize}
+\item {\color{PineGreen}{Are very immune to bias.}}
+\item {\color{PineGreen}{Do not suffer from boundary problems.}}
+\item {\color{PineGreen}{Require less statistic to work then likelihood fit.}}
+\item {\color{PineGreen}{They always have a Gaussian error}}.
+\item {\color{Red}{Estimator has a larger uncertainty.}}
+
+
+\end{itemize}
+
+\end{small}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Method of Moments, uncertainty estimator}
+\begin{small}
+\begin{columns}
+\column{0.1in}
+{~}
+\column{2.5in}
+$\color{PineGreen}\Rrightarrow$ It can be proven that Method of Moments estimator converges slower then the maximum likelihood fit.
+\column{2.5in}
+\includegraphics[width=0.95\textwidth]{images/S7.png}
+\end{columns}
+
+\begin{columns}
+\column{0.1in}
+{~}
+\column{2.5in}
+\includegraphics[width=0.95\textwidth]{images/S8F_950.png}
+\column{2.5in}
+\includegraphics[width=0.95\textwidth]{images/S8.png}
+\end{columns}
+
+\end{small}
+\end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Other application of MC - testing your analysis}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$  Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows:
+\begin{itemize}
+\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure.
+\item Simulate an assembly of simulation events for different values of parameters you want to measure.
+\item Do the analysis on this pseudo data.
+\item See if you are getting back what you have simulated.
+\end{itemize}
+
+\end{small}
+\end{frame}
+
+\begin{frame}\frametitle{Testing your analysis,  $\rm \color{RubineRed}{Lecture2/Test\_met}$}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$  Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows:
+\begin{itemize}
+\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure.
+\item Simulate an assembly of simulation events for different values of parameters you want to measure.
+\item Do the analysis on this pseudo data.
+\item See if you are getting back what you have simulated.
+\end{itemize}
+\includegraphics[width=0.4\textwidth]{{images/mean_estimator}.png}
+\end{small}
+\end{frame}
+
+
+\begin{frame}\frametitle{Wrap up}
+\begin{small}
+$\color{PineGreen}\Rrightarrow$ Things to remember:
+\begin{itemize}
+\item Computer cannot produce random numbers, only pseudorandom numbers.
+\item We use pseudorandon numbers as random numbers if they are statistically acting the same as random numbers.
+\item Linear generators are not commonly used nowadays.
+\item State of the art generators are the ones based on Kolomogorows theorem.
+\item MC methods used to simulate physics process, detector response and validating the estimators. 
+\end{itemize}
+\end{small}
+\end{frame}
+
+
+
 \backupbegin   
 
 \begin{frame}\frametitle{Backup}
diff --git a/Lectures_my/MC_2016/Lecture4/Makefile b/Lectures_my/MC_2016/Lecture4/Makefile
new file mode 100644
index 0000000..e4cf8e8
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/Makefile
@@ -0,0 +1,9 @@
+all: mchrzasz.pdf
+
+mchrzasz.pdf: mchrzasz.tex
+	xelatex mchrzasz.tex
+#	bibtex mchrzasz
+	xelatex mchrzasz.tex
+#	xelatex mchrzasz.tex
+clean:
+	rm *.aux *.bbl *.blg *.log *.out *.pdf *.sh *.xml *.snm *.toc *.nav 
diff --git a/Lectures_my/MC_2016/Lecture4/beamercolorthemeFlip.sty b/Lectures_my/MC_2016/Lecture4/beamercolorthemeFlip.sty
new file mode 100755
index 0000000..d6b9c50
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/beamercolorthemeFlip.sty
@@ -0,0 +1,179 @@
+% Color style file for the BEAMER FLIP THEME
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+
+
+
+\mode<presentation>
+\newif\ifbeamer@Flip@dark
+\beamer@Flip@darkfalse
+\DeclareOptionBeamer{dark}{\beamer@Flip@darktrue}
+\ProcessOptionsBeamer
+
+
+% These are a few colors that I like.
+
+\definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+\definecolor{darkcharcoal}{RGB}{25,25,25}		% Darker gray
+\definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+\definecolor{ash}{RGB}{100,100,100}			% medium gray
+\definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+\definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+\definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+\definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+\definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+\definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% Define some "invariant colors" so that a color that doen't work with a
+% light/dark background will automatically be substituted with one that does.
+% These will be rewritten in beamercolorthemeFlipDark
+
+\ifbeamer@Flip@dark
+	\definecolor{FlipSand}{RGB}{204,204,102}		% paleale, Only for dark BG
+	\definecolor{FlipGreen}{RGB}{0,255,0}			% Neon-like green, only for dark BG
+\else
+	\definecolor{FlipSand}{RGB}{140,110,10}			% lager, Only for light BG
+	\definecolor{FlipGreen}{RGB}{51,153,0}			% turtlegreen, works for both
+\fi
+
+%% Future revision: It would be nice to define these in terms of the palette.
+
+
+\ifbeamer@Flip@dark
+	\setbeamercolor{normal text}{fg=white, bg=black}		
+	\setbeamercolor{structure}{fg=gray, bg=black}		% fg controls bullets
+	\setbeamercolor*{footerlike}{fg=white,bg=black}		% Controls the footerline
+\else
+	% Default to black on white
+	\setbeamercolor*{footerlike}{fg=white,bg=charcoal}	% for charcoal footer
+	\setbeamercolor{structure}{fg=charcoal, bg=white}	% fg controls bullets
+	% \setbeamercolor{normal text}{fg=white, bg=black}	
+\fi
+
+
+
+
+
+% These defined colors can be used in \textcolor{colorname}{text in that color}.
+
+% Colors for Keynote background: DARK
+\definecolor{keynotebottom}{rgb}{0.32,0.3,0.38}
+\definecolor{keynotemiddle}{rgb}{0.08,0.08,0.16}
+\definecolor{keynotetop}{rgb}{0,0,0}
+%%% Example of usage:
+%% \setbeamertemplate{background canvas}[vertical shading]
+%% [bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+
+
+% ALERT AND COMMENT COLORS (just copies of the above colors)
+% (I don't know how ot do this more elegantly)
+\ifbeamer@Flip@dark
+	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{alert}{RGB}{204,204,102}		% paleale		
+	\definecolor{charcoal}{RGB}{51,51,51}		% charcoal
+	\definecolor{comment}{RGB}{80,80,80}		% light charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\else
+%	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{ALERT}{RGB}{28,134,238}		% DodgerBlue2
+%	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{Alert}{RGB}{255,127,36}		% chocolate1
+%	\definecolor{alert}{RGB}{140,110,10}		% lager
+	\definecolor{alert}{RGB}{0,205,102} 		% SpringGreen3
+	\definecolor{charcoal}{RGB}{80,80,80}		% light charcoal
+	\definecolor{comment}{RGB}{51,51,51}		% charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\fi
+
+
+% \definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+% \definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+% \definecolor{ash}{RGB}{100,100,100}			% medium gray
+% \definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+% \definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+% \definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+% \definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+% \definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+% \definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% A few useful colors
+\setbeamercolor{alerted text}{fg=alert}
+
+%% I would like ot use these colors, but there seems to be a bug in Beamer
+%% where the footnote color infects the normal text color
+%% See:  http://tex.stackexchange.com/questions/8264/beamer-changing-the-footnote-color-also-changes-the-normal-text-color/8268#8268
+
+% \setbeamercolor{footnote}{parent=normal text, fg=FlipSand}
+% \setbeamercolor{footnote mark}{parent=normal text, fg=FlipSand}
+
+% Blocks
+\setbeamercolor{block title}{fg=gray!30!white,bg=charcoal}
+\setbeamercolor{block body}{parent=normal text,bg=gray, fg=white}
+
+\setbeamercolor{block title example}{fg=turtlegreen!30!white,bg=turtlegreen!40!black}
+\setbeamercolor{block body example}{parent=normal text,bg=turtlegreen, fg=white}
+% 
+\setbeamercolor{block title alerted}{fg=crimsonred!30!white,bg=crimsonred!40!black}
+\setbeamercolor{block body alerted}{parent=normal text,bg=crimsonred, fg=white}
+
+
+
+% These "setbeamercolor" commands are more standardized. 
+
+
+%% From the Beamer User Guide:
+% Outer themes base the color of navigational elements and, possibly, also of other elements, on the 
+% four palette colors. The “primary” palette should be used for the most important navigational elements, 
+% which are usually the ones that change most often and hence require the most attention by the audience. 
+% The “secondary” and “tertiary” are less important, the “quaternary” one is least important. 
+% By default, the palette colors do not have a background and the foreground ranges from structure.fg to black. 
+% For the sidebar, there is an extra set of palette colors, see palette sidebar primary.
+
+\setbeamercolor*{palette primary}{fg=crimsonred,bg=charcoal}
+\setbeamercolor*{palette secondary}{fg=paleale,bg=lager}
+\setbeamercolor*{palette tertiary}{fg=turtlegreen,bg=jeans}
+\setbeamercolor*{palette quaternary}{fg=paleblue,bg=regal}
+
+% \setbeamercolor*{palette primary}{fg=crimsonred,bg=white}
+% \setbeamercolor*{palette secondary}{fg=charcoal,bg=white}
+% \setbeamercolor*{palette tertiary}{fg=paleblue,bg=white}
+% \setbeamercolor*{palette quaternary}{fg=turtlegreen,bg=white}
+
+
+
+%% These are some structures that I define for use in the outer theme.
+
+\setbeamercolor*{crimsontown}{fg=paleblue,bg=crimsonred}
+\setbeamercolor*{bluetown}{fg=white,bg=DodgerBlue2}
+%\setbeamercolor*{titlelike}{fg=crimsonred}
+\setbeamercolor*{titlelike}{fg=DodgerBlue2}
+\setbeamercolor*{topbarlike}{fg=white, bg=charcoal}
+
+
+
+
+
+%% Some old definitions that I might want to get back to
+%
+% \setbeamercolor*{frametitle}{bg=white,fg=charcoal}
+% \setbeamercolor*{part title}{bg=white,fg=charcoal}
+% \setbeamercolor*{item}{fg=charcoal, bg=white}
+% 
+% \setbeamercolor*{separation line}{}
+% \setbeamercolor*{fine separation line}{}
+
+
+
+
+
+\mode
+<all>
diff --git a/Lectures_my/MC_2016/Lecture4/beamerinnerthemeFlip.sty b/Lectures_my/MC_2016/Lecture4/beamerinnerthemeFlip.sty
new file mode 100755
index 0000000..9910f4d
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/beamerinnerthemeFlip.sty
@@ -0,0 +1,95 @@
+% Inner style file for the BEAMER FLIP THEME
+% Copyright 2010 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerinnerthemeFlip by Marco Barisione
+
+\mode<presentation>
+
+% % Use alternative title page style.
+% \DeclareOptionBeamer{alternativetitlepage}[true]{\def\beamer@Flip@alternativetitlepage{#1}}
+
+% Logo to use in the alternative title page.
+% \def\beamer@Flip@titlepagelogo{}
+% \DeclareOptionBeamer{titlepagelogo}{\def\beamer@Flip@titlepagelogo{#1}}
+
+% Bullet shape.
+\DeclareOptionBeamer{bullet}{\def\beamer@Flip@bullet{#1}}
+\DeclareOptionBeamer{shadow}[true]{\def\beamer@themerounded@shadow{#1}}
+\ExecuteOptionsBeamer{shadow=false}
+
+
+% \ExecuteOptionsBeamer{alternativetitlepage=false,bullet=square}
+\ProcessOptionsBeamer
+
+
+
+% Margins.
+\newlength{\beamer@Flip@normalmargin}
+\setlength{\beamer@Flip@normalmargin}{.06\paperwidth}
+\setbeamersize{text margin left=\beamer@Flip@normalmargin}
+\setbeamersize{text margin right=\beamer@Flip@normalmargin}
+\setlength\leftmargini{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginii{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginiii{.6\beamer@Flip@normalmargin}
+
+% Normal title page.
+\defbeamertemplate*{title page normal}{Flip theme}[1][]
+{
+  \vbox{}
+  \vfill
+  \begin{centering}
+    \begin{beamercolorbox}[wd=\paperwidth,sep=8pt,center,#1]{title page header}
+      \usebeamerfont{title}\inserttitle\par%
+      \ifx\insertsubtitle\@empty%
+      \else%
+        \vskip0.25em%
+        {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
+      \fi%     
+    \end{beamercolorbox}%
+    \vskip1em\par
+    \begin{beamercolorbox}[sep=8pt,center,#1]{author}
+      \usebeamerfont{author}\insertauthor
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{institute}
+      \usebeamerfont{institute}\insertinstitute
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{date}
+      \usebeamerfont{date}\insertdate
+    \end{beamercolorbox}\vskip0.5em
+    {\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
+  \end{centering}
+  \vfill
+}
+
+
+% Items.
+\defbeamertemplate{itemize item}{squarealt}%
+{\tiny\raise.5ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+\defbeamertemplate{itemize subitem}{squarealt}%
+{\tiny\raise.4ex\hbox{\donotcoloroutermaths$\square$}}
+\defbeamertemplate{itemize subsubitem}{squarealt}%
+{\tiny\raise.3ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+% 
+\defbeamertemplate{itemize item}{circlealt}%
+{\small\raise.2ex\hbox{\donotcoloroutermaths$\bullet$}}
+\defbeamertemplate{itemize subitem}{circlealt}%
+{\small\raise.1ex\hbox{\donotcoloroutermaths$\circ$}}
+\defbeamertemplate{itemize subsubitem}{circlealt}%
+{\scriptsize\raise.1ex\hbox{\donotcoloroutermaths$\bullet$}}
+\setbeamertemplate{items}[circlealt]
+% 
+\def\circletext{circle}
+\ifx\beamer@Flip@bullet\circletext
+  \setbeamertemplate{items}[circlealt]
+\else
+  \setbeamertemplate{items}[squarealt]
+\fi
+
+\setbeamertemplate{blocks}[rounded][shadow=\beamer@themerounded@shadow]
+
+
+\mode<all>
+
diff --git a/Lectures_my/MC_2016/Lecture4/beamerouterthemeFlip.sty b/Lectures_my/MC_2016/Lecture4/beamerouterthemeFlip.sty
new file mode 100755
index 0000000..d12f6d6
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/beamerouterthemeFlip.sty
@@ -0,0 +1,189 @@
+% Outer style file for the BEAMER FLIP THEME 212
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerouterthemesplit by Till Tantau
+
+
+\mode<presentation>
+
+% Watermark
+\DeclareOptionBeamer{watermark}{\def\beamer@Flip@watermark{#1}}
+\ExecuteOptionsBeamer{watermark=} % Default value is empty
+
+
+\newif\ifbeamer@Flip@bigpagenumber
+\beamer@Flip@bigpagenumberfalse
+\DeclareOptionBeamer{bigpagenumber}{\beamer@Flip@bigpagenumbertrue}
+
+
+\def\beamer@Flip@truetext{true} % Kludge for true/false so I can use ifx
+
+\DeclareOptionBeamer{topline}[true]{\def\beamer@Flip@topline{#1}}
+\ExecuteOptionsBeamer{topline=false} % Default
+
+
+
+% \ExecuteOptionsBeamer{topline=} % Default value is empty
+
+
+\ProcessOptionsBeamer
+
+
+\usepackage{tikz}	% For fancy decorations
+
+
+% I should fix these up to be more robust against color changes
+\setbeamercolor{section in head/foot}{parent=footerlike}
+\setbeamercolor{subsection in head/foot}{parent=footerlike}
+\setbeamercolor{author in head/foot}{parent=footerlike}
+\setbeamercolor{title in head/foot}{parent=footerlike}
+
+
+% \setbeamercolor{author in head/foot}{parent=section in head/foot}
+% \setbeamercolor{title in head/foot}{parent=subsection in head/foot}
+
+%\setbeamercolor{topbar}{parent=crimsontown}
+\setbeamercolor{topbar}{parent=bluetown}
+
+
+
+
+\usesectionheadtemplate
+  {\hfill\insertsectionhead}
+  {\hfill\color{fg!50!bg}\insertsectionhead}
+
+
+%%%%%%%%%%%%%%%%%%
+% The footer bar %
+%%%%%%%%%%%%%%%%%%
+
+\defbeamertemplate*{footline}{Flip theme}
+{%
+  \leavevmode% 
+% OBSERVATION: the "%" symbols inside hbox are all very important here.
+% The \hbox enviroment will insert spaces whenever there's whitespace
+% Adding a % at the END of each line ensures that any hard returns aren't
+% interpreted as white space. This allows the color boxes to be flush against
+% one another. If a faint white line appears one could probably do something like
+% hspace{-1px} and add an additional pixel-or-two wide beamercolorbox.	
+
+\hbox{%
+\begin{beamercolorbox}[wd=.4\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm, rightskip=.3cm plus1fil]{author in head/foot}%
+	\usebeamerfont{author in head/foot}\insertshortauthor  \end{beamercolorbox}%
+%
+\begin{beamercolorbox}[wd=.6\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}%
+    \usebeamerfont{title in head/foot}{\textit \insertshorttitle}%
+    \hskip2ex plus1fill%
+		% \insertpagenumber\,/\,\insertpresentationendpage %% Want FRAME, not SLIDE number
+		\insertframenumber/\inserttotalframenumber
+ \end{beamercolorbox}%
+}%
+
+% This is a complete kludge for the faint white line which appears
+% between the beamer color boxes above. (It doesnt appear in presentation mode
+% but sometimes looks funny when the pdfs are viewed in a window)
+\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[thick] (0,3.625ex) -- (\paperwidth,3.625ex);
+	\draw (.4\paperwidth,0) -- (.4\paperwidth,3.625ex);
+	\draw[thick] (0,0) -- (\paperwidth,0);
+\end{tikzpicture}
+
+
+
+\ifbeamer@Flip@bigpagenumber
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (6ex);
+	\usebeamercolor[fg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (5ex);
+	\usebeamercolor[bg]{author in head/foot}
+	\node at (\paperwidth-5ex,4ex) {\small$^\text{\insertframenumber}/_\text{\inserttotalframenumber}$};
+	\end{tikzpicture}
+\fi
+}
+
+
+
+\defbeamertemplate*{frametitle}{Flip theme}[1][left]
+{
+% \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+\@tempdima=\textwidth%
+\advance\@tempdima by\beamer@leftmargin%
+\advance\@tempdima by\beamer@rightmargin%
+
+\vbox{}\vskip-3.5ex%
+%
+\begin{beamercolorbox}[sep=0.3cm,#1,wd=\the\@tempdima]{titlelike}
+	\usebeamerfont{frametitle}%
+	\vbox{}\vskip-1ex%
+	\if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+	% \strut{\bf\insertframetitle}\strut\par% 					%% Include the \bf
+	\strut{\insertframetitle}\strut\par%
+	{%
+  		\ifx\insertframesubtitle\@empty%
+  		\else
+			{\usebeamerfont{framesubtitle}
+			 \usebeamercolor[fg]{framesubtitle}
+			\insertframesubtitle\strut\par}%
+  		\fi
+ 	}%
+	%
+  	\vskip-1ex%
+  	\if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
+\end{beamercolorbox}%
+%%
+
+% \def\beamer@fteright{\vskip0.35cm\advance\leftskip by 1.7cm\advance\rightskip by1.7cm}
+}
+
+
+
+
+% Sidebar right is important: anything placed here will be
+% visible *behind* the main text. This is where to put any
+% interesting watermarks.
+%
+% 
+\defbeamertemplate*{sidebar right}{Flip theme}
+{
+% \llap{\includegraphics[width=50px]{footdecoration}}
+% \llap{\includegraphics[width=\paperwidth,height=\paperheight]{upperleft_watermark_transp}}
+%
+% THIS WOULD BE A NICE WAY TO PUT THE PARTICLE BUBBLE CHAMBER BG
+% ... make a png of the background with TRANSPARENT BG
+% ... then put the image here (use \hfill to get it on the left)
+% Now one can FURTHER apply a slight gradient in the BG
+%
+% 
+% % Top Line 
+\ifx\beamer@Flip@topline\beamer@Flip@truetext
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[fg]{titlelike}	% alternately, topbarlike would be a good beamercolor
+%	\usebeamercolor[fg]{topbarlike}	% alternately, topbarlike would be a good beamercolor
+	\draw[fill] (-\paperwidth,0) rectangle (0,-3pt);
+	\end{tikzpicture}
+\fi
+%
+% 
+  \vskip.2\beamer@leftmargin%
+   %\llap{\insertlogo\hskip.5\beamer@leftmargin}%
+   \llap{\insertlogo\hskip.08\beamer@leftmargin}%
+    \vfill%
+  \if \beamer@Flip@watermark\@empty
+	% This is kind of a kludge since if I put anything in here, I automatically get errors
+	% Somehow LaTeX wants to read the \else part!
+	\else%
+	\llap{\includegraphics[width=\paperwidth,height=\paperheight]{\beamer@Flip@watermark}}
+  \fi
+}
+
+
+
+\mode<all>
+\endinput
+
+
diff --git a/Lectures_my/MC_2016/Lecture4/beamerouterthemedecolines.sty b/Lectures_my/MC_2016/Lecture4/beamerouterthemedecolines.sty
new file mode 100755
index 0000000..669e493
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/beamerouterthemedecolines.sty
@@ -0,0 +1,168 @@
+% Copyright 2007 by Marco Barisione
+%
+% This file may be distributed and/or modified
+%
+% 1. under the LaTeX Project Public License and/or
+% 2. under the GNU Public License.
+
+\mode<presentation>
+
+% String used between the current page and the total page count.
+\def\beamer@decolines@pageofpages{/}
+\DeclareOptionBeamer{pageofpages}{\def\beamer@decolines@pageofpages{#1}}
+
+% Show a line below the frame title.
+\DeclareOptionBeamer{titleline}[true]{\def\beamer@decolines@titleline{#1}}
+
+% Image used for the watermark.
+\def\beamer@decolines@watermarkorig{}
+\DeclareOptionBeamer{watermark}{\def\beamer@decolines@watermarkorig{#1}}
+
+% Height of the watermark.
+\def\beamer@decolines@watermarkheight{100px}
+\DeclareOptionBeamer{watermarkheight}{\def\beamer@decolines@watermarkheight{#1}}
+
+% The original image height is watermarkheightmult * watermarkheight.
+\def\beamer@decolines@watermarkheightmult{1}
+\DeclareOptionBeamer{watermarkheightmult}{\def\beamer@decolines@watermarkheightmult{#1}}
+
+\ExecuteOptionsBeamer{titleline=false}
+\ProcessOptionsBeamer
+
+% Enable/disable the watermark.
+\def\watermarkon{%
+  \def\beamer@decolines@watermark{\beamer@decolines@watermarkorig}%
+}
+\def\watermarkoff{\def\beamer@decolines@watermark{}}
+
+% Initially enable the watermark.
+\watermarkon
+
+% Colors.
+\setbeamercolor*{lineup}{parent=palette primary}
+\setbeamercolor*{linemid}{parent=palette secondary}
+\setbeamercolor*{linebottom}{parent=palette tertiary}
+\setbeamercolor*{page header}{parent=titlelike}
+
+% Lengths
+\newlength{\headerheight}
+\setlength{\headerheight}{.045\paperheight}
+\newlength{\beamer@decolines@lineup}
+\setlength{\beamer@decolines@lineup}{.025\paperheight}
+\newlength{\beamer@decolines@linemid}
+\setlength{\beamer@decolines@linemid}{.015\paperheight}
+\newlength{\beamer@decolines@linebottom}
+\setlength{\beamer@decolines@linebottom}{.01\paperheight}
+
+% The height of the watermark part below the 3 bottom lines.
+\newlength{\beamer@decolines@watermarkheightbottom}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@lineup}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linemid}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linebottom}
+
+% The height of the watermark part over the 3 bottom lines before shrinking.
+\newlength{\beamer@decolines@watermarkheightupperorig}
+\setlength{\beamer@decolines@watermarkheightupperorig}{\beamer@decolines@watermarkheight}
+\addtolength{\beamer@decolines@watermarkheightupperorig}{-\beamer@decolines@watermarkheightbottom}
+\multiply\beamer@decolines@watermarkheightupperorig by \beamer@decolines@watermarkheightmult
+
+% % Footer.
+% \defbeamertemplate*{footline}{decolines theme}
+% {
+%   \leavevmode%
+%   % Page number.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=0ex,dp=0ex,center]{}%
+%     \usebeamerfont{palette primary}\insertframenumber{} \beamer@decolines@pageofpages{} \inserttotalframenumber%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=0ex,dp=0ex]{}%
+%   \end{beamercolorbox}%
+%   } %
+%   % First line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{lineup}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Second line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=\paperwidth,ht=\beamer@decolines@linemid,dp=0pt]{linemid}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Third line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.1\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.9\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{linebottom}%
+%   \end{beamercolorbox}%
+%   }%
+%   % This seems to fix some alignment problems with the watermark. It has to be
+%   % always applied if you do not want to see the footer moving up and down when
+%   % moving from a page with watermark to a page without or vice versa.
+%   \vskip-.5px%
+%   % Watermark.
+%   \if\beamer@decolines@watermark\@empty\else%
+%     \vskip-\beamer@decolines@watermarkheightbottom%
+%     \llap{\includegraphics[height=\beamer@decolines@watermarkheightbottom,clip=true,%
+%       trim=0pt 0pt 0pt \beamer@decolines@watermarkheightupperorig]{\beamer@decolines@watermark}\hskip-\paperwidth}%
+%   \fi%
+% }
+
+\defbeamertemplate*{headline}{decolines theme}
+{
+  \leavevmode%
+  \hbox{%
+  \begin{beamercolorbox}[wd=\paperwidth,ht=\headerheight,dp=0pt]{page header}%
+  \end{beamercolorbox}%
+  } %
+  \vskip0pt%
+}
+
+\defbeamertemplate*{frametitle}{decolines theme}[1][left]
+{
+  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+  \@tempdima=\textwidth%
+  \advance\@tempdima by\beamer@leftmargin%
+  \advance\@tempdima by\beamer@rightmargin%
+  \vbox{}\vskip-.5\beamer@leftmargin%
+  \begin{beamercolorbox}[sep=\beamer@leftmargin,#1,wd=\the\@tempdima]{}
+    \usebeamerfont{frametitle}\usebeamercolor[bg]{framesubtitle}%
+    \vbox{}\vskip0ex%
+    \if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+    \strut\insertframetitle\strut\par%
+    {%
+      \ifx\insertframesubtitle\@empty%
+      \else%
+      {\usebeamerfont{framesubtitle}\usebeamercolor[bg]{framesubtitle}\insertframesubtitle\strut\par}%
+      \fi
+    }%
+    \vskip-1ex%
+    \if@tempswa\else\vskip-\beamer@leftmargin\fi
+  \end{beamercolorbox}%
+  \def\beamer@decolines@truetext{true}%
+  \ifx\beamer@decolines@titleline\beamer@decolines@truetext%
+    \vskip-.5\beamer@leftmargin%
+    \begin{beamercolorbox}[wd=\textwidth,ht=.1ex,dp=0ex]{linemid}%
+    \end{beamercolorbox}%
+  \fi
+}
+
+% Frame title continuations, default
+\defbeamertemplate*{frametitle continuation}{decolines theme}{(\insertcontinuationcount)}
+
+
+
+\defbeamertemplate*{sidebar right}{decolines theme}
+{
+  \vskip.1\beamer@leftmargin%
+  \llap{\insertlogo\hskip.5\beamer@leftmargin}%
+  \vfill%
+  \if\beamer@decolines@watermark\@empty\else%
+  \llap{\includegraphics[height=\beamer@decolines@watermarkheight]{\beamer@decolines@watermark}}%
+  \vskip-\beamer@decolines@watermarkheightbottom%
+  \fi
+}
+
+\mode
+<all>
diff --git a/Lectures_my/MC_2016/Lecture4/beamerthemeFlip.sty b/Lectures_my/MC_2016/Lecture4/beamerthemeFlip.sty
new file mode 100755
index 0000000..a615ff5
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/beamerthemeFlip.sty
@@ -0,0 +1,59 @@
+% Main style file for the BEAMER FLIP THEME 2012
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Comments: This is still a work in progress.
+% A good way to present: http://www.cs.hmc.edu/~oneill/freesoftware/pdftokeynote.html
+
+% \usepackage{beamerthemesplit}
+\mode<presentation>
+
+
+%% These are the options that are fed in through the driver file
+%% Some of them get passed on to the other theme files
+\DeclareOptionBeamer{bullet}{\PassOptionsToPackage{bullet=#1}{beamerinnerthemeFlip}}
+\DeclareOptionBeamer{bigpagenumber}{\PassOptionsToPackage{bigpagenumber}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{topline}[true]{\PassOptionsToPackage{topline=#1}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{shadow}[false]{\PassOptionsToPackage{shadow=#1}{beamerinnerthemeFlip}}
+
+\DeclareOptionBeamer{watermark}{\PassOptionsToPackage{watermark=#1}{beamerouterthemeFlip}}
+
+\ProcessOptionsBeamer
+
+\useinnertheme{Flip}	% Calls beamerinnerthemeFlip.sty
+\useoutertheme{Flip}	% Calls beamerouterthemeFlip.sty
+\usecolortheme{Flip}	% Calls beamercolorthemeFlip.sty
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+% Background  Gradient %
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%% Usually I would leave this on
+%% However, there seems to be problems with XeLaTeX
+%% http://tex.stackexchange.com/questions/29497/xelatex-preventing-beamer-from-using-different-backgrounds
+\setbeamertemplate{background canvas}[vertical shading][bottom=blue!.1, middle=white, top=white]
+
+%% HOWEVER: it seems to make it impossible to change this later on in the document
+
+
+%% For a dark background: 
+% \setbeamertemplate{background canvas}[vertical shading][bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+%% An alternate way: \beamertemplateshadingbackground{blue!.1}{red!2}
+%% In general: you want the gradient to be subtle!
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+% Navigation  symbols %
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\setbeamertemplate{navigation symbols}{} 	% Turns off PDF navigation symbols
+
+\mode<all>
+\endinput
\ No newline at end of file
diff --git a/Lectures_my/MC_2016/Lecture4/images/BG_lower.png b/Lectures_my/MC_2016/Lecture4/images/BG_lower.png
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diff --git a/Lectures_my/MC_2016/Lecture4/images/uzh-transp.pdf b/Lectures_my/MC_2016/Lecture4/images/uzh-transp.pdf
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diff --git a/Lectures_my/MC_2016/Lecture4/mchrzasz+mp.sh b/Lectures_my/MC_2016/Lecture4/mchrzasz+mp.sh
new file mode 100644
index 0000000..1a24852
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/mchrzasz+mp.sh
@@ -0,0 +1 @@
+#!/bin/sh
diff --git a/Lectures_my/MC_2016/Lecture4/mchrzasz-blx.bib b/Lectures_my/MC_2016/Lecture4/mchrzasz-blx.bib
new file mode 100644
index 0000000..9e405b3
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/mchrzasz-blx.bib
@@ -0,0 +1,11 @@
+@Comment{$ biblatex control file $}
+@Comment{$ biblatex version 2.3 $}
+Do not modify this file!
+
+This is an auxiliary file used by the 'biblatex' package.
+This file may safely be deleted. It will be recreated as
+required.
+
+@Control{biblatex-control,
+  options = {2.3:0:0:1:0:0:1:1:0:0:0:0:1:1:3:1:79:+},
+}
diff --git a/Lectures_my/MC_2016/Lecture4/mchrzasz.aux b/Lectures_my/MC_2016/Lecture4/mchrzasz.aux
new file mode 100644
index 0000000..55cf10a
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/mchrzasz.aux
@@ -0,0 +1,136 @@
+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldcontentsline\contentsline
+\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\contentsline\oldcontentsline
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand*\HyPL@Entry[1]{}
+\bibstyle{biblatex}
+\bibdata{mchrzasz-blx,bib}
+\citation{biblatex-control}
+\HyPL@Entry{0<</P(1)>>}
+\providecommand \oddpage@label [2]{}
+\@writefile{toc}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
+\@writefile{lof}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
+\@writefile{lot}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax }
+\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\beamer@endinputifotherversion {3.24pt}}
+\@writefile{nav}{\defcounter {refsection}{0}\relax }\@writefile{nav}{\beamer@endinputifotherversion {3.24pt}}
+\select@language{english}
+\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\select@language{english}}
+\@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\select@language{english}}
+\@writefile{lot}{\defcounter {refsection}{0}\relax }\@writefile{lot}{\select@language{english}}
+\@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}}
+\@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}}
+\pgfsyspdfmark {pgfid3}{0}{0}
+\pgfsyspdfmark {pgfid4}{0}{0}
+\HyPL@Entry{1<</P(2)>>}
+\pgfsyspdfmark {pgfid5}{23867907}{17900937}
+\@writefile{nav}{\headcommand {\slideentry {0}{0}{2}{2/2}{}{0}}}
+\@writefile{nav}{\headcommand {\beamer@framepages {2}{2}}}
+\pgfsyspdfmark {pgfid6}{0}{0}
+\pgfsyspdfmark {pgfid7}{0}{0}
+\HyPL@Entry{2<</P(3)>>}
+\pgfsyspdfmark {pgfid8}{23867907}{17900937}
+\@writefile{nav}{\headcommand {\slideentry {0}{0}{3}{3/3}{}{0}}}
+\@writefile{nav}{\headcommand {\beamer@framepages {3}{3}}}
+\pgfsyspdfmark {pgfid9}{0}{0}
+\pgfsyspdfmark {pgfid10}{0}{0}
+\HyPL@Entry{3<</P(4)>>}
+\pgfsyspdfmark {pgfid11}{23867907}{17900937}
+\@writefile{nav}{\headcommand {\slideentry {0}{0}{4}{4/4}{}{0}}}
+\@writefile{nav}{\headcommand {\beamer@framepages {4}{4}}}
+\pgfsyspdfmark {pgfid12}{0}{0}
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+\usetikzlibrary{trees}
+\usetikzlibrary{matrix,arrows} 				% For commutative diagram
+% http://www.felixl.de/commu.pdf
+\usetikzlibrary{positioning}				% For "above of=" commands
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+
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+	% >=stealth', %%  Uncomment for more conventional arrows
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+
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+
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+\bibliography{bib}
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+
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+
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+}
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+}
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+
+% For shapo's formulas:
+\def\lsi{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
+\def\gsi{\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
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+\def\be{\begin{equation}}
+\def\ee{\end{equation}}
+\def\ba{\begin{eqnarray}}
+\def\ea{\end{eqnarray}}
+\def\d{\partial}
+\def\l{\left(}
+\def\r{\right)}
+\def\la{\langle}
+\def\ra{\rangle}
+\def\e{{\rm e}}
+\def\Br{{\rm Br}}
+\def\fixme{{\color{red} FIXME!}}
+\def\mc{{\color{Magenta}{MC}}}
+\def\pdf{{\rm p.d.f.}}
+
+\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
+\institute{UZH}
+\title[Random number generators and application]{Random number generators and application}
+\date{\fixme}
+
+
+\begin{document}
+\tikzstyle{every picture}+=[remember picture]
+
+{
+\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
+\begin{frame}[c]%{\phantom{title page}} 
+\begin{center}
+\begin{center}
+	\begin{columns}
+		\begin{column}{0.9\textwidth}
+			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Applications of Monte Carlo methods}
+		\end{column}
+		\begin{column}{0.2\textwidth}
+		  %\includegraphics[width=\textwidth]{SHiP-2}
+		\end{column}
+	\end{columns}
+\end{center}
+	\quad
+	\vspace{3em}
+\begin{columns}
+\begin{column}{0.44\textwidth}
+\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}
+
+\end{column}
+\begin{column}{0.53\textwidth}
+\includegraphics[height=1.3cm]{uzh-transp}
+\end{column}
+\end{columns}
+
+\vspace{1em}
+%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
+\vspace{0.5em}
+	\textcolor{normal text.fg!50!Comment}{Experimental Methods in Particle Physics, \\ 26 November, 2015}
+\end{center}
+\end{frame}
+}
+
+
+
+
+
+\begin{frame}\frametitle{Markov Chain MC}
+\begin{itemize}
+\item Consider a finite possible states: $S_1$, $S_2$, ...
+\item And the time steps of time, labelled as $1$, $2$, ...
+\item At time $t$ the state is denoted $X_t$.
+\item The conditional probability is defined as:
+\end{itemize}
+\begin{equation}
+P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) \nonumber
+\end{equation}
+\begin{itemize}
+\item The Markov chain is then if the probability depends only on previous step.
+\end{itemize}
+\begin{equation}
+P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) =  P(X_t=S_j \vert X_{t-1}=S_{j-1} )\nonumber
+\end{equation}
+\begin{itemize}
+\item For this reason this reason MCMC is also knows as drunk sailor walk.
+\item Very powerful method. Used to solve linear eq. systems, invert matrix, solve differential equations, etc.
+\end{itemize}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Linear Equations}
+\begin{itemize}
+\item Lets say we have a linear equation system:
+\end{itemize}
+\begin{equation}
+\begin{array}{lcl} X & = & pY + (1-p) A \\ Y & = & qX + (1-q)B \end{array} \nonumber
+\end{equation}
+\begin{itemize}
+\item We know $A,B,p,q$; $X$ and $Y$ are meant to be determined.
+\item Algorithm:
+\begin{enumerate}
+\item We choose first element of the first equation with probability $p$ and second with probability $1-p$.
+\item We we choose the second one, the outcome of this MCMC is $W=A$.
+\item If we choose the first we go to second equation and choose the first element with probability $q$ and the second with $1-q$.
+\item We we choose the second one, the outcome of this MCMC is $W=B$.
+\item If we choose the first we go to the first equation back again.
+\item We repeat the procedure.
+\end{enumerate}
+\item We can estimate the solution of this system:
+\end{itemize}
+\begin{equation}
+\hat{X} = \dfrac{1}{N}\sum_{i=1} W_i{~}{~}{~}{~}{~} \hat{\sigma_X}=\dfrac{1}{\sqrt{N-1}}\sqrt{\dfrac{1}{N} \sum_{i=1}^N W_i^2-\hat{X}^2} \nonumber
+\end{equation}
+
+\end{frame}
+
+\begin{frame}\frametitle{Neumann-Ulam method}
+\begin{itemize}
+\item Let's try apply the basic MCMC method to solve a simple linear equation system:
+\end{itemize}
+\begin{equation}
+A \overrightarrow{x} = \overrightarrow{b} \nonumber
+\end{equation}
+\begin{itemize}
+\item The above system can be (always, see linear algebra lecture) translated into system:
+\end{itemize}
+\begin{equation}
+\overrightarrow{x} = \overrightarrow{a} + H \overrightarrow{x} \nonumber
+\end{equation}
+\begin{itemize}
+\item For this method we assume that the norm of the matrix is:
+\end{itemize}
+\begin{equation}
+\Vert H \Vert =  \underset{1 \leq i \leq n}{max} \sum_{j=1}^n \vert h_{ij} \vert <1 \nonumber \end{equation}
+\begin{itemize}
+\item Which we can write in a form:
+\end{itemize}
+\begin{equation}
+(1 -H)\overrightarrow{x}=\overrightarrow{a} \nonumber
+\end{equation}
+\end{frame}
+
+
+
+
+\begin{frame}\frametitle{Neumann-Ulam method}
+\begin{itemize}
+\item The solution would be then:
+\end{itemize}
+\begin{equation}
+\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} \nonumber
+\end{equation}
+\begin{itemize}
+\item We can Taylor expend this:
+\end{itemize}
+\begin{equation}
+\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} = \overrightarrow{a} + H \overrightarrow{a} + H^2 \overrightarrow{a} + H^3 \overrightarrow{a} +.... \nonumber
+\end{equation}
+\begin{itemize}
+\item For the $i$-th component of the $\overrightarrow{x}$ vector:
+\end{itemize}
+\begin{equation}
+x_0^i= a_i + \sum_{j=1}^n h_{ij} a_{j_1} + \sum_{j_1=1}^n \sum_{j_2=1}^n  h_{ij_1} h_{ij_2}  a_{j_2} +  \sum_{j_1=1}^n \sum_{j_2=1}^n \sum_{j_3=1}^n  h_{ij_1} h_{ij_2} h_{ij_3} a_{j_3} + ...\nonumber
+\end{equation}
+\begin{itemize}
+\item One can construct probabilistic behaviour of a system that follows the path of equation above.
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Neumann-Ulam method}
+\begin{itemize}
+\item To do so we add to our matrix an additional column of the matrix:
+\end{itemize}
+\begin{equation}
+h_{i,0} = 1-\sum_{j=1}^n h_{ij} > 0 \nonumber
+\end{equation}
+\begin{itemize}
+\item The system has states: $\lbrace 0,1,2...,n\rbrace$
+\item State at $t$ time is denoted as $i_t$.
+\item We make a random walk accordingly to to the following rules:
+\begin{itemize}
+\item At the begging of the walk ($t=0$) we are at $i_0$. 
+\item In the $t$ moment we are in the $i_t$ position then in $t+1$ time stamp we move to state $i_{t+1}$ with the probability $h_{i_t i_{t+1}}$. 
+\item We stop walking if we are in state $0$. 
+\end{itemize}
+\item The path $X(\gamma) = (i_0, i_1, i_2, ..., i_k, 0)$ is called trajectory.
+\item It can be proven that $x_i^0 =E \lbrace X (\gamma) \vert i_0=j \rbrace$.
+\end{itemize}
+
+
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Neumann-Ulam method, $\rm \color{RubineRed}{Lecture3/Markov}$}
+\begin{itemize}
+\item For example lets try to solve this equation system:
+\end{itemize}
+\begin{equation}
+\overrightarrow{x} = 
+\left(\begin{array}{c}
+ 1.5  \\
+-1.0\\
+0.7  \end{array} \right) 
++
+\left(\begin{array}{ccc}
+0.2 &  0.3 & 0.1  \\
+0.4 &  0.3 & 0.2 \\
+0.3 &  0.1 & 0.1  \end{array} \right) \overrightarrow{x}
+ \nonumber
+\end{equation}
+\begin{itemize}
+\item The solution is $\overrightarrow{x}_0 = (2.154303, 0.237389, 1.522255)$.
+\end{itemize}
+\begin{columns}
+
+\column{0.1in}
+
+\column{2.5in}
+\begin{itemize}
+\item The propability matrix $h_{ij}$ has the shape:
+\end{itemize}
+\begin{tabular}{|c|cccc|}
+\hline
+$i/j$ &  1 & 2 & 3 & 4  \\ \hline
+1 & 0.2 & 0.3 & 0.1 & 0.4 \\
+2 & 0.4 & 0.3 & 0.2 & 0.1 \\
+3 & 0.3 & 0.1 & 0.1 & 0.5 \\ \hline
+\end{tabular}
+
+\column{2.5in}
+\begin{itemize}
+\item An example solution:
+\end{itemize}
+\includegraphics[width=0.95\textwidth]{images/mark.png}
+
+\end{columns}
+
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Neumann-Ulam dual method}
+\begin{itemize}
+\item The problem with Neumann-Ulam method is that you need to repeat it for each of the coordinates of the $\overrightarrow{x}_0$ vector.
+\item The dual method calculates the whole $\overrightarrow{x}_0$ vector.
+\item The algorithm:
+\begin{itemize}
+\item On the indexes: $\lbrace0,1,...,n\rbrace$ we set a probability distribution:\\ $q_1, q_2,..., q_n$, $q_i>0$ and $\sum_i=1^n q_i=1$. 
+\item The starting point we select from $q_i$ distribution.
+\item If in $t$ time we are in $i_t$ state then with probability $p(i_{t+1} \vert i_t) = h_{i_{t+1},i_{t}}$ in $t+1$ we will be in state $i_1$. For  $i_{t+1}=0$ we define the probability: $h_{0,i_{t}}=1-\sum_{j=1}^n h_{j,i_{t}}$. Here we also assume that $h_{j,i_{t}} > 0$.
+\item NOTE: there the matrix is transposed compared to previous method: $H^{T}$.
+\item Again we end our walk when we are at state $0$. 
+\item For the trajectory: $\gamma = (i_0, i_1,...,i_k, 0)$, we assign the vector:
+\end{itemize}
+\begin{equation}
+\overrightarrow{Y}(\gamma) = \dfrac{a_{i_0}}{ q_{i_{0}} p(0 \vert i_k)   }  \widehat{e}_{i_{k}} \in \mathcal{R}^n \nonumber
+\end{equation}
+\item The solution will be : $\overrightarrow{x}^0 = \dfrac{1}{N} \sum \overrightarrow{Y}(\gamma)$
+\end{itemize}
+
+
+\end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Neumann-Ulam dual method, $\rm \color{RubineRed}{Lecture3/Markov2}$}
+\begin{itemize}
+\item Let's try to solve the equation system:
+\end{itemize}
+\begin{equation}
+\overrightarrow{x} = 
+\left(\begin{array}{c}
+ 1.5  \\
+-1.0\\
+0.7  \end{array} \right) 
++
+\left(\begin{array}{ccc}
+0.2 &  0.3 & 0.1  \\
+0.4 &  0.3 & 0.2 \\
+0.1 &  0.1 & 0.1  \end{array} \right) \overrightarrow{x}
+ \nonumber
+\end{equation}
+\begin{itemize}
+\item The solution is: $\overrightarrow{x}_0 = (2.0, 0.0, 1.0)$.
+\item Let's put the initial probability as constant:
+\end{itemize}
+\begin{equation}
+q_1=q_2=q_3=\dfrac{1}{3}  \nonumber
+\end{equation}
+\begin{columns}
+
+\column{0.1in}
+
+\column{2.5in}
+\begin{itemize}
+\item The propability matrix $h_{ij}$ has the shape:
+\end{itemize}
+\begin{tabular}{|c|cccc|}
+\hline
+$i/j$ &  1 & 2 & 3 & 4  \\ \hline
+1 & 0.2 & 0.4 & 0.1 & 0.3 \\
+2 & 0.3 & 0.3 & 0.1 & 0.3 \\
+3 & 0.1 & 0.2 & 0.1 & 0.6 \\ \hline
+\end{tabular}
+
+\column{2.5in}
+\begin{itemize}
+\item An example solution:
+\end{itemize}
+\includegraphics[width=0.95\textwidth]{images/mark2.png}
+
+\end{columns}
+\end{frame}
+
+
+
+
+\begin{frame}\frametitle{Look elsewhere effect, $\rm \color{RubineRed}{Lecture3/LEE}$}
+\begin{itemize}
+\item Look elsewhere effect addresses the following problem:
+\begin{itemize}
+\item Imagine you observed a $3\sigma$ deviation in one of the observable that you measured. 
+\item Before you get excited one needs to understand if given the fact that you had so many measurements this might happen!
+\end{itemize}
+\item Example: Let's say we have measured 50 observables. What is the probability to observed 1 that is $3\sigma$ away from theory prediction?
+\item Let's simulate 50 Gaussian distribution centred at 0 and width of 1. We count how simulations where at least one of the 50 numbers have the absolute value $>3$.
+\item More complicated example: what if you observed 3 in a row $2\sigma$ fluctuations among 50 measurements?
+\item This kind of studies are the best solvable by MC simulations.
+\end{itemize}
+
+
+\end{frame}
+
+
+
+
+
+
+\begin{frame}\frametitle{Travelling Salesman Problem}
+\begin{itemize}
+\item Salesman starting from his base has to visit $n-1$ other locations and return to base headquarters. The problem is to find the shortest way.
+\item For large $n$ the problem can't be solver by brutal force as the complexity of the problem is $(n-1)!$
+\item There exist simplified numerical solutions assuming factorizations. Unfortunately even those require anonymous computing power.
+\item Can MC help? YES :)
+\item The minimum distance $l$ has to depend on 2 factors: $P$ the area of the city the Salesman is travelling and the density of places he wants to visit: $\dfrac{n}{P}$
+\item Form this we can assume:
+\end{itemize}
+\begin{equation}
+l \sim P^a (\dfrac{n}{P})^b=P^{a-b}n^b. \nonumber
+\end{equation}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+
+
+\begin{frame}\frametitle{Traveling Salesman Problem}
+\begin{itemize}
+\item From dimension analysis:
+\end{itemize}
+\begin{equation}
+a-b=\dfrac{1}{2}. \nonumber
+\end{equation}
+\begin{itemize}
+\item To get $l$ we need square root of area.
+\item From this it's obvious:
+\end{itemize}
+\begin{equation}
+l \sim P^a (\dfrac{n}{P})^b=P^{0.5}n^{a-0.5}. \nonumber
+\end{equation}
+\begin{itemize}
+\item Now we can multiply the area by alpha factor that keeps the density constant then:
+\end{itemize}
+\begin{equation}
+l \sim \alpha^0.5 \alpha6{a-0.5} = \alpha^a \nonumber
+\end{equation}
+\begin{itemize}
+\item In this case the distance between the clients will not change, but the number of clients will increase by $\alpha$ so:
+\end{itemize}
+\begin{equation}
+l \sim \alpha \nonumber
+\end{equation}
+\begin{itemize}
+\item In the end we get: $a=1$
+\end{itemize}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Traveling Salesman Problem}
+\begin{itemize}
+\item In total:
+\end{itemize}
+\begin{equation}
+l \sim k (nP)^{0.5}\nonumber
+\end{equation}
+\begin{itemize}
+\item Of course the k depends on the shape of the area and locations of client. However for large $n$ the k starts loosing the dependency. It's an asymptotically free estimator.
+\item To use the above formula we need to somehow calculate k.
+\item How to estimate this? Well make a TOY MC: take a square put uniformly $n$ points. Then we can calculate $l$. Then it's trivial:
+\end{itemize}
+\begin{equation}
+k= l(nP)^{-0.5} \nonumber
+\end{equation}
+\end{frame}
+
+
+\begin{frame}\frametitle{Traveling Salesman Problem}
+\begin{itemize}
+\item This kind of MC experiment might require large CPU power and time. The adventage is that once we solve the problem we can use the obtained k for other cases (it's universal constant!).
+\item It turns out that:
+\end{itemize}
+\begin{equation}
+k \sim \dfrac{3}{4} \nonumber
+\end{equation}
+\begin{itemize}
+\item Ok, but in this case we can calculate $l$ but not the actual shortest way! Why the hell we did this exercise?!
+\item Turns out that for most of the problems we are looking for the solution that is close to smallest $l$ not the exact minimum.
+\end{itemize}
+\end{frame}
+
+\begin{frame}\frametitle{War Games}
+
+\begin{itemize}
+\item S. Andersoon 1966 simulated for Swedish government how would a tank battle look like.
+\item Each of the sides has  15 tanks. that they allocate on the battle field. 
+\item The battle is done in time steps.
+\item Each tank has 5 states:
+\begin{itemize}
+\item OK
+\item Tank can only shoot
+\item Tank can only move
+\item Tank is destroyed
+\item Temporary states
+\end{itemize}
+\item This models made possible to check different fighting strategies.
+
+\end{itemize}
+\end{frame}
+
+
+
+\begin{frame}
+
+\begin{center}
+\begin{Huge}
+ Q \& A
+\end{Huge}
+\end{center}
+
+
+\end{frame}
+
+
+
+
+
+
+
+
+\backupbegin   
+
+\begin{frame}\frametitle{Backup}
+
+
+\end{frame}
+
+\backupend			
+
+\end{document}
diff --git a/Lectures_my/MC_2016/Lecture4/mchrzasz.toc b/Lectures_my/MC_2016/Lecture4/mchrzasz.toc
new file mode 100644
index 0000000..429917f
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/mchrzasz.toc
@@ -0,0 +1,5 @@
+\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax 
+\defcounter {refsection}{0}\relax 
+\beamer@endinputifotherversion {3.24pt}
+\defcounter {refsection}{0}\relax 
+\select@language {english}
diff --git a/Lectures_my/MC_2016/Lecture4/tikzfeynman.sty b/Lectures_my/MC_2016/Lecture4/tikzfeynman.sty
new file mode 100755
index 0000000..521fc65
--- /dev/null
+++ b/Lectures_my/MC_2016/Lecture4/tikzfeynman.sty
@@ -0,0 +1,43 @@
+% Tikz Feynman Diagrams
+% by Flip Tanedo
+% 4 January 2011, work in progress
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% TIKZ - for drawing Feynman diagrams %%%%
+%%% ... use with pdflatex   			%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\usepackage{tikz}
+\usetikzlibrary{arrows,shapes}
+\usetikzlibrary{trees}
+\usetikzlibrary{matrix,arrows} 				% For commutative diagram
+											% http://www.felixl.de/commu.pdf
+\usetikzlibrary{positioning}				% For "above of=" commands
+\usetikzlibrary{calc,through}				% For coordinates
+\usetikzlibrary{decorations.pathreplacing}  % For curly braces
+\usepackage[tikz]{bclogo} 					% For cute logo boxes
+\usepackage{pgffor}							% For repeating patterns
+
+\usetikzlibrary{decorations.pathmorphing}	% For Feynman Diagrams
+\usetikzlibrary{decorations.markings}
+\tikzset{
+	% >=stealth', %% Different kind of arrows
+    vector/.style={decorate, decoration={snake}, draw},
+    fermion/.style={postaction={decorate},
+        decoration={markings,mark=at position .55 with {\arrow{>}}}},
+    fermionbar/.style={draw, postaction={decorate},
+        decoration={markings,mark=at position .55 with {\arrow{<}}}},
+    fermionnoarrow/.style={},
+    gluon/.style={decorate,
+        decoration={coil,amplitude=4pt, segment length=5pt}},
+    scalar/.style={dashed, postaction={decorate},
+        decoration={markings,mark=at position .55 with {\arrow{>}}}},
+    scalarbar/.style={dashed, postaction={decorate},
+        decoration={markings,mark=at position .55 with {\arrow{<}}}},
+    scalarnoarrow/.style={dashed,draw},
+%
+%% 	Special vectors (when you need to fine-tune wiggles)
+	provector/.style={decorate, decoration={snake,amplitude=2.5pt}, draw},
+	antivector/.style={decorate, decoration={snake,amplitude=-2.5pt}, draw},
+}
+