diff --git a/Lectures_my/NumMet/Lecture2/Makefile b/Lectures_my/NumMet/Lecture2/Makefile
new file mode 100644
index 0000000..e4cf8e8
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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/NumMet/Lecture2/beamercolorthemeFlip.sty b/Lectures_my/NumMet/Lecture2/beamercolorthemeFlip.sty
new file mode 100755
index 0000000..d6b9c50
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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/NumMet/Lecture2/beamerinnerthemeFlip.sty b/Lectures_my/NumMet/Lecture2/beamerinnerthemeFlip.sty
new file mode 100755
index 0000000..8dc12a0
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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}%
+    %\vskip0.5em\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/NumMet/Lecture2/beamerouterthemeFlip.sty b/Lectures_my/NumMet/Lecture2/beamerouterthemeFlip.sty
new file mode 100755
index 0000000..d12f6d6
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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/NumMet/Lecture2/beamerouterthemedecolines.sty b/Lectures_my/NumMet/Lecture2/beamerouterthemedecolines.sty
new file mode 100755
index 0000000..669e493
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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}
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+
+% % Footer.
+% \defbeamertemplate*{footline}{decolines theme}
+% {
+%   \leavevmode%
+%   % Page number.
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+%   \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]{}%
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+%   } %
+%   % First line.
+%   \hbox{%
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+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{lineup}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Second line.
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+%   \begin{beamercolorbox}[wd=\paperwidth,ht=\beamer@decolines@linemid,dp=0pt]{linemid}%
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+%   \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%
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+%       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}%
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+  } %
+  \vskip0pt%
+}
+
+\defbeamertemplate*{frametitle}{decolines theme}[1][left]
+{
+  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
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+  \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/NumMet/Lecture2/beamerthemeFlip.sty b/Lectures_my/NumMet/Lecture2/beamerthemeFlip.sty
new file mode 100755
index 0000000..a615ff5
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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/NumMet/Lecture2/mchrzasz-blx.bib b/Lectures_my/NumMet/Lecture2/mchrzasz-blx.bib
new file mode 100644
index 0000000..9e405b3
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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/NumMet/Lecture2/mchrzasz.tex b/Lectures_my/NumMet/Lecture2/mchrzasz.tex
new file mode 100644
index 0000000..8aa4012
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/mchrzasz.tex
@@ -0,0 +1,527 @@
+\documentclass[11 pt,xcolor={dvipsnames,svgnames,x11names,table}]{beamer}
+
+\usepackage[english]{babel} 
+\usepackage{polski}         
+
+
+\usetheme[
+	bullet=circle,		% Other option: square
+	bigpagenumber,		% circled page number on lower right
+	topline=true,			% colored bar at the top of the frame 
+	shadow=false,			% Shading for beamer blocks
+	watermark=BG_lower,	% png file for the watermark
+	]{Flip}
+
+%\logo{\kern+1.em\includegraphics[height=1cm]{SHiP-3_LightCharcoal}}
+                            
+
+\usepackage[lf]{berenis}
+\usepackage[LY1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{cases}
+\usepackage{mathtools}
+\usepackage{emerald}
+\usefonttheme{professionalfonts}
+\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
+\setsansfont{Gillius ADF}			% This is the font that beamer will use by default
+% \setmainfont{Gill Sans Light}		% Prettier, but harder to read
+
+\setbeamerfont{title}{family=\fontspec{Gillius ADF}}
+
+\input t1augie.fd
+
+%\newcommand{\handwriting}{\fontspec{augie}} % From Emerald City, free font
+%\newcommand{\handwriting}{\usefont{T1}{fau}{m}{n}} % From Emerald City, free font
+% \newcommand{\handwriting}{}	% If you prefer no special handwriting font or don't have augie
+
+%% Gill Sans doesn't look very nice when boldfaced
+%% This is a hack to use Helvetica instead
+%% Usage: \textbf{\forbold some stuff}
+%\newcommand{\forbold}{\fontspec{Arial}}
+
+\usepackage{graphicx}
+\usepackage[export]{adjustbox}
+
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{bm}
+\usepackage{colortbl}
+\usepackage{mathrsfs} 			% For Weinberg-esque letters
+\usepackage{cancel}				% For "SUSY-breaking" symbol
+\usepackage{slashed}            % for slashed characters in math mode
+\usepackage{bbm}                % for \mathbbm{1} (unit matrix)
+\usepackage{amsthm}				% For theorem environment
+\usepackage{multirow}			% For multi row cells in table
+\usepackage{arydshln} 			% For dashed lines in arrays and tables
+\usepackage{siunitx}
+\usepackage{xhfill}
+\usepackage{grffile}
+\usepackage{textpos}
+\usepackage{subfigure}
+\usepackage{tikz}
+\usepackage{hyperref}
+%\usepackage{hepparticles}    
+\usepackage[italic]{hepparticles}     
+
+\usepackage{hepnicenames} 
+
+% Drawing a line
+\tikzstyle{lw} = [line width=20pt]
+\newcommand{\topline}{%
+  \tikz[remember picture,overlay] {%
+    \draw[crimsonred] ([yshift=-23.5pt]current page.north west)
+             -- ([yshift=-23.5pt,xshift=\paperwidth]current page.north west);}}
+
+
+
+% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
+\usepackage{tikzfeynman}		% For Feynman diagrams
+\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
+% http://www.math.ucla.edu/~getreuer/tikz.html
+\usepackage{pgffor}							% For repeating patterns
+
+\usetikzlibrary{decorations.pathmorphing}	% For Feynman Diagrams
+\usetikzlibrary{decorations.markings}
+\tikzset{
+	% >=stealth', %%  Uncomment for more conventional arrows
+	vector/.style={decorate, decoration={snake}, draw},
+	provector/.style={decorate, decoration={snake,amplitude=2.5pt}, draw},
+	antivector/.style={decorate, decoration={snake,amplitude=-2.5pt}, draw},
+	fermion/.style={draw=gray, postaction={decorate},
+		decoration={markings,mark=at position .55 with {\arrow[draw=gray]{>}}}},
+	fermionbar/.style={draw=gray, postaction={decorate},
+		decoration={markings,mark=at position .55 with {\arrow[draw=gray]{<}}}},
+	fermionnoarrow/.style={draw=gray},
+	gluon/.style={decorate, draw=black,
+		decoration={coil,amplitude=4pt, segment length=5pt}},
+	scalar/.style={dashed,draw=black, postaction={decorate},
+		decoration={markings,mark=at position .55 with {\arrow[draw=black]{>}}}},
+	scalarbar/.style={dashed,draw=black, postaction={decorate},
+		decoration={markings,mark=at position .55 with {\arrow[draw=black]{<}}}},
+	scalarnoarrow/.style={dashed,draw=black},
+	electron/.style={draw=black, postaction={decorate},
+		decoration={markings,mark=at position .55 with {\arrow[draw=black]{>}}}},
+	bigvector/.style={decorate, decoration={snake,amplitude=4pt}, draw},
+}
+
+% TIKZ - for block diagrams, 
+% from http://www.texample.net/tikz/examples/control-system-principles/
+% \usetikzlibrary{shapes,arrows}
+\tikzstyle{block} = [draw, rectangle, 
+minimum height=3em, minimum width=6em]
+
+
+
+
+\usetikzlibrary{backgrounds}
+\usetikzlibrary{mindmap,trees}	% For mind map
+\newcommand{\degree}{\ensuremath{^\circ}}
+\newcommand{\E}{\mathrm{E}}
+\newcommand{\Var}{\mathrm{Var}}
+\newcommand{\Cov}{\mathrm{Cov}}
+\newcommand\Ts{\rule{0pt}{2.6ex}}       % Top strut
+\newcommand\Bs{\rule[-1.2ex]{0pt}{0pt}} % Bottom strut
+
+\graphicspath{{images/}}	% Put all images in this directory. Avoids clutter.
+
+% 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{\Alert}[1]{\textcolor{Alert}{#1}} % louder alert
+\newcommand{\ALERT}[1]{\textcolor{ALERT}{#1}} % loudest alert
+%% "\alert" is already a beamer pre-defined
+\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
+
+\def\Put(#1,#2)#3{\leavevmode\makebox(0,0){\put(#1,#2){#3}}}
+
+\usepackage{gmp}
+\usepackage[final]{feynmp-auto}
+
+\usepackage[backend=bibtex,style=numeric-comp,firstinits=true]{biblatex}
+\bibliography{bib}
+\setbeamertemplate{bibliography item}[text]
+
+\makeatletter\let\frametextheight\beamer@frametextheight\makeatother
+
+% suppress frame numbering for backup slides
+% you always need the appendix for this!
+\newcommand{\backupbegin}{
+	\newcounter{framenumberappendix}
+	\setcounter{framenumberappendix}{\value{framenumber}}
+}
+\newcommand{\backupend}{
+	\addtocounter{framenumberappendix}{-\value{framenumber}}
+	\addtocounter{framenumber}{\value{framenumberappendix}} 
+}
+
+
+\definecolor{links}{HTML}{2A1B81}
+%\hypersetup{colorlinks,linkcolor=,urlcolor=links}
+
+% For shapo's formulas:
+\def\lsi{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
+\def\gsi{\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
+\newcommand{\lsim}{\mathop{\lsi}}
+\newcommand{\gsim}{\mathop{\gsi}}
+\newcommand{\wt}{\widetilde}
+%\newcommand{\ol}{\overline}
+\newcommand{\Tr}{\rm{Tr}}
+\newcommand{\tr}{\rm{tr}}
+\newcommand{\eqn}[1]{&\hspace{-0.7em}#1\hspace{-0.7em}&}
+\newcommand{\vev}[1]{\rm{$\langle #1 \rangle$}}
+\newcommand{\abs}[1]{\rm{$\left| #1 \right|$}}
+\newcommand{\eV}{\rm{eV}}
+\newcommand{\keV}{\rm{keV}}
+\newcommand{\GeV}{\rm{GeV}}
+\newcommand{\im}{\rm{Im}}
+\newcommand{\disp}{\displaystyle}
+\def\be{\begin{equation}}
+\def\ee{\end{equation}}
+\def\ba{\begin{eqnarray}}
+\def\ea{\end{eqnarray}}
+\def\d{\partial}
+\def\l{\left(}
+\def\r{\right)}
+\def\la{\langle}
+\def\ra{\rangle}
+\def\e{{\rm e}}
+\def\Br{{\rm Br}}
+\def\fixme{{\color{red} FIXME!}}
+\def\mc{{\color{Magenta}{MC}}\xspace}
+\def\pdf{{\rm p.d.f.}}
+\def\ARROW{{\color{JungleGreen}{$\Rrightarrow$}}\xspace}
+\def\ARROWR{{\color{WildStrawberry}{$\Rrightarrow$}}\xspace} 
+
+\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz, Danny van Dyk} (UZH)}
+\institute{UZH}
+\title[Function interpolation]{Function Interpolation}
+\date{\fixme}
+
+
+\begin{document}
+\tikzstyle{every picture}+=[remember picture]
+
+{
+\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
+\begin{frame}[c]%{\phantom{title page}} 
+\begin{center}
+\begin{center}
+	\begin{columns}
+		\begin{column}{0.9\textwidth}
+			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Function Interpolation}
+		\end{column}
+		\begin{column}{0.2\textwidth}
+		  %\includegraphics[width=\textwidth]{SHiP-2}
+		\end{column}
+	\end{columns}
+\end{center}
+	\quad
+	\vspace{3em}
+\begin{columns}
+\begin{column}{0.6\textwidth}
+\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \large Marcin Chrząszcz, Dany van Dyk\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}, \href{mailto:dany.van.dyk@gmail.com}{danny.van.dyk@gmail.com}}
+
+\end{column}
+\begin{column}{0.4\textwidth}
+\includegraphics[height=1.3cm]{uzh-transp}
+\end{column}
+\end{columns}
+
+\vspace{1em}
+%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
+\vspace{0.5em}
+	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ \fixme September, 2016}
+\end{center}
+\end{frame}
+}
+
+
+
+\begin{frame}\frametitle{Interpolation - graphical interpretation}
+\begin{small}
+
+
+\ARROW Let's assume we have a given function in a tabular form:
+\begin{tabular}{c||c|c|c|c|c}
+$x_i$ & $x_1$ & $x_2$ & $x_3$ & ... & $x_n$ \\ \hline
+$f_i=f(x_i)$ & $f_1$ & $f_2$ & $f_3$ & ... & $f_n$ \\
+\end{tabular}\\
+\ARROWR The $x_i$ are data points.\\
+\ARROW The problem is how to calculate the function values between the points ({\it intepolation}) or outside ({\it extrapolation}).
+\end{small}
+\begin{center}
+\includegraphics[angle=-90,width=0.5\textwidth]{images/mass_resolution_Bconstr_afterSel_errBand_scaled.pdf}
+\end{center}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Interpolation - formal formulation}
+\begin{small}
+
+\ARROW On the interval: $\left[ a, b \right] \subset \mathbb{R}$ and set of points: $D=(x_i, y_i)$, $x_i \neq x_j~ \Leftrightarrow~ i \neq j$.\\
+\ARROW The interpolation is to find a function $F:\left[ a, b \right] \rightarrow  \mathbb{R}$ that satisfies:
+\begin{align*}
+\forall (x_i,y_i) \in D:~~~ F(x_i)=y_i
+\end{align*}
+\ARROW The above equation is called the interpolation equation.\\
+\ARROW The interpolation error:
+\begin{align*}
+\epsilon(x)=f(x)-F(x),
+\end{align*}
+where $f(x)$ is the function that is being interpolated.\\
+\ARROW Each interpolation method has different errors, each of it can be estimated.
+\end{small}
+
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Linear Interpolation}
+\begin{small}
+
+\ARROW Let's say we have $n+1$ interpolation points: $(y_i, x_i)$.\\
+\ARROW The linear interpolation is given with with the equation:
+\begin{align*}
+F(x)=\sum_{j=0}^n a_j \psi_j(x)=a_0 \psi_0(x) + a_1 \psi_1(x) + a_2 \psi_2(x)+ ... + a_n \psi_n(x),
+\end{align*}
+where:
+\begin{itemize}
+\item $a_j$ - parameter,
+\item $\psi_j(x)$ - base function.
+\end{itemize}
+\ARROW Now the only thing one needs to do is to find the $a_i$ coefficients. Now there are infinite number such solutions so we need to assume that the number interpolation points is equal to number of base functions. 
+
+
+\end{small}
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Linear Interpolation}
+\begin{small}
+
+\ARROW Now to find all the $a_i$ one just needs to solve the linear equation system:
+\begin{align*}
+\begin{pmatrix}
+\psi_0(x_0) & \psi_1(x_0) & ... & \psi_n(x_0) \\
+\psi_0(x_1) & \psi_1(x_1) & ... & \psi_n(x_1) \\
+& ... \\
+\psi_0(x_0) & \psi_1(x_0) & ... & \psi_n(x_2) 
+\end{pmatrix}
+\begin{pmatrix}
+a_0\\
+a_1\\
+...\\
+a_n
+\end{pmatrix}=
+\begin{pmatrix}
+y_0\\
+y_1\\
+...\\
+y_n
+\end{pmatrix}
+\end{align*}
+\ARROW The only necessary condition to solve the equation is that the:
+\begin{align*}
+\det \left[ \psi_i(x_j) \right] \neq 0
+\end{align*}
+\ARROW The polynomial of the order $n$ has $n+1$ base functions, so system has only one solution.\\
+\ARROW We can choose the base functions on many different ways. This depends on the input points and our own judgement. Often use are trigonometric functions and polynomials.\\
+\ARROW The proper selection of the base function can hugely simplify the problem by making the matrix triangular or even diagonal.
+
+\end{small}
+
+\end{frame}
+
+\begin{frame}\frametitle{Linear Interpolation, simplest base}
+\begin{small}
+\begin{exampleblock}{Theorem:}
+If all of the interpolation points $x_0,x_1,...,x_n$ are pair different there exists one and only one solution of the interpolation function of the order max. $n$. 
+\end{exampleblock}
+
+The simplest base functions one can imagine to use is the natural base:
+\begin{align*}
+\psi_0=1,~~~\psi_1=x,~~~\psi_2=x^2,~~~...~~~, \psi_n=x^n
+\end{align*}
+\begin{alertblock}{Attention/Ahtung:}
+This kind of interpolation is wrongly conditioned: small changed of input parameters will cause large output differences.
+\end{alertblock}
+
+\ARROW Because of the above it is used almost not used in practice. \\
+\ARROW For education purposes we will analyse it. 
+\end{small}
+
+\end{frame}
+
+\begin{frame}\frametitle{Linear Interpolation, simplest base}
+\begin{small}
+\ARROW When using natural base the interpolation polynomial will have the structure:
+\begin{align*}
+F(x)=a_0+a_1 x + a_2 x^2 + a_3 x^3 ... a_n x^n
+\end{align*}
+\ARROW Now base matrix (so-called Vandermonde matrix) has the form:
+\begin{align*}
+V=\begin{pmatrix}
+1 & x_0 & x_0^2 & ... & x_0^n  \\
+1 & x_1 & x_1^2 & ... & x_1^n  \\
+& ... \\
+1 & x_n & x_n^2 & ... & x_n^n  \\
+\end{pmatrix}
+\end{align*}
+\ARROW Now the only thing one needs to do is to solve:
+\begin{align*}
+\begin{pmatrix}
+1 & x_0 & x_0^2 & ... & x_0^n  \\
+1 & x_1 & x_1^2 & ... & x_1^n  \\
+& ... \\
+1 & x_n & x_n^2 & ... & x_n^n  \\
+\end{pmatrix}
+\begin{pmatrix}
+a_0\\
+a_1\\
+...\\
+a_n
+\end{pmatrix}=
+\begin{pmatrix}
+y_0\\
+y_1\\
+...\\
+y_n
+\end{pmatrix}
+\end{align*}
+\ARROW Because the natural base is wrongly conditioned the increase of the interpolation points significantly increases the condition parameter.
+
+\end{small}
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Lagrange interpolation}
+\begin{small}
+\ARROW Now let's choose a better base function that are given by:
+\begin{align}
+\label{eq:langrange}
+l_i(x)= \prod_{j=0,~j \neq i}^n \frac{x-x_j}{x_i-x_j} = \frac{(x-x_0)(x-x_1)...(x-x_n)}{(x_i-x_0)(x_i-x_1)...(x_i-x_n)}
+\end{align}
+\ARROW It is easy to notice:
+\begin{align*}
+l_i(x_j)=\begin{cases}
+0~~~i \neq j \\
+1~~~i=j
+\end{cases}
+\end{align*}
+\ARROW Now putting this into base matrix we get:
+\begin{align*}
+\begin{pmatrix}
+1 & 0 &  0 & ... & 0  \\
+0 & 1 & 0 & ... & 0  \\
+& ... \\
+0 & 0 & 0 & ... & 1  \\
+\end{pmatrix}
+\begin{pmatrix}
+a_0\\
+a_1\\
+...\\
+a_n
+\end{pmatrix}=
+\begin{pmatrix}
+y_0\\
+y_1\\
+...\\
+y_n
+\end{pmatrix}
+\end{align*}
+\ARROW Now this is something we like :) The interpolating function then takes the form:
+\begin{align*}
+F(x)=L_n(x)=\sum_{i=0}^n y_i l_i(x)
+\end{align*}
+
+\end{small}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Lagrange interpolation - the algorithm}
+\begin{small}
+\begin{enumerate}
+\item Get the input data $(x_i,y_i),~i=0,...,n$. 
+\item The coefficients of the polynomial are $y_i$. 
+\item Calculate the base functions from Eq.~\ref{eq:langrange}.
+\item Put everything together to get the $L_n$ Lagrange polynomial.
+\end{enumerate}
+\begin{alertblock}{Disadvantage:}
+The problem with this method is the fact that if you add another interpolation point you need to start over and recalculate everything.
+\end{alertblock}
+
+\begin{exampleblock}{Advantege}
+\only<1>{
+It is very popular and simple method. It is used in many different places: differential equations, numerical integrations, etc.  Because of this the precision of the method depends on the precision of the interpolation is self:
+\begin{align*}
+\epsilon(x) =  f(x) - F(x) = \frac{f^{(n+1}(\xi) }{(n+1)!} \omega_n(x)
+\end{align*}
+}
+\only<2>{
+where:\\
+$\omega_n(x)=(x-x_0)(x-x_1)...(x-x_n)$\\
+$f$ function of class $C^{n+1}$ on $\left [a,b\right]$\\
+$\xi$ rzedna wartosci srednie.
+}
+\end{exampleblock}
+
+
+\end{small}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Newton interpolation}
+\begin{small}
+\ARROW Sir Isaac Newton proposed a different base for the problem:
+\begin{align*}
+p_0(x) & =1\\
+p_1(x) & =(x-x_0)\\
+p_2(x) & =(x-x_0)(x-x_1)\\
+...\\
+p_n(x) & =(x-x_0)(x-x_1)...(x-x_n)
+\end{align*}
+\ARROW Using the above functions the basis matrix takes a nice form:
+\begin{align*}
+V=\begin{pmatrix}
+1 & 0 & 0 & ... & 0  \\
+1 & p_1(x_1) &  0 & ... & 0  \\
+& ... \\
+1 & p_1(x_n) & p_2(x_n) & ... & p_n(x_n)  \\
+\end{pmatrix}
+\end{align*}
+
+
+
+\end{small}
+\end{frame}
+
+
+\backupbegin   
+
+\begin{frame}\frametitle{Backup}
+
+
+\end{frame}
+
+
+
+
+\backupend			
+\end{document}
diff --git a/Lectures_my/NumMet/Lecture2/tikzfeynman.sty b/Lectures_my/NumMet/Lecture2/tikzfeynman.sty
new file mode 100755
index 0000000..521fc65
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture2/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},
+}
+