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LaTeX Info: Redefining \addtolength on input line 77. -\calc@Ccount=\count313 +\calc@Ccount=\count321 \calc@Cskip=\skip73 ) (/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty @@ -1951,27 +1975,27 @@ Package: textpos 2014/01/03 v1.7j Package: textpos 2014/01/03 1.7j, absolute positioning of text on the page -\TP@textbox=\box91 -\TPHorizModule=\dimen320 -\TPVertModule=\dimen321 -\TP@margin=\dimen322 -\TP@absmargin=\dimen323 +\TP@textbox=\box93 +\TPHorizModule=\dimen324 +\TPVertModule=\dimen325 +\TP@margin=\dimen326 +\TP@absmargin=\dimen327 Grid set 16 x 16 = 22.7622pt x 17.07166pt -\TPboxrulesize=\dimen324 -\TP@tbargs=\toks53 -\TP@prevdepth=\dimen325 +\TPboxrulesize=\dimen328 +\TP@tbargs=\toks54 +\TP@prevdepth=\dimen329 ) (/usr/share/texlive/texmf-dist/tex/latex/subfigure/subfigure.sty Package: subfigure 2002/03/15 v2.1.5 subfigure package \subfigtopskip=\skip74 \subfigcapskip=\skip75 -\subfigcaptopadj=\dimen326 +\subfigcaptopadj=\dimen330 \subfigbottomskip=\skip76 -\subfigcapmargin=\dimen327 +\subfigcapmargin=\dimen331 \subfiglabelskip=\skip77 -\c@subfigure=\count314 -\c@lofdepth=\count315 -\c@subtable=\count316 -\c@lotdepth=\count317 +\c@subfigure=\count322 +\c@lofdepth=\count323 +\c@subtable=\count324 +\c@lotdepth=\count325 **************************************** * Local config file subfigure.cfg used * @@ -2002,7 +2026,7 @@ (/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryarrows.code.tex File: pgflibraryarrows.code.tex 2008/10/27 v2.10 (rcs-revision 1.9) -\arrowsize=\dimen328 +\arrowsize=\dimen332 )) (/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshape s.code.tex @@ -2051,26 +2075,26 @@ (/usr/share/texmf/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.c ode.tex File: pgflibraryshapes.multipart.code.tex 2010/01/07 v2.10 (rcs-revision 1.2) -\pgfnodepartlowerbox=\box92 -\pgfnodeparttwobox=\box93 -\pgfnodepartthreebox=\box94 -\pgfnodepartfourbox=\box95 -\pgfnodeparttwentybox=\box96 -\pgfnodepartnineteenbox=\box97 -\pgfnodeparteighteenbox=\box98 -\pgfnodepartseventeenbox=\box99 -\pgfnodepartsixteenbox=\box100 -\pgfnodepartfifteenbox=\box101 -\pgfnodepartfourteenbox=\box102 -\pgfnodepartthirteenbox=\box103 -\pgfnodeparttwelvebox=\box104 -\pgfnodepartelevenbox=\box105 -\pgfnodeparttenbox=\box106 -\pgfnodepartninebox=\box107 -\pgfnodeparteightbox=\box108 -\pgfnodepartsevenbox=\box109 -\pgfnodepartsixbox=\box110 -\pgfnodepartfivebox=\box111 +\pgfnodepartlowerbox=\box94 +\pgfnodeparttwobox=\box95 +\pgfnodepartthreebox=\box96 +\pgfnodepartfourbox=\box97 +\pgfnodeparttwentybox=\box98 +\pgfnodepartnineteenbox=\box99 +\pgfnodeparteighteenbox=\box100 +\pgfnodepartseventeenbox=\box101 +\pgfnodepartsixteenbox=\box102 +\pgfnodepartfifteenbox=\box103 +\pgfnodepartfourteenbox=\box104 +\pgfnodepartthirteenbox=\box105 +\pgfnodeparttwelvebox=\box106 +\pgfnodepartelevenbox=\box107 +\pgfnodeparttenbox=\box108 +\pgfnodepartninebox=\box109 +\pgfnodeparteightbox=\box110 +\pgfnodepartsevenbox=\box111 +\pgfnodepartsixbox=\box112 +\pgfnodepartfivebox=\box113 ))) (/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytrees .code.tex @@ -2097,16 +2121,16 @@ (/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor ations.code.tex (/usr/share/texmf/tex/generic/pgf/modules/pgfmoduledecorations.code.tex -\pgfdecoratedcompleteddistance=\dimen329 -\pgfdecoratedremainingdistance=\dimen330 -\pgfdecoratedinputsegmentcompleteddistance=\dimen331 -\pgfdecoratedinputsegmentremainingdistance=\dimen332 -\pgf@decorate@distancetomove=\dimen333 -\pgf@decorate@repeatstate=\count318 -\pgfdecorationsegmentamplitude=\dimen334 -\pgfdecorationsegmentlength=\dimen335 +\pgfdecoratedcompleteddistance=\dimen333 +\pgfdecoratedremainingdistance=\dimen334 +\pgfdecoratedinputsegmentcompleteddistance=\dimen335 +\pgfdecoratedinputsegmentremainingdistance=\dimen336 +\pgf@decorate@distancetomove=\dimen337 +\pgf@decorate@repeatstate=\count326 +\pgfdecorationsegmentamplitude=\dimen338 +\pgfdecorationsegmentlength=\dimen339 ) -\tikz@lib@dec@box=\box112 +\tikz@lib@dec@box=\box114 ) (/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.p athreplacing.code.tex)) @@ -2137,13 +2161,13 @@ Invalid UTF-8 byte or sequence at line 108 replaced by U+FFFD. Invalid UTF-8 byte or sequence at line 108 replaced by U+FFFD. Invalid UTF-8 byte or sequence at line 127 replaced by U+FFFD. -\c@bclogocompteur=\count319 +\c@bclogocompteur=\count327 \bcval@epBarre=\skip81 \bcval@framesep=\skip82 \bcval@tailleOndu=\skip83 \bcval@borderWidth=\skip84 -\bc@envbox=\box113 -\bc@invbox=\box114 +\bc@envbox=\box115 +\bc@invbox=\box116 \bc@invboxh=\skip85 \bc@invboxd=\skip86 ) @@ -2154,8 +2178,8 @@ (/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackg rounds.code.tex File: tikzlibrarybackgrounds.code.tex 2010/02/08 v2.10 (rcs-revision 1.2) -\pgf@layerbox@background=\box115 -\pgf@layerboxsaved@background=\box116 +\pgf@layerbox@background=\box117 +\pgf@layerboxsaved@background=\box118 ) (/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymindm ap.code.tex @@ -2165,10 +2189,10 @@ \gmp@shellout=\write6 \openout6 = `mchrzasz+mp.sh'. -\gmp@packages=\toks54 -\gmp@commands=\toks55 +\gmp@packages=\toks55 +\gmp@commands=\toks56 \gmp@out=\write7 -\c@gmp@count=\count320 +\c@gmp@count=\count328 ) (/usr/share/texlive/texmf-dist/tex/latex/feynmp-auto/feynmp-auto.sty Package: feynmp-auto 2013/05/03 v1.1 Automatic processing of feynmp graphics @@ -2176,8 +2200,8 @@ (/usr/share/texmf/tex/latex/feynmf/feynmp.sty Package: feynmp 1996/12/02 v1.08 LaTeX/MetaPost Feynman Diagram Package (ohl) \@outfmf=\write8 -\fmfbuf@=\toks56 -\c@fmfgraph=\count321 +\fmfbuf@=\toks57 +\c@fmfgraph=\count329 )) (/usr/share/texlive/texmf-dist/tex/latex/biblatex/biblatex.sty Package: biblatex 2013/11/25 v2.8a programmable bibliographies (PK/JW/AB) @@ -2188,46 +2212,46 @@ (/usr/share/texlive/texmf-dist/tex/latex/logreq/logreq.sty Package: logreq 2010/08/04 v1.0 xml request logger -\lrq@indent=\count322 +\lrq@indent=\count330 (/usr/share/texlive/texmf-dist/tex/latex/logreq/logreq.def File: logreq.def 2010/08/04 v1.0 logreq spec v1.0 )) -\c@tabx@nest=\count323 -\c@listtotal=\count324 -\c@listcount=\count325 -\c@liststart=\count326 -\c@liststop=\count327 -\c@citecount=\count328 -\c@citetotal=\count329 -\c@multicitecount=\count330 -\c@multicitetotal=\count331 -\c@instcount=\count332 -\c@maxnames=\count333 -\c@minnames=\count334 -\c@maxitems=\count335 -\c@minitems=\count336 -\c@citecounter=\count337 -\c@savedcitecounter=\count338 -\c@uniquelist=\count339 -\c@uniquename=\count340 -\c@refsection=\count341 -\c@refsegment=\count342 -\c@maxextrayear=\count343 -\c@maxextraalpha=\count344 -\c@abbrvpenalty=\count345 -\c@highnamepenalty=\count346 -\c@lownamepenalty=\count347 -\c@maxparens=\count348 -\c@parenlevel=\count349 -\blx@tempcnta=\count350 -\blx@tempcntb=\count351 -\blx@tempcntc=\count352 -\blx@maxsection=\count353 -\blx@maxsegment=\count354 -\blx@notetype=\count355 -\blx@parenlevel@text=\count356 -\blx@parenlevel@foot=\count357 +\c@tabx@nest=\count331 +\c@listtotal=\count332 +\c@listcount=\count333 +\c@liststart=\count334 +\c@liststop=\count335 +\c@citecount=\count336 +\c@citetotal=\count337 +\c@multicitecount=\count338 +\c@multicitetotal=\count339 +\c@instcount=\count340 +\c@maxnames=\count341 +\c@minnames=\count342 +\c@maxitems=\count343 +\c@minitems=\count344 +\c@citecounter=\count345 +\c@savedcitecounter=\count346 +\c@uniquelist=\count347 +\c@uniquename=\count348 +\c@refsection=\count349 +\c@refsegment=\count350 +\c@maxextrayear=\count351 +\c@maxextraalpha=\count352 +\c@abbrvpenalty=\count353 +\c@highnamepenalty=\count354 +\c@lownamepenalty=\count355 +\c@maxparens=\count356 +\c@parenlevel=\count357 +\blx@tempcnta=\count358 +\blx@tempcntb=\count359 +\blx@tempcntc=\count360 +\blx@maxsection=\count361 +\blx@maxsegment=\count362 +\blx@notetype=\count363 +\blx@parenlevel@text=\count364 +\blx@parenlevel@foot=\count365 \labelnumberwidth=\skip87 \labelalphawidth=\skip88 \shorthandwidth=\skip89 @@ -2239,75 +2263,75 @@ \bibhang=\skip95 \blx@auxin=\read4 \blx@auxout=\write9 -\c@mincomprange=\count358 -\c@maxcomprange=\count359 -\c@mincompwidth=\count360 -\c@labelname=\count361 -\c@savedlabelname=\count362 -\c@author=\count363 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+\c@origpublisher=\count417 +\c@savedorigpublisher=\count418 +\c@pageref=\count419 +\c@savedpageref=\count420 +\c@publisher=\count421 +\c@savedpublisher=\count422 +\c@lista=\count423 +\c@savedlista=\count424 +\c@listb=\count425 +\c@savedlistb=\count426 +\c@listc=\count427 +\c@savedlistc=\count428 +\c@listd=\count429 +\c@savedlistd=\count430 +\c@liste=\count431 +\c@savedliste=\count432 +\c@listf=\count433 +\c@savedlistf=\count434 Package biblatex Info: Trying to load compatibility code... Package biblatex Info: ... file 'blx-compat.def' found. @@ -2319,13 +2343,13 @@ (/usr/share/texlive/texmf-dist/tex/latex/biblatex/biblatex.def File: biblatex.def -\c@textcitecount=\count427 -\c@textcitetotal=\count428 -\c@textcitemaxnames=\count429 -\c@biburlnumpenalty=\count430 -\c@biburlucpenalty=\count431 -\c@biburllcpenalty=\count432 -\c@smartand=\count433 +\c@textcitecount=\count435 +\c@textcitetotal=\count436 +\c@textcitemaxnames=\count437 +\c@biburlnumpenalty=\count438 +\c@biburlucpenalty=\count439 +\c@biburllcpenalty=\count440 +\c@smartand=\count441 ) Package biblatex Info: Trying to load bibliography style 'numeric-comp'... Package biblatex Info: ... file 'numeric-comp.bbx' found. @@ -2342,16 +2366,16 @@ (/usr/share/texlive/texmf-dist/tex/latex/biblatex/bbx/standard.bbx File: standard.bbx 2013/11/25 v2.8a biblatex bibliography style (PK/JW/AB) -\c@bbx:relatedcount=\count434 -\c@bbx:relatedtotal=\count435 +\c@bbx:relatedcount=\count442 +\c@bbx:relatedtotal=\count443 ))) Package biblatex Info: Trying to load citation style 'numeric-comp'... Package biblatex Info: ... file 'numeric-comp.cbx' found. (/usr/share/texlive/texmf-dist/tex/latex/biblatex/cbx/numeric-comp.cbx File: numeric-comp.cbx 2013/11/25 v2.8a biblatex citation style (PK/JW/AB) -\c@cbx@tempcnta=\count436 -\c@cbx@tempcntb=\count437 +\c@cbx@tempcnta=\count444 +\c@cbx@tempcntb=\count445 Package biblatex Info: Redefining '\cite'. Package biblatex Info: Redefining '\parencite'. Package biblatex Info: Redefining '\footcite'. @@ -2372,7 +2396,7 @@ ))) Package hyperref Warning: Token not allowed in a PDF string (Unicode): -(hyperref) removing `\fontspec' on input line 206. +(hyperref) removing `\fontspec' on input line 207. Package biblatex Info: Trying to load language 'english'... Package biblatex Info: ... file 'english.lbx' found. @@ -2387,45 +2411,45 @@ Package biblatex Warning: 'babel/polyglossia' detected but 'csquotes' missing. 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(/usr/share/texmf/tex/latex/tipa/t3cmr.fd File: t3cmr.fd 2001/12/31 TIPA font definitions ) -LaTeX Font Info: ... okay on input line 212. +LaTeX Font Info: ... okay on input line 213. *geometry* driver: auto-detecting *geometry* detected driver: xetex @@ -2462,8 +2486,8 @@ * (1in=72.27pt=25.4mm, 1cm=28.453pt) ABD: EveryShipout initializing macros -\AtBeginShipoutBox=\box117 -Package hyperref Info: Link coloring OFF on input line 212. +\AtBeginShipoutBox=\box119 +Package hyperref Info: Link coloring OFF on input line 213. 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(./mchrzasz.out) (./mchrzasz.out) \@outlinefile=\write10 @@ -2506,11 +2530,12 @@ tionary/translator-theorem-dictionary-English.dict Dictionary: translator-theorem-dictionary, Language: English ) -LaTeX Info: Redefining \includegraphics on input line 212. +LaTeX Info: Redefining \includegraphics on input line 213. *** you should *not* be loading the inputenc package *** XeTeX expects the source to be in UTF8 encoding *** some features of other encodings may conflict, resulting in poor output. +\c@lstlisting=\count449 Now handling font encoding TS1 ... ... processing UTF-8 mapping file for font encoding TS1 (/usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.dfu @@ -2602,15 +2627,15 @@ Package biblatex Info: Trying to load bibliographic data... Package biblatex Info: ... file 'mchrzasz.bbl' not found. 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+\includegraphics[width=0.8\textwidth]{images/shit.png} + +\end{frame} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{Linear generators} @@ -372,42 +381,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Linear generators; examples} - -$\color{PineGreen}\Rrightarrow$ 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: @@ -469,7 +442,23 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} @@ -477,31 +466,286 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} -\end{small} -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Non linear generators} -\begin{small} - - +$\color{PineGreen}\Rrightarrow$ 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_{nāˆ’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} + + + + + + + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%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}