diff --git a/Lectures_my/KT2_2017/Ex2/binary_half.png b/Lectures_my/KT2_2017/Ex2/binary_half.png new file mode 100644 index 0000000..05b667c --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/binary_half.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/header.tex b/Lectures_my/KT2_2017/Ex2/header.tex new file mode 100644 index 0000000..0df90f3 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/header.tex @@ -0,0 +1,200 @@ +% +% MMP Ex Sheet Template +% +% Rafael Kueng +% originally based on the template from the Jetzter group +% + +% +% USAGE: +% create a new tex file for each new exercise. use like this: +% +% \input{./_header} +% \input{./_shortcuts} +% +% \newcommand{\sheetnr}{1} +% \newcommand{\issued}{16.09.2014} +% \newcommand{\dueUni}{22.09.2014 9:00} +% +% \startsheet +% +% \exercise[Nr Pts]{Descripntion} +% \begin{subtasks} +% \task blablabla blasdf asdf dgsf +% \task sadfkj sdf asdf +% \end{subtasks} +% \turnpage + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +% +% Imports +% + +\documentclass[11pt,a4paper]{article} +% debug +%\documentclass[11pt,a4paper,draft]{article} +%\usepackage{showframe} + +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage[T1]{fontenc} +\usepackage{lmodern} + +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{mathtools} +\usepackage{siunitx} % help with si units, like: \si{\kilogram\metre\per\second} and \num{4.458757e-12} +\usepackage{upgreek} % upright greek letters, like: $\uppi$ +\usepackage{xparse} + +%\usepackage{psfrag} +%\usepackage{enumerate} +\usepackage{enumitem} +\usepackage{hyperref} +\usepackage{geometry} +\usepackage{xifthen} +\usepackage{verbatim} % used for the \comment used to hide solutions +\usepackage{xcolor} +\usepackage{framed} +\usepackage{cancel} +\usepackage{ulem} % for different underlines http://texdoc.net/texmf-dist/doc/generic/ulem/ulem.pdf +\usepackage{braket} +\usepackage{slashed} + +% +% Setup the layout +% + +\linespread{1.15} +\geometry{left=2cm,right=2cm,top=2cm,bottom=3cm} + +\setlength{\parindent}{0em} +\setlength{\parskip}{0.5em} + +\pagestyle{empty} + + +% +% Set the authors ect.. +% + +\newcommand{\thelecture}{Kern- und Teilchenphysik II} +\newcommand{\thelectureshort}{NM} +\newcommand{\thelectureURL}{\url{ http://www.physik.uzh.ch/de/lehre/PHY213/FS2017.html}} +\newcommand{\sheettype}{Exercise} +\newcommand{\thesemester}{HS 16} +\newcommand{\thelecturer}{Prof. Nicola Serra, \\Dr. Annapaola De Cosa} +\newcommand{\theassistants}{Dr. Marcin Chrzaszcz} + + + +% +% DEFINITIONS & NEW COMMANDS +% + +\newcounter{exercise} +\setcounter{exercise}{1} + +% define switches +\newboolean{turnpagesw} +\setboolean{turnpagesw}{true} + +\newboolean{showsol} +\setboolean{showsol}{false} + + + +% starts the doc and draws the header / banner +\newcommand{\startsheet}{ + \begin{document} + \begin{minipage}{0.25\textwidth}{ + \begin{flushleft}\includegraphics[width=2.5cm]{./unisiegel.pdf}\end{flushleft} + }\end{minipage} + \hfill + \begin{minipage}{0.40\textwidth}{ + \begin{center}\bf\LARGE% + \textrm{\thelecture}\\ + \LARGE\textrm{\sheettype \, Sheet \sheetnr}\end{center} + }\end{minipage} + \hfill + \begin{minipage}{0.25\textwidth}{ + \begin{flushright} + \textrm{\thesemester}\\ + \textrm{\thelecturer} \\ + \end{flushright} + }\end{minipage} + + \begin{minipage}{0.6\textwidth}{ + \theassistants\\ + \thelectureURL + }\end{minipage} + \hfill + \begin{minipage}{0.3\textwidth}{ + \begin{flushright} + Issued: \issued \ifthenelse{\isundefined{\version}}{}{ {\tiny(v\version)}}\\ + Due: \dueUni + \end{flushright} + }\end{minipage} + + \rule{\textwidth}{1pt} +} + +% use this to break a page +% creates a turn over note and creates the headline on the new page +\newcommand{\turnpage}{ + \ifthenelse{\boolean{turnpagesw}}{ + \vfill\hfill -- please turn over -- \newpage + {Exercises for \thelectureshort} \hfill Sheet \sheetnr\vspace{-0.2cm}\\ + \rule{\textwidth}{1pt} + }{}} + +% use this to start a new exercise +\newcommand{\exercise}[2][?]{ + \vspace{1.5em} + \textbf{Exercise \theexercise:} #2 (#1 Pts.) % + \stepcounter{exercise} +} + +% use this to create subtaks: +\newenvironment{subtasks}{ + \newcommand{\task}{\item } + \setlength{\itemsep}{1pt} + \setlength{\parskip}{0pt} + \setlength{\parsep}{0pt} + \begin{enumerate}[label=\alph*)] +} +{ + \end{enumerate} +} + + + +% set this "flag" at beginning if you want to print the solutions as well +\newcommand{\showsolutions}{ + \setboolean{showsol}{true} + \setboolean{turnpagesw}{false} % don't create pageturns if solutions are shown +} + + +% mark solutions with this envirionment +\newenvironment{solution}{ + \ifthenelse{\boolean{showsol}} + { + \begin{framed} + \textbf{SOLUTION:} + } + {\expandafter\comment} +} +{ + \ifthenelse{\boolean{showsol}} + { + \end{framed} + \newpage + } + {\expandafter\endcomment} +} + + diff --git a/Lectures_my/KT2_2017/Ex2/instal_1.png b/Lectures_my/KT2_2017/Ex2/instal_1.png new file mode 100644 index 0000000..528e605 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_1.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/instal_2.png b/Lectures_my/KT2_2017/Ex2/instal_2.png new file mode 100644 index 0000000..069774d --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_2.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/instal_3.png b/Lectures_my/KT2_2017/Ex2/instal_3.png new file mode 100644 index 0000000..34ca4b7 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_3.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/instal_4.png b/Lectures_my/KT2_2017/Ex2/instal_4.png new file mode 100644 index 0000000..ba7ee21 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_4.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/instal_5.png b/Lectures_my/KT2_2017/Ex2/instal_5.png new file mode 100644 index 0000000..85e89a4 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_5.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/instal_6.png b/Lectures_my/KT2_2017/Ex2/instal_6.png new file mode 100644 index 0000000..ae71911 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/instal_6.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/install_geant.txt b/Lectures_my/KT2_2017/Ex2/install_geant.txt deleted file mode 100644 index ba529ef..0000000 --- a/Lectures_my/KT2_2017/Ex2/install_geant.txt +++ /dev/null @@ -1,19 +0,0 @@ -# the path you need to adopt to your system. -cd /storage -mkdir Geant4 -cd Geant4 -wget http://cern.ch/geant4/support/source/geant4.10.03.p01.tar.gz -tar -xvf geant4.10.03.p01.tar.gz -mkdir geant4.10.3-build/ -cmake -DCMAKE_INSTALL_PREFIX=/storage/Geant4/geant4.10.3-install /storage/Geant4/geant4.10.03.p01 -DGEANT4_INSTALL_DATA=ON -make -jN -sudo make install - -# now here we finished instalation -# lets run example -source /storage/Geant4/geant4.10.3-install/share/Geant4-10.3.1/geant4make/geant4make.sh - -cd /storage/Geant4/geant4.10.3-install/share/Geant4-10.3.1/examples/basic/B2/B2a -make -exampleB2a run1.mac -exampleB2a run2.mac diff --git a/Lectures_my/KT2_2017/Ex2/sheet02.aux b/Lectures_my/KT2_2017/Ex2/sheet02.aux new file mode 100644 index 0000000..ee57387 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/sheet02.aux @@ -0,0 +1,22 @@ +\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\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\select@language{english} +\@writefile{toc}{\select@language{english}} +\@writefile{lof}{\select@language{english}} +\@writefile{lot}{\select@language{english}} diff --git a/Lectures_my/KT2_2017/Ex2/sheet02.log b/Lectures_my/KT2_2017/Ex2/sheet02.log new file mode 100644 index 0000000..191fad7 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/sheet02.log @@ -0,0 +1,1348 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.1.21) 25 MAR 2017 11:10 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**sheet02.tex +(./sheet02.tex +LaTeX2e <2016/02/01> +Babel <3.9q> and hyphenation patterns for 81 language(s) loaded. +(./header.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls +Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 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+\setcounter{exercise}{1} +\exercise[3.5]{Conservation laws} + + +Under which interactions are the following interactions possible (please not that some of the processes) are not possible at all): + +\begin{itemize} +\item $\pi^- p \to \pi^0 n$ +\item $\pi^0 \to \gamma \gamma \gamma$ +\item $\pi^0 \to \gamma \gamma$ +\item $\pi^+ \to \mu^+ \nu_{\mu}$ +\item $\pi^+ \to \mu^+ \bar{\nu}_{\mu}$ +\item $p \bar{p} \to \Lambda_0 \Lambda_0$ +\item $p \bar{p} \to \gamma$ +\end{itemize} + +\setcounter{exercise}{2} +\exercise[2.0]{Forbade decays} + + +Which conservation laws forbade the following decays: + +\begin{itemize} +\item $n \to p e^-$ +\item $n \to \pi^+ e^-$ +\item $n \to p \pi^-$ +\item $n \to p \gamma^-$ +\end{itemize} + + +\setcounter{exercise}{3} +\exercise[2.0]{Suppress decays} + + +Which conservation laws forbade or suppress the following processes: + +\begin{itemize} +\item $p n \to p \Lambda^0$ +\item $K^+ \to \pi^- \pi^+ \pi^- \pi^+ \pi^+ \pi^0 $ +\item $\Lambda_0 \to K^0 \pi^0$ +\item $K \to \pi \gamma$ +\item $K^- \to \pi^0 e^-$ +\item $K^+ \to \pi^+ \pi^- \pi^0$ +\end{itemize} + + + +\setcounter{exercise}{4} +\exercise[16]{Pion decay} + + +Please calculate the matrix element of the dacay $\pi^- \to \mu \nu_{\mu}$. The form factor for the pion has the form of $F^{\mu} = p_{\mu} f_{\pi}$, where $f_{\pi}$ is so called pion decay constant and is calculated on lattice to be $f_{\pi}=130~\rm MeV$. Using the matrix element calculate the $\Gamma$. Calculate also the $\Gamma$ for the $\pi^- \to e \nu_{e}$. Why is the electron mode different then the muon one? + +Please read about the ''Casimir trick'' in the D. Griffiths handbook (Sec. 7.7). + + +\setcounter{exercise}{5} +\exercise[16]{Muon decay} + +Calculate the matrix element for the dacay of the muon: $\mu^- \to e^- \nu_{\mu} \bar{\nu_e}$. Using ''Golden Rule'' calculate the calculate the $\Gamma$ and the lifetime of the muon. + +\setcounter{exercise}{6} +\exercise[12]{Muon decay simulation} + +Please simulate the muon decay from exercise 5 using ROOT. Please assume for the moment flat phase space (aka matrix element =1). The example can be found: + +\url{https://root.cern.ch/root/html/tutorials/physics/PhaseSpace.C.html} + +having simulate this decay please calculate the electron energy in the muon central of mass and draw it for your simulated events. Simulate at least 100.000 events. + + + +\clearpage +\iffalse +\begin{solution} +\begin{subtasks} + +\task The answers are: +\begin{center} +\begin{tabular}{| l | l | l |} +\hline +Decimal: & Hexadecimal: & Binary: \\ \hline +$12$ & C & 1100 \\ \hline +$53$ & 35 & 110101 \\ \hline +$123$ & 7B & 1111011 \\ \hline +$431$ & 1AF & 110101111 \\ \hline +\end{tabular} +\end{center} + +A simple algorithm to convert an integer $N$ from decimal to hexadecimal is the following: +\begin{enumerate} +\item[1.] set $i=1$. +\item[2.] Compute $h_i = N_{\Mod 16}$, i.e.~the remainder of the integer (Euclidean) division $N\div16$. +\item[3.] Use the hexadecimal “dictionary" to convert $h_i$: + +\begin{tabular}{| l | l | l | l | l | l | l | l | l | l | l | l | l | l | l | l | l |} +\hline +$h$: & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ +\hline +hex: & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & A & B & C & D & E & F \\ +\hline +\end{tabular} +\item[4.] Replace $N \rightarrow (N - h_i)/16$, set $i=i+1$ and go back to step 2 until $N=0$. +\item[5.] The number in hexadecimal representation is: $h_n h_{n-1} \cdots h_2 h_1$ +\end{enumerate} +This algorithm is actually valid for every base change. + +Let us apply the algorithm above. + +$N=12$: +\begin{align*} +& i=1; & \\ +& h_1 = 12_{\Mod 16} = 12 \rightarrow \boxed{C}; & \\ +& N \rightarrow (N - 12)/16 =0; \\ +&{\rm end:~} 12_{10} = \boxed{C_{16}} +\end{align*} +% +$N=53$: +\begin{align*} +& i=1 && i=2 \\ +& h_1 = 53_{\Mod 16} = 5 \rightarrow \boxed{5}; & & h_2 = 3_{\Mod 16} = 3 \rightarrow \boxed{3} \\ +& N \rightarrow (N - 5)/16 =3; && N \rightarrow (N - 3)/16 =0; \\ +&&&{\rm end:~} 53_{10} = \boxed{35_{16}} +\end{align*} +% +$N=123$: +\begin{align*} +& i=1 && i=2 \\ +& h_1 = 123_{\Mod 16} = 11 \rightarrow \boxed{B}; & & h_2 = 7_{\Mod 16} = 7 \rightarrow \boxed{7} \\ +& N \rightarrow (N - 11)/16 =7; && N \rightarrow (N - 7)/16 =0; \\ +&&&{\rm end:~} 123_{10} = \boxed{7B_{16}} +\end{align*} +% +$N=431$: +\begin{align*} +& i=1 && i=2 &&i=3\\ +& h_1 = 431_{\Mod 16} = 15 \rightarrow \boxed{F}; +&& h_2 = 26_{\Mod 16} = 10 \rightarrow \boxed{A} +&& h_3 = 1_{\Mod 16} = 1 \rightarrow \boxed{1} +\\ +& N \rightarrow (N - 11)/16 =26; && N \rightarrow (N - 10)/16 =1; && N \rightarrow (N - 1)/16 =0; \\ +&&&&&{\rm end:~} 431_{10} = \boxed{1AF_{16}} +\end{align*} +% + +The former algorithm can be applied also for the conversion in binary. One just need to change $\Mod 16 \rightarrow \Mod 2$ in step 2 and to divide by $2$ (not by $16$) in step 4. + +$N=12$: +\begin{align*} +& i=1 && i=2 && i=3 && i=4 \\ +& h_1 = 12_{\Mod 2} = \boxed{0} && h_2 = 6_{\Mod 2} = \boxed{0} +&& h_3 = 3_{\Mod 2} = \boxed{1} && h_4 = 1_{\Mod 2} = \boxed{1} \\ +& N \rightarrow (N - 0)/2 =6; && N \rightarrow (N - 0)/2 =3; +&& N \rightarrow (N - 1)/2 =1; && N \rightarrow (N - 1)/2 =0; \\ +&&&&&&&{\rm end:~} 12_{10} = \boxed{1100_{2}} +\end{align*} +% +$N=53$: +\begin{align*} +& i=1 && i=2 && i=3 && i=4 \\ +& h_1 = 53_{\Mod 2} = \boxed{1} && h_2 = 26_{\Mod 2} = \boxed{0} +&& h_3 = 13_{\Mod 2} = \boxed{1} && h_4 = 6_{\Mod 2} = \boxed{0} \\ +& N \rightarrow (N - 1)/2 =26; && N \rightarrow (N - 0)/2 =13; +&& N \rightarrow (N - 1)/2 =6; && N \rightarrow (N - 0)/2 =3; \\ +% +\\ +% +& i=5 && i=6 \\ +& h_5 = 3_{\Mod 2} = \boxed{1} && h_6 = 1_{\Mod 2} = \boxed{1} \\ +& N \rightarrow (N - 1)/2 =1; && N \rightarrow (N - 1)/2 =0; \\ +&&&{\rm end:~} 53_{10} = \boxed{110101_{2}} +\end{align*} +% +$N=123$: +\begin{align*} +& i=1; && i=2 && i=3 && i=4 \\ +& h_1 = 123_{\Mod 2} = \boxed{1} && h_2 = 61_{\Mod 2} = \boxed{1} +&& h_3 = 30_{\Mod 2} = \boxed{0} && h_4 = 15_{\Mod 2} = \boxed{1} \\ +& N \rightarrow (N - 1)/2 =61; && N \rightarrow (N - 1)/2 =30; +&& N \rightarrow (N)/2 =15; && N \rightarrow (N - 1)/2 =7; \\ +% +\\ +% +& i=5 && i=6 && i=7\\ +& h_5 = 7_{\Mod 2} = \boxed{1} && h_6 = 3_{\Mod 2} = \boxed{1} && h_7 = 1_{\Mod 2} = \boxed{1} \\ +& N \rightarrow (N - 1)/2 =3; && N \rightarrow (N-1)/2 =1; && N \rightarrow (N - 1)/2 =0; \\ +&&&{\rm end:~} 53_{10} = \boxed{1111011_{2}} +\end{align*} +% +$N=431$: +\begin{align*} +& i=1 && i=2 && i=3 && i=4,5,6,7,8,9 \\ +& h_1 = 431_{\Mod 2} = \boxed{1} && h_2 = 215_{\Mod 2} = \boxed{1} +&& h_3 = 107_{\Mod 2} = \boxed{1} && \dots ({\rm see~}N=53) \\ +& N \rightarrow (N - 1)/2 =215; && N \rightarrow (N - 1)/2 = 107; +&& N \rightarrow (N )/2 =53; && \dots \\ +\\ +& {\rm end:~} 431_{10} = \boxed{110101111_{2}} +\end{align*} +% + +\task One needs to understand that the decimal representation of an integer is just a short-cut notation. With $124_{10}$ one really means: +\begin{equation*} +124_{10} = 1 \cdot 10^2 + 2 \cdot 10^1 + 4 \cdot 10^0 ~. +\end{equation*} +% +This is valid in \emph{every} basis. Thus is easy to find: +% +\begin{align*} +10011_{2} &= (1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0)_{10} + = (16+2+1)_{10} = 19_{10} +\\ +1101_{2} &=(1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0)_{10} + = (8+4+1)_{10} = 13_{10} +\\ +A2_{16} &=(A \cdot 16^1 + 2 \cdot 16^0)_{10} + = (10 \cdot 16 + 2)_{10} = 162_{10} +\\ +1AD_{16} &=(1 \cdot 16^2 + A \cdot 16^1 + D \cdot 16^0)_{10} + = (1 \cdot 256 + 10 \cdot 16 + 13)_{10} = 429_{10} +\end{align*} +% + + +\task Before converting a value into a floating number, one should convert it into a binary expression. We already know how to convert $-431_{10}$. To convert a non-integer number $N$ into binary, one can use an extension of the algorithm used in exercise (1a). +\begin{enumerate} +\item[1.] i=0. +\item[2.] Take $\lfloor N \rfloor$ (i.e.~integer part of $N$) and convert it into a binary expression +(call it $h_i$). +\item[3.] Substitute $N \rightarrow 2(N - \lfloor N \rfloor)$. +\item[4.] Set $i=i+1$ and go back to step 2 until $N=0$ or you have reached the desired precision. +\item[5.] Your number is $h_0. h_1 h_2 \dots h_n$ +\end{enumerate} +% + +Let us convert $0.3125$ with the mentioned algorithm: +% +\begin{align*} +& i=0 && i=1 && i=2 \\ +& h_0 = \lfloor 0.3125 \rfloor = \boxed{0} && h_1 = \lfloor 0.625 \rfloor = \boxed{0} +&& h_2 = \lfloor 1.25 \rfloor = \boxed{1} \\ +& N \rightarrow 2(N - \lfloor N \rfloor) = 0.625; && N \rightarrow 2(N - \lfloor N) \rfloor) = 1.25; +&& N \rightarrow 2(N - \lfloor N) \rfloor) = 0.5; \\ +\\ +& i=3 && i=4 \\ +& h_3 = \lfloor 0.5 \rfloor = \boxed{0} && h_1 = \lfloor 1 \rfloor = \boxed{1}\\ +& N \rightarrow 2(N - \lfloor N \rfloor) = 1; && N \rightarrow 2(N - \lfloor N) \rfloor = 0; + \\ +&&& {\rm end:~} 0.3125_{10} = \boxed{0.0101_{2}} +\end{align*} +% +Thus $0.3125_{10} = 0.0101_{2} = (1.01 \cdot 2^{-2})_2$. Thus: +% +\begin{itemize} +\item[-] The sign is $+ \rightarrow 0$ +\item[-] The exponent is $-2$. Exponent+bias: $-2+15=13=01101_2$. +\item[-] The mantissa is $1.010$. The first digit ($1.$) is not stored, and we store $0100000000$. +\end{itemize} +% +Thus we have: +% +\begin{equation*} +0.3125_{10} \rightarrow \underbrace{0}_{\rm sign} + \underbrace{01101}_{\rm exp+bias} + \underbrace{0100000000}_{\rm mantissa} + \rightarrow \underbrace{00110101}_{\rm byte~1} + ~ \underbrace{00000000}_{\rm byte~2} + \rightarrow \underbrace{35~~00}_{\rm hexadecimal} +\end{equation*} +% + +For $-431_{10}$ the exercise is similar. We have already computed $431_{10}=110101111 = 1.10101111 \cdot 2^{8}$. Thus: +% +\begin{itemize} +\item[-] The sign is $- \rightarrow 1$ +\item[-] The exponent is $8$. Exponent+bias: $8+15=23=10111_2$. +\item[-] The mantissa is $1.10101111$. The first digit ($1.$) is not stored, and we store $1010111100$. +\end{itemize} +% +Thus we have: +% +\begin{equation*} +-431_{10} \rightarrow \underbrace{1}_{\rm sign} + \underbrace{10111}_{\rm exp+bias} + \underbrace{1010111100}_{\rm mantissa} + \rightarrow \underbrace{11011110}_{\rm byte~1} + ~ \underbrace{10111100}_{\rm byte~2} + \rightarrow \underbrace{\rm DE~~BC}_{\rm hexadecimal} +\end{equation*} +% + + +\task We have already found the hexadecimal bytes representation for our floating numbers: $0.3125_{10}=35~00$ and $-431_{10}={\rm DE~BC}$. +In big endian, these bytes would be sequentially stored in the memory “from left to right", while in little endian the sequential storage would go in the other direction. Explicitly: +% +\begin{align*} +0.3125_{10} \rightarrow ~& {\rm big~endian:~} + \underbrace{\boxed{35}}_{\rm slot~i}~\underbrace{\boxed{00}}_{\rm slot~i+1} +& +-431_{10} \rightarrow ~& {\rm big~endian:~} + \underbrace{\boxed{\rm DE}}_{\rm slot~i}~\underbrace{\boxed{\rm BC}}_{\rm slot~i+1} +\\ + \rightarrow ~& {\rm little~endian:~} + \underbrace{\boxed{00}}_{\rm slot~i}~\underbrace{\boxed{35}}_{\rm slot~i+1} +& + \rightarrow ~& {\rm little~endian:~} + \underbrace{\boxed{\rm BC}}_{\rm slot~i}~\underbrace{\boxed{\rm DE}}_{\rm slot~i+1} +\end{align*} +% + + +\task +The number $2050$ expressed in binary is $100000000010$. + +In {\it binary16} its representation is: +% +\begin{equation} + \underbrace{0}_{\rm sign} + \underbrace{11010}_{\rm exp+bias} + \underbrace{0000000001}_{\rm mantissa} + ~~\Rightarrow~~ + + \; + 2^{26 - 15} \times + 1.0000000001 + = + 1.0000000001 \times 2^{11} +\end{equation} +% +From this representation is already clear that the {\it binary16} precision doesn't allow to store numbers like $2049$ or $2051$ at all, since this would require a 12 digit precision. + +Thus, both in INPUT A and in INPUT B, the code is requiring the machine a precision it cannot reach. As it should be clear from how the number $2050$ is stored, the machine has reached it maximal precision and it cannot store the number $2051$ whatsoever. This means that at each step in which the number $2051$ is met, the machine truncate it consistently with its maximum working precision, returning $2050$. + +INPUT C is well written and works fine. Differently from INPUT B, this time the machine works \emph{at first} with numbers of order 1, which are manipulated without difficulties. For example, the {\it binary16} representation of $5$ would be: +% +\begin{equation} + \underbrace{0}_{\rm sign} + \underbrace{10001}_{\rm exp+bias} + \underbrace{0100000000}_{\rm mantissa -1} + ~~\Rightarrow~~ + + \; + 2^{17 - 15} \times + 1.0100000000 + = + 1.01 \times 2^2 = 101_2 +\end{equation} +% +where indeed $101_2=5_{10}$. After having dealt with all order-1 numbers, it finally sums them to 2050, getting 2060 which is a perfectly storable number. + +In conclusion, one should be aware of the bad consequences of requiring the machine to work with more significant digits than its maximum. + +\end{subtasks} +\end{solution} + + + +\fi + + +\end{document} diff --git a/Lectures_my/KT2_2017/Ex2/shortcuts.tex b/Lectures_my/KT2_2017/Ex2/shortcuts.tex new file mode 100644 index 0000000..2f62b70 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/shortcuts.tex @@ -0,0 +1,83 @@ +% +% Define here some handy shortcuts +% +% to be consistent and backwards compatible, try not to delete +% or replace existing ones +% + + +% \renewcommand{\d}{\mathrm{d}} +% \renewcommand{\labelenumi}{(\roman{enumi})} + +% \newcommand{\bm}{\boldmath} +% \newcommand{\dps}{\displaystyle} +% \newcommand{\e}{\mbox{e}} +% \newcommand{\del}{\partial} +% \newcommand{\arctanh}{\mathrm{arctanh}} +% \newcommand{\prim}[1]{{#1^{\prime}}} +% \newcommand{\tr}{\mathrm{Tr}} +% \newcommand{\naeher}{\!\!\!} +% \newcommand{\vect}[1]{\vec #1} +% \newcommand{\ddd}[1]{\d\vect{#1} } + +\newcommand{\IN}{\mathbb{N}} +\newcommand{\INO}{\mathbb{N}_0} +\newcommand{\IR}{\mathbb{R}} +\newcommand{\IC}{\mathbb{C}} + +\renewcommand{\Re}{\operatorname{Re}} +\renewcommand{\Im}{\operatorname{Im}} + +\DeclareMathOperator{\arsinh}{arsinh} +\DeclareMathOperator{\arcosh}{arcosh} +\DeclareMathOperator{\artanh}{artanh} +\DeclareMathOperator{\arcoth}{arcoth} + +% commonly misstyped... +\DeclareMathOperator{\arcsinh}{arsinh} +\DeclareMathOperator{\arccosh}{arcosh} +\DeclareMathOperator{\arctanh}{artanh} +\DeclareMathOperator{\arccoth}{arcoth} + +\DeclareMathOperator*{\Res}{Res} % residuum +\DeclareMathOperator{\Ang}{Ang} % Angle + +\newcommand{\PI}{\uppi} % consult http://tex.stackexchange.com/questions/54166/upright-lowercase-pi +\newcommand{\I}{\textrm{i}} %\upiota} % http://tex.stackexchange.com/questions/223759/how-to-add-a-dot-above-imath +\newcommand{\E}{\mathrm{e}} + +\DeclarePairedDelimiter\abs{\lvert}{\rvert} + +\newcommand{\Order}[1]{\mathcal{O}\!\left(#1\right)} +\newcommand{\D}{\mathrm{d}} +\newcommand{\pderr}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\integ}[2]{\int \D #2 \; \left[#1\right]} + +% sofisticated integral.. +% \Int[lower][upper]{int_variable}{term} +% \Int*[lower][upper]{int_variable}{term} --> without brackets around term +\NewDocumentCommand{\Int}{ s O{} O{} m m }{ + \IfBooleanTF {#1} + {\int_{#2}^{#3} \D #4 \; #5} + {\int_{#2}^{#3} \D #4 \; \left[#5\right]} +} + +% handy derrivative +% \DD[power]{variable} +\NewDocumentCommand{\DD}{ O{x} O{} }{ + \frac{\D^{#2}}{\D#1^{#2}} +} +\NewDocumentCommand{\DDx}{ O{} }{\DD[x][#1]} +\NewDocumentCommand{\DDy}{ O{} }{\DD[x][#1]} +\NewDocumentCommand{\DDt}{ O{} }{\DD[x][#1]} + + +\newcommand{\half}{\frac{1}{2}} +\newcommand{\oneover}[1]{\frac{1}{#1}} + +\newcommand{\textoversymb}[2]{\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily #2}}}{#1}}} + + +%\DeclarePairedDelimiter\parents{\left(}{\right)} +%\DeclarePairedDelimiter\brackets{\left\[}{\right\]} +%\DeclarePairedDelimiter\braces{\left\{}{\right\}} \ No newline at end of file diff --git a/Lectures_my/KT2_2017/Ex2/unisiegel.pdf b/Lectures_my/KT2_2017/Ex2/unisiegel.pdf new file mode 100644 index 0000000..24e9582 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/unisiegel.pdf Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/virtual_box_1.png b/Lectures_my/KT2_2017/Ex2/virtual_box_1.png new file mode 100644 index 0000000..eb68197 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/virtual_box_1.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/virtual_box_2.png b/Lectures_my/KT2_2017/Ex2/virtual_box_2.png new file mode 100644 index 0000000..0f99655 --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/virtual_box_2.png Binary files differ diff --git a/Lectures_my/KT2_2017/Ex2/virtual_box_3.png b/Lectures_my/KT2_2017/Ex2/virtual_box_3.png new file mode 100644 index 0000000..89a11ea --- /dev/null +++ b/Lectures_my/KT2_2017/Ex2/virtual_box_3.png Binary files differ