From c2029dadc68a3916a1a2ad03e01f0464655b06f8 Mon Sep 17 00:00:00 2001 From: Bazsalanszky Date: Wed, 6 May 2020 09:16:07 +0200 Subject: [PATCH] Init --- .gitignore | 282 ++++++++++++++++++++++++++++++++ main.tex | 466 +++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 748 insertions(+) create mode 100644 .gitignore create mode 100644 main.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..3ba16fe --- /dev/null +++ b/.gitignore @@ -0,0 +1,282 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf +*.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..756b0dd --- /dev/null +++ b/main.tex @@ -0,0 +1,466 @@ +\documentclass[12pt,a4paper,twoside]{report} +%\usepackage{geometry} + + +%\headheight 0mm +%\headsep 10mm + +\linespread{1.1} + +%nyelvtani specialitasok +\usepackage{t1enc} +\usepackage[utf8]{inputenc} +\usepackage[magyar,english]{babel} +\usepackage{times} +\usepackage{amsfonts} + +%matematikai csomagok +\usepackage{amsmath} +\usepackage{amssymb} + + +%abrak, grafika +\usepackage{graphicx} +%\usepackage[draft]{graphicx} +\usepackage{caption} +\usepackage{epstopdf} +\usepackage[dvipsnames]{xcolor} +\usepackage{appendix} +\usepackage{titlesec} +\usepackage{float} +\usepackage{hyperref} +\usepackage{fancyvrb} +\usepackage{color} +\usepackage{setspace} +\usepackage{verbatim} +\usepackage{tikz} +\usepackage{array} +\usepackage{fixltx2e} +% Starred variant +\titleformat{name=\section,numberless} +{\normalfont\Large\bfseries} +{} +{0pt} +{} + +\newcommand{\bibentry}[7]{ + {\textsc{#1}}%%author + {#2}:~%%year + {\textit{#3}}%%title + {#4}%%others + {#5}%%others + {#6}%%others + {#7}.%%others +} + + +\makeatletter +\if0\magyar@opt@@figurecaptions\@@magyar@skiplong\fi +\if1\magyar@opt@@figurecaptions +\def\@@magyar@fnum@figure{\textit{\thesection-\thefigure.~\figurename}}% +\else \def\@@magyar@fnum@figure{\figurename\nobreakspace\thefigure}\fi +\expandafter\addto\csname extras\CurrentOption\endcsname{% + \babel@save\fnum@figure\let\fnum@figure\@@magyar@fnum@figure} +\@gobble +{^} +\makeatother + +\frenchspacing +\sloppy + +\renewcommand{\thesection}{\arabic{section}} + +\setcounter{tocdepth}{3} +\setcounter{secnumdepth}{5} + + + + +%\usepackage{titlesec} + +%\titlespacing*{\section} %\titlespacing*{}{}{}{} +%{0pt}{18pt plus 1ex minus 1ex}{6pt plus 1ex minus 1ex} +%\titlespacing*{\subsection} +%{0pt}{18pt plus 1ex minus 1ex}{6pt plus 1ex minus 1ex} +%\titlespacing*{\subsubsection} +%{0pt}{12pt plus 1ex minus 1ex}{6pt plus 1ex minus 1ex} +%\titlespacing*{\paragraph} +%{0pt}{6pt plus 1ex minus 1ex}{3pt plus 1ex minus 1ex} + +\usepackage[nottoc,numbib]{tocbibind} +\usepackage[sectionbib]{chapterbib} + + +\def\re#1{(\ref{#1})} %% Note: AMSTeX's \eqref also does (\ref{#1}) +\def\are#1{\az+\re{#1}} \def\Are#1{\Az+\re{#1}} %% these three lines +\def\tre#1#2{\told\re{#1}+#2{}} %% for Hungarian texts +\def\atre#1#2{\atold\re{#1}+#2{}} \def\Atre#1#2{\Atold\re{#1}+#2{}} + + + +\newcommand\nc{\newcommand*} \nc\longnc{\newcommand} + +%% Shorthands, to save space, typing and mistyping: +\let\X\hskip \let\Y\vskip \let\Z\kern %% positioning +\def\x#1{\X#1em} \def\y#1{\Y#1ex} +\def\XX#1{\HB to#1{\ }} \def\YY#1{\setbox1\HB to0em{\ }\RB{#1}} +\def\xx#1{\XX{#1em}} \def\yy#1{\YY{#1ex}} +\let\xph\hphantom \let\yph\vphantom \let\ph\phantom +\let\HB\hbox \def\SB{\setbox1\HB} \def\CB{\copy1} %% for temporary +\def\SC{\setbox2\HB} \def\CC{\copy2} %% boxes +\def\RB#1{\raise#1\CB} \def\XB{\wd1} \def\YB{\ht1} \def\ZB{\dp1} +\def\RC#1{\raise#1\CC} \def\XC{\wd2} \def\YC{\ht2} \def\ZC{\dp2} +\def\UB{\Z-\XB} \def\VB{\CB\UB} \def\WB#1{\RB{#1}\UB} %% puts out and +\def\UC{\Z-\XC} \def\VC{\CC\UC} \def\WC#1{\RC{#1}\UC} %% steps back +\newcount\n \newdimen\w \newdimen\h %% for numbers, widths, heights +\def\mathsizes#1#2#3{\mathchoice{#1}{#1}{#2}{#3}} %% the 3 math sizes + +%% Sizes: +\nc\tS[1]{\ifcase#1\tiny\or %% type sizes: \tS0 = \tiny, ... + \scriptsize\or\footnotesize\or\small\or\normalsize\or\large\or + \Large\or\LARGE\or\huge\or\Huge\else\ifnum#1<0\tiny\else\Huge\fi\fi} +\makeatletter %% ideas credited to relsize.sty +\nc\cS{\ifx\@currsize\normalsize %% current type size, as 0 .. 9 + 4\else\ifx\@currsize\small 3\else\ifx\@currsize\footnotesize + 2\else\ifx\@currsize\large 5\else\ifx\@currsize\Large + 6\else\ifx\@currsize\LARGE 7\else\ifx\@currsize\scriptsize + 1\else\ifx\@currsize\tiny 0\else\ifx\@currsize\huge + 8\else\ifx\@currsize\Huge 9\else 4\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi} +\DeclareRobustCommand\rS[1]{\ifmmode\@nomath\rS\else %% increases type + \@tempcnta\cS\advance\@tempcnta#1\relax\tS\@tempcnta} %% size by #1 +\makeatother +\def\mS#1{\ifcase#1\displaystyle\or %% \ms0 = \displaystyle, ... + \textstyle\or\scriptstyle\or\scriptscriptstyle\else\textstyle\fi} +\nc\dS[1]{\csname\ifcase#1relax\or %% delimiter sizes, \big if 2, ... + relax\or big\or Big\or bigg\or Bigg\fi\endcsname} + +%% Fonts: +\nc\textinmath[1]{{\mathsizes %% for texts and text + {\HB{#1}}{\HB{\tS1#1}}{\HB{\tS0#1}}}} %% fonts within formulas +\nc\txt[3]{\mskip#1mu %% #1, #2: space before and + \textinmath{#3}\mskip#2mu\relax } %% after, #3: text to put out +\nc\mathcl\mathcal %% standard LaTeX version +\def\mathBf#1{{\mathsizes{\HB{\boldmath %% bf nonletters, bf it letters + {$#1$}}}{\HB{\boldmath{$\mS2#1$}}}{\HB{\boldmath{$\mS3#1$}}}}} +\nc\mathBF[1]{{\h=.03ex{\mathsizes + {\w=.020em\SB{$ #1$}\WB\h\Z\w\VB\WB{2\h}\Z2\w\VB\WB{2\h}\Z\w\RB\h} + {\w=.018em\SB{$\mS2#1$}\WB\h\Z\w\VB\WB{2\h}\Z2\w\VB\WB{2\h}\Z\w\RB\h} + {\w=.016em\SB{$\mS3#1$}\WB\h\Z\w\VB\WB{2\h}\Z2\w\VB\WB{2\h}\Z\w\RB\h}}}} + +%% Text writing: +%\nc\re[1]{(\ref{#1})} %% shorter to type than the amsmath \eqref +\nc\rp\pageref \nc\rb\cite +\nc\chap[2]{\chapter{#2}\label{#1}} +\nc\chaP[3]{\chapter[#3]{#2}\label{#1}} +\nc\sect[2]{{\section{#2}\label{#1}}} +\nc\secT[3]{\section[#3]{#2}\label{#1}} +\nc\ssect[2]{\subsection{#2}\label{#1}} +\nc\ssecT[3]{\subsection[#3]{#2}\label{#1}} +\nc\sssect[2]{\subsubsection{#2}\label{#1}} +\nc\sssecT[3]{\subsubsection[#3]{#2}\label{#1}} +\longnc\quot[1]{`#1'} \longnc\quott[1]{``#1''} %% English quotes +\longnc\idezz[1]{,,#1''} %% Hungarian quote +\longnc\idezzz[1]{\raisebox{.22ex} %% quote as >>something<< + {$\mS3\gg$}#1\raisebox{.22ex}{$\mS3\ll$}} +\nc\emp\textit %% emphasizing +\nc\lat\textit %% latin: i.e. e.g. in situ +\nc\ie{\lat{i.e.,\ }} \nc\etal{\lat{et al.\ }} \nc\etc{\lat{etc.\ }} +\nc\eg{\lat{e.g.,\ }} \nc\insitu{\lat{in situ}} \nc\QED{\lat{Q.E.D.}} +\nc\cf{cf.\ } \nc\wrt{w.r.t.\ } +%\nc\lhs{l.h.s.\ } \nc\rhs{r.h.s.\ } +\nc\lhs{lhs} \nc\rhs{rhs} +\nc\st[1]{\overset{\bitt\raisebox{-.2ex}[0ex][0ex]{$\mS2*$}}{#1}} + +%% Abrak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\unitlength=.5pt %% pici: egesz tobbszoroseivel lehessen finomhangolni +%% Aritmetika, szamlalok helyett parancs valtozokkal: +%% pl. \cca , \ccb , ... valtozoneveket hasznalni; +\def\set#1#2{\xdef#1{#2}} %\newcount\n +\def\add#1#2{\n=#1\advance \n by #2\xdef#1{\the\n}} +\def\sub#1#2{\n=#1\advance \n by-#2\xdef#1{\the\n}} +\def\mul#1#2{\n=#1\multiply\n by #2\xdef#1{\the\n}} +\def\div#1#2{\n=#1\divide \n by #2\xdef#1{\the\n}} +\def\setadd#1#2#3{\set{#1}{#2}\add{#1}{#3}} +\def\setsub#1#2#3{\set{#1}{#2}\sub{#1}{#3}} +\def\setmul#1#2#3{\set{#1}{#2}\mul{#1}{#3}} +\def\setdiv#1#2#3{\set{#1}{#2}\div{#1}{#3}} +\def\addadd#1#2#3{\add{#1}{#2}\add{#1}{#3}} +\def\addsub#1#2#3{\add{#1}{#2}\sub{#1}{#3}} +\def\subsub#1#2#3{\sub{#1}{#2}\sub{#1}{#3}} +\def\muladd#1#2#3{\mul{#1}{#2}\add{#1}{#3}} +\def\mulsub#1#2#3{\mul{#1}{#2}\sub{#1}{#3}} +\def\muldiv#1#2#3{\mul{#1}{#2}\div{#1}{#3}} +\def\divadd#1#2#3{\div{#1}{#2}\add{#1}{#3}} +\def\divsub#1#2#3{\div{#1}{#2}\sub{#1}{#3}} +\def\setaddadd#1#2#3#4{\setadd{#1}{#2}{#3}\add{#1}{#4}} +\def\setaddsub#1#2#3#4{\setadd{#1}{#2}{#3}\sub{#1}{#4}} +\def\setsubsub#1#2#3#4{\setsub{#1}{#2}{#3}\sub{#1}{#4}} +\def\setmuladd#1#2#3#4{\setmul{#1}{#2}{#3}\add{#1}{#4}} +\def\setmulsub#1#2#3#4{\setmul{#1}{#2}{#3}\sub{#1}{#4}} +\def\setmuldiv#1#2#3#4{\setmul{#1}{#2}{#3}\div{#1}{#4}} +\def\setdivadd#1#2#3#4{\setdiv{#1}{#2}{#3}\add{#1}{#4}} +\def\setdivsub#1#2#3#4{\setdiv{#1}{#2}{#3}\sub{#1}{#4}} +\let\bez\qbezier %% Now comes an alternative convention to \qbezier : +%% instead of the second point, the tangent vectors at the first +%% and third points are to be given (with integer coordinates): +%% Usage: \beztan{x1}{y1}{ux}{uy}{x3}{y3}{vx}{vy} +\def\beztan#1#2#3#4#5#6#7#8{\setsub{\nP}{#5}{#1}\setsub{\nQ}{#6}{#2}% + \setmul{\nR}{\nP}{#8}\setmul{\nS}{\nQ}{#7}\setsub{\nP}{\nR}{\nS}\setmul + {\nR}{#3}{#8}\setmul{\nS}{#4}{#7}\setsub{\nQ}{\nR}{\nS}\setmul{\nR}{#3} + {\nP}\div{\nR}{\nQ}\setmul{\nS}{#4}{\nP}\div{\nS}{\nQ}\add{\nR}{#1}\add + {\nS}{#2}\bez(#1,#2)(\nR,\nS)(#5,#6)} +%% The same, adding \qbezier 's optional number of points: +%% Usage: \beztann{numpoints}{x1}{y1}{ux}{uy}{x3}{y3}{vx}{vy} +\def\beztann#1#2#3#4#5#6#7#8#9{\setsub{\nP}{#6}{#2}\setsub{\nQ}{#7}{#3}% + \setmul{\nR}{\nP}{#9}\setmul{\nS}{\nQ}{#8}\setsub{\nP}{\nR}{\nS}\setmul + {\nR}{#4}{#9}\setmul{\nS}{#5}{#8}\setsub{\nQ}{\nR}{\nS}\setmul{\nR}{#4} + {\nP}\div{\nR}{\nQ}\setmul{\nS}{#5}{\nP}\div{\nS}{\nQ}\add{\nR}{#2}\add + {\nS}{#3}\bez[#1](#2,#3)(\nR,\nS)(#6,#7)} +\def\mut{\multiput} +\def\nb{\makebox(0,0)} %% \put needs a box: \put(9,7){\nb[t]{$x$}}} + +%% Formula handling: +\nc\m[1]{\scase=0$ #1 $} %% space around an in-text formula +\nc\mm[1]{\m{ \, #1 \, }} %% Tip: always use \m{...} +\nc\mmm[1]{\m{ \,\, #1 \,\, }} %% instead of $...$, thus later +\nc\mmmm[1]{\m{ \,\,\, #1 \,\,\, }} %% you can add space easily. +\nc\M[3]{\scase=0$ %% finer and unequal spaces + \mskip#1mu#3\mskip#2mu$} %% around an in-text formula + + +%% Some journals don't accept amsmath, some others recommend it... +%% Making the switch between amsmath and non-amsmath easier: +\makeatletter\@ifpackageloaded{amsmath}{ + %% Equations: amsmath definitions: + \def\eq#1#2{ \scase=1 \begin{alignn} \elabel{#1} #2 \end{alignn}} + %% eq/array, #1: label, #2: formula. + \def\eqa{\eq} %% same, only for compatibility with non-amsmath + \def\eqn#1#2{ \scase=1 \begin{alignn} \elabel{#1} \non #2 \end{alignn}} + %% unnumbered equation (still let's give it a label!) + \def\eqan{\eqn} %% same, only for compatibility with non-amsmath + \def\lel#1{ \\ \elabel{#1} } %% line break within equation + \def\leln#1{\\ \elabel{#1} \non} %% same, non-numbered line (still label it!) + \def\tagg{\tag*{}} %% auxiliary, see above + %% Shorthands for some other amsmath macros: + \def\mat#1{\begin{matrix} #1 \end{matrix}} + \def\smat#1{\begin{smallmatrix} #1 \end{smallmatrix}} +}{ + %% Equations: non-amsmath definitions: + \def\eq#1#2{ \scase=0\begin{equation} \elabel{#1} #2 \end{equation}} + \def\eqa#1#2{ \scase=2\begin{eqnarray} \elabel{#1} #2 \end{eqnarray}} + \def\eqn#1#2{ \scase=0\begin{displaymath} \elabel{#1} #2 \end{displaymath}} + \def\eqan#1#2{\scase=2\begin{eqnarray} \elabel{#1} \non #2 \end{eqnarray}} + \def\lel#1{\ifnum\scase=2\else\erroreqaneeded\fi \\ \elabel{#1}} + \def\leln#1{\ifnum\scase=2\else\erroreqaneeded\fi \\ \elabel{#1} \non} + \def\tagg{\nonumber} %% auxiliary, see above. + %% Other macros: non-amsmath version: + \def\lvert{|} \def\rvert{|} \def\lVert{\|} \def\rVert{\|} + \def\mat#1{{\def\\{\cr}\matrix{#1}}} %% not elegant, just struggling + \def\smat#1{\hbox{\scriptsize{$\mat{#1}$}}} %% also a minimal solution +}\makeatother %% whether amsmath was loaded. + +%% Write \s= instead of = in equations. Its meaning will be: +%% = in \m, \mm, \mmm, \mmmm and \eq as non-amsmath equation, +%% &= in \eq and \eqa as amsmath alignn +%% &=& in \eqa as non-amsmath eqnarray +%% (Naturally, it works for < > etc. as well.) +\newcount\scase \def\7{&} \def\s#1{\ifcase\scase#1\or\7#1\or\7#1\7\fi} + +\def\smatup{\yy{1.9 }} \def\smatdn{\yy{-.8 }} %% in \smat, these add +\def\smatupdn{\smatup\smatdn} %% distance between lines + +%% Brackets: size and shape +%% \0 = \left(...\right) but with no extra spaces around +%% \1 = (...) +%% \2 = \big(...\big) \3 = \Big(...\Big) +%% \4 = \bigg(...\bigg) \5 = \Bigg(...\Bigg) +%% \9 = \left(...\right) (ordinary) +%% #1 = shape: 0 no bracket 2 [ ] 4 < > 6 | | +%% 1 ( ) 3 \{ \} 5 \langle \rangle 7 \| \| +\nc\0[2]{\ifcase#1{#2}\or\lt(#2\rt)\or\lt[{#2}\rt]\or\lt\{{#2}\rt\}\or + \mathord<{#2}\mathord>\or\lt\langle{#2}\rt\rangle\or\lt\lvert{#2}\rt + \rvert\or\lt\lVert{#2}\rt\rVert\fi} +\nc\1[2]{\ifcase#1{#2}\or(#2)\or[#2]\or\{#2\}\or\mathord<{#2}\mathord + >\or\langle{#2}\rangle\or\lvert{#2}\rvert\or\lVert{#2}\rVert\fi} +\nc\2[2]{\ifcase#1{#2}\or\big(#2\big)\or\big[#2\big]\or\big + \{#2\big\}\or\big<#2\big>\or\big\langle#2\big\rangle\or\big + \lvert#2\big\rvert\or\big\lVert#2\big\rVert\fi} +\nc\3[2]{\ifcase#1{#2}\or\Big(#2\Big)\or\Big[#2\Big]\or\Big\{#2\Big + \}\or\Big<#2\Big>\or\Big\langle#2\Big\rangle\or\Big\lvert#2\Big + \rvert\or\Big\lVert#2\Big\rVert\fi} +\nc\4[2]{\ifcase#1{#2}\or\bigg(#2\bigg)\or\bigg[#2\bigg]\or\bigg + \{#2\bigg\}\or\bigg<#2\bigg>\or\bigg\langle#2\bigg\rangle\or\bigg + \lvert#2\bigg\rvert\or\bigg\lVert#2\bigg\rVert\fi} +\nc\5[2]{\ifcase#1{#2}\or\Bigg(#2\Bigg)\or\Bigg[#2\Bigg]\or\Bigg + \{#2\Bigg\}\or\Bigg<#2\Bigg>\or\Bigg\langle#2\Bigg\rangle\or\Bigg + \lvert#2\Bigg\rvert\or\Bigg\lVert#2\Bigg\rVert\fi} +\nc\9[2]{\ifcase#1{#2}\or\left(#2\right)\or\left[#2\right]\or\left + \{#2\right\}\or\left\langle{#2}\right\rangle\or\left\langle{#2}\right + \rangle\or\left\lvert{#2}\right\rvert\or\left\lVert{#2}\right\rVert\fi} +\nc\lt{\mathopen{}\mathclose\bgroup\left} \nc\rt{\aftergroup\egroup\right} + +\nc\bi\relax %% spacing finer than \! \, \: \; +\nc\bit{ \mskip1mu} \nc\biT{ \mskip-1mu} %% Note: +\nc\bitt{ \mskip2mu} \nc\biTT{ \mskip-2mu} %% \! = -3mu, +\nc\bittt{ \mskip3mu} \nc\biTTT{ \mskip-3mu} %% \, = 3mu, +\nc\bitttt{ \mskip4mu} \nc\biTTTT{ \mskip-4mu} %% \: = 4mu, +\nc\bittttt{\mskip5mu} \nc\biTTTTT{\mskip-5mu} %% \; = 5mu. + +\nc\f\frac %% fraction styles +\nc\F[5]{\1#3{\1#1{#4}/\1#2{#5}}} %% e.g., \F012{a}{b} = [a/(b)] +\nc\ff{\largerfrac{-1}} \nc\fF{\largerfrac{+1}} %% smaller +\nc\fff{\largerfrac{-2}} \nc\fFF{\largerfrac{+2}} %% and +\nc\ffff{\largerfrac{-3}} \nc\fFFF{\largerfrac{+3}} %% larger +\nc\fffff{\largerfrac{-4}} \nc\fFFFF{\largerfrac{+4}} %% fractions +\nc\largerfrac[3]{\mathchoice %% \frac at a type size larger by #1 + {\SB{$\mS0\vcenter{}$}\w=\YB\SB{\rS{#1}$\mS0\vcenter{}$} + \advance\w by-\YB\raise\w\HB{\rS{#1}$\mS0\frac{#2}{#3}$}} + {\SB{$ \vcenter{}$}\w=\YB\SB{\rS{#1}$ \vcenter{}$} + \advance\w by-\YB\raise\w\HB{\rS{#1}$ \frac{#2}{#3}$}} + {\SB{$\mS2\vcenter{}$}\w=\YB\SB{\rS{#1}$\mS2\vcenter{}$} + \advance\w by-\YB\raise\w\HB{\rS{#1}$\mS2\frac{#2}{#3}$}} + {\SB{$\mS3\vcenter{}$}\w=\YB\SB{\rS{#1}$\mS3\vcenter{}$} + \advance\w by-\YB\raise\w\HB{\rS{#1}$\mS3\frac{#2}{#3}$}}} + +\nc\restr[2]{{\lt.#1\rt|}_{#2}} %% restriction; value at #2 +\nc\tr{\mathop{\txt00{tr}}} + +%% Math symbols: +\nc\e{\mathrm{e}} %% e = 2.718281828459... +\nc\dd{\mathrm{d}} \nc\ddd{\bit\d} %% differential d +\nc\pd\partial +\def\lta#1{{\overset{{\scriptscriptstyle \leftarrow}}{#1}}} +\def\rta#1{{\overset{{\scriptscriptstyle \rightarrow}}{#1}}} +\def\nablal{\lta{\nabla}} +\def\nablar{\rta{\nabla}} +\nc\ql{\lambda} +\nc\xv{\underline{x}} +\nc\yv{\underline{y}} +\nc\rn{\mathbb{R}^n} +\nc\rk{\mathbb{R}^k} +\nc\Ker{\textrm{Ker }} +\nc\im{\textrm{Im }} +\nc\biz{\subsubsection*{Bizonyítás}} + +\usepackage{tkz-berge} + +\usepackage{mathtools} +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} +%%%%%%%%%% \mathbf{}=vastagít \mathrm{}=felállít +\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]} +\let\endchangemargin=\endlist + +\title{Bevezetés a számelméletbe 2 jegyzet} +\author{Toldi Balázs Ádám } +\date{Február 2020} + +\begin{document} + +\maketitle +\tableofcontents +\newpage +\section{Kombinatorika alapjai} +\subsection{Permutáció} +\subsubsection{Ismétlés nélküli permutáció} +$k$ különbőző dolog sorrenjeinek száma, ismétlés nélkül. +Kiszámítása: $k!$ +\subsubsection{Ismétléses permutáció} +$k$ különbőző dolog sorrenjeinek száma, ismétléssel. +Kiszámítása: $\frac{(k_1+k_2+..+k_r)!}{k_1!k_2!...k_R!}$ +\subsection{Variáció} +\subsubsection{Ismétlés nélküli variáció} +$n$ különbőző dologból választunk $k$ különbözőt és számít a sorrend. +Kiszámítása:$\frac{n!}{(n-k)!}$ +\subsubsection{Ismétléses variáció} +$n$ különbőző dolog közül választunk $k$ darab, nem feltétlenül különböző dolgot és számít hogy milyen sorrendben. +Kitszámítása: $n^k$ +\subsection{Kombináció} +\subsubsection{Ismétlés nélküli kombináció} +$n$ különböző dolog közül kiválasztunk $k$ darab különbőző dolgot sorrendtől függetlenül. +Kiszámítása: +\begin{equation*} +\begin{pmatrix} + n\\ + k +\end{pmatrix} +=\frac{n!}{(n-k)!k!} +\end{equation*} +\subsubsection*{Megjegyzés} +A $\bigl( \begin{smallmatrix} n\\k \end{smallmatrix}\bigl)$ számokat binominális együtthatónak nevezzük. +\subsubsection{Ismétléses kombináció} +$n$ kükönböző dologból kiválasztunk $k$ darab, nem feltétlen különböző dolgot és a sorrend nem számít. +Kiszámítása: +\begin{equation*} +\begin{pmatrix} + (n-1)+k\\ + k +\end{pmatrix} +\end{equation*} +\subsubsection{Fontos tudnivaló a binomiális együtthatókról} +\begin{equation*} +\begin{pmatrix} + n\\ + k +\end{pmatrix} += +\begin{pmatrix} + n\\ + n-k +\end{pmatrix} +\end{equation*} +\subsection{Pascal-háromszög} +\begin{tabular}{>{$n=}l<{$\hspace{12pt}}*{13}{c}} +0 &&&&&&&1&&&&&&\\ +1 &&&&&&1&&1&&&&&\\ +2 &&&&&1&&2&&1&&&&\\ +3 &&&&1&&3&&3&&1&&&\\ +4 &&&1&&4&&6&&4&&1&&\\ +5 &&1&&5&&10&&10&&5&&1&\\ +6 &1&&6&&15&&20&&15&&6&&1 +\end{tabular} +\subsubsection{Elemei} +Minden $n$ sor, $k$. eleme megegyezik $\bigl( \begin{smallmatrix*}n\\k\end{smallmatrix*}\bigl)$-val. +\subsubsection{Binomiális tétel} +$$(a+b)^n=\sum_{i=0}^{n}a^i\cdot b^{n-i}\cdot\bigl( \begin{smallmatrix*}n\\i\end{smallmatrix*}\bigl)$$ +\subsubsection{Tétel} +Minden $n$. sor elemeinek összege $2^n$. +\biz +$$(1+1)^n=2^n=\sum_{i=0}^{n}1^i\cdot 1^{n-i}\cdot\bigl( \begin{smallmatrix*}n\\i\end{smallmatrix*}\bigl)=\sum_{i=0}^{n}\bigl( \begin{smallmatrix*}n\\i\end{smallmatrix*}\bigl)$$ +\section{Gráfelmélet} +\subsection{Egyszerű gráfok} +Olyan gráf,amely nem tartalmaz hurok- és párhuzamos éleket. +\subsection{Részgráf} +$G'(V',G')$ gráf részgráfja $G(V,E)$-nek,ha $V'\leq V$,$E'\leq E$ és minedn $E'$-beli él vlgpontja $V'$ elemei. +\subsection{Állítás} +A fokok összege az élek számának kétszerese. +\subsection{Teljes gráf} +Bármely két különböző csúcs össze van kötve. +\subsection{Komplementer gráf} +Ugyanazon pontokból áll, teljes gráf $-$ gráf élei +\subsection{Izomorf gráf} +Két gráfot akkor nevezünk izomorfnak, ha pontjaik és éleik kölcsönösen egyértelműen és illeszkedéstartóan megfeleltethetők egymásnak. +\subsection{Út} +Olyan élsorozat,amelyben minden csúcs különböző +\subsection{Kör} +Olyan út,amelynek kezdőpontja és végpontja megegyezik +\subsection{Összefüggő gráfok} +Két gráf akkor összefüggő ha bármely két csúcsa közt létezik élsorozat/út +\subsection{Komponens} +Összefüggő feszített részgráf,amelyből nem emgy ki él.Nem bővíthető tovább összefüggő pontal. +\subsection{Állítás} +Ha $G$ egy $n$ csúcsú összefüggő gráf, akkor minimum $n-1$ éle van. +\biz +TODO +\subsection{Definíció} +Az összefüggő,körmentes gráfokat fának nevezzük. +\subsection{Állítás} +Minden n csúcsú fának pont n- éle van. +\subsection{Lemma} +G körmentes, n csúcsú gráf. Ekkor legfeljebb n-1 éle van G-nek. +\biz TODO +\subsection{Állítás} +Minden legalább 2 csúcsú fának van levele. +\subsection{Feszítőfa} +$G$-nek $F$ feszítőfája, ha $F$ fa és $F$ részgráfja G-nek,$F$ minden csúcsot tartalmaz. +\end{document}