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## 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

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\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*{<command>}{<left>}{<before-sep>}{<after-sep>}
%{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
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\usepackage{tkz-berge}
\usepackage{mathtools}
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
%%%%%%%%%% \mathbf{}=vastagít \mathrm{}=felállít
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\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}