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diff --git a/Master/texmf-dist/doc/latex/jamtimes/mathsample.tex b/Master/texmf-dist/doc/latex/jamtimes/mathsample.tex deleted file mode 100644 index 2c63fae0a12..00000000000 --- a/Master/texmf-dist/doc/latex/jamtimes/mathsample.tex +++ /dev/null @@ -1,953 +0,0 @@ -\documentclass{article} -\usepackage{ifpdf} -\ifpdf\pdfmapfile{+jtm.map}\fi -%\usepackage[T1]{fontenc} -\usepackage{jamtimes} -\usepackage{lipsum,textcomp,amsmath,url,amsfonts,longtable} -\DeclareMathSymbol{\dit}{\mathord}{letters}{`d} -\DeclareMathSymbol{\dup}{\mathord}{operators}{`d} -\def\test#1{#1} - -\def\testnums{% - \test 0 \test 1 \test 2 \test 3 \test 4 \test 5 \test 6 \test 7 - \test 8 \test 9 } -\def\testupperi{% - \test A \test B \test C \test D \test E \test F \test G \test H - \test I \test J \test K \test L \test M } -\def\testupperii{% - \test N \test O \test P \test Q \test R \test S \test T \test U - \test V \test W \test X \test Y \test Z } -\def\testupper{% - \testupperi\testupperii} - -\def\testloweri{% - \test a \test b \test c \test d \test e \test f \test g \test h - \test i \test j \test k \test l \test m } -\def\testlowerii{% - \test n \test o \test p \test q \test r \test s \test t \test u - \test v \test w \test x \test y \test z - \test\imath \test\jmath } -\def\testlower{% - \testloweri\testlowerii} - -\def\testupgreeki{% - \test A \test B \test\Gamma \test\Delta \test E \test Z \test H - \test\Theta \test I \test K \test\Lambda \test M } -\def\testupgreekii{% - \test N \test\Xi \test O \test\Pi \test P \test\Sigma \test T - \test\Upsilon \test\Phi \test X \test\Psi \test\Omega -} -\def\testupgreek{% - \testupgreeki\testupgreekii} - -\def\testlowgreeki{% - \test\alpha \test\beta \test\gamma \test\delta \test\epsilon - \test\zeta \test\eta \test\theta \test\iota \test\kappa \test\lambda - \test\mu } -\def\testlowgreekii{% - \test\nu \test\xi \test o \test\pi \test\rho \test\sigma \test\tau - \test\upsilon \test\phi \test\chi \test\psi \test\omega } -\def\testlowgreekiii{% - \test\varepsilon \test\vartheta \test\varpi \test\varrho - \test\varsigma \test\varphi \test\ell \test\wp} -\def\testlowgreek{% - \testlowgreeki\testlowgreekii\testlowgreekiii} -\begin{document} - -\section{Sebastian's math test} - - - -The default math mode font is $Math\ Italic$. This should not be -confused with ordinary \emph{Text Italic} -- notice the different spacing\,! -\verb|\mathbf| produces bold roman letters: $ \mathbf{abcABC} $. -If you wish to embolden complete formulas, -use the \verb|\boldmath| command \emph{before} going into math mode. -This changes the default math fonts to bold. - -\begin{tabular}{ll} -\texttt{normal} & $ x = 2\pi \Rightarrow x \simeq 6.28 $\\ -\texttt{mathbf} & $\mathbf{x} = 2\pi \Rightarrow \mathbf{x} \simeq 6.28 $\\ -\texttt{boldmath} & {\boldmath $x = \mathbf{2}\pi \Rightarrow x - \simeq{\mathbf{6.28}} $}\\ -\end{tabular} -\smallskip - -Greek is available in upper and lower case: -$\alpha,\beta \dots \Omega$, and there are special -symbols such as $ \hbar$ (compare to $h$). -Digits in formulas $1, 2, 3\dots$ may differ from those in text: 4, 5, -6\dots - -There is Sans Serif alphabet $\mathsf{abcdeABCD}$ selected by -\verb|\mathsf| and Typewriter math $\mathtt{abcdeABCD}$ selected by -\verb|\mathtt|. - -There is a calligraphic alphabet \verb|\mathcal| for upper case letters -$ \mathcal{ABCDE}\dots $, and there are letters for number sets: $\mathbb{A\dots Z} $, -which are produced using \verb|\mathbb|. There are Fraktur letters -$\mathfrak{abcdeABCDE}$ produced using \verb|\mathfrak| - -\begin{equation} - \sigma(t)=\frac{1}{\sqrt{2\pi}} - \int^t_0 e^{-x^2/2} dx -\end{equation} - -\begin{equation} - \prod_{j\geq 0} - \left(\sum_{k\geq 0}a_{jk} z^k\right) -= \sum_{k\geq 0} z^n - \left( \sum_{{k_0,k_1,\ldots\geq 0} - \atop{k_0+k_1+\ldots=n} } - a{_0k_0}a_{1k_1}\ldots \right) -\end{equation} - -\begin{equation} -\pi(n) = \sum_{m=2}^{n} - \left\lfloor \left(\sum_{k=1}^{m-1} - \lfloor(m/k)/\lceil m/k\rceil - \rfloor \right)^{-1} - \right\rfloor -\end{equation} - -\begin{equation} -\{\underbrace{% - \overbrace{\mathstrut a,\ldots,a}^{k\ a's}, - \overbrace{\mathstrut b,\ldots,b}^{l\ b's}} - _{k+l\ \mathrm{elements}} \} -\end{equation} - -\[ -\mbox{W}^+\ -\begin{array}{l} -\nearrow\raise5pt\hbox{$\mu^+ + \nu_{\mu}$}\\ -\rightarrow \pi^+ +\pi^0 \\[5pt] -\rightarrow \kappa^+ +\pi^0 \\ -\searrow\lower5pt\hbox{$\mathrm{e}^+ - +\nu_{\scriptstyle\mathrm{e}}$} -\end{array} -\] - -\[ -\frac{\pm -\left|\begin{array}{ccc} -x_1-x_2 & y_1-y_2 & z_1-z_2 \\ -l_1 & m_1 & n_1 \\ -l_2 & m_2 & n_2 -\end{array}\right|}{ -\sqrt{\left|\begin{array}{cc}l_1&m_1\\ -l_2&m_2\end{array}\right|^2 -+ \left|\begin{array}{cc}m_1&n_1\\ -n_1&l_1\end{array}\right|^2 -+ \left|\begin{array}{cc}m_2&n_2\\ -n_2&l_2\end{array}\right|^2}} -\] - - - - -\section{Math Tests} -\label{sec:mthtests} - - - -Math test are taken from\cite{Schmidt04:PSNFSS9.2}. - -\parindent 0pt -%\mathindent 1em - - -\subsection{Math Alphabets} - -Math Italic (\texttt{\string\mathnormal}) -\def\test#1{\mathnormal{#1},} -\begin{eqnarray*} -% && {\testnums}\\ - && {\testupper}\\ - && {\testlower}\\ - && {\testupgreek}\\ - && {\testlowgreek} -\end{eqnarray*}% - -Math Roman (\texttt{\string\mathrm}) -\def\test#1{\mathrm{#1},} -\begin{eqnarray*} - && {\testnums}\\ - && {\testupper}\\ - && {\testlower}\\ - && {\testupgreek}\\ - && {\testlowgreek} -\end{eqnarray*}% - -%Math Italic Bold -%\def\test#1{\mathbm{#1},} -%\begin{eqnarray*} -% && {\testnums}\\ -% && {\testupper}\\ -% && {\testlower}\\ -% && {\testupgreek}\\ -% && {\testlowgreek} -%\end{eqnarray*}% - -Math Bold (\texttt{\string\mathbf}) -\def\test#1{\mathbf{#1},} -\begin{eqnarray*} - && {\testnums}\\ - && {\testupper}\\ - && {\testlower}\\ -% && {\testupgreek} -\end{eqnarray*}% - -Math Sans Serif (\texttt{\string\mathsf}) -\def\test#1{\mathsf{#1},} -\begin{eqnarray*} - && {\testnums}\\ - && {\testupper}\\ - && {\testlower}\\ -% && {\testupgreek} -\end{eqnarray*}% - - - -Caligraphic (\texttt{\string\mathcal}) -\def\test#1{\mathcal{#1},} -\begin{eqnarray*} - && {\testupper} -\end{eqnarray*}% - -%Script (\texttt{\string\mathscr}) -%\def\test#1{\mathscr{#1},} -%\begin{eqnarray*} -% && {\testupper} -%\end{eqnarray*}% - -Fraktur (\texttt{\string\mathfrak}) -\def\test#1{\mathfrak{#1},} -\begin{eqnarray*} - && {\testupper}\\ - && {\testlower} -\end{eqnarray*}% - -Blackboard Bold (\texttt{\string\mathbb}) -\def\test#1{\mathbb{#1},} -\begin{eqnarray*} - && {\testupper} -\end{eqnarray*}% - - -\clearpage -\subsection{Character Sidebearings} - -\def\test#1{|#1|+} -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*}% -% -\def\test#1{|\mathrm{#1}|+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% -% -%\def\test#1{|\mathbm{#1}|+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii}\\ -% && {\testlowgreeki}\\ -% && {\testlowgreekii}\\ -% && {\testlowgreekiii} -%\end{eqnarray*}% -%% -%\def\test#1{|\mathbf{#1}|+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii} -%\end{eqnarray*}% -% -\def\test#1{|\mathcal{#1}|+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii} -\end{eqnarray*}% - - -\clearpage -\subsection{Superscript positioning} - -\def\test#1{#1^{2}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*}% -% -\def\test#1{\mathrm{#1}^{2}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% -% -%\def\test#1{\mathbm{#1}^{2}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii}\\ -% && {\testlowgreeki}\\ -% && {\testlowgreekii}\\ -% && {\testlowgreekiii} -%\end{eqnarray*}% -% -%\def\test#1{\mathbf{#1}^{2}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii} -%\end{eqnarray*} -% -\def\test#1{\mathcal{#1}^{2}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii} -\end{eqnarray*}% - - -\clearpage -\subsection{Subscript positioning} - -\def\test#1{\mathnormal{#1}_{i}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*}% -% -\def\test#1{\mathrm{#1}_{i}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% -% -%\def\test#1{\mathbm{#1}_{i}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii}\\ -% && {\testlowgreeki}\\ -% && {\testlowgreekii}\\ -% && {\testlowgreekiii} -%\end{eqnarray*} -%% -%\def\test#1{\mathbf{#1}_{i}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii} -%\end{eqnarray*}% -% -\def\test#1{\mathcal{#1}_{i}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii} -\end{eqnarray*}% - - -\clearpage -\subsection{Accent positioning} - -\def\test#1{\hat{#1}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*}% -% -\def\test#1{\hat{\mathrm{#1}}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii} -\end{eqnarray*}% -% -%\def\test#1{\hat{\mathbm{#1}}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii}\\ -% && {\testlowgreeki}\\ -% && {\testlowgreekii}\\ -% && {\testlowgreekiii} -%\end{eqnarray*}% -%% -%\def\test#1{\hat{\mathbf{#1}}+}% -%\begin{eqnarray*} -% && {\testupperi}\\ -% && {\testupperii}\\ -% && {\testloweri}\\ -% && {\testlowerii}\\ -% && {\testupgreeki}\\ -% && {\testupgreekii} -%\end{eqnarray*} -% -\def\test#1{\hat{\mathcal{#1}}+}% -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii} -\end{eqnarray*}% - - -\clearpage -\subsection{Differentials} - -\begin{eqnarray*} -\gdef\test#1{\dit #1+}% - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii}\\ -\gdef\test#1{\dit \mathrm{#1}+}% - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% -% -\begin{eqnarray*} -\gdef\test#1{\dup #1+}% - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii}\\ -\gdef\test#1{\dup \mathrm{#1}+}% - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% -% -\begin{eqnarray*} -\gdef\test#1{\partial #1+}% - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii}\\ -\gdef\test#1{\partial \mathrm{#1}+}% - && {\testupgreeki}\\ - && {\testupgreekii} -\end{eqnarray*}% - - -\clearpage -\subsection{Slash kerning} - -\def\test#1{1/#1+} -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*} - -\def\test#1{#1/2+} -\begin{eqnarray*} - && {\testupperi}\\ - && {\testupperii}\\ - && {\testloweri}\\ - && {\testlowerii}\\ - && {\testupgreeki}\\ - && {\testupgreekii}\\ - && {\testlowgreeki}\\ - && {\testlowgreekii}\\ - && {\testlowgreekiii} -\end{eqnarray*} - - -\clearpage -\subsection{Big operators} - -\def\testop#1{#1_{i=1}^{n} x^{n} \quad} -\begin{displaymath} - \testop\sum - \testop\prod - \testop\coprod - \testop\int - \testop\oint -\end{displaymath} -\begin{displaymath} - \testop\bigotimes - \testop\bigoplus - \testop\bigodot - \testop\bigwedge - \testop\bigvee - \testop\biguplus - \testop\bigcup - \testop\bigcap - \testop\bigsqcup -% \testop\bigsqcap -\end{displaymath} - - -\subsection{Radicals} - -\begin{displaymath} - \sqrt{x+y} \qquad \sqrt{x^{2}+y^{2}} \qquad - \sqrt{x_{i}^{2}+y_{j}^{2}} \qquad - \sqrt{\left(\frac{\cos x}{2}\right)} \qquad - \sqrt{\left(\frac{\sin x}{2}\right)} -\end{displaymath} - -\begingroup -\delimitershortfall-1pt -\begin{displaymath} - \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{x+y}}}}}}} -\end{displaymath} -\endgroup % \delimitershortfall - - -\subsection{Over- and underbraces} - -\begin{displaymath} - \overbrace{x} \quad - \overbrace{x+y} \quad - \overbrace{x^{2}+y^{2}} \quad - \overbrace{x_{i}^{2}+y_{j}^{2}} \quad - \underbrace{x} \quad - \underbrace{x+y} \quad - \underbrace{x_{i}+y_{j}} \quad - \underbrace{x_{i}^{2}+y_{j}^{2}} \quad -\end{displaymath} - - -\subsection{Normal and wide accents} - -\begin{displaymath} - \dot{x} \quad - \ddot{x} \quad - \vec{x} \quad - \bar{x} \quad - \overline{x} \quad - \overline{xx} \quad - \tilde{x} \quad - \widetilde{x} \quad - \widetilde{xx} \quad - \widetilde{xxx} \quad - \hat{x} \quad - \widehat{x} \quad - \widehat{xx} \quad - \widehat{xxx} \quad -\end{displaymath} - - -\subsection{Long arrows} - -\begin{displaymath} - \leftarrow \mathrel{-} \rightarrow \quad - \leftrightarrow \quad - \longleftarrow \quad - \longrightarrow \quad - \longleftrightarrow \quad - \Leftarrow = \Rightarrow \quad - \Leftrightarrow \quad - \Longleftarrow \quad - \Longrightarrow \quad - \Longleftrightarrow \quad -\end{displaymath} - - -\subsection{Left and right delimters} - -\def\testdelim#1#2{ - #1 f #2 - } -\begin{displaymath} - \testdelim() - \testdelim[] - \testdelim\lfloor\rfloor - \testdelim\lceil\rceil - \testdelim\langle\rangle - \testdelim\{\} -\end{displaymath} - -\def\testdelim#1#2{ - \left#1 f \right#2 - } -\begin{displaymath} - \testdelim() - \testdelim[] - \testdelim\lfloor\rfloor - \testdelim\lceil\rceil - \testdelim\langle\rangle - \testdelim\{\} -% \testdelim\lgroup\rgroup -% \testdelim\lmoustache\rmoustache -\end{displaymath} -\begin{displaymath} - \testdelim)( - \testdelim][ - \testdelim// - \testdelim\backslash\backslash - \testdelim/\backslash - \testdelim\backslash/ -\end{displaymath} - - -\clearpage -\subsection{Big-g-g delimters} - -\def\testdelim#1#2{% - - \left#1\left#1\left#1\left#1\left#1\left#1\left#1\left#1 - - \right#2\right#2\right#2\right#2\right#2\right#2\right#2\right#2 -} - -\begingroup -\delimitershortfall-1pt -\begin{displaymath} - \testdelim\lfloor\rfloor - \qquad - \testdelim() -\end{displaymath} -\begin{displaymath} - \testdelim\lceil\rceil - \qquad - \testdelim\{\} -\end{displaymath} -\begin{displaymath} - \testdelim[] - \qquad - \testdelim\lgroup\rgroup -\end{displaymath} -\begin{displaymath} - \testdelim\langle\rangle - \qquad - \testdelim\lmoustache\rmoustache -\end{displaymath} -\begin{displaymath} - \testdelim\uparrow\downarrow \quad - \testdelim\Uparrow\Downarrow \quad -\end{displaymath} -\endgroup % \delimitershortfall - -\subsection{Symbols} -\label{sec:symbols} - -This is from~\cite{Eijkhout07:TeXbyTopic} - -\begin{longtable}{lllll} -Symbol & Control Sequence & mathcode & Family & Hex Position \\ -$\partial$&partial& "0140&1&40\\ -$\flat$&flat& "015B&1&5B\\ -$\natural$&natural& "015C&1&5C\\ -$\sharp$&sharp& "015D&1&5D\\ -$\ell$&ell& "0160&1&60\\ -$\imath$&imath& "017B&1&7B\\ -$\jmath$&jmath& "017C&1&7C\\ -$\wp$&wp& "017D&1&7D\\ -$\prime$&prime& "0230&2&30\\ -$\infty$&infty& "0231&2&31\\ -$\triangle$&triangle& "0234&2&34\\ -$\forall$&forall& "0238&2&38\\ -$\exists$&exists& "0239&2&39\\ -$\neg$&neg& "023A&2&3A\\ -$\emptyset$&emptyset& "023B&2&3B\\ -$\Re$&Re& "023C&2&3C\\ -$\Im$&Im& "023D&2&3D\\ -$\top$&top& "023E&2&3E\\ -$\bot$&bot& "023F&2&3F\\ -$\aleph$&aleph& "0240&2&40\\ -$\nabla$&nabla& "0272&2&72\\ -$\clubsuit$&clubsuit& "027C&2&7C\\ -$\diamondsuit$&diamondsuit& "027D&2&7D\\ -$\heartsuit$&heartsuit& "027E&2&7E\\ -$\spadesuit$&spadesuit& "027F&2&7F\\ -$\smallint \displaystyle\smallint$& - smallint& "1273&2&73\cr -$\bigsqcup \displaystyle\bigsqcup$& - bigsqcup& "1346&3&46\cr -$\ointop \displaystyle\ointop$& - ointop& "1348&3&48\cr -$\bigodot \displaystyle\bigodot$& - bigodot& "134A&3&4A\cr -$\bigoplus \displaystyle\bigoplus$& - bigoplus& "134C&3&4C\cr -$\bigotimes \displaystyle\bigotimes$& - bigotimes& "134E&3&4E\cr -$\sum \displaystyle\sum$& - sum& "1350&3&50\cr -$\prod \displaystyle\prod$& - prod& "1351&3&51\cr -$\intop \displaystyle\intop$& - intop& "1352&3&52\cr -$\bigcup \displaystyle\bigcup$& - bigcup& "1353&3&53\cr -$\bigcap \displaystyle\bigcap$& - bigcap& "1354&3&54\cr -$\biguplus \displaystyle\biguplus$& - biguplus& "1355&3&55\cr -$\bigwedge \displaystyle\bigwedge$& - bigwedge& "1356&3&56\cr -$\bigvee \displaystyle\bigvee$& - bigvee& "1357&3&57\cr -$\coprod \displaystyle\coprod$& - coprod& "1360&3&60\cr -$\triangleright$&triangleright& "212E&1&2E\cr -$\triangleleft$&triangleleft& "212F&1&2F\cr -$\star$&star& "213F&1&3F\cr -$\cdot$&cdot& "2201&2&01\cr -$\times$×& "2202&2&02\cr -$\ast$&ast& "2203&2&03\cr -$\div$&div& "2204&2&04\cr -$\diamond$&diamond& "2205&2&05\cr -$\pm$&pm& "2206&2&06\cr -$\mp$&mp& "2207&2&07\cr -$\oplus$&oplus& "2208&2&08\cr -$\ominus$&ominus& "2209&2&09\cr -$\otimes$&otimes& "220A&2&0A\cr -$\oslash$ø& "220B&2&0B\cr -$\odot$&odot& "220C&2&0C\cr -$\bigcirc$&bigcirc& "220D&2&0D\cr -$\circ$&circ& "220E&2&0E\cr -$\bullet$&bullet& "220F&2&0F\cr -$\bigtriangleup$&bigtriangleup& "2234&2&34\cr -$\bigtriangledown$&bigtriangledown& "2235&2&35\cr -$\cup$&cup& "225B&2&5B\cr -$\cap$&cap& "225C&2&5C\cr -$\uplus$&uplus& "225D&2&5D\cr -$\wedge$&wedge& "225E&2&5E\cr -$\vee$&vee& "225F&2&5F\cr -$\setminus$&setminus& "226E&2&6E\cr -$\wr$&wr& "226F&2&6F\cr -$\amalg$&amalg& "2271&2&71\cr -$\sqcup$&sqcup& "2274&2&74\cr -$\sqcap$&sqcap& "2275&2&75\cr -$\dagger$&dagger& "2279&2&79\cr -$\ddagger$&ddagger& "227A&2&7A\cr -$\leftharpoonup$&leftharpoonup& "3128&1&28\cr -$\leftharpoondown$&leftharpoondown& "3129&1&29\cr -$\rightharpoonup$&rightharpoonup& "312A&1&2A\cr -$\rightharpoondown$&rightharpoondown& "312B&1&2B\cr -$\smile$&smile& "315E&1&5E\cr -$\frown$&frown& "315F&1&5F\cr -$\asymp$&asymp& "3210&2&10\cr -$\equiv$&equiv& "3211&2&11\cr -$\subseteq$&subseteq& "3212&2&12\cr -$\supseteq$&supseteq& "3213&2&13\cr -$\leq$&leq& "3214&2&14\cr -$\geq$&geq& "3215&2&15\cr -$\preceq$&preceq& "3216&2&16\cr -$\succeq$&succeq& "3217&2&17\cr -$\sim$&sim& "3218&2&18\cr -$\approx$&approx& "3219&2&19\cr -$\subset$&subset& "321A&2&1A\cr -$\supset$&supset& "321B&2&1B\cr -$\ll$&ll& "321C&2&1C\cr -$\gg$&gg& "321D&2&1D\cr -$\prec$&prec& "321E&2&1E\cr -$\succ$&succ& "321F&2&1F\cr -$\leftarrow$&leftarrow& "3220&2&20\cr -$\rightarrow$&rightarrow& "3221&2&21\cr -$\leftrightarrow$&leftrightarrow& "3224&2&24\cr -$\nearrow$&nearrow& "3225&2&25\cr -$\searrow$&searrow& "3226&2&26\cr -$\simeq$&simeq& "3227&2&27\cr -$\Leftarrow$&Leftarrow& "3228&2&28\cr -$\Rightarrow$&Rightarrow& "3229&2&29\cr -$\Leftrightarrow$&Leftrightarrow& "322C&2&2C\cr -$\nwarrow$&nwarrow& "322D&2&2D\cr -$\swarrow$&swarrow& "322E&2&2E\cr -$\propto$&propto& "322F&2&2F\cr -$\in$&in& "3232&2&32\cr -$\ni$&ni& "3233&2&33\cr -$\not$¬& "3236&2&36\cr -$\mapstochar$&mapstochar& "3237&2&37\cr -$\perp$&perp& "323F&2&3F\cr -$\vdash$&vdash& "3260&2&60\cr -$\dashv$&dashv& "3261&2&61\cr -$\mid$&mid& "326A&2&6A\cr -$\parallel$¶llel& "326B&2&6B\cr -$\sqsubseteq$&sqsubseteq& "3276&2&76\cr -$\sqsupseteq$&sqsupseteq& "3277&2&77\cr -\end{longtable} - - -\subsection{Miscellanneous formulae} - -Taken from~\cite{Downes04:amsart} - -\label{sec:misc} -\begin{displaymath} - \hbar\nu=E -\end{displaymath} - -Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The -corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from -$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the -degree of its corresponding vertex; i.e., the $i$th diagonal entry is -identified with the degree of the $i$th vertex. It is well known that -\begin{equation} -\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$}, -\quad i=1,\dots,n -\end{equation} -where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of -$\mathbf{K}$. - -\newcommand{\abs}[1]{\left\lvert#1\right\rvert} -\newcommand{\wh}{\widehat} -Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge -$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j -C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a -subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det -\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$. Define multiplication for the elements of $\wh X$ by -\begin{equation}\label{multdef} -\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad -i,j=1,\dots,n. -\end{equation} -Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat -k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the -relation -\begin{equation}\label{H-cycles} -\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det -\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n. -\end{equation} -The task here is to express \eqref{H-cycles} -in a form free of any $\hat x_i$, -$i=1,\dots,n$. The result also leads to the resolution of enumeration of -Hamiltonian paths in a graph. - -It is well known that the enumeration of Hamiltonian cycles and paths -in a complete graph $K_n$ and in a complete bipartite graph -$K_{n_1n_2}$ can only be found from \textit{first combinatorial - principles}. One wonders if there exists a formula which can be used -very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently, using -Lagrangian methods, Goulden and Jackson have shown that $H_c$ can be -expressed in terms of the determinant and permanent of the adjacency -matrix. However, the formula of Goulden and -Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this -paper, using an algebraic method, we parametrize the adjacency matrix. -The resulting formula also involves the determinant and permanent, but -it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we -eliminate the permanent from $H_c$ and show that $H_c$ can be -represented by a determinantal function of multivariables, each -variable with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be -written by number of spanning trees of subgraphs. Finally, we apply -the formulas to a complete multigraph $K_{n_1\dots n_p}$. - -The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in -this paper. All formulas can be extended to a digraph simply by -multiplying $H_c$ by 2. - -The boundedness, property of $\Phi_ 0$, then yields -\[\int_{\mathcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha -\int_{\mathcal{D}}\abs{u}^2e^{\alpha\abs{z}^2} -+c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\] - -Let $B(X)$ be the set of blocks of $\Lambda_{X}$ -and let $b(X) = \abs{B(X)}$. If $\phi \in Q_{X}$ then -$\phi$ is constant on the blocks of $\Lambda_{X}$. -\begin{equation}\label{far-d} - P_{X} = \{ \phi \in M \mid \Lambda_{\phi} = \Lambda_{X} \}, -\qquad -Q_{X} = \{\phi \in M \mid \Lambda_{\phi} \geq \Lambda_{X} \}. -\end{equation} -If $\Lambda_{\phi} \geq \Lambda_{X}$ then -$\Lambda_{\phi} = \Lambda_{Y}$ for some $Y \geq X$ so that -\[ Q_{X} = \bigcup_{Y \geq X} P_{Y}. \] -Thus by M\"obius inversion -\[ \abs{P_{Y}}= \sum_{X\geq Y} \mu (Y,X)\abs{Q_{X}}.\] -Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$. -In particular $\abs{Q_{X}} = w^{b(X)}$. - - -\renewcommand{\arraystretch}{2.2} -\[W(\Phi)= \begin{Vmatrix} -\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ -\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& -\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ -\hdotsfor{5}\\ -\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& -\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& -\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& -\dfrac{\varphi}{(\varphi_n,\varepsilon_n)} -\end{Vmatrix}\] - - - -\bibliography{jamtimes} -\bibliographystyle{unsrt} - - - -\end{document} |