\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} \label{sec:misc} Taken from~\cite{Downes04:amsart} \begin{displaymath} \hbar\nu=E,\qquad \hbar\ne\pi, \qquad \partial j, \qquad x^j, \qquad x^l \end{displaymath} Some other other equations: $\sum^J a'$, $r^a$ and $D^k$. 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}