diff options
Diffstat (limited to 'fonts/newcomputermodern')
50 files changed, 2895 insertions, 60 deletions
diff --git a/fonts/newcomputermodern/README b/fonts/newcomputermodern/README index 4a99ea9c8d..cc68608d5c 100644 --- a/fonts/newcomputermodern/README +++ b/fonts/newcomputermodern/README @@ -1,7 +1,7 @@ New Computer Modern Fonts - (Package version 5.1, GustFLv1 or later) + (Package version 7.0.0, GustFLv1 or later) Antonis Tsolomitis Samos, Greece, 2019--2024 @@ -26,12 +26,20 @@ or OR +\usepackage[sansdefault]{fontsetup} (loads the Regular Sans weight) +or +\usepackage[sansdefault]{newcomputermodern} (loads the Regular Sans weight) + + + +OR + \usepackage{fontspec} \setmainfont{NewCM10-Book} \setsansfont{NewCMSans10-Book} \setmonofont{NewCMMono10-Book} -and similarly for the Regular weight. +and similarly for the Regular weight or the Sans family. ======================================================================= The fonts contain Latin and accented latin letters and combinations, @@ -45,7 +53,11 @@ As of version 5.0 the Devanagari script is supported for Languages Hindi (default), Sanskrit, Marathi and Nepali in Regular/Book 08/10pt and Bold 10t. -The Math fonts (Regular and Book) support fully all Unicode blocks +As of version 6.0.0 a Bold Math font has been added. + +As of version 7.0.0 a Sans Math font has been added. + +The Math fonts (Regular, Book, Bold and Sans) support fully all Unicode blocks of Mathematics providing the widest possible Mathematical coverage. The fonts have their own protrusion settings through the microtype package @@ -103,6 +115,72 @@ xelatex newcm-doc.tex Latest additions/improvements: ------------------------------ +Version 7.0.0 + +Added NewCMSansMath-Regular + +Sans letters used for math variables are a new design +(and not the original with a slant) so they really look as variables! + +NewCM mathbb/symbb symbols in Sans have been reworked to match the Sans design +in both style and wight. + +Same goes for mathcal/symcal letters. + +Math Kerning has been introduced in ALL math fonts to compensate +for bad spacing especially of the index of mathcal/symcal letters. + +Added ss06 and ss07 to be selected for auto +middle beta and theta in Greek. + +The kerning of Greek Oblique block has been greatly improved; "theorem" +statements now look much better. + +testmath-newcm.tex/pdf has been added to the documentation from the AMS bundle +to showcase the new SansMath font. + + + +Version 6.0.0 + +Added Bold Math font + +NewCM mathbb symbols (in ss03) have been reworked and they now +match the design of ComputerModern fonts (the AMSbb remain the default) + +AMSbb and NewCMbb are now provided in SmallCaps for special applications +(see the documentation) in ss05 + +An additional operator similar to \sum and \int has been added +to the Math fonts. Namely now it is possible to convolute many +function in the usual shorthand notation mathematicians use. +The new operator is called: \convolution + +Added Currency Symbols font in all text fonts (except Uncials) + +Fixed several typos in Lookup tables thanks to Eric Biedert + +Fixed CapHeight parameter in OS2 tables for some of the fonts +thanks to Max Kortge + + + + +Version 5.2.0 + +Added Cherokee in the Sans font too + +Added Canadian Aboriginal Syllabics to Sans + +Added Allergies symbols to Sans Regular and Book + +Added twoheadhookrightarrow and twoheadhookleftarrow +to the Math fonts in Private USe Area, +since they are needed and do not exist in Unicode. + +Fixed minor IPA tilde overlapping bugs + + Version 5.1 Added Aegean Numbers in the Sans font diff --git a/fonts/newcomputermodern/doc/mathkern-example.pdf b/fonts/newcomputermodern/doc/mathkern-example.pdf Binary files differnew file mode 100644 index 0000000000..564665901b --- /dev/null +++ b/fonts/newcomputermodern/doc/mathkern-example.pdf diff --git a/fonts/newcomputermodern/doc/mathkern-example.tex b/fonts/newcomputermodern/doc/mathkern-example.tex new file mode 100644 index 0000000000..a3ed1a2076 --- /dev/null +++ b/fonts/newcomputermodern/doc/mathkern-example.tex @@ -0,0 +1,24 @@ +\documentclass{standalone} +\pagestyle{empty} +\usepackage{fontspec} +\usepackage{unicode-math} +\usepackage[olddefault]{fontsetup} +\begin{document} + + + +% \begin{center} +\renewcommand{\arraystretch}{2} + \begin{tabular}{c|c} + Latin Modern & NewCM-Regular\\ \hline +{\setmathfont{latinmodern-math.otf}\Large $F_x$ $Y_x$ $\symscr N_x$ $\symcal J_x$ $\symcal G_x$ $\Gamma_x$ } & {\setmathfont{NewCMMath-Regular.otf}\Large $F_x$ $Y_x$ $\symscr N_x$ $\symcal J_x$ $\symcal G_x$ $\Gamma_x$ } + \end{tabular} +%\end{center} + + + +\end{document} + +\Large +$F_x$ $J_x$ $Y_x$ $\mathcal J_x$ $\mathcal G_x$ $\Gamma_x$ +\end{document} diff --git a/fonts/newcomputermodern/doc/newcm-doc.pdf b/fonts/newcomputermodern/doc/newcm-doc.pdf Binary files differindex 953e01a753..a2102f5b48 100644 --- a/fonts/newcomputermodern/doc/newcm-doc.pdf +++ b/fonts/newcomputermodern/doc/newcm-doc.pdf diff --git a/fonts/newcomputermodern/doc/newcm-doc.tex b/fonts/newcomputermodern/doc/newcm-doc.tex index 7e20605e67..69ce5eed3c 100644 --- a/fonts/newcomputermodern/doc/newcm-doc.tex +++ b/fonts/newcomputermodern/doc/newcm-doc.tex @@ -8,6 +8,7 @@ \setotherlanguage[variant=polytonic]{greek} \usepackage[hidelinks,pdfa]{hyperref} %\usepackage{xgreek} +\usepackage{pstricks,hvlogos} \usepackage[default,varnothing]{fontsetup} \usepackage{unicodefonttable,graphicx,wrapfig,xcolor,calc} \newfontfamily\lmboldsans{lmsans10-bold.otf} @@ -20,7 +21,11 @@ \newfontfamily\showtiefont[CharacterVariant=5]{NewCM10-Book.otf} \newfontfamily\ipafont[%Renderer = {Harfbuzz}, StylisticSet = {05},ItalicFont=NewCM10-BookItalic]{NewCM10-Book.otf} -% +\newfontfamily\middlegreekon[StylisticSet=7]{NewCM10-Book.otf} +\newfontfamily\middlegreekonsans[StylisticSet=6,StylisticSet=7]{NewCMSans10-Book.otf} +\newfontfamily{\mathastext}{NewCMMath-Book.otf} +\newfontfamily{\mathastextnewcm}[StylisticSet=3]{NewCMMath-Book.otf} +% %%%%%%%%%%%%%%%%%%%%%%%%%%%% Devanagari text %%%% \newfontfamily\hinditext[% Script=Devanagari,% @@ -90,19 +95,22 @@ Language=Nepali, % \Umathcode #1 = "0 ~ \use:c{ symdevletters } ~ #1 %} %\ExplSyntaxOff - + \renewcommand{\arraystretch}{1.4} -\title{The New Computer Modern FontFamily\\ version 5.1} +\title{The New Computer Modern FontFamily\\ version 7.0.0} \author{Antonis Tsolomitis} %\address{Department of Mathematics\\ University of the Aegean\\ Karlovassi, 832\,00 Samos\\ Greece} %\netaddress{atsol (at) aegean dot gr} %\personalURL{https://myria.math.aegean.gr/~atsol/} %%% End of metadata %%% -\usepackage{pstricks} \begin{document} +\mathversion{bold} +\newsavebox{\mathcalscrA} +\savebox{\mathcalscrA}{$\symcal A$ $\symbfscr A$} +\mathversion{normal} % \rput(-2,-3){\devgray ल}% @@ -174,17 +182,8 @@ Language=Nepali, \null\thispagestyle{empty} -%\vfill -% -%\ttxstackedup \qquad -%{\ttxstackeddown} -% -%\vfill -% - \newpage -%\end{document} \maketitle \tableofcontents @@ -195,8 +194,8 @@ the number of additional glyphs) of the \verb|lm| fonts. It is not just a family adding random missing glyphs but it adds support for several more languages and shapes needed for academic (and not only) work. Currently it supports among others, Greek\footnote{from Claudio Beccari's Greek.}, -Cyrillic\footnote{from the \texttt{cmu} package.}, Hebrew, Cherokee and -Coptic. Since it supports +Cyrillic\footnote{from the \texttt{cmu} package.}, Devanagari, Hebrew, Coptic, Cherokee and +Canadian Aboriginal. Since it supports diacritics stacking the number of languages that use the Latin alphabet is greatly expanded. Diacritics stacking is also needed for Greek for papyrological work and this is also supported. @@ -214,17 +213,27 @@ that Unicode has forgotten to encode). \textit{What follows is a sequence of commands and results so as to show how to access all features of the fonts. Character tables are also included}. +\textsc{Important:} If you want to provide patches for the fonts +please contact me before you create them. The fonts evolve quickly +and you may not have the latest development version and your patches +may not apply if created for the published version. + + \section{How to load the fonts} The simpler way to load the fonts is through the \verb|fontsetup| package. The command \verb|\usepackage[default]{fontsetup}| -\noindent will load the Book weight of the NewCM family, and +\noindent will load the Book weight of the NewCM family, \verb|\usepackage[olddefault]{fontsetup}| -\noindent will load the Regular weight. +\noindent will load the Regular weight, and + +\verb|\usepackage[sansdefault]{fontsetup}| + +\noindent will load the Sans Serif NewCM family. Also notice that the fonts support the microtype package for fine typographic tuning. See the documentation of microtype for this. @@ -316,7 +325,7 @@ U10307, U10310, U10312, U10314, U1031F and U1032F are {\sffamily\char"10307\char"10310\char"10312\char"10314\char"1031F\char"1032F}. \subsection{Diacritics Stacking} -\marginpar{\begin{center} +\marginpar{\begin{center} \ \\[-1cm] {\color{red}$\rightarrow$}\ \xstacked\quad{\sffamily\xstacked}\quad{\ttfamily\xstacked}\ {\color{red}$\leftarrow$} \end{center}} Diacritics---the full block U+0300 to U+036F---and diacritics stacking @@ -336,7 +345,7 @@ these accents you can use the \verb|\char| command or just copy-paste from the f Please note that stacking is by default supported with xetex. With luatex you have to add the option \texttt{Renderer=Harfbuzz}, say by - \noindent\verb|\addfontfeature{Renderer=Harfbuzz}| + \noindent\verb|\addfontfeatures{Renderer=Harfbuzz}| Also notice that your text editor may not support stacking. The editor may show the accents one after the other, but the pdf produced by xetex or luatex will have the accents stacked. @@ -344,8 +353,8 @@ accents one after the other, but the pdf produced by xetex or luatex will have t \subsubsection{Coloring diacritics} If one wants to use color for diacritics, different from the color of the base character -this does not work with Xe\LaTeX\ (the commands of the \verb|color| package -break the stacking mechanism). It works though with Lua\LaTeX\ using the \verb|luacolor| +this does not work with \XeLaTeX\ (the commands of the \verb|color| package +break the stacking mechanism). It works though with \LuaLaTeX\ using the \verb|luacolor| package. However, there is a problem when the base glyph and the first diacritic above exist in the font as a precomposed character. For example, this is the case with aacute (á) (U+00E1). Such characters are treated as one by Lua and they can not @@ -393,6 +402,36 @@ The full Unicode Greek block is supported, which is Pamphilian digamma (ͷ) etc, have been added. For example, it is now possible to write \centerline{βιϐλίο instead of βιβλίο.} + + In order to auto select this conversion for middle beta and theta the StylisticSets + ss07 must be enabled with, say, + + \noindent\verb|\addfontfeatures{StylisticSet=7}|, + + \noindent but for the Sans and Mono fonts \texttt{StylisticSet=6} is also needed, with + + \noindent\verb|\addfontfeatures{StylisticSet=6,StylisticSet=7}|. + + To disable this feature + you can do + + \noindent\verb|\addfontfeature{RawFeature={-ss07}}| + + or + + \noindent\verb|\addfontfeature{RawFeature={-ss06},RawFeature={-ss07}}| + + for the Sans and Mono families. + + \begin{center} + \begin{tabular}{|c|c|c|} \hline + \texttt{Source} & \texttt{βιβλίο} & \texttt{θυμήθηκα}\\ \hline + \texttt{ss07} enabled & {\middlegreekon βιβλίο} & {\middlegreekon θυμήθηκα}\\ \hline + \texttt{ss06} and \texttt{ss07} enabled for Sans & {\middlegreekonsans βιβλίο}& {\middlegreekonsans θυμήθηκα}\\ \hline + \texttt{ss06} and \texttt{ss07} disabled & βιβλίο, \textsf{βιβλίο} &θυμήθηκα, \textsf{θυμήθηκα} \\ \hline + \end{tabular} + \end{center} + \item U1F00--U1FFF for polytonic, and \item U10140--U1018F for ancient Greek numbers. @@ -669,13 +708,105 @@ A few letters from Coptic and Epact numbers follow: \section{Cherokee} Both Unicode blocks -U13A0--13FF and UAB70--UABBF for Cherokee are supported. A few letters are: +U13A0--13FF and UAB70--UABBF for Cherokee are supported. +% A few letters are: %\begin{center} - ᎣᎤᎹᏊᏐ ꭳꭴꭷꮂꮔꮿ +% ᎣᎤᎹᏊᏐ ꭳꭴꭷꮂꮔꮿ %\end{center} +The samples below were kindly provided by Sedi Eastwood: +\medskip + +\noindent Sample 1: +\begin{center} +\begin{tabular}{ll} + \begin{minipage}{5cm} ᎬᏣᎵᏨᏗᏣᏓᏲᎯᏍᏗ ᎢᏤᎮᏍᏗ\\ + \textsf{ᎬᏣᎵᏨᏗᏣᏓᏲᎯᏍᏗ ᎢᏤᎮᏍᏗ} +\end{minipage} + & + \begin{minipage}{6cm} + Live/exist in a manner that there is never a reason or purpose to let go of one another.\\ + (gvjalijvdijadayohisdi ijehesdi\\ +\hphantom{(}gvtsalitsvditsadayohisdi itsehesdi) + \end{minipage} +\end{tabular} +\end{center} +\medskip + +\noindent Sample 2: +\begin{center} +\begin{tabular}{ll} +\begin{minipage}{4cm} ᏕᏣᏓᎨᏳᏎᏍᏗ\\ \textsf{ᏕᏣᏓᎨᏳᏎᏍᏗ} +\end{minipage} + & + \begin{minipage}{6cm} + Value the existence of one another.\\ + (dejadageyusesdi\\ +\hphantom{(}detsadageyusesdi) + \end{minipage} +\end{tabular} +\end{center} + +\medskip + +\noindent Sample 3: +\begin{center} +\begin{tabular}{ll} +\begin{minipage}{4cm} ᎥᎵᏍᎨᏗ ᏕᏣᏓᏰᎸᏎᏍᏗ\\ \textsf{ᎥᎵᏍᎨᏗ ᏕᏣᏓᏰᎸᏎᏍᏗ} +\end{minipage} + & + \begin{minipage}{6cm} + Think of one another as sacred or hold the existence of others sacred.\\ + (vlisgedi dejadayelvsesdi) + \end{minipage} +\end{tabular} +\end{center} + + +\section{Canadian Aboriginal Syllabics} + +\newfontfamily{\textsfca}[BoldFont=NewCMSans10-Bold.otf,% +ItalicFont=NewCMSans10-BookOblique.otf]{NewCMSans10-Book.otf} + +Canadian Aboriginal Syllabics are supported in the sans font. +The full Unicode blocks are covered, which are +U1400--U167F, U18B0--U18F5, and U11AB0--U11ABF. + +Some examples in Blackfoot Siksiká: + +\noindent Blackfoot-speaking real people: {\textsfca ᖹᐟᒧᐧᒣᑯ} +Siksiká: {\textsfca ᓱᖽᐧᖿ}\\ +Siksikáíʼpowahsin: {\textsfca ᓱᘁᓱᘁᖳᐟᑲᖷᑊᓱᐡ} In bold: {\textsfca\bfseries ᓱᘁᓱᘁᖳᐟᑲᖷᑊᓱᐡ}\\ +Niitsipowahsin: {\textsfca ᖹᐨᓱᑲᖷᑊᓱᐡ} In oblique: {\textsfca\itshape ᖹᐨᓱᑲᖷᑊᓱᐡ} + +\medskip + +Some more in Cree: + +\begin{tabular}{l|l} +\begin{minipage}{5.5cm} +\noindent{\textsfca ᐌᐘᑯ ᐅᒪ ᐊᓯᓂᐏᒣᓂᑲᐣ ᑭᒥᓇᐘᐠ}\\ +{\textsfca ᐃᓂᓂᐘᐠ ᒪᓂᑐᐸ ᑲᐊᔭᒋᐠ᙮ ᐁᐘᑿᓂᐠ ᐅᑭ}\\ +{\textsfca ᑲᓄᒋᐦᑕᒋᐠ ᐊᓯᓂᐏᐊᑐᐢᑭᐃᓂᓂᐤ ᑲᑭᒥᓂᐦᒋᐠ}\\ +{\textsfca ᐅᒣᓂᐤ᙮ ᐊᑿᓂ ᒥᑕᐦᑐᒥᑕᓇᐤ ᐊᐢᑭᐩ ᐊᓴᐩ}\\ +{\textsfca ᐁᐊᑐᐢᑫᒋᐠ ᐅᑕ ᒪᓂᑐᐸ᙮} +\end{minipage} + & +\begin{minipage}{4.8cm} +{\sffamily The masonry for the wall through time + has been given to people of Manitoba + by the international union of bricklayers + and allied craftsmen local one Manitoba + to commemorate 100 years of service, by + Manitoba masonry contractors and by Manitoba + masonry suppliers.} +\end{minipage} +\end{tabular} + + + \section{Devanagari} Devanagari script is supported for the serifed font in Regular (10pt/8pt), Book (10pt/8pt), @@ -686,7 +817,7 @@ must include \noindent \verb|Script=Devanagari, Language=XXXX| where \verb|XXXX| must be replaced with one of \verb|Hindi|, \verb|Sanskrit|, \verb|Marathi|, \verb|Nepali|. If the \verb|Language| parameter is not set then the default is \verb|Hindi|. -For Lua\LaTeX\ the parameter \verb|Renderer=Harfbuzz| must also be included. +For \LuaLaTeX\ the parameter \verb|Renderer=Harfbuzz| must also be included. So if say Marathi is needed as the default font document then one can use the following: \begin{verbatim} @@ -698,7 +829,11 @@ Renderer=Harfbuzz]{NewCM10Devanagari-Book.otf} The Devanagari fonts were developed with the help of {\devanagaritext निरंजन} (Niranjan) whose name appears in the copyright section of the fonts and I also thank him for -providing the samples below. +providing the samples below. It should also be noted that the design is original +and based on old handwritten books. In old civilizations, such as the Indian one, +it is only natural that the design be affected by how the handwritten +book look. Ink creates ``drop''-like serifs on paper that absorbs it, and the +tool used to write also affects the look of the script. All these were taken into account. \noindent A Sanskrit sample from {\sanskrittext बृहदारण्यकोपनिषद्} (bṛhadāraṇyakopaniṣad) follows: @@ -930,6 +1065,56 @@ Next we give a real example of using \acro{\textsc{ipa}} symbols. The quote belo {\ipafont [t̪y]} ``you.'' \end{quote} +A recent addition for \textsc{ipa} is the support of the Unicode block U1D80--U1DBF which +includes more letters with palatal and retroflex hook and more modifier small letters. +Eg. +\begin{center} + \begin{tabular}{|c|c|}\hline +Upright & Italic\\ \hline + ᶀ ᶂ ᶈ ᶉ ᶋ ᶌ ᶕ ᶔ ᶙ ᶚ ᶛ ᶞ ᶽ ᶿ &\textit{ᶀ ᶂ ᶈ ᶉ ᶋ ᶌ ᶕ ᶔ ᶙ ᶚ ᶛ ᶞ ᶽ ᶿ}\\ \hline + Sans Upright & Sans Oblique\\ \hline + \textsf{ᶀ ᶂ ᶈ ᶉ ᶋ ᶌ ᶕ ᶔ ᶙ ᶚ ᶛ ᶞ ᶽ ᶿ}& \textsf{\itshape ᶀ ᶂ ᶈ ᶉ ᶋ ᶌ ᶕ ᶔ ᶙ ᶚ ᶛ ᶞ ᶽ ᶿ}\\ + \hline + \end{tabular} +\end{center} + + +\section{Currency Symbols} + +Support for currency symbols block U20A0--U20C0 in all text fonts (except uncials). +\begin{center} + \begin{tabular}{rl} +Roman:& ₠₡₢₣₤₥₦₧₨₩₪₫€₭₮₯₰₱₲₳₴₵₶₷₸₹₺₻₼₽₾₿⃀\\ +Sans:& \textsf{₠₡₢₣₤₥₦₧₨₩₪₫€₭₮₯₰₱₲₳₴₵₶₷₸₹₺₻₼₽₾₿⃀}\\ + Mono:& \texttt{₠₡₢₣₤₥₦₧₨₩₪₫€₭₮₯₰₱₲₳₴₵₶₷₸₹₺₻₼₽₾₿⃀} + \end{tabular} +\end{center} +Interesting is the design of Spanish Peseta in the Mono family +(seen enlarged in the margin) were we have +to fit essentially three letters in the width of the Mono font. +\marginpar{\scalebox{2}{\texttt{\huge ₧}}} +%\begin{center} +%{Roman: \huge ₧}\hspace{1cm}{Mono: \texttt{\huge ₧}} +%\end{center} + + +Especially for the Rupee sign (U20B9) the upright serifed +non-Devanagari fonts have the ``latinized'' +version, but to avoid changing fonts they also provide the Devanagari +version in CharacterVariant 7. So the code +\begin{verbatim} +\begin{LARGE} +\begin{center} + 100\,₹ \hspace{2cm} {\addfontfeatures{CharacterVariant=7}100\,₹} +\end{center} +\end{LARGE} +\end{verbatim} +gives +\begin{LARGE} +\begin{center} + 100\,₹ \hspace{2cm} {\addfontfeatures{CharacterVariant=7}100\,₹} +\end{center} +\end{LARGE} @@ -945,6 +1130,36 @@ NewCM fixes that and provides a true BoldSans: \end{tabular} +\section{Food Allergies} + +Food allergy symbols have long been proposed to be accepted to +Unicode standard but there has not been any progress up to now. +The Sans 10 Regular and Book include standard allergy symbols +in the Private Use Area. Each glyph is named after what it represents. +For example, the soya symbol is named ``soya'' so copying the symbol +from a pdf file and pasting elsewhere you will get its name, that is ``soya''. +The symbols are in positions U+E033 to U+E040 and can be accessed using the +commands of the next table. +\begin{sffamily} + \newcommand{\fdal}[2]{\begin{minipage}[c]{2.5cm}\begin{center}\ \ \\[1.5ex] \scalebox{4}{\char"#1}\\ \textsc{#2}\\[-1ex] \ \ \end{center}\end{minipage}} +% \newcommand\crustaceans{\begin{minipage}[c]{1cm}\begin{center}\scalebox{4}{\char"E033}\\ \textsc{crustaceans}\end{center}\end{minipage}} +%\newcommand\eggs{\begin{minipage}[c]{1cm}\scalebox{4}{\char"E034}\\ \textsc{eggs}\end{minipage}} +\begin{center} + \begin{tabular}{|l|c||l|c|}\hline + \verb|\char"E033| & \fdal{E033}{crustaceans} & \verb|\char"E034| & \fdal{E034}{eggs}\\ \hline + \verb|\char"E035| & \fdal{E035}{gluten} & \verb|\char"E036| & \fdal{E036}{fish}\\ \hline + \verb|\char"E037| & \fdal{E037}{lupin} & \verb|\char"E038| & \fdal{E038}{milk}\\ \hline + \verb|\char"E039| & \fdal{E039}{molluscs} & \verb|\char"E03A| & \fdal{E03A}{mustard}\\ \hline + \end{tabular} + \end{center} +\begin{center} + \begin{tabular}{|l|c||l|c|}\hline + \verb|\char"E03B| & \fdal{E03B}{peanut} & \verb|\char"E03C| & \fdal{E03C}{sesame}\\ \hline + \verb|\char"E03D| & \fdal{E03D}{soya} & \verb|\char"E03E| & \fdal{E03E}{treenuts}\\ \hline + \verb|\char"E03F| & \fdal{E03F}{celery} & \verb|\char"E040| & \fdal{E040}{SO$_2$}\\ \hline + \end{tabular} +\end{center} +\end{sffamily} \section{Unicode Math coverage and options} @@ -952,7 +1167,7 @@ NewCM provides full Unicode math support, that is all Mathematics Unicode Slots presented in \url{http://www.unicode.org/charts/} - in both Math weights, Regular and Book. These blocks are: + in the Math weights, Regular, Book and Bold. These blocks are: \hspace*{-5em} \begin{description} \item[Mathematical Symbols] \ \\ @@ -999,6 +1214,97 @@ For example, \verb|\char"2BDA| will give the Hygeia symbol that appeared in TUGboat (see \cite{1}), being more realistic will be moved to a new font in the future with ornaments. + +\subsection{Bold Math} + +A complete math font, such as NewCM, contains all alphabetic characters in bold too. +These characters are typically accessed using the \verb|\mathbf| command. However, +this is not true for bold versions of symbols. This creates difficulties +when a user has some math in chapter or section titles, or when a user wants to +create a poster with colored background. In the later case the Regular and Book +weights look too light (especially with dark backgrounds) and one is in need +of a real Bold Math font, that has everything in Bold. + +NewCM, starting from version 6.0.0 provides an independent Bold Math font for the +first time for a Computer Modern font family. +Let us compare: + +Book inline: $\sqrt[3]{x+y\pm 1}=\sum_{n=1}^\infty \int_{K}f_n(x)\,dx$ and the same in display +$$\sqrt[3]{x+y\pm 1}=\sum_{n=1}^\infty \int_{K}f_n(x)\,dx.$$ + +\setmathfont{NewCMMath-Bold.otf} %[version=bold] +%\mathversion{bold} +\setmathfont[range={\mathscr,\mathbfscr},StylisticSet=1]{NewCMMath-Book.otf} + + +Bold inline: $\sqrt[3]{x+y\pm 1}=\sum_{n=1}^\infty \int_{K}f_n(x)\,dx$ and the same in display +$$\sqrt[3]{x+y\pm 1}=\sum_{n=1}^\infty \int_{K}f_n(x)\,dx.$$ + +Bold Math with colors inverted: + +\medskip + + +\begin{pspicture}(0,0)(10,2) +\psframe[fillstyle=solid,fillcolor=black](1.5,0)(9.5,2) +\rput(5.5,1){{\color{white}$\displaystyle\sqrt[3]{x+y\pm 1}=\sum_{n=1}^\infty \int_{K}f_n(x)\,dx.$}} +\end{pspicture} + +In order to use the Bold font for chapter and section titles, when using the \verb|default| +or \verb|olddefault| options of the \verb|fontsetup| package, change the math version +to bold with \verb|\mathversion{bold}| \textit{before} the commands for chapter and section +and switch back to normal with \verb|\mathversion{normal}| \textit{afterwards}. + +The \verb|unicode-math| package, according to it's documentation +has still some troubles with ``versions'' and the \verb|range| options. These seem to affect +at least the calligraphic and script math alphabets. In normal version for example +the commands \verb|\symcal| and \verb|\symscr| work as expected, but when one switches +to the bold version the \verb|\symscr| fails. In this case one can use +\verb|\symbfscr|: \usebox{\mathcalscrA} (which was \verb|$\symcal A$ $\symbfscr A$|). + +%\mathversion{normal} + +\subsection{Sans Serif Math} + +\setmathfont{NewCMSansMath-Regular.otf} +\newfontfamily{\sansAmSfont}{NewCMSansMath-Regular.otf} +\renewcommand{\AmS}{{\sansAmSfont $\symcal A$\kern-.1667em\lower.5ex\hbox{$\symcal M$}\kern-.125em$\symcal S$}} + + +As of version 7.0.0, the family includes a full-featured Math Sans font. Up to now such a font +did not exist in the CM family (although partial solutions existed), and it +posed a serious problem for scientific writing especially in the preparation of presentations. +The font supplied with NewCM covers all Unicode math slots but it also provides some new features. +The lowercase Latin alphabet has been completely re-worked so that it really looks +as it should when used for Math variables. The letters are + +$$|a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|q|r|s|t|u|v|w|x|y|z|$$ +In Large +\begin{Large} +$$|a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|q|r|s|t|u|v|w|x|y|z|$$ +\end{Large} +Moreover the calligraphic capitals have been re-worked to match +better with the sans serif style (see subsection~\ref{mathscript} for how to select them) +$$\symcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ + +Same goes for the \verb|\mathbb/\symbb| capital letters. They have been adjusted to match +the Sans design, both in style and weight: +$$\symbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ +It is also worth noticing that the SansSerifed and the Serifed letters have swapped slots +in the font. So if using the MathSans font and you write for example \verb|$\symsf{ABCDabcd}$| +you will get the Serifed version(!): $\symsf{ABCDabcd}$. This choice facilitates +converting a document with Serifed fonts using Sans for emphasis or differentiation +to keep these characteristics when changed to Sans. + +One can see the Sans Serif Math font in action in the provided file +\verb|testmath-newcm.pdf| which comes form the \AmS-{\sffamily L\kern-.5ex\raise.5ex\hbox{\textsc{a}}\kern-.2exT\kern-.4ex\lower.6ex\hbox{E}X} +bundle (here the logo is written in NewCMSans). + + + + +\setmathfont{NewCMMath-Book.otf} + \subsection{Optical sizes for more glyphs} So far the fonts provided optical sizes for 1st and 2nd order exponents for letters. This was not true though for binary operators and some symbols commonly used in mathematics. @@ -1014,9 +1320,9 @@ Zoom or print and compare with \texttt{lat\-in\-modern-math} font: \end{center} -\subsection{Math Script} +\subsection{Math Script}\label{mathscript} -Calligraphic letters are accessed as usual with \verb|\mathcal| producing +Calligraphic letters are accessed as usual with \verb|\mathcal| or \verb|\symcal|, producing $$\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ However, mathematicians often need a second level of ``scriptness''. The fonts @@ -1046,29 +1352,41 @@ $$ \setmathfont{NewCMMath-Book.otf}$$\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ +\subsection{Math Kerning} + +Math kerning has been added to all NewCM Math fonts. This feature +greatly improves Math typesetting, especially for the calligraphic letters but for regular +letters as well, such as, $Y$ or $\Gamma$. + + +\begin{center} + \includegraphics{mathkern-example.pdf} +\end{center} + + + \subsection{Blackboard Bold} -In version 5.0 of the fonts a new NewCM blackboard bold was introduced -in the place of the \textsc{ams} blackboard -bold letters. There were many complains from Mathematicians for this choice. -I have to make a statement here: it seems that although \textsc{ams} blackboard -bold are not matching with the computer modern design the long time -Mathematicians use them had its effect. People (including myself) -got used to it and find it difficult to feel their non-matching design. -Moreover, the new design can not be metrically equivalent with the past -so there will be slight changes in the older documents if re-run. -With this in mind and the fact that the new design should be considered -``beta'', as there are characters that need improvement, -I decided bring back to default the \textsc{ams} design and keep the -new design for new documents for the StylisticSet 3. - - - -The \textsc{ams} blackboard bold are: +The fonts contain as default the \textsc{ams} blackboard bold. These are: + $$\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ $$\mathbb{abcdefghijklmnopqrstuvwxyz}$$ -$$\mathbb{0123456789\ \pi\gamma\Gamma\Pi\Sigma\mitBbbD\mitBbbd\mitBbbe\mitBbbi\mitBbbj}$$ +$$\mathbb{0123456789\ \pi\gamma\Gamma\Pi\textrm{\mathastext\char"2140}\mitBbbD\mitBbbd\mitBbbe\mitBbbi\mitBbbj}$$ -To access the new ones needs to load the math font enabling the +They also contain a blackboard bold that matches the design of Computer Modern +but respecting the fact that most users have been used for a long period to +the AMS bb design. Compare +the +default \begin{center}{\LARGE $\textup{R}\in\mathbb R\qquad \textup{Q}\in\mathbb Q$} +\end{center} +with the new design +\setmathfont[StylisticSet=3]{NewCMMath-Book.otf} +\begin{center} +{\LARGE $\textup{R}\in\mathbb R\qquad \textup{Q}\in\mathbb Q$} +\end{center} + +\setmathfont{NewCMMath-Book.otf} + +To access this design one needs to load the math font enabling the \verb|ss03| stylistic set using for example \noindent\verb|\setmathfont[StylisticSet=3]{NewCMMath-Book.otf}| @@ -1076,14 +1394,53 @@ Then the above blackboard bold design changes to \setmathfont[StylisticSet=3]{NewCMMath-Book.otf} $$\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$ $$\mathbb{abcdefghijklmnopqrstuvwxyz}$$ -$$\mathbb{0123456789\ \pi\gamma\Gamma\Pi\Sigma\mitBbbD\mitBbbd\mitBbbe\mitBbbi\mitBbbj}$$ +$$\mathbb{0123456789\ \pi\gamma\Gamma\Pi\textrm{\mathastextnewcm\char"2140}\mitBbbD\mitBbbd\mitBbbe\mitBbbi\mitBbbj}$$ \setmathfont{NewCMMath-Book.otf} +\newsavebox{\emptysetdefault} +\savebox{\emptysetdefault}{$\emptyset$} -If using the latest \verb|fontsetup| (to be released soon) +\setmathfont[CharacterVariant=1]{NewCMMath-Book.otf} + +If using the latest \verb|fontsetup| then you can choose the NewCM blackboard bold with the option \verb|newcmbb|. +\subsubsection{Small Caps for Blackboard Bold} + +It happens one to need Blackboard Bold in Small Caps. For example, +this may arise when running heads +are in small caps and contain a Blackboard Bold symbol. Consider for example +the case when a chapter is named ``The Lebesgue measure on $\mathbb R$''. +Then the running head in small caps will look awkward: +\begin{center} +\textsc{the lebesgue measure on} $\mathbb R$ +\end{center} +In such cases, a small caps version of the number sets is needed. NewCM provides +the needed glyphs in \verb|ss05|. So setting + +\noindent\verb|\setmathfont[StylisticSet=5]{NewCMMath-Book.otf}| + +\noindent the above +running head becomes: +\setmathfont[StylisticSet=5]{NewCMMath-Book.otf} +\begin{center} +\textsc{the lebesgue measure on} $\mathbb R$ +\end{center} +\setmathfont{NewCMMath-Book.otf} +These glyphs are provided in the CM style (instead of the AMS bb) +by enabling \verb|ss03| too. So the command + +\noindent\verb|\setmathfont[StylisticSet=3,StylisticSet=5]{NewCMMath-Book.otf}| + +\noindent will produce: +\setmathfont[StylisticSet=3,StylisticSet=5]{NewCMMath-Book.otf} +\begin{center} +\textsc{the lebesgue measure on} $\mathbb R$ +\end{center} +\setmathfont[CharacterVariant=1]{NewCMMath-Book.otf} +The same is true for the Regular and Bold Math fonts. + \subsection{Upright and extensible integrals} The Math fonts (both Regular and Book weights) include upright integrals in the ss02 StylisticSet. @@ -1113,7 +1470,7 @@ for the regular weight. Moreover, extensible integrals are supported by the fonts but \textit{NOT} by the Unicode TeX engines. The following code is a trick so that extensible integrals can be -constructed using Lua\LaTeX. The result is shown at the end +constructed using \LuaLaTeX. The result is shown at the end of the article. What the code below does, is that it defines the slot uni222B (integral) as a delimiter. And then this is extended as a delimiter with the mechanism that @@ -1144,20 +1501,30 @@ $ \end{tabular} \subsection{Additional and alternative characters in Math} + +\subsubsection{Alternative symbols} + The Math fonts provide the character \verb|\varnothing| ($\varnothing$), as -an alternative to \verb|\emptyset| (a slashed zero), +an alternative to \verb|\emptyset| (\usebox{\emptysetdefault}), through Character Variant \verb|cv01|. The \verb|fontsetup| package provides the option `\verb|varnothing|' to easily switch to the alternative character. -It also provides two more symbols that correspond to the commands + +\subsubsection{Additional symbols} + +%\newcommand{\twoheadhookrightarrow}{\char"E034} +%\newcommand{\twoheadhookleftarrow}{\char"E035} + +It also provides four more arrows that correspond to the commands \medskip - \verb|\nrightrightarrows| ($\nrightrightarrows$) + \verb|\nrightrightarrows| ($\nrightrightarrows$) \quad \verb|\nleftleftarrows| ($\nleftleftarrows$) \noindent and -\verb|\nleftleftarrows| ($\nleftleftarrows$) +\verb|\twoheadhookrightarrow| ($\twoheadhookrightarrow$)\quad \verb|\twoheadhookleftarrow| ($\twoheadhookleftarrow$) + \medskip @@ -1166,6 +1533,45 @@ and supported by the \verb|default| and \verb|olddefault| options of the \verb|f These symbols are not in the Unicode Standard and so they are added in the Private Area of the fonts. +\subsubsection{Additional operators} + +An operator for convolution of functions seems to have long been forgotten from both +Unicode and the \TeX\ world. +The \verb|default| and \verb|olddefault| options of the \verb|fontsetup| package +define a new command \verb|\convolution| which behaves just like the \verb|\sum| and +\verb|\int| operators. +The convolution of $N$ functions in inline mode: $\convolution_{n=1}^N f_n$ and the same +in display mode: $$\convolution_{n=1}^N f_n.$$ + +\subsubsection{``Smoother'' changing radicals} + +One more radical size has been added that improves the way radical sizes change. Compare +the previous state +\setmathfont{latinmodern-math.otf} +$$\sqrt{q}\quad\sqrt{q_1^2}\quad\sqrt{q_1^{2^3}}\quad\sqrt{\frac12}$$ with the new one +\setmathfont[CharacterVariant=1]{NewCMMath-Book.otf} +$$\sqrt{q}\quad\sqrt{q_1^2}\quad\sqrt{q_1^{2^3}}\quad\sqrt{\frac12}.$$ + + +\subsection{Notes on Mathematics} + +Extensible tildes and hats produce different results +with \XeLaTeX\ and \LuaLaTeX\ because these unicode +engines treat differently the width of characters. In particular, \XeLaTeX\ handles italic +correction as part of the character width but \LuaLaTeX\ does this only if a character +follows. This affects how extensible accents like \verb|\widehat| or \verb|\widetilde| +select the proper size. With \XeLaTeX, \verb|\widetilde{Y}| \verb|\widetilde{X}| will give the +expected result $\widetilde{Y}\widetilde{X}$; but with \LuaLaTeX\ the letter $Y$ is +narrower (and its italic correction is not taken into account if it stands alone) +and gets the plain tilde size, as in $\tilde{Y}$. To bypass this with \LuaLaTeX\ one +has to use a zero width character after a letter such as $Y$ so that +\LuaLaTeX\ takes into account the italic correction of $Y$. For example with +\begin{center}\verb|\widetilde{Y\Uchar"200D}| +\end{center} +the result will be correct (U200D is the ``Zero Width Joiner'' in Unicode). + +I thank Ulrike Fischer for this solution. + \section{The Medieval Latin and Uncial Greek glyph complement} @@ -1210,8 +1616,6 @@ Private Area of the fonts. - - \begin{thebibliography}{9} \bibitem[\textsc{at}]{1} Antonis Tsolomitis, \textit{The NewComputerModern font family}, \textsc{tug}boat Vol.~\textsc{42}, No.~\textsc{1}, \textsc{2021}. \bibitem[\textsc{ipa}rev]{2} Council actions on revisions of the \textsc{ipa}, Phonetic Representation: b) Revision of the \textsc{ipa}, Journal of the International Phonetic Association, Volume \textsc{23}, Issue \textsc{1}, diff --git a/fonts/newcomputermodern/doc/testmath-newcm-sans.pdf b/fonts/newcomputermodern/doc/testmath-newcm-sans.pdf Binary files differnew file mode 100644 index 0000000000..9ff206ea70 --- /dev/null +++ b/fonts/newcomputermodern/doc/testmath-newcm-sans.pdf diff --git a/fonts/newcomputermodern/doc/testmath-newcm-sans.tex b/fonts/newcomputermodern/doc/testmath-newcm-sans.tex new file mode 100644 index 0000000000..44ee168372 --- /dev/null +++ b/fonts/newcomputermodern/doc/testmath-newcm-sans.tex @@ -0,0 +1,2321 @@ +%%% ==================================================================== +%%% filename = testmath.tex +%%% version = 2.0a +%%% date = 2023/08/24 +%%% author = American Mathematical Society +%%% copyright = Copyright 1995, 1999 American Mathematical Society +%%% 2023 LaTeX Project. +%%% License = https://www.latex-project.org/lppl/lppl-1-3c +%%% keywords = latex, amsmath, examples, documentation +%%% abstract = This is a test file containing extensive examples of +%%% mathematical constructs supported by the amsmath +%%% package." +%%% ==================================================================== +\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) +[1994/12/01]% LaTeX date must December 1994 or later +\documentclass[draft]{article} +\synctex=1 +\pagestyle{headings} + +\title{Sample Paper for the \pkg{amsmath} Package\\ +File name: \fn{testmath.tex}} +\author{American Mathematical Society} +\date{Version 2.0a, 2023/08/24} + +\usepackage{amsmath,amsthm} +\usepackage[sansdefault]{fontsetup} +\newfontfamily{\sansAmSfont}{NewCMSansMath-Regular.otf} +\renewcommand{\AmS}{{\sansAmSfont $\symcal A$\kern-.1667em\lower.5ex\hbox{$\symcal M$}\kern-.125em$\symcal S$}} +% \input fspdefault.tex + +% Some definitions useful in producing this sort of documentation: +\chardef\bslash=`\\ % p. 424, TeXbook +% Normalized (nonbold, nonitalic) tt font, to avoid font +% substitution warning messages if tt is used inside section +% headings and other places where odd font combinations might +% result. +\newcommand{\ntt}{\normalfont\ttfamily} +% command name +\newcommand{\cn}[1]{{\protect\ntt\bslash#1}} +% LaTeX package name +\newcommand{\pkg}[1]{{\protect\ntt#1}} +% File name +\newcommand{\fn}[1]{{\protect\ntt#1}} +% environment name +\newcommand{\env}[1]{{\protect\ntt#1}} +\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc + +% Theorem environments + +%% \theoremstyle{plain} %% This is the default +\newtheorem{thm}{Theorem}[section] +\newtheorem{cor}[thm]{Corollary} +\newtheorem{lem}[thm]{Lemma} +\newtheorem{prop}[thm]{Proposition} +\newtheorem{ax}{Axiom} + +\theoremstyle{definition} +\newtheorem{defn}{Definition}[section] + +\theoremstyle{remark} +\newtheorem{rem}{Remark}[section] +\newtheorem*{notation}{Notation} + +%\numberwithin{equation}{section} + +\newcommand{\thmref}[1]{Theorem~\ref{#1}} +\newcommand{\secref}[1]{\S\ref{#1}} +\newcommand{\lemref}[1]{Lemma~\ref{#1}} + +\newcommand{\bysame}{\mbox{\rule{3em}{.4pt}}\,} + +% Math definitions + +\newcommand{\A}{{\symcal A}} +\newcommand{\B}{{\symcal B}} +\newcommand{\st}{\sigma} +\newcommand{\XcY}{{(X,Y)}} +\newcommand{\SX}{{S_X}} +\newcommand{\SY}{{S_Y}} +\newcommand{\SXY}{{S_{X,Y}}} +\newcommand{\SXgYy}{{S_{X|Y}(y)}} +\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}} +\newcommand{\G}{{G(X|Y)}} +\newcommand{\PY}{{P_{\symcal Y}}} +\newcommand{\X}{{\symcal X}} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} + +\DeclareMathOperator{\per}{per} +\DeclareMathOperator{\cov}{cov} +\DeclareMathOperator{\non}{non} +\DeclareMathOperator{\cf}{cf} +\DeclareMathOperator{\add}{add} +\DeclareMathOperator{\Cham}{Cham} +\DeclareMathOperator{\IM}{Im} +\DeclareMathOperator{\esssup}{ess\,sup} +\DeclareMathOperator{\meas}{meas} +\DeclareMathOperator{\seg}{seg} + +% \interval is used to provide better spacing after a [ that +% is used as a closing delimiter. +\newcommand{\interval}[1]{\mathinner{#1}} + +% Notation for an expression evaluated at a particular condition. The +% optional argument can be used to override automatic sizing of the +% right vert bar, e.g. \eval[\biggr]{...}_{...} +\newcommand{\eval}[2][\right]{\relax + \ifx#1\right\relax \left.\fi#2#1\rvert} + +% Enclose the argument in vert-bar delimiters: +\newcommand{\envert}[1]{\left\lvert#1\right\rvert} +\let\abs=\envert + +% Enclose the argument in double-vert-bar delimiters: +\newcommand{\enVert}[1]{\left\lVert#1\right\rVert} +\let\norm=\enVert + +\begin{document} +\maketitle +\markboth{Sample paper for the {\protect\ntt\lowercase{amsmath}} package} +{Sample paper for the {\protect\ntt\lowercase{amsmath}} package} +\renewcommand{\sectionmark}[1]{} + +\section{Introduction} + +This paper contains examples of various features from \AmS-\LaTeX{}. + + +\section{Enumeration of Hamiltonian paths in a graph} + +Let $\symbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The +corresponding Kirchhoff matrix $\symbf{K}=(k_{ij})$ is obtained from +$\symbf{A}$ by replacing in $-\symbf{A}$ 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\symbf{K}(i|i)=\text{ the number of spanning trees of $G$}, +\quad i=1,\dots,n +\end{equation} +where $\symbf{K}(i|i)$ is the $i$th principal submatrix of +$\symbf{K}$. +\begin{verbatim} +\det\symbf{K}(i|i)=\text{ the number of spanning trees of $G$}, +\end{verbatim} + +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 $k_{ii}\det +\symbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$. +\begin{verbatim} +$\wh X=\{\hat x_1,\dots,\hat x_n\}$ +\end{verbatim} +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 \cite{liuchow:formalsum} +\begin{equation}\label{H-cycles} +\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det +\wh{\symbf{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} +\cite{hapa:graphenum}. 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 \cite{gouja:lagrmeth}. 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. + +\section{Main Theorem} +\label{s:mt} + +\begin{notation} For $p,q\in P$ and $n\in\omega$ we write +$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$. +\begin{verbatim} +\begin{notation} For $p,q\in P$ and $n\in\omega$ +... +\end{notation} +\end{verbatim} +\end{notation} + +Let $\symbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\symbf{n}=\{1, +\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen +that + +\begin{lem}\label{lem-per} +\begin{equation} +\prod_{i\in\symbf{n}} +\biggl(\sum_{\,j\in\symbf{n}}b_{ij}\hat x_i\biggr) +=\biggl(\prod_{\,i\in\symbf{n}}\hat x_i\biggr)\per \symbf{B} +\end{equation} +where $\per \symbf{B}$ is the permanent of $\symbf{B}$. +\end{lem} + +Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication +for the elements of $\wh Y$ by +\begin{equation} +\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n. +\end{equation} +Then, it follows that +\begin{lem}\label{lem-det} +\begin{equation}\label{detprod} +\prod_{i\in\symbf{n}} +\biggl(\sum_{\,j\in\symbf{n}}b_{ij}\hat y_j\biggr) +=\biggl(\prod_{\,i\in\symbf{n}}\hat y_i\biggr)\det\symbf{B}. +\end{equation} +\end{lem} + +Note that all basic properties of determinants are direct consequences +of Lemma~\ref{lem-det}. Write +\begin{equation}\label{sum-bij} +\sum_{j\in\symbf{n}}b_{ij}\hat y_j=\sum_{j\in\symbf{n}}b^{(\lambda)} +_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y +\end{equation} +where +\begin{equation} +b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij}, +\quad i\not=j. +\end{equation} +Let $\symbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod} +and \eqref{sum-bij}, it is +straightforward to show the following +result: +\begin{thm}\label{thm-main} +\begin{equation}\label{detB} +\det\symbf{B}= +\sum^n_{l =0}\sum_{I_l \subseteq n} +\prod_{i\in I_l}(b_{ii}-\lambda_i) +\det\symbf{B}^{(\lambda)}(I_l |I_l ), +\end{equation} +where $I_l =\{i_1,\dots,i_l \}$ and $\symbf{B}^{(\lambda)}(I_l |I_l )$ +is the principal submatrix obtained from $\symbf{B}^{(\lambda)}$ +by deleting its $i_1,\dots,i_l $ rows and columns. +\end{thm} + +\begin{rem} +Let $\symbf{M}$ be an $n\times n$ matrix. The convention +$\symbf{M}(\symbf{n}|\symbf{n})=1$ has been used in \eqref{detB} and +hereafter. +\end{rem} + +Before proceeding with our discussion, we pause to note that +\thmref{thm-main} yields immediately a fundamental formula which can be +used to compute the coefficients of a characteristic polynomial +\cite{mami:matrixth}: +\begin{cor}\label{BI} +Write $\det(\symbf{B}-x\symbf{I})=\sum^n_{l =0}(-1) +^l b_l x^l $. Then +\begin{equation}\label{bl-sum} +b_l =\sum_{I_l \subseteq\symbf{n}}\det\symbf{B}(I_l |I_l ). +\end{equation} +\end{cor} +Let +\begin{equation} +\symbf{K}(t,t_1,\dots,t_n) +=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ +-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ +\hdotsfor[2]{4}\\ +-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}, +\end{equation} +\begin{verbatim} +\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ +-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ +\hdotsfor[2]{4}\\ +-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix} +\end{verbatim} +where +\begin{equation} +D_i=\sum_{j\in\symbf{n}}a_{ij}t_j,\quad i=1,\dots,n. +\end{equation} + +Set +\begin{equation*} +D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\symbf{K}(t,t_1,\dots,t_n) +}_{t=1}. +\end{equation*} +Then +\begin{equation}\label{sum-Di} +D(t_1,\dots,t_n) +=\sum_{i\in\symbf{n}}D_i\det\symbf{K}(t=1,t_1,\dots,t_n; i|i), +\end{equation} +where $\symbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal +submatrix of $\symbf{K}(t=1,t_1,\dots,t_n)$. + +Theorem~\ref{thm-main} leads to +\begin{equation}\label{detK1} +\det\symbf{K}(t_1,t_1,\dots,t_n) +=\sum_{I\in\symbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}} +\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\symbf{A} +^{(\lambda t)}(\overline{I}|\overline I). +\end{equation} +Note that +\begin{equation}\label{detK2} +\det\symbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\symbf{n}}(-1)^{\envert{I}} +\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\symbf{A} +^{(\lambda)}(\overline{I}|\overline{I})=0. +\end{equation} + +Let $t_i=\hat x_i,i=1,\dots,n$. Lemma~\ref{lem-per} yields +\begin{multline} +\biggl(\sum_{\,i\in\symbf{n}}a_{l _i}x_i\biggr) +\det\symbf{K}(t=1,x_1,\dots,x_n;l |l )\\ +=\biggl(\prod_{\,i\in\symbf{n}}\hat x_i\biggr) +\sum_{I\subseteq\symbf{n}-\{l \}} +(-1)^{\envert{I}}\per\symbf{A}^{(\lambda)}(I|I) +\det\symbf{A}^{(\lambda)} +(\overline I\cup\{l \}|\overline I\cup\{l \}). +\label{sum-ali} +\end{multline} +\begin{verbatim} +\begin{multline} +\biggl(\sum_{\,i\in\symbf{n}}a_{l _i}x_i\biggr) +\det\symbf{K}(t=1,x_1,\dots,x_n;l |l )\\ +=\biggl(\prod_{\,i\in\symbf{n}}\hat x_i\biggr) +\sum_{I\subseteq\symbf{n}-\{l \}} +(-1)^{\envert{I}}\per\symbf{A}^{(\lambda)}(I|I) +\det\symbf{A}^{(\lambda)} +(\overline I\cup\{l \}|\overline I\cup\{l \}). +\label{sum-ali} +\end{multline} +\end{verbatim} + +By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have +\begin{prop}\label{prop:eg} +\begin{equation} +H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l} +D_{l}, +\end{equation} +where +\begin{equation}\label{delta-l} +D_{l}=\eval[2]{\sum_{I_{l}\subseteq \symbf{n}} +D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix} +0,& \text{if }i\in I_{l}\quad\\% \quad added for centering +1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}. +\end{equation} +\end{prop} + +\section{Application} +\label{lincomp} + +We consider here the applications of Theorems~\ref{th-info-ow-ow} +and~\ref{th-weak-ske-owf} to a complete +multipartite graph $K_{n_1\dots n_p}$. It can be shown that the +number of spanning trees of $K_{n_1\dots n_p}$ +may be written +\begin{equation}\label{e:st} +T=n^{p-2}\prod^p_{i=1} +(n-n_i)^{n_i-1} +\end{equation} +where +\begin{equation} +n=n_1+\dots+n_p. +\end{equation} + +It follows from Theorems~\ref{th-info-ow-ow} and~\ref{th-weak-ske-owf} that +\begin{equation}\label{e:barwq} +\begin{split} +H_c&=\frac1{2n} +\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2} +\sum_{l _1+\dots+l _p=l}\prod^p_{i=1} +\binom{n_i}{l _i}\\ +&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot +\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split} +\end{equation} +\begin{verbatim} +... \binom{n_i}{l _i}\\ +\end{verbatim} +and +\begin{equation}\label{joe} +\begin{split} +H_c&=\frac12\sum^{n-1}_{l =0} +(-1)^{l}(n-l )^{p-2} +\sum_{l _1+\dots+l _p=l} +\prod^p_{i=1}\binom{n_i}{l _i}\\ +&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i} +\left(1-\frac{l _p}{n_p}\right) +[(n-l )-(n_p-l _p)]. +\end{split} +\end{equation} + +The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be +carried out by Theorem~\ref{thm-H-param} or~\ref{thm-asym} +together with the algebraic method of \eqref{multdef}. +Some elegant representations may be obtained. For example, $H_c$ in +a $K_{n_1n_2n_3}$ graph may be written +\begin{equation}\label{j:mark} +\begin{split} +H_c=& +\frac{n_1!\,n_2!\,n_3!} +{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i} +\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\ +&+\left.\binom{n_1-1}{i} +\binom{n_2-1}{n_3-n_1+i} +\binom{n_3-1}{n_3-n_2+i}\right].\end{split} +\end{equation} + +\section{Secret Key Exchanges} +\label{SKE} + +Modern cryptography is fundamentally concerned with the problem of +secure private communication. A Secret Key Exchange is a protocol +where Alice and Bob, having no secret information in common to start, +are able to agree on a common secret key, conversing over a public +channel. The notion of a Secret Key Exchange protocol was first +introduced in the seminal paper of Diffie and Hellman +\cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete +implementation of a Secret Key Exchange protocol, dependent on a +specific assumption (a variant on the discrete log), specially +tailored to yield Secret Key Exchange. Secret Key Exchange is of +course trivial if trapdoor permutations exist. However, there is no +known implementation based on a weaker general assumption. + +The concept of an informationally one-way function was introduced +in \cite{imlelu:oneway}. We give only an informal definition here: + +\begin{defn} A polynomial time +computable function $f = \{f_k\}$ is informationally +one-way if there is no probabilistic polynomial time algorithm which +(with probability of the form $1 - k^{-e}$ for some $e > 0$) +returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$. +\end{defn} +In the non-uniform setting \cite{imlelu:oneway} show that these are not +weaker than one-way functions: +\begin{thm}[\cite{imlelu:oneway} (non-uniform)] +\label{th-info-ow-ow} +The existence of informationally one-way functions +implies the existence of one-way functions. +\end{thm} +We will stick to the convention introduced above of saying +``non-uniform'' before the theorem statement when the theorem +makes use of non-uniformity. It should be understood that +if nothing is said then the result holds for both the uniform and +the non-uniform models. + +It now follows from \thmref{th-info-ow-ow} that + +\begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE +implies the existence of a one-way function. +\end{thm} + +More recently, the polynomial-time, interior point algorithms for linear +programming have been extended to the case of convex quadratic programs +\cite{moad:quadpro,ye:intalg}, certain linear complementarity problems +\cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity +problem \cite{komiyo:unipfunc}. The connection between these algorithms +and the classical Newton method for nonlinear equations is well +explained in \cite{komiyo:lincomp}. + +\section{Review} +\label{computation} + +We begin our discussion with the following definition: + +\begin{defn} + +A function $H\colon \Re^n \to \Re^n$ is said to be +\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz +continuous in a neighborhood of $z$, and (ii)~there exists a positive +homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the +\emph{B-derivative} of $H$ at $z$, such that +\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \] +The function $H$ is \textit{B-differentiable in set $S$} if it is +B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said +to be \textit{strong} if +\[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v + -v')}{\enVert{v - v'}} = 0. \] +\end{defn} + + +\begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$ +defined for $\abs{z}>1-2a$ satisfying the following properties\textup{:} +\begin{enumerate} +\renewcommand{\labelenumi}{(\roman{enumi})} +\item $\psi_0(z)$ is bounded above and below by positive constants +$c_1\leq \psi_0(z)\leq c_2$. +\item If $\abs{z}>1$, then $\psi_0(z)=1$. +\item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$. +\item If $1-2a<\abs{z}<1-a$, then $\Delta_0\ln \psi_0\geq +c_3>0$. +\end{enumerate} +\end{lem} + +\begin{proof} +We choose $\psi_0(z)$ to be a radial function depending only on $r=\abs{z}$. +Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$ +for $1-2a<\abs{z}<1-a$, and $h(r)=0$ for $\abs{z}>1-\tfrac a2$. The radial +Laplacian +\[\Delta_0\ln\psi_0(r)=\left(\frac {d^2}{dr^2}+\frac +1r\frac d{dr}\right)\ln\psi_0(r)\] +has smooth coefficients for $r>1-2a$. Therefore, we may +apply the existence and uniqueness theory for ordinary differential +equations. Simply let $\ln \psi_0(r)$ be the solution of the differential +equation +\[\left(\frac{d^2}{dr^2}+\frac 1r\frac d{dr}\right)\ln \psi_0(r)=h(r)\] +with initial conditions given by $\ln \psi_0(1)=0$ and +$\ln\psi_0'(1)=0$. + +Next, let $D_\nu$ be a finite collection of pairwise disjoint disks, +all of which are contained in the unit disk centered at the origin in +$C$. We assume that $D_\nu=\{z\mid \abs{z-z_\nu}<\delta\}$. Suppose that +$D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z\mid +\abs{z-z_\nu}\leq (1-2a)\delta\}$. We define a smooth weight function +$\Phi_0(z)$ for $z\in C-\bigcup_\nu D_\nu(a)$ by setting $\Phi_ +0(z)=1$ when $z\notin \bigcup_\nu D_\nu$ and $\Phi_ +0(z)=\psi_0((z-z_\nu)/\delta)$ when $z$ is an element of $D_\nu$. It +follows from \lemref{limbog} that $\Phi_ 0$ satisfies the properties: +\begin{enumerate} +\renewcommand{\labelenumi}{(\roman{enumi})} +\item \label{boundab}$\Phi_ 0(z)$ is bounded above and below by +positive constants $c_1\leq \Phi_ 0(z)\leq c_2$. +\item \label{d:over}$\Delta_0\ln\Phi_ 0\geq 0$ for all +$z\in C-\bigcup_\nu D_\nu(a)$, +the domain where the function $\Phi_ 0$ is defined. +\item \label{d:ad}$\Delta_0\ln\Phi_ 0\geq c_3\delta^{-2}$ +when $(1-2a)\delta<\abs{z-z_\nu}<(1-a)\delta$. +\end{enumerate} +Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<\abs{z-z_\nu}<(1-a) +\delta \}$, and set $A=\bigcup_\nu A_\nu$. The +properties (\ref{d:over}) and (\ref{d:ad}) of $\Phi_ 0$ +may be summarized as $\Delta_0\ln \Phi_ 0\geq c_3\delta^{-2}\chi_A$, +where $\chi _A$ is the characteristic function of $A$. +\end{proof} + +Suppose that $\alpha$ is a nonnegative real constant. We apply +Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha\abs{z}^2}$. If +$u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\symcal{D}$ +is a bounded domain containing the support of $u$ and $A\subset +\symcal{D}\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives +\[\int_{\symcal{D}}\abs{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\abs{z}^2} +\geq c_4\alpha\int_{\symcal{D}}\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2} ++c_5\delta^{-2}\int_ A\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}.\] + +The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields +\[\int_{\symcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha +\int_{\symcal{D}}\abs{u}^2e^{\alpha\abs{z}^2} ++c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\] + +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 $\abs{Q_{X}} = w^{b(X)}$. + +Next note that $b(X)=\dim X$. We see this by choosing a +basis for $X$ consisting of vectors $v^{k}$ defined by +\[v^{k}_{i}= +\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ +0 &\text{otherwise.} \end{cases} +\] +\begin{verbatim} +\[v^{k}_{i}= +\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ +0 &\text{otherwise.} \end{cases} +\] +\end{verbatim} + +\begin{lem}\label{p0201} +Let $\A$ be an arrangement. Then +\[ \chi (\A,t) = \sum_{\B \subseteq \A} +(-1)^{\abs{\B}} t^{\dim T(\B)}. \] +\end{lem} + +In order to compute $R''$ recall the definition +of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$, +$\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then +$\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then +\begin{equation}\label{E_SXgYy} +\begin{split} +R''&= \sum_{H\in \B \subseteq \A} (-1)^{\abs{\B}} +t^{\dim T(\B)}\\ +&= \sum_{Y \in L''} \sum_{\B \in S(H,Y)} +(-1)^{\abs{\B}}t^{\dim Y} \\ +&= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^ +{\abs{\B - \A_{H}}} t^{\dim Y} \\ +&= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\ +&= -\chi (\A '',t). +\end{split} +\end{equation} + +\begin{cor}\label{tripleA} +Let $(\A,\A',\A'')$ be a triple of arrangements. Then +\[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \] +\end{cor} + +\begin{defn} +Let $(\A,\A',\A'')$ be a triple with respect to +the hyperplane $H \in \A$. Call $H$ a \textit{separator} +if $T(\A) \not\in L(\A')$. +\end{defn} + +\begin{cor}\label{nsep} +Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$. +\begin{enumerate} +\renewcommand{\labelenumi}{(\roman{enumi})} +\item +If $H$ is a separator then +\[ \mu (\A) = - \mu (\A'') \] +and hence +\[ \abs{\mu (\A)} = \abs{ \mu (\A'')}. \] + +\item If $H$ is not a separator then +\[\mu (\A) = \mu (\A') - \mu (\A'') \] +and +\[ \abs{\mu (\A)} = \abs{\mu (\A')} + \abs{\mu (\A'')}. \] +\end{enumerate} +\end{cor} + +\begin{proof} +It follows from \thmref{th-info-ow-ow} that $\pi(\A,t)$ +has leading term +\[(-1)^{r(\A)}\mu (\A)t^{r(\A)}.\] +The conclusion +follows by comparing coefficients of the leading +terms on both sides of the equation in +Corollary~\ref{tripleA}. If $H$ is a separator then +$r(\A') < r(\A)$ and there is no contribution +from $\pi (\A',t)$. +\end{proof} + +The Poincar\'e polynomial of an arrangement +will appear repeatedly +in these notes. It will be shown to equal the +Poincar\'e polynomial +of the graded algebras which we are going to +associate with $\A$. It is also the Poincar\'e +polynomial of the complement $M(\A)$ for a +complex arrangement. Here we prove +that the Poincar\'e polynomial is the chamber +counting function for a real arrangement. The +complement $M(\A)$ is a disjoint union of chambers +\[M(\A) = \bigcup_{C \in \Cham(\A)} C.\] +The number +of chambers is determined by the Poincar\'e +polynomial as follows. + +\begin{thm}\label{th-realarr} +Let $\A_{\symbf{R}}$ be a real arrangement. Then +\[ \abs{\Cham(\A_{\symbf{R}})} = \pi (\A_{\symbf{R}},1). \] +\end{thm} + +\begin{proof} +We check the properties required in Corollary~\ref{nsep}: +(i) follows from $\pi (\Phi_{ l},t) = 1$, and (ii) is a +consequence of Corollary~\ref{BI}. +\end{proof} + +\begin{figure} +\vspace{5cm} +\caption[]{$Q(\A_{1}) = xyz(x-z)(x+z)(y-z)(y+z)$} +\end{figure} + +\begin{figure} +\vspace{5cm} +\caption[]{$Q(\A_{2})= xyz(x+y+z)(x+y-z)(x-y+z)(x-y-z)$} +\end{figure} + + +\begin{thm} +\label{T_first_the_int} +Let $\phi$ be a protocol for a random pair $\XcY$. +If one of $\st_\phi(x',y)$ and $\st_\phi(x,y')$ is a prefix of the other +and $(x,y)\in\SXY$, then +\[ +\langle \st_j(x',y)\rangle_{j=1}^\infty +=\langle \st_j(x,y)\rangle_{j=1}^\infty +=\langle \st_j(x,y')\rangle_{j=1}^\infty . +\] +\end{thm} +\begin{proof} +We show by induction on $i$ that +\[ +\langle \st_j(x',y)\rangle_{j=1}^i +=\langle \st_j(x,y)\rangle_{j=1}^i +=\langle \st_j(x,y')\rangle_{j=1}^i. +\] +The induction hypothesis holds vacuously for $i=0$. Assume it holds for +$i-1$, in particular +$[\st_j(x',y)]_{j=1}^{i-1}=[\st_j(x,y')]_{j=1}^{i-1}$. Then one of +$[\st_j(x',y)]_{j=i}^{\infty}$ and $[\st_j(x,y')]_{j=i}^{\infty}$ is a +prefix of the other which implies that one of $\st_i(x',y)$ and +$\st_i(x,y')$ is a prefix of the other. If the $i$th message is +transmitted by $P_\X$ then, by the separate-transmissions property and +the induction hypothesis, $\st_i(x,y)=\st_i(x,y')$, hence one of +$\st_i(x,y)$ and $\st_i(x',y)$ is a prefix of the other. By the +implicit-termination property, neither $\st_i(x,y)$ nor $\st_i(x',y)$ +can be a proper prefix of the other, hence they must be the same and +$\st_i(x',y)=\st_i(x,y)=\st_i(x,y')$. If the $i$th message is +transmitted by $\PY$ then, symmetrically, $\st_i(x,y)=\st_i(x',y)$ by +the induction hypothesis and the separate-transmissions property, and, +then, $\st_i(x,y)=\st_i(x,y')$ by the implicit-termination property, +proving the induction step. +\end{proof} + +If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct +inputs in $\SXY$, then, by the correct-decision property, +$\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle +\st_j(x',y)\rangle_{j=1}^\infty$. + +Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$ +to be the set of possible $X$ values when $Y=y$. +The last corollary implies that for all $y\in\SY$, +the multiset% +\footnote{A multiset allows multiplicity of elements. +Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.} +of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free. + +\section{One-Way Complexity} +\label{S_Cp1} + +$\Cw1$, the one-way complexity of a random pair $\XcY$, +is the number of bits $P_\X$ must transmit in the worst case +when $\PY$ is not permitted to transmit any feedback messages. +Starting with $\SXY$, the support set of $\XcY$, we define $\G$, +the \textit{characteristic hypergraph} of $\XcY$, and show that +\[ +\Cw1=\lceil\,\log\chi(\G)\rceil\ . +\] + +Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of +$Y$, Equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible +$x$ values when $Y=y$. The \textit{characteristic hypergraph} $\G$ of +$\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each +$y\in\SY$. + + +We can now prove a continuity theorem. +\begin{thm}\label{t:conl} +Let $\Omega \subset\symbf{R}^n$ be an open set, let +$u\in BV(\Omega ;\symbf{R}^m)$, and let +\begin{equation}\label{quts} +T^u_x=\left\{y\in\symbf{R}^m: + y=\tilde u(x)+\left\langle \frac{Du}{\abs{Du}}(x),z +\right\rangle \text{ for some }z\in\symbf{R}^n\right\} +\end{equation} +for every $x\in\Omega \backslash S_u$. Let $f\colon \symbf{R}^m\to +\symbf{R}^k$ be a Lipschitz continuous function such that $f(0)=0$, and +let $v=f(u)\colon \Omega \to \symbf{R}^k$. Then $v\in BV(\Omega +;\symbf{R}^k)$ and +\begin{equation} +Jv=\eval{(f(u^+)-f(u^-))\otimes \nu_u\cdot\, +\symcal{H}_{n-1}}_{S_u}. +\end{equation} +In addition, for $\abs{\wt{D}u}$-almost every $x\in\Omega $ the +restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde +u(x)$ and +\begin{equation} +\wt{D}v=\nabla (\eval{f}_{T^u_x})(\tilde u) +\frac{\wt{D}u}{\abs{\wt{D}u}}\cdot\abs{\wt{D}u}.\end{equation} +\end{thm} + +Before proving the theorem, we state without proof three elementary +remarks which will be useful in the sequel. +\begin{rem}\label{r:omb} +Let $\omega\colon \left]0,+\infty\right[\to \left]0,+\infty\right[$ +be a continuous function such that $\omega (t)\to 0$ as $t\to +0$. Then +\[\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to +0^+}g(h)=L\] +for any function $g\colon \left]0,+\infty\right[\to \symbf{R}$. +\end{rem} +\begin{rem}\label{r:dif} +Let $g \colon \symbf{R}^n\to \symbf{R}$ be a Lipschitz +continuous function and assume that +\[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\] +exists for every $z\in\symbf{Q}^n$ and that $L$ is a linear function of +$z$. Then $g$ is differentiable at 0. +\end{rem} +\begin{rem}\label{r:dif0} +Let $A \colon \symbf{R}^n\to \symbf{R}^m$ be a linear function, and +let $f \colon \symbf{R}^m\to \symbf{R}$ be a function. Then the +restriction of $f$ to the range of $A$ is differentiable at 0 if and +only if $f(A)\colon \symbf{R}^n\to \symbf{R}$ is differentiable at 0 +and +\[\nabla(\eval{f}_{\IM(A)})(0)A=\nabla (f(A))(0).\] +\end{rem} + +\begin{proof} + We begin by showing that $v\in BV(\Omega;\symbf{R}^k)$ and +\begin{equation}\label{e:bomb} +\abs{Dv}(B)\le K\abs{Du}(B)\qquad\forall B\in\symbf{B}(\Omega ), +\end{equation} +where $K>0$ is the Lipschitz constant of $f$. By \eqref{sum-Di} and by +the approximation result quoted in \secref{s:mt}, it is possible to find +a sequence $(u_h)\subset C^1(\Omega ;\symbf{R}^m)$ converging to $u$ in +$L^1(\Omega ;\symbf{R}^m)$ and such that +\[\lim_{h\to +\infty}\int_\Omega \abs{\nabla u_h}\,dx=\abs{Du}(\Omega ).\] +The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega +$, and the definition of differential implies that $\abs{\nabla v_h}\le +K\abs{\nabla u_h}$ almost everywhere in $\Omega $. The lower semicontinuity +of the total variation and \eqref{sum-Di} yield +\begin{equation} +\begin{split} +\abs{Dv}(\Omega )\le\liminf_{h\to +\infty}\abs{Dv_h}(\Omega) & +=\liminf_{h\to +\infty}\int_\Omega \abs{\nabla v_h}\,dx\\ +&\le K\liminf_{h\to +\infty}\int_\Omega +\abs{\nabla u_h}\,dx=K\abs{Du}(\Omega). +\end{split}\end{equation} +Since $f(0)=0$, we have also +\[\int_\Omega \abs{v}\,dx\le K\int_\Omega \abs{u}\,dx;\] +therefore $u\in BV(\Omega ;\symbf{R}^k)$. Repeating the same argument +for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every +$B\in\symbf{B}(\Omega)$, because $\abs{Dv}$, $\abs{Du}$ are Radon measures. To +prove \lemref{limbog}, first we observe that +\begin{equation}\label{e:SS} +S_v\subset S_u,\qquad\tilde v(x)=f(\tilde u(x))\qquad \forall x\in\Omega +\backslash S_u.\end{equation} +In fact, for every $\varepsilon >0$ we have +\[\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>\varepsilon \}\subset \{y\in +B_\rho(x): \abs{u(y)-\tilde u(x)}>\varepsilon /K\},\] +hence +\[\lim_{\rho\to 0^+}\frac{\abs{\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}> +\varepsilon \}}}{\rho^n}=0\] +whenever $x\in\Omega \backslash S_u$. By a similar argument, if $x\in +S_u$ is a point such that there exists a triplet $(u^+,u^-,\nu_u)$ +satisfying \eqref{detK1}, \eqref{detK2}, then +\[ +(v^+(x)-v^-(x))\otimes \nu_v=(f(u^+(x))-f(u^-(x)))\otimes\nu_u\quad +\text{if }x\in S_v +\] +and $f(u^-(x))=f(u^+(x))$ if $x\in S_u\backslash S_v$. Hence, by (1.8) +we get +\begin{equation*}\begin{split} +Jv(B)=\int_{B\cap S_v}(v^+-v^-)\otimes \nu_v\,d\symcal{H}_{n-1}&= +\int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\symcal{H}_{n-1}\\ +&=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\symcal{H}_{n-1} +\end{split}\end{equation*} +and \lemref{limbog} is proved. +\end{proof} + +To prove \eqref{e:SS}, it is not restrictive to assume that $k=1$. +Moreover, to simplify our notation, from now on we shall assume that +$\Omega = \symbf{R}^n$. The proof of \eqref{e:SS} is divided into two +steps. In the first step we prove the statement in the one-dimensional +case $(n=1)$, using \thmref{th-weak-ske-owf}. In the second step we +achieve the general result using \thmref{t:conl}. + +\subsection*{Step 1} +Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij} +yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that +\eqref{e:st} and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is +the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and +singular part with respect to $\abs{\wt{D} u}$. By +\thmref{th-weak-ske-owf}, we have +\begin{equation*} +\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+} +\frac{Dv(\interval{\left[t,s\right[})} +{\abs{\wt{D}u}(\interval{\left[t,s\right[})},\qquad +\frac{\wt{D}u}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+} +\frac{Du(\interval{\left[t,s\right[})} +{\abs{\wt{D}u}(\interval{\left[t,s\right[})} +\end{equation*} +$\abs{\wt{D}u}$-almost everywhere in $\symbf{R}$. It is well known +(see, for instance, \cite[2.5.16]{ste:sint}) that every one-dimensional +function of bounded variation $w$ has a unique left continuous +representative, i.e., a function $\hat w$ such that $\hat w=w$ almost +everywhere and $\lim_{s\to t^-}\hat w(s)=\hat w(t)$ for every $t\in +\symbf{R}$. These conditions imply +\begin{equation} +\hat u(t)=Du(\interval{\left]-\infty,t\right[}), +\qquad \hat v(t)=Dv(\interval{\left]-\infty,t\right[})\qquad +\forall t\in\symbf{R} +\end{equation} +and +\begin{equation}\label{alimo} +\hat v(t)=f(\hat u(t))\qquad\forall t\in\symbf{R}.\end{equation} +Let $t\in\symbf{R}$ be such that +$\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and +assume that the limits in \eqref{joe} exist. By \eqref{j:mark} and +\eqref{far-d} we get +\begin{equation*}\begin{split} +\frac{\hat v(s)-\hat +v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat +u(s))-f(\hat u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\ +&=\frac{f(\hat u(s))-f(\hat +u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}u +}(\interval{\left[t,s\right[}))}% +{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\ +&+\frac +{f(\hat u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D} +u}(\interval{\left[t,s\right[}))-f(\hat +u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} +\end{split}\end{equation*} +for every $s>t$. Using the Lipschitz condition on $f$ we find +{\setlength{\multlinegap}{0pt} +\begin{multline*} +\left\lvert\frac{\hat v(s)-\hat +v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat +u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t) +\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat +u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert\\ +\le K\left\lvert +\frac{\hat u(s)-\hat u(t)} + {\abs{\wt{D}u}(\interval{\left[t,s\right[})} +-\frac{\wt{D}u}{\abs{ +\wt{D}u}}(t)\right\rvert.\end{multline*} +}% end of group with \multlinegap=0pt +By \eqref{e:bomb}, the function $s\to +\abs{\wt{D}u}(\interval{\left[t,s\right[})$ is continuous and +converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the +previous inequality imply +\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0^+} +\frac{f(\hat u(t)+h\dfrac{\wt{D}u}{\abs{\wt{D}u}} +(t))-f(\hat u(t))}h\quad\abs{\wt{D}u}\text{-a.e. in }\symbf{R}.\] +By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every +$x\in\symbf{R}\backslash S_u$; moreover, applying the same argument to +the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get +\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0} +\frac{f(\tilde u(t) ++h\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t))-f(\tilde u(t))}{h} +\qquad\abs{\wt{D}u}\text{-a.e. in }\symbf{R}\] +and our statement is proved. + +\subsection*{Step 2} + +Let us consider now the general case $n>1$. Let $\nu\in \symbf{R}^n$ be +such that $\abs{\nu}=1$, and let $\pi_\nu=\{y\in\symbf{R}^n: \langle +y,\nu\rangle =0\}$. In the following, we shall identify $\symbf{R}^n$ +with $\pi_\nu\times\symbf{R}$, and we shall denote by $y$ the variable +ranging in $\pi_\nu$ and by $t$ the variable ranging in $\symbf{R}$. By +the just proven one-dimensional result, and by \thmref{thm-main}, we get +\[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\wt{D}u_y}{\abs{ +\wt{D}u_y}}(t))-f(\tilde u(y+t\nu))}h=\frac{\wt{D}v_y}{\abs{ +\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\symbf{R}\] +for $\symcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that +\begin{equation} +\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle +}}(y+t\nu)=\frac{\wt{D}u_y} +{\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\symbf{R} +\end{equation} +for $\symcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, by +\eqref{sum-ali} and \eqref{delta-l} we get +\begin{multline*} +\int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y +}\,d\symcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\symcal{H}_{n-1}(y)\\ +=\langle \wt{D}u,\nu\rangle =\frac +{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}\cdot +\abs{\langle \wt{D}u,\nu\rangle }=\int_{\pi_\nu}\frac{ +\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }} +(y+\cdot \nu)\cdot\abs{\wt{D}u_y}\,d\symcal{H}_{n-1}(y) +\end{multline*} +and \eqref{far-d} follows from \eqref{sum-Di}. By the same argument it +is possible to prove that +\begin{equation} +\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle +}}(y+t\nu)=\frac{\wt{D}v_y}{\abs{\wt{D}u_y}}(t)\qquad\abs{ +\wt{D}u_y}\text{-a.e. in }\symbf{R}\end{equation} +for $\symcal{H}_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d} +and \eqref{E_SXgYy} we get +\[ +\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \wt{D} +u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde +u(y+t\nu))}{h} +=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle +\wt{D}u,\nu\rangle }}(y+t\nu)\] +for $\symcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again +\eqref{detK1}, \eqref{detK2} we get +\[ +\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle +\wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde +u(x))}{h}=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu +\rangle }}(x) +\] +$\abs{\langle \wt{D}u,\nu\rangle}$-a.e. in $\symbf{R}^n$. + +Since the function $\abs{\langle \wt{D}u,\nu\rangle }/\abs{\wt{D}u}$ +is strictly positive $\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere, +we obtain also +\begin{multline*} +\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\abs{\langle +\wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\dfrac{\langle \wt{D} +u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde u(x))}{h}\\ +=\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\frac +{\langle \wt{D}v,\nu\rangle }{\abs{\langle +\wt{D}u,\nu\rangle }}(x) +\end{multline*} +$\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere in $\symbf{R}^n$. + +Finally, since +\begin{align*} +&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}} +\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}} +=\frac{\langle \wt{D}u,\nu\rangle }{\abs{\wt{D}u}} +=\left\langle \frac{\wt{D}u}{\abs{\wt{D}u}},\nu\right\rangle + \qquad\abs{\wt{D}u}\text{-a.e. in }\symbf{R}^n\\ +&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}} +\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}} +=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\wt{D}u}} +=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}},\nu\right\rangle + \qquad\abs{\wt{D}u}\text{-a.e. in }\symbf{R}^n +\end{align*} +and since both sides of \eqref{alimo} +are zero $\abs{\wt{D}u}$-almost everywhere +on $\abs{\langle \wt{D}u,\nu\rangle }$-negligible sets, we conclude that +\[ +\lim_{h\to 0}\frac{f\left( +\tilde u(x)+h\left\langle \dfrac{\wt{D} +u}{\abs{\wt{D}u}}(x),\nu\right\rangle \right)-f(\tilde u(x))}h +=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}}(x),\nu\right\rangle, +\] +$\abs{\wt{D}u}$-a.e. in $\symbf{R}^n$. +Since $\nu$ is arbitrary, by Remarks \ref{r:dif} and~\ref{r:dif0} +the restriction of $f$ to +the affine space $T^u_x$ is differentiable at $\tilde u(x)$ for $\abs{\wt{D} +u}$-almost every $x\in \symbf{R}^n$ and \eqref{quts} holds.\qed + +It follows from \eqref{sum-Di}, \eqref{detK1}, and \eqref{detK2} that +\begin{equation}\label{Dt} +D(t_1,\dots,t_n)=\sum_{I\in\symbf{n}}(-1)^{\abs{I}-1}\abs{I} +\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\symbf{A}^{(\lambda)} +(\overline I|\overline I). +\end{equation} +Let $t_i=\hat x_i$, $i=1,\dots,n$. Lemma 1 leads to +\begin{equation}\label{Dx} +D(\hat x_1,\dots,\hat x_n)=\prod_{i\in\symbf{n}}\hat x_i +\sum_{I\in\symbf{n}}(-1)^{\abs{I}-1}\abs{I}\per \symbf{A} +^{(\lambda)}(I|I)\det\symbf{A}^{(\lambda)}(\overline I|\overline I). +\end{equation} +By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx}, +we have the following result: +\begin{thm}\label{thm-H-param} +\begin{equation}\label{H-param} +H_c=\frac{1}{2n}\sum^n_{l =1}l (-1)^{l -1}A_{l} +^{(\lambda)}, +\end{equation} +where +\begin{equation}\label{A-l-lambda} +A^{(\lambda)}_l =\sum_{I_l \subseteq\symbf{n}}\per \symbf{A} +^{(\lambda)}(I_l |I_l )\det\symbf{A}^{(\lambda)} +(\overline I_{l}|\overline I_l ),\abs{I_{l}}=l . +\end{equation} +\end{thm} + +It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is +similar to the coefficients $b_l $ of the characteristic polynomial of +\eqref{bl-sum}. It is well known in graph theory that the coefficients +$b_l $ can be expressed as a sum over certain subgraphs. It is +interesting to see whether $A_l $, $\lambda=0$, structural properties +of a graph. + +We may call \eqref{H-param} a parametric representation of $H_c$. In +computation, the parameter $\lambda_i$ plays very important roles. The +choice of the parameter usually depends on the properties of the given +graph. For a complete graph $K_n$, let $\lambda_i=1$, $i=1,\dots,n$. +It follows from \eqref{A-l-lambda} that +\begin{equation}\label{compl-gr} +A^{(1)}_l =\begin{cases} n!,&\text{if }l =1\\ +0,&\text{otherwise}.\end{cases} +\end{equation} +By \eqref{H-param} +\begin{equation} +H_c=\frac 12(n-1)!. +\end{equation} +For a complete bipartite graph $K_{n_1n_2}$, let $\lambda_i=0$, $i=1,\dots,n$. +By \eqref{A-l-lambda}, +\begin{equation} +A_l = +\begin{cases} -n_1!n_2!\delta_{n_1n_2},&\text{if }l =2\\ +0,&\text{otherwise }.\end{cases} +\label{compl-bip-gr} +\end{equation} +Theorem~\ref{thm-H-param} +leads to +\begin{equation} +H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}. +\end{equation} + +Now, we consider an asymmetrical approach. Theorem \ref{thm-main} leads to +\begin{multline} +\det\symbf{K}(t=1,t_1,\dots,t_n;l |l )\\ +=\sum_{I\subseteq\symbf{n}-\{l \}} +(-1)^{\abs{I}}\prod_{i\in I}t_i\prod_{j\in I} +(D_j+\lambda_jt_j)\det\symbf{A}^{(\lambda)} +(\overline I\cup\{l \}|\overline I\cup\{l \}). +\end{multline} + +By \eqref{H-cycles} and \eqref{sum-ali} we have the following asymmetrical +result: +\begin{thm}\label{thm-asym} +\begin{equation} +H_c=\frac12\sum_{I\subseteq\symbf{n}-\{l \}} +(-1)^{\abs{I}}\per\symbf{A}^{(\lambda)}(I|I)\det +\symbf{A}^{(\lambda)} +(\overline I\cup\{l \}|\overline I\cup\{l \}) +\end{equation} +which reduces to Goulden--Jackson's formula when $\lambda_i=0,i=1,\dots,n$ +\cite{mami:matrixth}. +\end{thm} + +\section{Various font features of the \pkg{amsmath} package} +\label{s:font} +\subsection{Bold versions of special symbols} + +In the \pkg{amsmath} package \cn{boldsymbol} is used for getting +individual bold math symbols and bold Greek letters---everything in +math except for letters of the Latin alphabet, +where you'd use \cn{symbf}. For example, +\begin{verbatim} +A_\infty + \pi A_0 \sim +\symbf{A}_{\boldsymbol{\infty}} \boldsymbol{+} +\boldsymbol{\pi} \symbf{A}_{\boldsymbol{0}} +\end{verbatim} +looks like this: +\[A_\infty + \pi A_0 \sim \symbf{A}_{\boldsymbol{\infty}} +\boldsymbol{+} \boldsymbol{\pi} \symbf{A}_{\boldsymbol{0}}\] + +\subsection{``Poor man's bold''} +If a bold version of a particular symbol doesn't exist in the +available fonts, +then \cn{boldsymbol} can't be used to make that symbol bold. +At the present time, this means that +\cn{boldsymbol} can't be used with symbols from +the \fn{msam} and \fn{msbm} fonts, among others. +In some cases, poor man's bold (\cn{pmb}) can be used instead +of \cn{boldsymbol}: +% Can't show example from msam or msbm because this document is +% supposed to be TeXable even if the user doesn't have +% AMSFonts. MJD 5-JUL-1990 +\[\frac{\partial x}{\partial y} +\pmb{\bigg\vert} +\frac{\partial y}{\partial z}\] +\begin{verbatim} +\[\frac{\partial x}{\partial y} +\pmb{\bigg\vert} +\frac{\partial y}{\partial z}\] +\end{verbatim} +So-called ``large operator'' symbols such as $\sum$ and $\prod$ +require an additional command, \cn{mathop}, +to produce proper spacing and limits when \cn{pmb} is used. +For further details see \textit{The \TeX book}. +\[\sum_{\substack{i<B\\\text{$i$ odd}}} +\prod_\kappa \kappa F(r_i)\qquad +\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}} +\mathop{\pmb{\prod}}_\kappa \kappa(r_i) +\] +\begin{verbatim} +\[\sum_{\substack{i<B\\\text{$i$ odd}}} +\prod_\kappa \kappa F(r_i)\qquad +\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}} +\mathop{\pmb{\prod}}_\kappa \kappa(r_i) +\] +\end{verbatim} + +\section{Compound symbols and other features} +\label{s:comp} +\subsection{Multiple integral signs} + +\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral signs +with the spacing between them nicely adjusted, in both text and +display style. \cn{idotsint} gives two integral signs with dots +between them. +\begin{gather} +\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A +f(x,y,z)\,dx\,dy\,dz\\ +\iiiint\limits_A +f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k) +\end{gather} + +\subsection{Over and under arrows} + +Some extra over and under arrow operations are provided in +the \pkg{amsmath} package. (Basic \LaTeX\ provides +\cn{overrightarrow} and \cn{overleftarrow}). +\begin{align*} +\overrightarrow{\psi_\delta(t) E_t h}& +=\underrightarrow{\psi_\delta(t) E_t h}\\ +\overleftarrow{\psi_\delta(t) E_t h}& +=\underleftarrow{\psi_\delta(t) E_t h}\\ +\overleftrightarrow{\psi_\delta(t) E_t h}& +=\underleftrightarrow{\psi_\delta(t) E_t h} +\end{align*} +\begin{verbatim} +\begin{align*} +\overrightarrow{\psi_\delta(t) E_t h}& +=\underrightarrow{\psi_\delta(t) E_t h}\\ +\overleftarrow{\psi_\delta(t) E_t h}& +=\underleftarrow{\psi_\delta(t) E_t h}\\ +\overleftrightarrow{\psi_\delta(t) E_t h}& +=\underleftrightarrow{\psi_\delta(t) E_t h} +\end{align*} +\end{verbatim} +These all scale properly in subscript sizes: +\[\int_{\overrightarrow{AB}} ax\,dx\] +\begin{verbatim} +\[\int_{\overrightarrow{AB}} ax\,dx\] +\end{verbatim} + +\subsection{Dots} + +Normally you need only type \cn{dots} for ellipsis dots in a +math formula. The main exception is when the dots +fall at the end of the formula; then you need to +specify one of \cn{dotsc} (series dots, after a comma), +\cn{dotsb} (binary dots, for binary relations or operators), +\cn{dotsm} (multiplication dots), or \cn{dotsi} (dots after +an integral). For example, the input +\begin{verbatim} +Then we have the series $A_1,A_2,\dotsc$, +the regional sum $A_1+A_2+\dotsb$, +the orthogonal product $A_1A_2\dotsm$, +and the infinite integral +\[\int_{A_1}\int_{A_2}\dotsi\]. +\end{verbatim} +produces +\begin{quotation} +Then we have the series $A_1,A_2,\dotsc$, +the regional sum $A_1+A_2+\dotsb$, +the orthogonal product $A_1A_2\dotsm$, +and the infinite integral +\[\int_{A_1}\int_{A_2}\dotsi\] +\end{quotation} + +\subsection{Accents in math} + +Double accents: +\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad +\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad +\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad +\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad +\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\] +\begin{verbatim} +\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad +\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad +\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad +\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad +\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\] +\end{verbatim} +This double accent operation is complicated +and tends to slow down the processing of a \LaTeX\ file. + + +\subsection{Dot accents} +\cn{dddot} and \cn{ddddot} are available to +produce triple and quadruple dot accents +in addition to the \cn{dot} and \cn{ddot} accents already available +in \LaTeX: +\[\dddot{Q}\qquad\ddddot{R}\] +\begin{verbatim} +\[\dddot{Q}\qquad\ddddot{R}\] +\end{verbatim} + +\subsection{Roots} + +In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust +the position of the root index of a radical: +\begin{verbatim} +\sqrt[\leftroot{-2}\uproot{2}\beta]{k} +\end{verbatim} +gives good positioning of the $\beta$: +\[\sqrt[\leftroot{0}\uproot{2}\beta]{k}\] + +\subsection{Boxed formulas} The command \cn{boxed} puts a box around its +argument, like \cn{fbox} except that the contents are in math mode: +\begin{verbatim} +\boxed{W_t-F\subseteq V(P_i)\subseteq W_t} +\end{verbatim} +\[\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}.\] + +\subsection{Extensible arrows} +\cn{xleftarrow} and \cn{xrightarrow} produce +arrows that extend automatically to accommodate unusually wide +subscripts or superscripts. The text of the subscript or superscript +are given as an optional resp.\@ mandatory argument: +Example: +\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1] + \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\] +\begin{verbatim} +\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1] + \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\] +\end{verbatim} + +\subsection{\cn{overset}, \cn{underset}, and \cn{sideset}} +Examples: +\[\overset{*}{X}\qquad\underset{*}{X}\qquad +\overset{a}{\underset{b}{X}}\] +\begin{verbatim} +\[\overset{*}{X}\qquad\underset{*}{X}\qquad +\overset{a}{\underset{b}{X}}\] +\end{verbatim} + +The command \cn{sideset} is for a rather special +purpose: putting symbols at the subscript and superscript +corners of a large operator symbol such as $\sum$ or $\prod$, +without affecting the placement of limits. +Examples: +\[\sideset{_*^*}{_*^*}\prod_k\qquad +\sideset{}{'}\sum_{0\le i\le m} E_i\beta x +\] +\begin{verbatim} +\[\sideset{_*^*}{_*^*}\prod_k\qquad +\sideset{}{'}\sum_{0\le i\le m} E_i\beta x +\] +\end{verbatim} + +\subsection{The \cn{text} command} +The main use of the command \cn{text} is for words or phrases in a +display: +\[\symbf{y}=\symbf{y}'\quad\text{if and only if}\quad +y'_k=\delta_k y_{\tau(k)}\] +\begin{verbatim} +\[\symbf{y}=\symbf{y}'\quad\text{if and only if}\quad +y'_k=\delta_k y_{\tau(k)}\] +\end{verbatim} + +\subsection{Operator names} +The more common math functions such as $\log$, $\sin$, and $\lim$ +have predefined control sequences: \verb=\log=, \verb=\sin=, +\verb=\lim=. +The \pkg{amsmath} package provides \cn{DeclareMathOperator} and +\cn{DeclareMathOperator*} +for producing new function names that will have the +same typographical treatment. +Examples: +\[\norm{f}_\infty= +\esssup_{x\in R^n}\abs{f(x)}\] +\begin{verbatim} +\[\norm{f}_\infty= +\esssup_{x\in R^n}\abs{f(x)}\] +\end{verbatim} +\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\} +=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\} +\quad \forall\alpha>0.\] +\begin{verbatim} +\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\} +=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\} +\quad \forall\alpha>0.\] +\end{verbatim} +\cn{esssup} and \cn{meas} would be defined in the document preamble as +\begin{verbatim} +\DeclareMathOperator*{\esssup}{ess\,sup} +\DeclareMathOperator{\meas}{meas} +\end{verbatim} + +The following special operator names are predefined in the \pkg{amsmath} +package: \cn{varlimsup}, \cn{varliminf}, \cn{varinjlim}, and +\cn{varprojlim}. Here's what they look like in use: +\begin{align} +&\varlimsup_{n\rightarrow\infty} + \symcal{Q}(u_n,u_n-u^{\#})\le0\\ +&\varliminf_{n\rightarrow\infty} + \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\ +&\varinjlim (m_i^\lambda\cdot)^*\le0\\ +&\varprojlim_{p\in S(A)}A_p\le0 +\end{align} +\begin{verbatim} +\begin{align} +&\varlimsup_{n\rightarrow\infty} + \symcal{Q}(u_n,u_n-u^{\#})\le0\\ +&\varliminf_{n\rightarrow\infty} + \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\ +&\varinjlim (m_i^\lambda\cdot)^*\le0\\ +&\varprojlim_{p\in S(A)}A_p\le0 +\end{align} +\end{verbatim} + +\subsection{\cn{mod} and its relatives} +The commands \cn{mod} and \cn{pod} are variants of +\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses, +whereas \cn{pod} omits the `mod' and retains the parentheses. +Examples: +\begin{align} +x&\equiv y+1\pmod{m^2}\\ +x&\equiv y+1\mod{m^2}\\ +x&\equiv y+1\pod{m^2} +\end{align} +\begin{verbatim} +\begin{align} +x&\equiv y+1\pmod{m^2}\\ +x&\equiv y+1\mod{m^2}\\ +x&\equiv y+1\pod{m^2} +\end{align} +\end{verbatim} + +\subsection{Fractions and related constructions} +\label{fracs} + +The usual notation for binomials is similar to the fraction concept, +so it has a similar command \cn{binom} with two arguments. Example: +\begin{equation} +\begin{split} +\sum_{\gamma\in\Gamma_C} I_\gamma& +=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\ +&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l} ++\dots+(-1)^k\\ +&=(2-1)^k=1 +\end{split} +\end{equation} +\begin{verbatim} +\begin{equation} +\begin{split} +[\sum_{\gamma\in\Gamma_C} I_\gamma& +=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\ +&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l} ++\dots+(-1)^k\\ +&=(2-1)^k=1 +\end{split} +\end{equation} +\end{verbatim} +There are also abbreviations +\begin{verbatim} +\dfrac \dbinom +\tfrac \tbinom +\end{verbatim} +for the commonly needed constructions +\begin{verbatim} +{\displaystyle\frac ... } {\displaystyle\binom ... } +{\textstyle\frac ... } {\textstyle\binom ... } +\end{verbatim} + +The generalized fraction command \cn{genfrac} provides full access to +the six \TeX{} fraction primitives: +\begin{align} +\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}& +\text{\cn{overwithdelims}: }& + \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\ +\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}& +\text{\cn{atopwithdelims}: }& + \genfrac{(}{)}{0pt}{}{n+1}{2}\\ +\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}& +\text{\cn{abovewithdelims}: }& + \genfrac{[}{]}{1pt}{}{n+1}{2} +\end{align} +\begin{verbatim} +\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}& +\text{\cn{overwithdelims}: }& + \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\ +\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}& +\text{\cn{atopwithdelims}: }& + \genfrac{(}{)}{0pt}{}{n+1}{2}\\ +\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}& +\text{\cn{abovewithdelims}: }& + \genfrac{[}{]}{1pt}{}{n+1}{2} +\end{verbatim} + +\subsection{Continued fractions} +The continued fraction +\begin{equation} +\cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+\dotsb +}}}}} +\end{equation} +can be obtained by typing +\begin{verbatim} +\cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+ + \cfrac{1}{\sqrt{2}+\dotsb +}}}}} +\end{verbatim} +Left or right placement of any of the numerators is accomplished by using +\cn{cfrac[l]} or \cn{cfrac[r]} instead of \cn{cfrac}. + +\subsection{Smash} + +In \pkg{amsmath} there are optional arguments \verb"t" and \verb"b" for +the plain \TeX\ command \cn{smash}, because sometimes it is advantageous +to be able to `smash' only the top or only the bottom of something while +retaining the natural depth or height. In the formula +$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cn{smash}\verb=[b]= has been +used to limit the size of the radical symbol. +\begin{verbatim} +$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ +\end{verbatim} +Without the use of \cn{smash}\verb=[b]= the formula would have appeared +thus: $X_j=(1/\sqrt{\lambda_j})X_j'$, with the radical extending to +encompass the depth of the subscript $j$. + +\subsection{The `cases' environment} +`Cases' constructions like the following can be produced using +the \env{cases} environment. +\begin{equation} +P_{r-j}= + \begin{cases} + 0& \text{if $r-j$ is odd},\\ + r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}. + \end{cases} +\end{equation} +\begin{verbatim} +\begin{equation} P_{r-j}= + \begin{cases} + 0& \text{if $r-j$ is odd},\\ + r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}. + \end{cases} +\end{equation} +\end{verbatim} +Notice the use of \cn{text} and the embedded math. + +\subsection{Matrix} + +Here are samples of the matrix environments, +\cn{matrix}, \cn{pmatrix}, \cn{bmatrix}, \cn{Bmatrix}, \cn{vmatrix} +and \cn{Vmatrix}: +\begin{equation} +\begin{matrix} +\vartheta& \varrho\\\varphi& \varpi +\end{matrix}\quad +\begin{pmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{pmatrix}\quad +\begin{bmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{bmatrix}\quad +\begin{Bmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{Bmatrix}\quad +\begin{vmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{vmatrix}\quad +\begin{Vmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{Vmatrix} +\end{equation} +% +\begin{verbatim} +\begin{matrix} +\vartheta& \varrho\\\varphi& \varpi +\end{matrix}\quad +\begin{pmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{pmatrix}\quad +\begin{bmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{bmatrix}\quad +\begin{Bmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{Bmatrix}\quad +\begin{vmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{vmatrix}\quad +\begin{Vmatrix} +\vartheta& \varrho\\\varphi& \varpi +\end{Vmatrix} +\end{verbatim} + +To produce a small matrix suitable for use in text, use the +\env{smallmatrix} environment. +\begin{verbatim} +\begin{math} + \bigl( \begin{smallmatrix} + a&b\\ c&d + \end{smallmatrix} \bigr) +\end{math} +\end{verbatim} +To show +the effect of the matrix on the surrounding lines of +a paragraph, we put it here: \begin{math} + \bigl( \begin{smallmatrix} + a&b\\ c&d + \end{smallmatrix} \bigr) +\end{math} +and follow it with enough text to ensure that there will +be at least one full line below the matrix. + +\cn{hdotsfor}\verb"{"\textit{number}\verb"}" produces a row of dots in a matrix +spanning the given number of columns: +\[W(\Phi)= \begin{Vmatrix} +\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ +\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& +\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ +\hdotsfor{5}\\ +\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& +\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& +\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& +\dfrac{\varphi}{(\varphi_n,\varepsilon_n)} +\end{Vmatrix}\] +\begin{verbatim} +\[W(\Phi)= \begin{Vmatrix} +\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ +\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& +\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ +\hdotsfor{5}\\ +\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& +\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& +\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& +\dfrac{\varphi}{(\varphi_n,\varepsilon_n)} +\end{Vmatrix}\] +\end{verbatim} +The spacing of the dots can be varied through use of a square-bracket +option, for example, \verb"\hdotsfor[1.5]{3}". The number in square brackets +will be used as a multiplier; the normal value is 1. + +\subsection{The \cn{substack} command} + +The \cn{substack} command can be used to produce a multiline +subscript or superscript: +for example +\begin{verbatim} +\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) +\end{verbatim} +produces a two-line subscript underneath the sum: +\begin{equation} +\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) +\end{equation} +A slightly more generalized form is the \env{subarray} environment which +allows you to specify that each line should be left-aligned instead of +centered, as here: +\begin{equation} +\sum_{\begin{subarray}{l} + 0\le i\le m\\ 0<j<n + \end{subarray}} + P(i,j) +\end{equation} +\begin{verbatim} +\sum_{\begin{subarray}{l} + 0\le i\le m\\ 0<j<n + \end{subarray}} + P(i,j) +\end{verbatim} + + +\subsection{Big-g-g delimiters} +Here are some big delimiters, first in \cn{normalsize}: +\[\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + \biggr) +\] +\begin{verbatim} +\[\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + \biggr) +\] +\end{verbatim} +and now in \cn{Large} size: +{\Large +\[\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + \biggr) +\]} +\begin{verbatim} +{\Large +\[\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + \biggr) +\]} +\end{verbatim} + +\newpage +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter + +%% This turns on vertical rules at the right and left margins, to +%% better illustrate the spacing for certain multiple-line equation +%% structures. +\def\@makecol{\ifvoid\footins \setbox\@outputbox\box\@cclv + \else\setbox\@outputbox + \vbox{\boxmaxdepth \maxdepth + \unvbox\@cclv\vskip\skip\footins\footnoterule\unvbox\footins}\fi + \xdef\@freelist{\@freelist\@midlist}\gdef\@midlist{}\@combinefloats + \setbox\@outputbox\hbox{\vrule width\marginrulewidth + \vbox to\@colht{\boxmaxdepth\maxdepth + \@texttop\dimen128=\dp\@outputbox\unvbox\@outputbox + \vskip-\dimen128\@textbottom}% + \vrule width\marginrulewidth}% + \global\maxdepth\@maxdepth} +\newdimen\marginrulewidth +\setlength{\marginrulewidth}{.1pt} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\appendix +\section{Examples of multiple-line equation structures} +\label{s:eq} + +\textbf{\large Note: Starting on this page, vertical rules are +added at the margins so that the positioning of various display elements +with respect to the margins can be seen more clearly.} + +\subsection{Split} +The \env{split} environment is not an independent environment +but should be used inside something else such as \env{equation} +or \env{align}. + +If there is not enough room for it, the equation number for a +\env{split} will be shifted to the previous line, when equation numbers are +on the left; the number shifts down to the next line when numbers are on +the right. +\begin{equation} +\begin{split} +f_{h,\varepsilon}(x,y) +&=\varepsilon\symbf{E}_{x,y}\int_0^{t_\varepsilon} +L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ +&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ +&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ +&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} + \biggl(\symbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} + \varphi(x)\,ds -\symbf{E}_{x,y}\int_0^{t_\varepsilon} + L_{x,y_\varepsilon(\varepsilon s)} + \varphi(x)\,ds\biggr)\biggr]\\ +&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), +\end{split} +\end{equation} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{equation} +\begin{split} +f_{h,\varepsilon}(x,y) +&=\varepsilon\symbf{E}_{x,y}\int_0^{t_\varepsilon} +L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ +&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ +&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ +&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} + \biggl(\symbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} + \varphi(x)\,ds -\symbf{E}_{x,y}\int_0^{t_\varepsilon} + L_{x,y_\varepsilon(\varepsilon s)} + \varphi(x)\,ds\biggr)\biggr]\\ +&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), +\end{split} +\end{equation} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +Unnumbered version: +\begin{equation*} +\begin{split} +f_{h,\varepsilon}(x,y) +&=\varepsilon\symbf{E}_{x,y}\int_0^{t_\varepsilon} +L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ +&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ +&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ +&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} + \biggl(\symbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} + \varphi(x)\,ds -\symbf{E}_{x,y}\int_0^{t_\varepsilon} + L_{x,y_\varepsilon(\varepsilon s)} + \varphi(x)\,ds\biggr)\biggr]\\ +&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), +\end{split} +\end{equation*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{equation*} +\begin{split} +f_{h,\varepsilon}(x,y) +&=\varepsilon\symbf{E}_{x,y}\int_0^{t_\varepsilon} +L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ +&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ +&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\symbf{E}_{y} + \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds + -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ +&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} + \biggl(\symbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} + \varphi(x)\,ds -\symbf{E}_{x,y}\int_0^{t_\varepsilon} + L_{x,y_\varepsilon(\varepsilon s)} + \varphi(x)\,ds\biggr)\biggr]\\ +&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), +\end{split} +\end{equation*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +If the option \env{centertags} is included in the options +list of the \pkg{amsmath} package, +the equation numbers for \env{split} environments will be +centered vertically on the height +of the \env{split}: +{\makeatletter\ctagsplit@true +\begin{equation} +\begin{split} + \abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)-\int_{\gamma(t)}^a + \frac{d\theta}{k(\theta,t)} + \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ +&\le C_6\left\lvert \left\lvert f\int_\Omega\left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split} +\end{equation}}% +Some text after to test the below-display spacing. + +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +Use of \env{split} within \env{align}: +{\delimiterfactor750 +\begin{align} +\begin{split}\abs{I_1} + &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ +&\le C_3\left[\int_\Omega\left(\int_{a}^x + g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ +&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} + \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} + c\Omega\right]^{1/2}\\ +&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\label{eq:A}\\ +\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) + -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} + \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ +&\le C_6\left\lvert \left\lvert f\int_\Omega + \left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split} +\end{align}}% +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{align} +\begin{split}\abs{I_1} + &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ +&\le C_3\left[\int_\Omega\left(\int_{a}^x + g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ +&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} + \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} + c\Omega\right]^{1/2}\\ +&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\label{eq:A}\\ +\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) + -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} + \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ +&\le C_6\left\lvert \left\lvert f\int_\Omega + \left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split} +\end{align} +\end{verbatim} + +%%%%%%%%%%%%%%%%%% + +\newpage +Unnumbered \env{align}, with a number on the second \env{split}: +\begin{align*} +\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ + &\le C_3\left[\int_\Omega\left(\int_{a}^x + g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ +&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} + \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} + c\Omega\right]^{1/2}\\ +&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\\ +\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) + -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} + \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ +&\le C_6\left\lvert \left\lvert f\int_\Omega + \left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\tag{\theequation$'$} +\end{align*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{align*} +\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ + &\le C_3\left[\int_\Omega\left(\int_{a}^x + g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ +&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} + \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} + c\Omega\right]^{1/2}\\ +&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\\ +\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) + -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} + \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ +&\le C_6\left\lvert \left\lvert f\int_\Omega + \left\lvert \wt{S}^{-1,0}_{a,-} + W_2(\Omega,\Gamma_l)\right\rvert\right\rvert + \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} + (\Omega;\Gamma_r,T)\right\rvert\right\rvert. +\end{split}\tag{\theequation$'$} +\end{align*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\subsection{Multline} +Numbered version: +\begin{multline}\label{eq:E} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline} +To test the use of \verb=\label= and +\verb=\ref=, we refer to the number of this +equation here: (\ref{eq:E}). + +\begin{verbatim} +\begin{multline}\label{eq:E} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Unnumbered version: +\begin{multline*} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{multline*} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\iffalse % bugfix needed, error message "Multiple \tag" + % [mjd,24-Jan-1995] +\newpage +And now an ``unnumbered'' version numbered with a literal tag: +\begin{multline*}\tag*{[a]} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{multline*}\tag*{[a]} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*} +\end{verbatim} + +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The same display with \verb=\multlinegap= set to zero. +Notice that the space on the left in +the first line does not change, because of the equation number, while +the second line is pushed over to the right margin. +{\setlength{\multlinegap}{0pt} +\begin{multline*}\tag*{[a]} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*}}% +Some text after to test the below-display spacing. + +\begin{verbatim} +{\setlength{\multlinegap}{0pt} +\begin{multline*}\tag*{[a]} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ + =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy +\end{multline*}} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% +\fi % matches \iffalse above [mjd,24-Jan-1995] + +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\subsection{Gather} +Numbered version with \verb;\notag; on the second line: +\begin{gather} +D(a,r)\equiv\{z\in\symbf{C}\colon \abs{z-a}<r\},\\ +\seg(a,r)\equiv\{z\in\symbf{C}\colon +\Im z= \Im a,\ \abs{z-a}<r\},\notag\\ +c(e,\theta,r)\equiv\{(x,y)\in\symbf{C} +\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ +C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). +\end{gather} +\begin{verbatim} +\begin{gather} +D(a,r)\equiv\{z\in\symbf{C}\colon \abs{z-a}<r\},\\ +\seg(a,r)\equiv\{z\in\symbf{C}\colon +\Im z= \Im a,\ \abs{z-a}<r\},\notag\\ +c(e,\theta,r)\equiv\{(x,y)\in\symbf{C} +\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ +C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). +\end{gather} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Unnumbered version. +\begin{gather*} +D(a,r)\equiv\{z\in\symbf{C}\colon \abs{z-a}<r\},\\ +\seg (a,r)\equiv\{z\in\symbf{C}\colon +\Im z= \Im a,\ \abs{z-a}<r\},\\ +c(e,\theta,r)\equiv\{(x,y)\in\symbf{C} + \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ +C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). +\end{gather*} +Some text after to test the below-display spacing. +\begin{verbatim} +\begin{gather*} +D(a,r)\equiv\{z\in\symbf{C}\colon \abs{z-a}<r\},\\ +\seg (a,r)\equiv\{z\in\symbf{C}\colon +\Im z= \Im a,\ \abs{z-a}<r\},\\ +c(e,\theta,r)\equiv\{(x,y)\in\symbf{C} + \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ +C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). +\end{gather*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\subsection{Align} +Numbered version: +\begin{align} +\gamma_x(t)&=(\cos tu+\sin tx,v),\\ +\gamma_y(t)&=(u,\cos tv+\sin ty),\\ +\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, + -\frac\beta\alpha\sin tu+\cos tv\right). +\end{align} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{align} +\gamma_x(t)&=(\cos tu+\sin tx,v),\\ +\gamma_y(t)&=(u,\cos tv+\sin ty),\\ +\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, + -\frac\beta\alpha\sin tu+\cos tv\right). +\end{align} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Unnumbered version: +\begin{align*} +\gamma_x(t)&=(\cos tu+\sin tx,v),\\ +\gamma_y(t)&=(u,\cos tv+\sin ty),\\ +\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, + -\frac\beta\alpha\sin tu+\cos tv\right). +\end{align*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{align*} +\gamma_x(t)&=(\cos tu+\sin tx,v),\\ +\gamma_y(t)&=(u,\cos tv+\sin ty),\\ +\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, + -\frac\beta\alpha\sin tu+\cos tv\right). +\end{align*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +A variation: +\begin{align} +x& =y && \text {by (\ref{eq:C})}\\ +x'& = y' && \text {by (\ref{eq:D})}\\ +x+x' & = y+y' && \text {by Axiom 1.} +\end{align} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{align} +x& =y && \text {by (\ref{eq:C})}\\ +x'& = y' && \text {by (\ref{eq:D})}\\ +x+x' & = y+y' && \text {by Axiom 1.} +\end{align} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\subsection{Align and split within gather} +When using the \env{align} environment within the \env{gather} +environment, one or the other, or both, should be unnumbered (using the +\verb"*" form); numbering both the outer and inner environment would +cause a conflict. + +Automatically numbered \env{gather} with \env{split} and \env{align*}: +\begin{gather} +\begin{split} \varphi(x,z) +&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ +&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n +\end{split}\\[6pt] +\begin{align*} +\zeta^0 &=(\xi^0)^2,\\ +\zeta^1 &=\xi^0\xi^1,\\ +\zeta^2 &=(\xi^1)^2, +\end{align*} +\end{gather} +Here the \env{split} environment gets a number from the outer +\env{gather} environment; numbers for individual lines of the +\env{align*} are suppressed because of the star. + +\begin{verbatim} +\begin{gather} +\begin{split} \varphi(x,z) +&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ +&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n +\end{split}\\[6pt] +\begin{align*} +\zeta^0 &=(\xi^0)^2,\\ +\zeta^1 &=\xi^0\xi^1,\\ +\zeta^2 &=(\xi^1)^2, +\end{align*} +\end{gather} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The \verb"*"-ed form of \env{gather} with the non-\verb"*"-ed form of +\env{align}. +\begin{gather*} +\begin{split} \varphi(x,z) +&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ +&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n +\end{split}\\[6pt] +\begin{align} \zeta^0&=(\xi^0)^2,\\ +\zeta^1 &=\xi^0\xi^1,\\ +\zeta^2 &=(\xi^1)^2, +\end{align} +\end{gather*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{gather*} +\begin{split} \varphi(x,z) +&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ +&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n +\end{split}\\[6pt] +\begin{align} \zeta^0&=(\xi^0)^2,\\ +\zeta^1 &=\xi^0\xi^1,\\ +\zeta^2 &=(\xi^1)^2, +\end{align} +\end{gather*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\subsection{Alignat} +Numbered version: +\begin{alignat}{3} +V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, + & \qquad U_i & = u_i, + \qquad \text{for $i\ne j$;}\label{eq:B}\\ +V_j & = v_j, & \qquad X_j & = x_j, + & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. +\end{alignat} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{alignat}{3} +V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, + & \qquad U_i & = u_i, + \qquad \text{for $i\ne j$;}\label{eq:B}\\ +V_j & = v_j, & \qquad X_j & = x_j, + & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. +\end{alignat} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Unnumbered version: +\begin{alignat*}3 +V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, + & \qquad U_i & = u_i, + \qquad \text{for $i\ne j$;} \\ +V_j & = v_j, & \qquad X_j & = x_j, + & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. +\end{alignat*} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{alignat*}3 +V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, + & \qquad U_i & = u_i, + \qquad \text{for $i\ne j$;} \\ +V_j & = v_j, & \qquad X_j & = x_j, + & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. +\end{alignat*} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +The most common use for \env{alignat} is for things like +\begin{alignat}{2} +x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\ +x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\ +x+x' & = y+y' && \qquad \text {by Axiom 1.} +\end{alignat} +Some text after to test the below-display spacing. + +\begin{verbatim} +\begin{alignat}{2} +x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\ +x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\ +x+x' & = y+y' && \qquad \text {by Axiom 1.} +\end{alignat} +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newpage +\setlength{\marginrulewidth}{0pt} + +\begin{thebibliography}{10} + +\bibitem{dihe:newdir} +W.~Diffie and E.~Hellman, \emph{New directions in cryptography}, IEEE +Transactions on Information Theory \textbf{22} (1976), no.~5, 644--654. + +\bibitem{fre:cichon} +D.~H. Fremlin, \emph{Cichon's diagram}, 1983/1984, presented at the +S{\'e}minaire Initiation {\`a} l'Analyse, G. Choquet, M. Rogalski, J. +Saint Raymond, at the Universit{\'e} Pierre et Marie Curie, Paris, 23e +ann{\'e}e. + +\bibitem{gouja:lagrmeth} +I.~P. Goulden and D.~M. Jackson, \emph{The enumeration of directed +closed {E}uler trails and directed {H}amiltonian circuits by +{L}angrangian methods}, European J. Combin. \textbf{2} (1981), 131--212. + +\bibitem{hapa:graphenum} +F.~Harary and E.~M. Palmer, \emph{Graphical enumeration}, Academic +Press, 1973. + +\bibitem{imlelu:oneway} +R.~Impagliazzo, L.~Levin, and M.~Luby, \emph{Pseudo-random generation +from one-way functions}, Proc. 21st STOC (1989), ACM, New York, +pp.~12--24. + +\bibitem{komiyo:unipfunc} +M.~Kojima, S.~Mizuno, and A.~Yoshise, \emph{A new continuation method +for complementarity problems with uniform p-functions}, Tech. Report +B-194, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information +Sciences. + +\bibitem{komiyo:lincomp} +\bysame, \emph{A polynomial-time algorithm for a class of linear +complementarity problems}, Tech. Report B-193, Tokyo Inst. of +Technology, Tokyo, 1987, Dept. of Information Sciences. + +\bibitem{liuchow:formalsum} +C.~J. Liu and Yutze Chow, \emph{On operator and formal sum methods for +graph enumeration problems}, SIAM J. Algorithms Discrete Methods +\textbf{5} (1984), 384--438. + +\bibitem{mami:matrixth} +M.~Marcus and H.~Minc, \emph{A survey of matrix theory and matrix +inequalities}, Complementary Series in Math. \textbf{14} (1964), 21--48. + +\bibitem{miyoki:lincomp} +S.~Mizuno, A.~Yoshise, and T.~Kikuchi, \emph{Practical polynomial time +algorithms for linear complementarity problems}, Tech. Report~13, Tokyo +Inst. of Technology, Tokyo, April 1988, Dept. of Industrial Engineering +and Management. + +\bibitem{moad:quadpro} +R.~D. Monteiro and I.~Adler, \emph{Interior path following primal-dual +algorithms, part {II}: Quadratic programming}, August 1987, Working +paper, Dept. of Industrial Engineering and Operations Research. + +\bibitem{ste:sint} +E.~M. Stein, \emph{Singular integrals and differentiability properties +of functions}, Princeton Univ. Press, Princeton, N.J., 1970. + +\bibitem{ye:intalg} +Y.~Ye, \emph{Interior algorithms for linear, quadratic and linearly +constrained convex programming}, Ph.D. thesis, Stanford Univ., Palo +Alto, Calif., July 1987, Dept. of Engineering--Economic Systems, +unpublished. + +\end{thebibliography} + +\end{document} +\endinput diff --git a/fonts/newcomputermodern/otf/NewCM08-Book.otf b/fonts/newcomputermodern/otf/NewCM08-Book.otf Binary files differindex 7ed4116f3d..3ce70d1c06 100644 --- a/fonts/newcomputermodern/otf/NewCM08-Book.otf +++ b/fonts/newcomputermodern/otf/NewCM08-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCM08-BookItalic.otf b/fonts/newcomputermodern/otf/NewCM08-BookItalic.otf Binary files differindex 473e3c3468..0ffe04ec08 100644 --- a/fonts/newcomputermodern/otf/NewCM08-BookItalic.otf +++ b/fonts/newcomputermodern/otf/NewCM08-BookItalic.otf diff --git a/fonts/newcomputermodern/otf/NewCM08-Italic.otf b/fonts/newcomputermodern/otf/NewCM08-Italic.otf Binary files differindex 5cf41ce959..dc63f591c2 100644 --- a/fonts/newcomputermodern/otf/NewCM08-Italic.otf +++ b/fonts/newcomputermodern/otf/NewCM08-Italic.otf diff --git a/fonts/newcomputermodern/otf/NewCM08-Regular.otf b/fonts/newcomputermodern/otf/NewCM08-Regular.otf Binary files differindex c7d99ad897..67449187a0 100644 --- a/fonts/newcomputermodern/otf/NewCM08-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCM08-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCM08Devanagari-Book.otf b/fonts/newcomputermodern/otf/NewCM08Devanagari-Book.otf Binary files differindex b02200f784..c00015f006 100644 --- a/fonts/newcomputermodern/otf/NewCM08Devanagari-Book.otf +++ b/fonts/newcomputermodern/otf/NewCM08Devanagari-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCM08Devanagari-Regular.otf b/fonts/newcomputermodern/otf/NewCM08Devanagari-Regular.otf Binary files differindex 78c7b8e1ea..4f8e94b9fa 100644 --- a/fonts/newcomputermodern/otf/NewCM08Devanagari-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCM08Devanagari-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-Bold.otf b/fonts/newcomputermodern/otf/NewCM10-Bold.otf Binary files differindex 4ce44956d2..e170424b44 100644 --- a/fonts/newcomputermodern/otf/NewCM10-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCM10-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-BoldItalic.otf b/fonts/newcomputermodern/otf/NewCM10-BoldItalic.otf Binary files differindex 4960299320..75dbe2bb12 100644 --- a/fonts/newcomputermodern/otf/NewCM10-BoldItalic.otf +++ b/fonts/newcomputermodern/otf/NewCM10-BoldItalic.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-Book.otf b/fonts/newcomputermodern/otf/NewCM10-Book.otf Binary files differindex 01cd69a60a..047920b634 100644 --- a/fonts/newcomputermodern/otf/NewCM10-Book.otf +++ b/fonts/newcomputermodern/otf/NewCM10-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-BookItalic.otf b/fonts/newcomputermodern/otf/NewCM10-BookItalic.otf Binary files differindex e6f3f86e8a..987eca62da 100644 --- a/fonts/newcomputermodern/otf/NewCM10-BookItalic.otf +++ b/fonts/newcomputermodern/otf/NewCM10-BookItalic.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-Italic.otf b/fonts/newcomputermodern/otf/NewCM10-Italic.otf Binary files differindex 4d1ca0cae5..d8668a3bd4 100644 --- a/fonts/newcomputermodern/otf/NewCM10-Italic.otf +++ b/fonts/newcomputermodern/otf/NewCM10-Italic.otf diff --git a/fonts/newcomputermodern/otf/NewCM10-Regular.otf b/fonts/newcomputermodern/otf/NewCM10-Regular.otf Binary files differindex 84c08a3bf5..deca16ead0 100644 --- a/fonts/newcomputermodern/otf/NewCM10-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCM10-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCM10Devanagari-Bold.otf b/fonts/newcomputermodern/otf/NewCM10Devanagari-Bold.otf Binary files differindex 6987e9619d..6a4add2226 100644 --- a/fonts/newcomputermodern/otf/NewCM10Devanagari-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCM10Devanagari-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCM10Devanagari-Book.otf b/fonts/newcomputermodern/otf/NewCM10Devanagari-Book.otf Binary files differindex be1cf51a30..b452d018e9 100644 --- a/fonts/newcomputermodern/otf/NewCM10Devanagari-Book.otf +++ b/fonts/newcomputermodern/otf/NewCM10Devanagari-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCM10Devanagari-Regular.otf b/fonts/newcomputermodern/otf/NewCM10Devanagari-Regular.otf Binary files differindex f185c316f0..7423521961 100644 --- a/fonts/newcomputermodern/otf/NewCM10Devanagari-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCM10Devanagari-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMMath-Bold.otf b/fonts/newcomputermodern/otf/NewCMMath-Bold.otf Binary files differnew file mode 100644 index 0000000000..7f066cc5df --- /dev/null +++ b/fonts/newcomputermodern/otf/NewCMMath-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCMMath-Book.otf b/fonts/newcomputermodern/otf/NewCMMath-Book.otf Binary files differindex e27c8b7831..0b16c7abed 100644 --- a/fonts/newcomputermodern/otf/NewCMMath-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMMath-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMMath-Regular.otf b/fonts/newcomputermodern/otf/NewCMMath-Regular.otf Binary files differindex fa03029f51..fbcb0b8ce3 100644 --- a/fonts/newcomputermodern/otf/NewCMMath-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMMath-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-Bold.otf b/fonts/newcomputermodern/otf/NewCMMono10-Bold.otf Binary files differindex 50ba877119..33e4e5ad7c 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-BoldOblique.otf b/fonts/newcomputermodern/otf/NewCMMono10-BoldOblique.otf Binary files differindex 5c4ba9e86a..51c5d16b14 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-BoldOblique.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-BoldOblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-Book.otf b/fonts/newcomputermodern/otf/NewCMMono10-Book.otf Binary files differindex ebd8ad581f..c48df75bd5 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-BookItalic.otf b/fonts/newcomputermodern/otf/NewCMMono10-BookItalic.otf Binary files differindex a04b75a355..2aa9ec408c 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-BookItalic.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-BookItalic.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-Italic.otf b/fonts/newcomputermodern/otf/NewCMMono10-Italic.otf Binary files differindex 99eda8c986..0e0bcf3c05 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-Italic.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-Italic.otf diff --git a/fonts/newcomputermodern/otf/NewCMMono10-Regular.otf b/fonts/newcomputermodern/otf/NewCMMono10-Regular.otf Binary files differindex ac4bf3aa9f..c0c2177119 100644 --- a/fonts/newcomputermodern/otf/NewCMMono10-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMMono10-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans08-Book.otf b/fonts/newcomputermodern/otf/NewCMSans08-Book.otf Binary files differindex c79936eb80..e21ad7a489 100644 --- a/fonts/newcomputermodern/otf/NewCMSans08-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMSans08-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans08-BookOblique.otf b/fonts/newcomputermodern/otf/NewCMSans08-BookOblique.otf Binary files differindex e4178cd80c..7b57e56a94 100644 --- a/fonts/newcomputermodern/otf/NewCMSans08-BookOblique.otf +++ b/fonts/newcomputermodern/otf/NewCMSans08-BookOblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans08-Oblique.otf b/fonts/newcomputermodern/otf/NewCMSans08-Oblique.otf Binary files differindex cf707402e1..a41ad065f8 100644 --- a/fonts/newcomputermodern/otf/NewCMSans08-Oblique.otf +++ b/fonts/newcomputermodern/otf/NewCMSans08-Oblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans08-Regular.otf b/fonts/newcomputermodern/otf/NewCMSans08-Regular.otf Binary files differindex 120b73717e..8d7e495eac 100644 --- a/fonts/newcomputermodern/otf/NewCMSans08-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMSans08-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-Bold.otf b/fonts/newcomputermodern/otf/NewCMSans10-Bold.otf Binary files differindex ca2bcdfebc..1b094dc38b 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-BoldOblique.otf b/fonts/newcomputermodern/otf/NewCMSans10-BoldOblique.otf Binary files differindex 62ab987327..1402569f6d 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-BoldOblique.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-BoldOblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-Book.otf b/fonts/newcomputermodern/otf/NewCMSans10-Book.otf Binary files differindex 43ddfb93b3..5f289aba7d 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-BookOblique.otf b/fonts/newcomputermodern/otf/NewCMSans10-BookOblique.otf Binary files differindex 056d616445..84aa0abc49 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-BookOblique.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-BookOblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-Oblique.otf b/fonts/newcomputermodern/otf/NewCMSans10-Oblique.otf Binary files differindex e8d4baad44..f623188ea2 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-Oblique.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-Oblique.otf diff --git a/fonts/newcomputermodern/otf/NewCMSans10-Regular.otf b/fonts/newcomputermodern/otf/NewCMSans10-Regular.otf Binary files differindex 96da57f58f..e4bde6b3c0 100644 --- a/fonts/newcomputermodern/otf/NewCMSans10-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMSans10-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMSansMath-Regular.otf b/fonts/newcomputermodern/otf/NewCMSansMath-Regular.otf Binary files differnew file mode 100644 index 0000000000..eb571b226e --- /dev/null +++ b/fonts/newcomputermodern/otf/NewCMSansMath-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial08-Bold.otf b/fonts/newcomputermodern/otf/NewCMUncial08-Bold.otf Binary files differindex d8508a473f..0974289bf6 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial08-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial08-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial08-Book.otf b/fonts/newcomputermodern/otf/NewCMUncial08-Book.otf Binary files differindex 40eee883e3..7051678d15 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial08-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial08-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial08-Regular.otf b/fonts/newcomputermodern/otf/NewCMUncial08-Regular.otf Binary files differindex 197896ad94..ceeea18eaa 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial08-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial08-Regular.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial10-Bold.otf b/fonts/newcomputermodern/otf/NewCMUncial10-Bold.otf Binary files differindex ee074fe4be..e4115d91ff 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial10-Bold.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial10-Bold.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial10-Book.otf b/fonts/newcomputermodern/otf/NewCMUncial10-Book.otf Binary files differindex ac7f816a2c..614ac51a5b 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial10-Book.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial10-Book.otf diff --git a/fonts/newcomputermodern/otf/NewCMUncial10-Regular.otf b/fonts/newcomputermodern/otf/NewCMUncial10-Regular.otf Binary files differindex 5ea6f2df1d..71eae5a0e7 100644 --- a/fonts/newcomputermodern/otf/NewCMUncial10-Regular.otf +++ b/fonts/newcomputermodern/otf/NewCMUncial10-Regular.otf diff --git a/fonts/newcomputermodern/source.txz b/fonts/newcomputermodern/source.txz Binary files differindex c631bc4687..99d6c998d9 100644 --- a/fonts/newcomputermodern/source.txz +++ b/fonts/newcomputermodern/source.txz diff --git a/fonts/newcomputermodern/tex/newcomputermodern.sty b/fonts/newcomputermodern/tex/newcomputermodern.sty index 22e39e68a1..d5e1db18c4 100644 --- a/fonts/newcomputermodern/tex/newcomputermodern.sty +++ b/fonts/newcomputermodern/tex/newcomputermodern.sty @@ -1,5 +1,5 @@ %\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{newcomputermodern}[v5.1 NewCM fonts by Antonis Tsolomitis] +\ProvidesPackage{newcomputermodern}[v7.0.0 NewCM fonts by Antonis Tsolomitis] \typeout{Loading NewCM fonts by Antonis Tsolomitis} \RequirePackage{fontspec} \RequirePackage{unicode-math} @@ -8,6 +8,10 @@ \DeclareOption{regular}{\@olddefaulttrue} \ProcessOptions\relax +\newif\if@sansdefault +\DeclareOption{sansdefault}{\@sansdefaulttrue} +\ProcessOptions\relax + %Next command has been moved to fspolddefault.tex and % fspdefault.tex %\let\prosgegrammeni=\undefined %Babel compatibility @@ -15,7 +19,11 @@ \if@olddefault \input{fspolddefault.tex} \else +\if@sansdefault +\input{fspsansdefault.tex} +\else \input{fspdefault.tex} \fi +\fi \endinput |