MathML Examples for Tralics

The Tralics documentation sometimes shows the translation of an expression as an image. These do not always correspond to the latest version of the software. We give here a variant, using MathML. For instance, the translation of $x$ is <formula type='inline'> <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>x</mi></math></formula>; the math element of the formula renders as $x$. In the case of environment like align, we have used the option -multi-math-label. This has as effect to add id and id-text attributes to parts of the math formula. Since id-text is not defined by MathML, we have removed these attributes. Example: on row1 and row3 for the eqnarray example; on row1 and row2 for the gather example; on row1 and row2 for the split example.

Accents

Example of math accents: $\hat{a} \acute{a} \bar{a} \dot{a} \breve{a} \check{a} \grave{a} \vec{a} \ddot{a} \tilde{a}$ translates as $\widehat{a}\acute{a}\bar{a}\dot{a}\breve{a}\check{a}\stackrel{`}{a}\overrightarrow{a}\ddot{a}\tilde{a}$ . Example of usual accents: \`e \'e \^e \"e \~a \.e \=e translates as è é ê ë ã ė ē. Example of other accents: \c c, \H o, \C o, \"o, \k a, \b b, \d a, \u a, \f a, \v a, \T e, \r a, \D a, \h a, \V e translates as ç, ő, ȍ, ö, ą, ḇ, ạ, ă, ȃ, ǎ, ḛ, å, ḁ, ả, ḙ. Example of double accents, translation of \={\.a} \={\"a}, \"{\'u} \"{\`u} \"{\v u} \c{\'c} \c{\u e} \d{\. s} \'{\^e} is ǡ ǟ, ǘ ǜ ǚ ḉ ḝ ṩ ế. The accentee is normally a 7bit character; there are exeptions so that \=\ae \'\ae \'\aa \'\o \'\i translates as ǣǽǻǿí.

\accentset

Translation of $\accentset xy = \accentset{\ast}{X} = \hat{\accentset{\star}{\hat h}} $ is. $\stackrel{x}{y}=\stackrel{*}{X}=\widehat{\stackrel{☆}{\widehat{h}}}$ .

Math accents

$\acute{x} \bar{x} \breve{x} \check{x} \ddddot{x} \dddot{x} \ddot{x} \dot{x} \grave{x} \hat{x} \mathring{x} \tilde{x} \vec{x} \widehat{xyz} \widetilde{xyz}$ gives $\acute{x}\bar{x}\breve{x}\check{x}\stackrel{⃜}{x}\stackrel{⃛}{x}\ddot{x}\dot{x}\stackrel{`}{x}\widehat{x}\mathring{x}\tilde{x}\overrightarrow{x}\widehat{xyz}\widetilde{xyz}$ .

$\widetilde{abc} \widehat{abc} \overleftarrow{abc} \overrightarrow{abc} \overline{abc} \underline{abc} \overbrace{abc} \underbrace{abc} \underleftarrow{abc} \underrightarrow{abc}$ gives $\widetilde{abc}\widehat{abc}\overleftarrow{abc}\overrightarrow{abc}\overline{abc}\underset{\_}{abc}\overbrace{abc}\underbrace{abc}\underleftarrow{abc}\underrightarrow{abc}$ .

$\stackrel{j}{\longrightarrow} \overset{*}{X} \underset{*}{X} \sqrt{abc} \sqrt[n]{abc} \root n \of{abc} \frac{abc}{xyz} \dfrac{abc}{xyz}$ gives $\stackrel{j}{⟶}\stackrel{*}{X}\underset{*}{X}\sqrt{abc}\sqrt[n]{abc}\sqrt[n]{abc}\frac{abc}{xyz}{\displaystyle \frac{abc}{xyz}}$

Alignments

\begin{align}
x^2+y^2&=1\\ x&=\sqrt{1-y^2}
\end{align}

\begin{equation}
\begin{aligned}[x]
x^2+y^2&=1& 1&=X^2+Y^2\\
x&=0.01&0.001=X
\end{aligned}
\end{equation}

\def\R{\mathbf{R}}
\begin{eqnarray*}
\left\{\begin{array}{lcl}
\dot{x} & = & Ax+g(x,u)\\
 y & = & Cx \\
 \multicolumn{3}{l}{x\in \R^n} 
\end{array}
    \right.
\end{eqnarray*}

\def\MAT#1{\begin{array}{c#1}1&22\\3&4\end{array}}

\[\left|
\begin{array}{lcr}
AAAAAAA&BBBBBCC&CCCCCCC\\
A&B&C\\
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{B}&\multicolumn{1}{c}{C}\\
\multicolumn{1}{r}{A}&\multicolumn{1}{r}{B}&\multicolumn{r}{c}{C}\\
\multicolumn{1}{l}{A}&\multicolumn{1}{l}{B}&\multicolumn{l}{c}{C}\\
\MAT l&\MAT c&\MAT r\\
\multicolumn{2}{c}{0123456789abcdef}&C\\
A&\multicolumn{2}{c}{0123456789abcdef}\\
\multicolumn{2}{r}{0123456789abcdef}&C\\
A&\multicolumn{2}{l}{0123456789abcdef}\\
A&B&C\\
\end{array}
\right|\]

\begin{eqnarray}
x & = &17y \\
y & > & a + b + c+d+e+f+g+h+i+j+ \nonumber\\
  &   & k+l+m+n+o+p
\end{eqnarray}

\begin{eqnarray*}
x & \ll & y_{1} + \cdots + y_{n} \\
  & \leq &z
\end{eqnarray*}

\begin{gather}
  (a + b)^2 = a^2 + 2ab + b^2          \\
  (a + b) \cdot (a - b) = a^2 - b^2
\end{gather}

\begin{multline}
\text{First line: left align}\\
\text{centered}\\
\text{This is the longest line of the table}\\
\shoveright{\text{shoved left}}\\
\shoveleft{\text{shoved right}}\\
\text{Last right}
\end{multline}

\begin{equation}
\begin{split}
(a+b)^4 &= (a+b)^ 2 (a+b)^2           \\
        &= (a^2+2ab+b^2)(a^2+2ab+b^2) \\
        &= a^4+4a^3b+6a^2b^2+4ab^3+b^4 \\
\end{split}
\end{equation}

Counters

\def\showcounter#1{%
\arabic{#1} \roman{#1} \Roman{#1} \alph{#1} \Alph{#1} \fnsymbol{#1}  $\fnsymbol{#1}$\\}
\newcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}
\stepcounter{ctr}\showcounter{ctr}

One can notice that \fnsymbol works in math mode or outside it, and the result is not always the same. The effect of \\ is to start a no-indented paragraph by adding an attribute to the <p> element. We have replaced it by a class attribute interpreted by the style sheet.

The default translation of a Greek letter like \alpha is an entity. There is an option -noentnames that replaces the entity names by character references &alpha; will be replaced by &#x3B1;. We show here the entities. $\alpha \beta \gamma \delta ϵϵ\zeta \eta \theta \iota \kappa \lambda \mu \nu \xi \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega \varpi \varrho \varsigma \varphi \varkappa \vartheta$ $\Gamma \Delta \Theta \Lambda \Xi \Sigma \Upsilon \Phi \Pi \Psi \Omega$, and the text version: α β γ δ ϵ ϵ ζ η θ ι κ λ μ ν ξ π ρ σ τ υ φ χ ψ ω ϖ ϱ ς ϕ ϰ ϑ Γ Δ Θ Λ Ξ Σ ϒ Φ Π Ψ Ω

Binom

 
\def\n{\mathchoice{a}{b}{c}{d}}
$$\binom{\frac{\n+m}{2}} \n \quad
\tbinom{\frac{\n+m}{2}} \n \quad
\dbinom{\frac{\n+m}{2}} \n\qquad
{}^{\binom{\frac{\n+m}{2}} \n \quad
\tbinom{\frac{\n+m}{2}} \n \quad
\dbinom{\frac{\n+m}{2}} \n}$$

${a\over b}\quad {a\above1pt b}\quad {a\atop b}$
${a\overwithdelims() b}\quad {a\abovewithdelims()1pt b}%
  \quad {a\atopwithdelims() b}$

Translates as $\frac{a}{b}\genfrac{}{}{1pt}{}{a}{b}\genfrac{}{}{0pt}{}{a}{b}$ and $\left(\frac{a}{b}\right)\left(\genfrac{}{}{1pt}{}{a}{b}\right)\left(\genfrac{}{}{0pt}{}{a}{b}\right)$.

Bmod

\begin{align*}
u& \equiv v+1 \mod{n^2} \\
u& \equiv v+1 \bmod{n^2} \\
u&   =    v+1 \pmod{n^2} \\
u&   =    v+1 \pod{n^2} \\
\end{align*}

\boxed

$x = \boxed{\frac12}+z$ translates as $x=\begin{array}{|c|}\hline \frac{1}{2}\\ \hline\end{array}+z$.

Cases

\[\text{The sign function: \ } 
\mathcal{S}(x) = \begin{cases}
-1&x<0 \\
0&x=0 \\
1&x>0
\end{cases}
\]

Math fonts

${\cal ABCDE FGHIJ KLMNO PQRST UVWXYZ}$ gives $\mathrm{𝒜ℬ𝒞𝒟ℰ}\mathrm{ℱ𝒢\mathscr{H}ℐ𝒥}\mathrm{𝒦ℒℳ𝒩𝒪}\mathrm{𝒫𝒬ℛ𝒮𝒯}\mathrm{𝒰𝒱𝒲𝒳𝒴𝒵}$ when using Unicode characters in the SMP and $\mathcal{ABCDE}\mathcal{FGHIJ}\mathcal{KLMNO}\mathcal{PQRST}\mathcal{UVWXYZ}$ when using 7-bit characters and mathvariant attribute. The two results should be identical.

Style

${\frac12}{x\displaystyle\frac 12} 
{x\textstyle\frac 12}{x\scriptstyle\frac 12}{x\scriptscriptstyle\frac 12} \qquad
{x\displaystyle a}+{x\textstyle a}+{x\scriptstyle a}+{x\scriptscriptstyle a}\qquad
\frac{x\displaystyle a}{x\textstyle b}+\frac{x\scriptstyle a}{x\scriptscriptstyle b}$

Translates to

Fractions

$$\frac12 \dfrac 12 \tfrac12 \qquad
{}^{\frac12 \dfrac 12 \tfrac12}$$

\def\N{\mathchoice{a}{b}{c}{d}}
$$\frac{\N\frac{\N+m}{2}} \N \quad
\tfrac{\N\frac{\N+m}{2}} \N \quad
\dfrac{\N\frac{\N+m}{2}} \N
$$

$$\genfrac{}{}{}{}12 % this is \frac 
\genfrac[]{2pt}{2}{aaa}{bbb} % brackets, small, thick 
\genfrac(){0cm}{0}nm % this is \dbinom 
$$

Math tables

$\smallint \int \land \wedge \lor \owns \ni \notin
\neg \lnot \gets \leftarrow \to \rightarrow
\gtrless \geqslant \leqslant \iff \backslash$

Translation $\int \int \wedge \wedge \vee \ni \ni \notin \neg \neg \leftarrow \leftarrow \to \to \gtrless ⩾⩽\iff \setminus$

 
$ \pm \mp \times \div \ast \star \circ \bullet \cdot 
\cap \cup \uplus \sqcap \sqcup \vee \wedge \setminus \wr
\diamond \bigtriangleup \bigtriangledown \triangleleft \triangleright
\oplus \ominus \otimes \oslash \odot \bigcirc \dagger \ddagger \amalg$

Table 8.4 $\pm \mp \times ÷*☆\circ •·\cap \cup \uplus \sqcap \bigsqcup \vee \wedge \setminus \wr \diamond △▽◃▹\oplus \ominus \otimes \oslash \odot ◯†‡⨿$

$\le \leq \succ \simeq \parallel \subseteq \sqsubset \doteq
\ge \geq \sim \mid \subset \subseteq \ne \neq \frown \vdash
\equiv \perp \ll \supset \cong \smile \in \dashv 
\models \preceq \gg \approx \Join \sqsubseteq \ni
\prec \succeq \asymp \bowtie \sqsupseteq \propto$

Table 8.5 $\le \le \succ \simeq \parallel \subseteq ⊏\doteq \ge \ge \sim \mid \subset \subseteq \ne \ne ⌢⊢\equiv \perp \ll \supset \cong ⌣\in ⊣\vDash ⪯\gg \approx \bowtie ⊑\ni \prec ⪰\asymp \bowtie ⊒\propto$

$\leftarrow\longleftarrow  \Leftarrow \Longleftarrow 
\rightarrow \longrightarrow \Rightarrow \Longrightarrow
\leftrightarrow \Leftrightarrow \longleftrightarrow\Longleftrightarrow
\mapsto \longmapsto \hookleftarrow \hookrightarrow
\leftharpoondown \leftharpoonup \rightharpoondown \rightharpoonup
\uparrow \Uparrow  \updownarrow \Updownarrow 
\downarrow \Downarrow \nearrow \searrow \swarrow \nwarrow
$

Table 8.6 $\leftarrow ⟵\Leftarrow ⟸\to ⟶\Rightarrow ⟹\leftrightarrow \iff ⟷⟺\mapsto ⟼\hookleftarrow \hookrightarrow \leftharpoondown \leftharpoonup \rightharpoondown \rightharpoonup \uparrow \Uparrow \updownarrow \Updownarrow \downarrow \Downarrow \nearrow \searrow \swarrow \nwarrow$

$\ldots \prime \exists \Diamond \top \bot \mho 
\cdots \forall \nabla \imath \flat \clubsuit \Re
\vdots \infty \surd \jmath \natural \diamondsuit \Im
\ddots \hbar \square \ell \sharp \heartsuit \angle
\aleph \emptyset \triangle \neg \wp \spadesuit \partial$

Table 8.7: $...\text{'}\exists \diamond \top \perp \mho \cdots \forall \nabla ı♭♣\Re ⋮\infty \surd j♮♦\Im \ddots \hslash \square \ell ♯♥\angle \aleph \varnothing ▵\neg \wp ♠\partial$

$\coprod \bigvee \bigwedge \biguplus \bigcap \bigcup \int \sum 
 \prod \bigotimes \bigoplus \oint \bigsqcup  \bigodot$

Table 8.8: $\coprod \bigvee \bigwedge \uplus \bigcap \bigcup \int \sum \prod ⨂⨁\oint ⨆⨀$

 
$$\arccos^2 (x),\, \arcsin^2(x),\,  \arctan^2(x),\,  \arg^2(x),\, 
  \cos^2(x),\,  \cosh^2(x)  ,\, \cot^2(x)$$
$$\coth^2(x),\, \csc^2(x),\,   \deg^2(x),\,   \dim^2(x),\, \exp^2(x),\, 
  \hom^2(x),\,  \ker^2(x) ,\, \lg^2(x)$$
$$\ln^2(x),\, \log^2(x),\,  \Pr^2(x),\, \sec^2(x),\, \sin^2(x) 
  ,\,\sinh^2(x),\,  \tan^2(x),\, \tanh^2(x) $$
$$\det_{x=1},\, \gcd_{x=1},\, \inf_{x=1},\, \injlim_{x=1},\, \lim_{x=1},\, 
\liminf_{x=1},\,\limsup_{x=1},\,\max_{x=1},\, \min_{x=1},\, 
 \projlim_{x=1},\, \sup_{x=1}
$$

Table 8.9

\def\kernel{\frac{x^1_2}{y^3_4}}
\def\test{
\left\lmoustache\left\uparrow\left\Uparrow\left\updownarrow\left\lgroup
\kernel
\right\rgroup\right\Updownarrow\right\Downarrow\right\downarrow\right\rmoustache}
$\test$
$\let\left\relax\let\right\relax \test$
$\def\kernel{\int_0^{\frac{x^1_2}{y^3_4}}} \test$

Translation is $⎰↑⇑↕〔\frac{x_2^1}{y_4^3}〕⇕⇓↓⎱$ and $⎰\uparrow \Uparrow \updownarrow 〔\frac{x_2^1}{y_4^3}〕\Updownarrow \Downarrow \downarrow ⎱$ and $⎰↑⇑↕〔{\int }_0^{\frac{x_2^1}{y_4^3}}〕⇕⇓↓⎱$

Delimiters

 
$
\def\kernel{\frac{x^1_2}{y^3_4}}
\left\Vert\left|\left[\left(\left\{ \kernel \right\}\right)\right]\right|\right\Vert
\left\vert\left|\left<\left\langle \kernel \right\rangle\right>\right|\right\vert
\left\lbrace\left\lceil\left\lfloor \kernel \right\rfloor\right\rceil\right\rbrace
\lbrack\rbrack
$

Translation $∥\left|\left[\left(\left\{\frac{x_2^1}{y_4^3}\right\}\right)\right]\right|∥\left|\left|〈〈\frac{x_2^1}{y_4^3}〉〉\right|\right|\left\{⌈⌊\frac{x_2^1}{y_4^3}⌋⌉\right\}\left[\right]$

 
\[ \left(\left[\left\lbrack \left\{\left\lbrace\left\lfloor 
\left\lceil\left\langle\left/ \left|\left\|\left\uparrow 
\left\downarrow\left\updownarrow\left<\left\lmoustache
\left\lgroup\frac 12\right\rgroup
\right\rmoustache\right>\right\Updownarrow\right\downarrow 
\right\Uparrow\right\Vert\right\vert \right\backslash\right\rangle
\right\rceil\right\rfloor\right\rbrace\right\} \right\rbrack\right]\right)\]

Math examples

\providecommand\operatorname[1]{%
  \mathmo{#1}%
  \mathattribute{form}{prefix}%
  \mathattribute{movablelimits}{true}%
}
\def\Dmin{\operatorname{dmin}}
$\min _xf(x) >\Dmin _xf(x)$.

Translation ${min}_{x}f\left(x\right)>{dmin}_{x}f\left(x\right)$

Limits

\def\A#1{\mathop a #1_b, \mathop a#1^c, \mathop a#1_b^c\qquad}
$\A{} \A\limits \A\nolimits \A\displaylimits $
\[\A{} \A\limits \A\nolimits \A\displaylimits \]

Compare the inline math formula $a_b,a^c,a_b^c\underset{b}{a},\stackrel{c}{a},\underset{b}{\overset{c}{a}}{a}_b,a^c,a_b^c{a}_b,a^c,a_b^c$ and the display math formula

\DeclareMathOperator{\Sin}{sin}
\DeclareMathOperator*{\Limsup}{lim \, sup}
%% the effect of * is visible only in display style
\[\Sin^2 = \sin^2 = \operatorname{sin}^2 \quad
\Limsup_x = \limsup_x = \operatorname*{lim\;sup}_x \]

Attributes

\begin{align}
\formulaattribute{tag}{8-2-3}
\thismathattribute{background}{yellow}
\tableattribute{mathcolor}{red}
\rowattribute{mathvariant}{bold} x^2 + y^2+100 &=  z^2 \\
\cellattribute{columnalign}{left}\cellattribute{class}{bl}  x^3 + y^3+1 &<  z^3
\cellattribute{mathbackground}{white}\cellattribute{mathvariant}{bold} 
\end{align}

This example produces illegal MathML (illegal attribute names) so that what you see below is a modification of the MathML code produced by Tralics. The attribute background is illegal, it was replaced by mathbackground. This attribute, as well as mathcolor and mathvariant can only be put in token elements. So we inserted two <mstyle> elements with the attributes, one that contains the table, one that contains the content of the last cell. It is not possible to add a mathvariant attribute to the first row since (a) one cannot put the row in a <mstyle> element and (b) the child of the row cannot be a <mstyle> element.

Math spacing

Math spacing\\
$xxxxx$\\
$x\,x\>x\;x\!x$\\
$a\,\,\,\,\,a\>\>\>\>\>a$\\
$a\;\;\;a\;\;\;\;a$\\
  

Math spacing
$xxxxx$
$xxxxx$
$aaa$
$aaa$

Sideset

$\sideset{}{'}\sum_i x_i$
$\sideset{^{a}_b}{_{D}^c}\sum_x^yw$

Translates as $\underset{i}{\sum ^{\text{'}}}{x}_i$ and $\underset{x}{\overset{y}{{}_b{}^a\sum _D^c}}w$; same formula in \displaystyle added ${\displaystyle \underset{i}{\sum ^{\text{'}}}{x}_i}$ and ${\displaystyle \underset{x}{\overset{y}{{}_b{}^a\sum _D^c}}w}$

\[\xleftarrow{U+u}  \xleftarrow[D+d]{} \xleftarrow[U+u+v]{D+d}
\xrightarrow{U+u}  \xrightarrow[D+d]{} \xrightarrow[U+u]{P+p+q}\]

Characters

The following tables were generated accorging to the code given in this file. We have regoupred all characters with Unicode betwenn U+1D400 and U+1D7FF in a single table. At position 1D551 there is \red\mathZopf where the first fommand is defined by \def\red#1{#1\mathattribute{mathcolor}{red}}. This means that the character shouls be there, but is in fact at position U+2124.

Other symbols

And more

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