Producing MathML with Tralics

French title: Produire du MathML avec Tralics

Author: José Grimm

Location: Sophia Antipolis

Inria Research Theme: THnum

Inria Research Report Number: 6181

Revision: 2

Team: Apics

Date: May 2007

Revised Date: 29 November 2007

Keywords: Tralics, XML, MathML, mathematics, LaTeX, HTML, Pdf.

French keywords: Tralics, XML, MathML, mathématiques, LaTeX, HTML, Pdf.

Note: The software and the documentation is available at http://www-sop.inria.fr/apics/tralics.

Author's email address: Jose.Grimm[at]sophia.inria.fr.

Abstract

The purpose of this paper is to show that it is possible to translate almost every mathematical formula from LaTeX syntax to XML. The document you are reading contains a great number of formulas extracted from the TeXbook, the LaTeX companion, the MathML recommendation, and translated into XML by Tralics; it is available in Pdf or HTML version.

The HTML version was produced using a very simple style sheet; mathematics are left unchanged, and you need a MathML-aware browser, like Firefox, or adequate plugins, and a set of fonts for the symbols or operators. The Pdf version was obtained by use of xmltex (a package by D. Carlisle that makes TeX an XML interpreter), and a great number of modifications to the file mathml2.xmt (that interprets elements in the MathML namespace). All files needed to produce this document are part of the Tralics bundle (version 2.10 or more).

This document was compiled with the experimantal version 2.10.8, in which examples in section 4.1 should work fine.

French Abstract

Le but de ce papier est de démontrer qu´il est possible de traduire presque toutes les formules de mathématiques de la syntaxe LaTeX vers le langage XML. Le document que vous lisez contient un grand nombre de formules extraites du TeXbook, du LaTeX companion, de la recommandation MathML, et traduites en XML par le logiciel Tralics. Il est disponible en HTML et en Pdf.

La version HTML est obtenue en utilisant une feuille de style assez simple, qui laisse les formules de mathématiques inchangées; pour la lire, il vous faut un navigateur ou un plugin qui sache interpréter le MathML et les fontes associées pour les symboles et les opérateurs. La version Pdf a été obtenue grâce au package xmltex de D. Carlisle (qui fait de TeX un interprète XML), et un grand nombre de modifications dans le fichier mathml2.xmt, qui contient le code d´interprétation des mathématiques. Tous les fichiers nécessaires pour obtenir ce document sont distribués avec le package Tralics à partir de la version 2.10.

Short Table of Contents

1. Introduction
2. Examples from the TEXbook
3. Higher Mathematics, TLC2
4. Other Examples
5. Font tests
6. Examples from the MathML recommendation
Bibliography
Table of Contents

1. Introduction

The technical reports [6] and [7] describe some features of the Tralics software, a LaTeX to XML translator, and associated tools. In particular, we explain how the XML file can be converted to HTML, or to Pdf (using the excellent work of D. Carlisle and S. Rahtz ([10], [3], [2]). One application is RalyX (Inria´s Annual Activity Report), where all math formulas are converted to images before inclusion in the HTML document.

Recently, the cedram (Centre de diffusion de revues académiques de mathématiques, http://www.cedram.org/) has decided to convert the metadata of some collections (including the Annales de l´institut Fourier) from LaTeX to HTML, [1]. The Tralics software has been adapted for this purpose, the idea being the following: there are nowadays some browsers with a high quality math renderer, where native MathML formulas are more readable than images; thus the web site presents each document in two versions, neither of which containing an image. By default you see something like $L^1\left(\mathbb{T}\right)$, and the alternate version contains $L^1(\mathbb{T})$. This looks like the source code, but non-trivial operations are performed, as explained below.

There is a torture file that comes with Tralics; the purpose is to make sure that no error occurs while compiling expressions like \cfrac12, but it is hard to check that the result is correctly translated. For this reason, we decided to create a sample file containing, not only all examples of the LaTeX companion second edition [9], but also those of the TeXbook [8], or the MathML recommendation [4], and to convert it to HTML. Translation is correct if the HTML page prints correctly. This file is available on the Web, we hope that it will convince you to use the Tralics software, and put your math documents on the Web. We have inserted comments explaining why some constructions do not work, if you have a solution or suggestions, please send mail to the author. Additional examples are welcome.

We also wanted to make sure that the RalyX still works: more than one hundred teams use Tralics once a year, asking sometimes silly questions, like: how to add color to the text, or a page break; the answer is generally either `not yet implemented, wait for next year´, or `incompatible with the Raweb semantics´. This document uses colors and page breaks, as explained below; no modification to the translator was needed, but, unless the style sheets are modified, these additions are ignored.

For these reasons we converted the whole document to Pdf. This was not trivial, especially because of a lack of font families (this will be discussed later). Note also the format of the document: in the HTML version all metadata (author, title, abstract, etc.) are placed at the start of the document by the XML to HTML processor. In the case of the Pdf version, this was not so easy: something as trivial as \newpage \null \vskip-2cm is impossible to express in XSL/FO (said otherwise, all my attempts failed), and I had to add a new element <vspace> for this purpose.

The source file testmath.tex is part of the Tralics distribution; translating it requires Tralics version 2.10, and some packages, like amsmath (do not try to include the file amsmath.sty, this is not possible; you must use amsmath.plt instead). In most cases we show the TeX source before the translation. There are some exceptions, for instance we leave as an exercise to find out how we solved the exercises of the TeXbook. Examples of the LaTeX companion [9] are available on the CD that comes with the book.

The XML file was converted into HTML, math formulas are left unchanged. We explain in some cases that what we see is not what we expected. Here FL means Firefox on Linux (unless specified, it is Firefox 2.0, on Fedora Core 3) and FM means Firefox on MacIntosh (unless specified, it is Firefox 2.0 on a PowerBook G4, MacOS 10.4). We also tested Amaya, bot AM (version 9.55 on Mac) and AL (version 9.99 on Linux FC5).

Installing math fonts for use with a web browser is not always easy: they are installed as explained on the Mozilla web page, but this page seems unclear to me: on one Linux machine, Firefox is happy with the fonts, but uses the wrong font when printing, on another one, Firefox says that fonts Math1, Math2 and Math4 are missing, but the printed result is correct.

1.1. Special features

You are not supposed to insert page breaks in your document, but if you put twenty tables in a row, you will get error messages of the form: to many unprocessed floats, and then you are in trouble. In this case, the only remedy is to insert some \clearpage commands; in this document, we used the \xbox command in order to produce an empty <clearpage> element in the XML document, and we adapted the style sheets.

There is a \clearpage before this section; you will not notice it because there is a natural page break there. On the other hand, it is likely that you will see the page breaks in Chapter three, but there is no easy solution: there is no text in the current chapter to fill the gaps. In order to improve the layout, we changed the ordering of the figures, starting with tables 3 and 4, that occupy a full page, and inserting smaller tables after that. Moreover, table 4 was too big to fit on a page (the baseline in this document is much larger than in the LaTeX companion) and we split it into two parts. We are happy since this gives 8 full pages with floats, and a single bad page, the last text page of the chapter.

Concerning colors, you can put any attribute you like to any object (there is no attempt to validate the document against some DTD). In the case of math formulas, identifiers, numbers and operators accept some common attributes including font information and colors, these are currently ignored when converting XSL/FO into Pdf (but this can easily be changed, it suffices to add some lines of code to the raweb-cfg.sty file. However, the color attribute is honored for the <mstyle> element (all symbols appearing in the LaTeX companion for which no Unicode element has been found are replaced by a red $X$ using this method). For the text case, the <inline> element is translated into <span> element in the HTML and a <fo:inline> element in the Pdf, all attributes are copied; in the previous sentence we have used

\xbox{inline}{\XMLaddatt{color}{#F00000}\XMLaddatt{style}{color:#E00000}red}
$\mathbox{mstyle}[color][\#D00000]{X}$
\def\UnimplementedOperator{\mathbox{mstyle}[color][red]{X}}

This example shows that you can use a named color, or one defined in RGB syntax, using six hexadecimal digits; use sharp or backslash sharp as indicated here.

1.2. Compiling the example file

These are the options you can use when compiling the file testmath.tex

-noentnames -trivialmath=7 -leftquote=2018 -rightquote=2019 -nozerowidthspace

The first option says that you prefer &#x21DA; to &Lleftarrow; (this is useless if you use an XSLT processor to convert the XML document). Option `trivialmath=7´ says that simple math expressions are translated as text; option `mathvariant´ says that a command like \mathbb should add a mathvariant attribute; these two options set initial values to some counters, they are not needed, because the associated file testmath.ult resets them to the values needed here. Following options are problematic. If you say `nostraightquotes´, this changes the translation of the apostrophe character to character U+B4, this option produces a nice HTML file, but the Pdf is not so nice. Here we modify left and right quotes to U+2018 and U+2019. If you say `zerowidthspace´, Tralics inserts an invisible character in verbatim mode, that is supposed to inhibit ligatures when converting the XML file into Pdf file, the trouble is that this character is shown as a normal space by my browser in certain cases. In Tralics version 2.10, the translation changed: the old behavior can be obtained by saying -nozerowidthelt. Otherwise an element is created, this element is transformed into a zerowidthspace character when the document is converted to Pdf, is omitted when the document is converted to HTML. This means that the option is useless.

The following files are needed for the compilation:

• a configuration file, that behaves like the default one; namely one that defines the root element to be <std> and the dtd file to be classes.dtd;

• the source file testmath.tex, and the associated testmath.ult file;

• the class file report.clt (and the auxiliary file std.clt);

• the standard math packages amsmath.plt, amscd.plt, delarray.plt, amsxtra.plt, amsgen.plt, amsbsy.plt, amsopn.plt;

• the file fancyvrb.plt for the special verbatim mode; and

• the file RR.plt for the meta data.

When converting the document to HTML, we use the `xsltproc´ program. The style sheet is testmathhtml.xsl; this is a short style sheet; but it needs cls.xsl and RR.xsl. Conversion to XSL/FO uses RRfo.xsl as style sheet, it needs clsfo.xsl, that includes raweb3-param.xsl, rrrafo3.xsl, clspages.xsl, RRfosimple.xsl, clsfotable.xsl. Note: this list is likely to change; we have to separate clearly the RA (activity report), the RR (research report) and CLS (standard classes). All these files are in the directory xml or styles of the Tralics distribution, but the XSLT processor wants to see them in the current directory.

Conversion from XSL/FO to pdf is achieved by compiling the file wtestmath.tex containing the following lines

1 \def\xmlfile{testmath.fo}
2 \def\LastDeclaredEncoding{T1}
3 \input{xmltex.tex}
4 \end{document}

It requires xmltex.tex and other files from the xmltex distribution, so that you must install these files first. There are some bugs or incompatibilities in the fotex distribution, so that you must use the two files fotex.xmt, fotex.sty as distributed in the xml directory. Additional required files are fotex-add.sty, raweb-uni.sty, and fotex.cfg. Finally the file wtestmath.cfg should be a symbolic link to raweb-cfg.sty.

1.3. Overview of math mode

The main change between Tralics version 2.9 and 2.10 concerns handling of math formulas. Two steps are required for processing them: first a tree is created, after that, it is converted. This mechanism is similar to the behavior of TeX, but some details are not yet implemented, and some information is lost (for instance, \mathbin is a command that says that the object that follows should be of type `binary´; this information is used by the program, but the resulting XML tree does not contain this information (we cannot simply add an attribute pair class=bin to the object).

When Tralics sees a math formula, it constructs recursively a tree, also called a math list. The action after a token has been read is the following: If the token is expandable (a user defined command for instance), expansion takes place, this can read some tokens, it can add new tokens to the stream (to be read again). If the token is a mode-independent command it will be executed (for instance, you can change the meaning of some commands). Otherwise, the token is added to the tree, but there are some exceptions. The easy case is when a whole subtree is read, for instance when the token is a left brace, a \begin command, etc. In some cases, the execution level is incremented (said otherwise, the behavior is the same as a group defined by braces in non-math mode). In a case of \begin, a token list is read, and the behavior depends on whether this is a user defined environment (normal expansion rules apply), or a built-in one. Most built-in environments are matrix-like, and each cell is evaluated in a group; this has the following consequence: if the math formula contains an unwanted ampersand character, an error will be signaled when that character tries to finish the cell-group (because the current group is of type math), and an error is signaled at the end of the math expression (because the ampersand character has added a second group after the first, a cell-group, where the end-of-math character expected a math-group). A third error will be signaled later: when the tree is converted into a MathML object, Tralics may complain about non-math tokens in the expression. A font change command like \mathbb changes the internal state and adds two tokens to the tree, one that selects the blackboard font (before the argument) and a token that selects the current font (after the argument).

In the case of $L^1(\mathbb{T})$ the tree has eight nodes. One is a subtree, containing the argument of the command. They are six characters, and two commands (select double-struck font, or select normal font). When the tree is converted into a MathML expression, some nodes are converted to basic XML elements (the letters), some are ignored (the font change commands), and others are used to construct an XML tree. We shall see later that parentheses, braces, and tokens like that can induce <mrow> elements; translation of the hat character is non-obvious, because MathML provides <msub> and <mover> as possible translations; a command like \nolimits placed after a token is an indication for the translation of the hat that follows. When the tree is converted the current style is looked at, this quantity depends on the position in the tree (the style of the numerator of a fraction or the style of an exponent after the hat is smaller than the current style, while the style of the numerator of a \cfrac is always text style) or the presence of style change commands. Some commands depend on the style (for instance \mathchoice) and are conditionally interpreted.

The characters in the formula above are of type letter (L, T), or type non-letter (the digit, or the parentheses), or other (the hat). An interesting question is: what happens if the single L (or any other character) is replaced by a double one? A first answer is that this just adds a new token to the tree (there is a special case: ^^1 is a non-obvious way to represents the character q). Spaces are ignored, so that doubling them has no effect. Two hat characters in a row signal an error (assume for instance that a space between them inhibits the double-hat feature mentioned above). Two consecutive new line characters are replaced by \par, this is illegal. Characters that are neither letters nor digits generally translate to a <mo> element. No attempt is made to convert two plus characters into a double plus character (some people use `:=´ as an operator, meaning `is equal by definition´, and expect no white space between the two characters). In a case like \sqrt\frac{1}{2}, where we have two consecutive commands that take arguments, the argument of the root operator is the fraction operator, not the full fraction (this expression is accepted by LaTeX, as an unexpected consequence of the implemented of the fraction operator that provides that braces that delimit the argument of the square root, it is rejected by Tralics).

The superscript and subscript operators (generally associated to hat and underscore characters) are hybrid commands: most commands, for instance \frac, are prefix commands (they come before their arguments), a few commands, like \over, are infix operators (the first argument is before the operator). In the case of hat, there is one argument after the operator, but the operator acts on the kernel that is to its left (in the example the kernel is L, but a kernel can have an index and an exponent, so that the order of tokens can be: kernel, underscore, index, hat, exponent). Translation of character L is a <mi> element; Tralics tries to converts LL or TT into a single element, according to the following rule. Assume that \T is defined to be \mathbb{T} and \TT is defined to be \mathbb{TT}. Then \TT produces a single element with two characters, while \T\T produces two elements with a single character. On the other hand, a strange rule of MathML says that the font used for a <mi> element (that lacks font attributes) depends on the number of characters in it; for this reason, in a formula like $xy$, the translation is a sequence of two <mi> elements, without attributes, containing a single character (implicit product). In a case like $17$, a single <mn> element is constructed, because this represents a number (in a case like $x_{12}$, an implicit comma could be added in the case where the expression refers to row 1 column 2 of the matrix $x$; however, do not expect such a behavior in a near future). The source code of the expression above is $x_{12}$. Without the braces, the translation is $x_{1}2$ (if you do not see the difference, there is a bug somewhere).

If the internal counter \@nomathml is negative, the tree is converted to a character string in a trivial manner; in a case like \cfrac12, you will see \cfrac{1}{2}, because we have a sub-tree with two nodes (the two arguments of the command). In reality, the command takes an optional argument, and its value is printed only if it is not empty. In a case like $L^1\left(\mathbb{T}\right)$, you will see two font changes as explained above. If you want to see the characters \mathbb there are two solutions: the easy one works only since version 2.10: Tralics inserts in the tree all commands that behave like \relax; these commands are ignored when converting the tree into a MathML expression, but the name is used otherwise; thus \let \mathbb \relax does the job. Otherwise, it suffices to redefine \mathbb in order to expand to the desired string (using \string for instance). Note: since version 2.10, a font change command that takes an argument defines a semi-simple group, by inserting \begingroup and \endgroup commands.

2. Examples from the TEXbook

2.1. Typing math formulas, TB 16

The TeX book starts slowly; first Knuth explains that math formulas are enclosed in special math brackets, dollar signs. He also says that spaces are ignored in math mode.

1 `$x$', `$2$', `$ x$', `$ 2 $',
2 `$(x + y)/(x - y)$', `$(x+y) / (x-y)$'

Easy formulas: `$x$´, `$2$´, `$x$´, `$2$´, `$(x+y)/(x-y)$´, `$(x+y)/(x-y)$´.

We show here some Greek letters, and some other symbols:

3 $\Gamma, \nu,\kappa$,
4 $\phi,\emptyset, \epsilon,\in,\approx,\mapsto$
5 $(\phi,\theta,\epsilon,\rho)$,
6 $(\varphi,\vartheta,\varepsilon,\varrho)$,
7 $$\alpha, \beta, \gamma, \delta.$$

Greek letters $\Gamma ,\nu ,\kappa$, and other symbols: $\phi ,\varnothing ,ϵ,\in ,\approx ,\mapsto$; standard Greek(note: ➳) letters, $(\phi ,\theta ,ϵ,\rho )$, and variants $(\varphi ,\vartheta ,\epsilon ,\varrho )$, a display math formula:

8 \begingroup
9 \catcode`\*=3
10 *x^2*, \( x_2\), \begin{math}\Sigma\end{math}
11 **A** \[ B\] \begin{displaymath} C \end{displaymath}
12 \endgroup

We show here that any character of category code 3 can be used as math delimiter: $x^2$, as well as two alternative ways introduced by LaTeX: $x_2$, $\Sigma$. We show that display math can be entered by doubling the character of category 3 (actual rules are more complicated, see the TeXBook), or using a LaTeX environment (additional environments will be explained in later chapters), or using brackets. Note that brackets are very useful, because they are so easy to type.

Exercise 16.1: $\gamma +\nu \in \Gamma$.

Exercise 16.2: \le,\ge,\ne, \leq,\geq, and \neq: $\le ,\ge ,\ne$, $\le ,\ge ,\ne$.

Complex formulas with superscripts^{(up high)} and subscripts _{(down low)}.

13 $x^2$, $x_2$, $2^x$, $x^2y^2$, $x ^ 2y ^ 2$, $x_2y_2$, $_2F_3$,
14 $x^{2y}$, $2^{2^x}$, $2^{2^{2^x}}$, $y_{x_2}$, $y_{x^2}$.

Translation $x^2$, $x_2$, $2^x$, $x^2{y}^2$, $x^2{y}^2$, $x_2{y}_2$, ${}_2{F}_3$, $x^{2y}$, $2^{2^x}$, $2^{2^{2^x}}$, $y_{x_2}$, $y_{x^2}$.

Consider now the following two formulas:

15 $((x^2)^3)^4$,  ${({(x^2)}^3)}^4$.

Translation of the first formula

16 <formula type='inline'>
17   <math xmlns='http://www.w3.org/1998/Math/MathML'>
18     <mrow>
19       <mo>(</mo>
20       <mo>(</mo>
21       <msup><mi>x</mi> <mn>2</mn> </msup>
22       <msup><mo>)</mo> <mn>3</mn> </msup>
23       <msup><mo>)</mo> <mn>4</mn> </msup>
24     </mrow>
25   </math>
26 </formula>

Translation of the second formula

27 <formula type='inline'>
28   <math xmlns='http://www.w3.org/1998/Math/MathML'>
29     <msup>
30       <mrow><mo>(</mo>
31         <msup>
32           <mrow>
33             <mo>(</mo>
34             <msup><mi>x</mi> <mn>2</mn> </msup>
35             <mo>)</mo>
36           </mrow>
37           <mn>3</mn>
38         </msup>
39         <mo>)</mo>
40       </mrow>
41       <mn>4</mn>
42     </msup>
43   </math>
44 </formula>

The current Tralics uses some hacks, so that the translation is actually different. You will get first $\left(\right(x^2{)}^3{)}^4$, second ${\left({\left(x^2\right)}^3\right)}^4$, third ${\left({\left(x^2\right)}^3\right)}^4$, and fourth ${\left({\left(x^2\right)}^3\right)}^4$. Here the first two formulas correspond to the XML fragment shown above (they were produced by prefixing each parenthesis with \mathord, and after that we have the actual translation. The MathML recommendation says that some operators are stretchy, so that the width and height can depend on the context. This is not the case in xmltex, so that, in the Pdf version, all parentheses have the same size. The \mathop prefix tells Tralics to not consider parentheses as stretchy, so no hack is applied. We shall not explain the hack here; moreover we modified it in version 2.10.8. We originally wrote All parentheses have the same size in the pdf version, the same is true for the HTML version in the first formula. Now, the <mrow> elements delimit a scope, so that inner parentheses are smaller than outer parentheses (but only for HTML, second formula); note that the placement of the exponent depend on the context, so that, in the second formula, all three superscripts at placed at different positions. In the case of the first formula, placement of the exponent depends only on the the size of the parentheses (hence, in the Pdf version, they should be aligned, in the HTML version, they are not). Firefox on Mac shows the following: all superscript are aligned. Big parentheses are used but for case 2. Amaya shows only small parentheses, scripts are not aligned, but this is hard to see.

The MathML recommendation says that the second alternative is better; but Knuth says: “The first alternative is preferable, because it is much easier to type, and it is just as easy to read.”

45 `${}_2F_3$', `${_2}F_3$', `${_2F_3}$'

Exercise 16.3: Three ways to have an empty kernel: `${}_2{F}_3$´, `${}_2{F}_3$´, `${}_2{F}_3$´. The translation is the same in all cases because of the following two rules: if a math list starts with a subscript or superscript operator, an empty math list is added before, it serves as kernel (thus, braces in the first example are useless); a math list is packaged by putting it in a <mrow> element, unless the list has one element (in the second example, there are two tokens between the braces, converted to a single <msub> element, the math list has a single element, braces are useless). In these two cases, the main math list has two <msub> elements, and a <mrow> is added. In the last case, the main math list has a single element, a <mrow>, thus, no <mrow> is added and braces are useless.

46 `$x+_2F_3$' and `$x+{}_2F_3$'.

Effect of braces after plus sign: `$x{+}_2{F}_3$´ and `$x+{}_2{F}_3$´. The MathML recommendation says that the distance between the plus sign and the letter F should be the same in both cases; in the first case, index 2 is attached to the plus sign, in the second case to the letter F (AM shows both formulas with index attached to F).

47 ${x_2}_3$, $\displaystyle {\sum}'$

Note that an <mrow> element is added for a math list with a single element in the case where it is followed by a subscript or a subscript. Reason one: in a case like ${x_2}_3$, you will get an error if braces are omitted; in the same fashion, an error is signaled (when converting to Pdf) if <mrow> is missing. Reason two, in a case like ${\displaystyle {\sum }^{\text{'}}}$, the apostrophe is not placed above the sum if there is a <mrow> (the sum operator has type Op, but not the group, unless there is a \mathop before it).

Exercise 16.4: Double superscript, ${x^y}^z$, $x^{y^z}$.

48 $x^2_3$, $x_3^2$, $x^{31415}_{92}+\pi$, $x_{y^a_b}^{z_c^d}$.
49 $P_2^2$ and $P{}_2^2$.

Simultaneous superscripts and subscripts $x_3^2$, $x_3^2$, $x_{92}^{31415}+\pi$, $x_{y_b^a}^{z_c^d}$.

Vertical alignment: $P_2^2$ and $P{}_2^2$. In the first case, scripts are attached to the letter P, and alignment can depend on the slant of the letter. (Amaya has extra space in the second formula).

50 `$\prime$', $y_1^\prime$, $y_2^{\prime\prime}$, $y_3^{\prime\prime\prime}$,
51 $f'[g(x)]g'(x)$, $y_1'+y_2''$, $y'_1+y''_2$, $y'''_3+g'^2$

Primes and shorthand: `$\prime$´, $y_1^{\prime }$, $y_2^{\prime \prime }$, $y_3^{\prime \prime \prime }$, $f^{\prime }\left[g\left(x\right)\right]g^{\prime }\left(x\right)$, $y_1^{\prime }+y_2^{\prime \prime }$, $y_1^{\prime }+y_2^{\prime \prime }$, $y_3^{\prime \prime \prime }+g^{\prime 2}$. In the code shown here, the character used as delimiter for XML attributes is character U+39, the quotes around the math formulas are U+60 and U+B4 (this character can be changed via an option of the program, in verbatim mode it is always U+39, straight quote). The prime character used in the formula is U+2032. This is not the right character. Thus, in Tralics 2.9.5, translation of prime changed, it is now character U+39. The same formula is now: `$\text{'}$´, $y_1^{\text{'}}$, $y_2^{\text{'}\text{'}}$, $y_3^{\text{'}\text{'}\text{'}}$, $f^{\text{'}}\left[g\left(x\right)\right]g^{\text{'}}\left(x\right)$, $y_1^{\text{'}}+y_2^{\text{'}\text{'}}$, $y_1^{\text{'}}+y_2^{\text{'}\text{'}}$, $y_3^{\text{'}\text{'}\text{'}}+g^{\text{'}2}$. It is unclear whether this is the good solution. It seems that Firefox uses the same font metrics as TeX, i.e., a very large prime character that has a normal size when used at script size, while Amaya uses a normal size prime, that is small when used as a superscript. In the case of f-prime, the slant of the letter is not taken into account by Amaya, and the prime sign is hard to see.

52 $x\varprime y^\varprime, x\Prime y^\Prime,
53 x\tprime y^\tprime,x\bprime y^\bprime,x\qprime y^\qprime$

The amsmath package provides the symbols shown above; as you can see (at least with FM), these characters are not meant to be used as an exponent: $x\prime y^{\prime },x″y^{″},x‴y^{‴},x‵y^{‵},x⁗y^{⁗}$. Note that \qprime is unkown to Amaya, and the prime subscript in the following exercice is hard to see.

Exercise 16.5: $F^{\text{'}}(w,z)=\partial F(w,z)/\partial z$ and $F_{\text{'}}(w,z)=\partial F(w,z)/\partial w$.

54 \let\none\mmlnone
55 $R_i{}^{jk}{}_l$ versus $\mathbox{mmultiscripts}{Ri\none\none jk \none l\none}$

Exercise 16.6: $R_i{}^{jk}{}_l$ versus $R_i{}^j{}_k{}_l$ (MathML example, section 3.4.7.2, only second index raised).

56 $\sqrt 2$, $\sqrt{x+2}$, $\underline4$, $\overline{x+y}$,
57 $\overline x+ \overline y$, $x^{\underline n}$, $x^{\overline{m+n}}$,
58 $\sqrt{x^3+\sqrt\alpha}$

Translation $\sqrt{2}$, $\sqrt{x+2}$, $\underline{4}$, $\overline{x+y}$, $\overline{x}+\overline{y}$, $x^{\underline{n}}$, $x^{\overline{m+n}}$, $\sqrt{x^3+\sqrt{\alpha }}$. There are problems with underline and overline on FM. Vertical position is not always good, and the length is sometimes incorrect. On the fifth formula, rules are sometimes invisible in the printed version. There is a text version in Tralics: $\underline{\text{Mfoo}}$, Tfoo, $\overline{\text{Mfoo}}$, Tfoo, $\overline{\underline{\text{Mfoo}}}$, Tfoo.

59 $\root 3 \of 2$, $\root n \of {x^n + y^n}$, $\root n+1 \of a$,
60 $\sqrt[3]{2}$, $\sqrt[n]{x^n + y^n}$, $\sqrt[n+1]a$, `$\sqrt[3]{~~}$'.

Translation $\sqrt[3]{2}$, $\sqrt[n]{x^n+y^n}$, $\sqrt[n+1]{a}$, $\sqrt[3]{2}$, $\sqrt[n]{x^n+y^n}$, $\sqrt[n+1]{a}$, `$\sqrt[3]{}$´.

61 $\sqrt{\mathstrut a} + \sqrt{\mathstrut d} + \sqrt{\mathstrut y}$,
62 $\sqrt{a} + \sqrt{d} + \sqrt{y}$,
63 $\overline{a} + \overline{d} + \overline{y}$,
64 $\overline{\mathstrut a} + \overline{\mathstrut d} + \overline{\mathstrut y}$.

Translation $\sqrt{\phantom{(}a}+\sqrt{\phantom{(}d}+\sqrt{\phantom{(}y}$, $\sqrt{a}+\sqrt{d}+\sqrt{y}$, $\overline{a}+\overline{d}+\overline{y}$, $\overline{\phantom{(}a}+\overline{\phantom{(}d}+\overline{\phantom{(}y}$.

Exercise 16.7: ${10}^{10}$, $2^{n+1}$, ${(n+1)}^2$, $\sqrt{1-x^2}$, $\overline{w+\overline{z}}$, $p_1^{e_1}$, $a_{b_{c_{d_e}}}$, $\sqrt[3]{h_n^{\text{'}\text{'}}\left(\alpha x\right)}$.

Exercise 16.8: If$x=y$,then $x$ is equal to $y.$ (this exercise says what you should not do).

Exercise 16.9: Deleting an element from an $n$-tuple leaves an $(n-1)$-tuple.

Exercise 16.10: Letters with descenders are Qfgjpqy.

65 $x+y-z$, $x+y*z$, $x*y/z$

Basic binary operators: $x+y-z$, $x+y*z$, $x*y/z$.

66 $x\times y\cdot z$, $x\circ y\bullet z$, $x\cup y\cap z$,
67 $x\sqcup y\sqcap z$, $x\vee y\wedge z$, $x\pm y \mp z$.
68 Aliases   $x\land y\lor z$.

Many more binary operators $x\times y·z$, $x\circ y•z$, $x\cup y\cap z$, $x\bigsqcup y\sqcap z$, $x\vee y\wedge z$, $x\pm y\mp z$. Aliases $x\wedge y\vee z$.

69 $x=+1$, $3.142-$, $(D*)$
70 $x=\mathmo[form][prefix]{+}1$
71 $3.142\mathmo[form][prefix]{-}$
72 $(D\mathmi[mathvariant][normal]{*})$
73 $(D\mathmo[lspace][0][rspace][0]{*})$

Binary as ordinary symbols: $x=+1$, $3.142-$, $(D*)$. The HTML version is slightly different from the TeX version. You can declare the plus sign as prefix operator, this gives $x=+1$, you can define the minus sign as prefix operator, this gives $3.142-$. Removing the space around the star is more complicated, a solution consists of using an identifier in upright variant like this $\left(D\mathrm{*}\right)$, or by setting the lspace and rspace attributes to zero: $(D*)$.

74 $K_n^+,K_n^-$, $z^*_{ij}$, $g^\circ\mapsto g^\bullet$, $f^*(x)\cap f_*(y)$

Binary operators in superscripts $K_n^{+},K_n^{-}$, $z_{ij}^{*}$, $g^{\circ }\mapsto g^{•}$, $f^{*}\left(x\right)\cap f_{*}\left(y\right)$.

Exercise 16.11: $z^{*2}$ and $h_{*}^{\text{'}}\left(z\right)$.

75 $x=y>z$, $x:=y$, $x\le y\ne z$, $x\sim y\simeq z$, $x\equiv y\not\equiv z$,
76 $x\subset y\subseteq z$

$x=y>z$, $x:=y$, $x\le y\ne z$, $x\sim y\simeq z$, $x\equiv yX\equiv z$, $x\subset y\subseteq z$. In the Pdf version there is no space between the colon and the equals sign, but the HTML version shows some, because the rules are not the same for TeX and MathML. The best solution would be to use a single operator instead of two consecutive ones; the user can define a command \coloneq that behaves like colon-eq in normal TeX, or Tralics could be modified in order to recognise sequences like this. Translation of \not is problematic: in this document we consider it as an undefined operator, and you see a red X. We should add rules like: \not= should give \ne.

77 $f(x,y;z)$, $f:A\to B$, $f\colon A\to B$
78 %\def\colon{\mathmo[lspace][0]{:}}

Punctuation $f(x,y;z)$, $f:A\to B$, $f:A\to B$. Note that \colon is a colon with lspace = `0pt´, but the attribute is ignored in the Pdf version. It seems that Amaya uses a zero lspace by default.

79 $12,345x$, $12{,}245x$, $\mathcn{12,345}x$, $\mathmn{12,345}x$

$12,345x$, $12,245x$, $12,345x$, $\mathrm{12,345}x$. The translation of the first two expressions is the same, braces are useless here. A silly bug of my F browser: the second digit of the first number disappears; this does not happen if the number is not the first word of a paragraph (this is why the sentence starts with a number). You should use one of the last two variants if you want the sequence of digits plus the comma to be considered as a number.

Exercise 16.12: $3·1416$, but \mathmn{3^^b71416} gives $\mathrm{3·1416}$ (less space in the HTML version).

80 $\hat a$, $\check a$, $\tilde a$, $\acute a$, $\grave a$, $\dot a$,
81 $\ddot a$, $\breve a$, $\bar a$, $\vec a$
82 \def\ihat{{\hat \imath}} \def\jhat{{\hat\jmath}} $\ihat$, $\jhat$

$\widehat{a}$, $\check{a}$, $\tilde{a}$, $\acute{a}$, $\stackrel{`}{a}$, $\dot{a}$, $\ddot{a}$, $\breve{a}$, $\overline{a}$, $\overrightarrow{a}$, $\widehat{ı}$, $\widehat{j}$. Character \jmath is a normal j, because there is no dotless j in most fonts. There is also a problem with the rendering of grave accents on FM.

83 $\hat{I+M}$, $\bar z+ \overline z$, $\widehat x, \widetilde x$,
84 $\widehat{xy}, \widetilde{xy}$
85 $\widehat{xy}, \widetilde{xy}$
86 $\widehat{xyz}, \widetilde{xyz}$
87 % $\ghat\in{(H^{\pi_1^{-1}})}' -> Ex16.13 below

Large accents $\widehat{I+M}$, $\overline{z}+\overline{z}$, $\widehat{x},\tilde{x}$, $\widehat{xy},\widetilde{xy}$, $\widehat{xy},\widetilde{xy}$, $\widehat{xyz},\widetilde{xyz}$. As you can see, there is no difference between wide and non-wide operators. My browser (AM as well as FM) shows a small hat, and a variable length tilde. For the Pdf version, we have decided to use the variable size variant.

Exercise 16.13: $e^{-x^2}$, $D\sim p^{\alpha }M+l$, $\widehat{g}\in {\left(H^{{\pi }_1^{-1}}\right)}^{\text{'}}$, (braces added), $\widehat{g}\in {\left(H^{{\pi }_1^{-1}}\right)}^{\text{'}}$ (without braces). In Tralics 2.11, there shuld be no difference between the version with and without braces.

2.2. More about Math, TB 17

88 $$ {1\over 2}\qquad {\rm and}\qquad {n+1 \over 3} \qquad {\rm and}\qquad {n+1
89   \choose 3}\qquad {\rm and}\qquad \sum_{n=1}^3Z_n^2. \label{eq17.1}$$
90 \[ \frac{1}{2}\qquad \text{and}\qquad \frac{n+1}{3} \qquad \text{and}\qquad
91 \binom{n+1}{3}\qquad \text{and}\qquad \sum_{n=1}^3Z_n^2.\label{eq17.2}\]

Example of vertical alignment. Translation of the first three expressions is a <mfrac> element, subexpressions are in text style (normal size). In the case of the last expression scripts are in script style (small size) and we have a <munderover> element for the sum (it is an operator with limits in display style) and a <msubsup> element for the Z (that is not an operator).

The same, using LaTeX syntax

92 \[{x+y^2\over k+1},\qquad {x+y^2\over k}+1,\qquad
93 x+{y^2\over k}+1,\qquad x+{y^2\over k+1},\qquad x+y^{2\over k+1}\]
94 \[\frac{x+y^2}{k+1},\qquad \frac{x+y^2}{k}+1,\qquad
95 x+\frac{y^2}{k}+1,\qquad x+\frac{y^2}{k+1},\qquad x+y^{\frac{2}{k+1}}\]

Single over. The only difference in these expressions is the placement of the braces.

LaTeX style: this uses a command with two arguments, that behaves without surprise.

96 \[{ {a \over b}\over 2} \qquad \text{and}\qquad { a\over {b\over 2}}
97 \qquad \text{and}\qquad { a/ b\over 2} \qquad \text{and}\qquad { a\over b/2}
98  \]
99 \[ \frac{\frac{a}{b}}{2} \qquad \text{and}\qquad \frac{a}{\frac{b}{2}}
100 \qquad \text{and}\qquad  \frac{a/b}{2} \qquad \text{and}\qquad \frac{a}{b/2} \]

Double over. It is an error if you say A over B over C without adding braces. As you can see, if a fraction is in text style, its numerator and denominator are in script style, so that it is sometimes better to use a slash.

LaTeX style (nothing special here).

Exercise 17.1: Compare $x+y^{\frac{2}{k+1}}$ with $x+y^{2/(k+1)}$. If a fraction is in script style, its numerator and denominator are in script script style, i.e. smaller. In TeX, there are four styles and three sizes (display style and text style have the same size). In MathML, the size is defined by a level, zero, one, or two, but larger levels are possible; thus more than three sizes are possible. There is however a minimal font size.

Exercise 17.2: Compare $\frac{a+1}{b+1}x$ with $\left(\right(a+1)/(b+1\left)\right)x$.

Exercise 17.3: Wrong use of \over in ${\displaystyle \frac{x=(y^2}{k+1)}}$.

Exercise 17.4: $7\frac{1}{2}¢$, using the \textcent command.

Exercise 17.5: Same as in the TeXbook, but not cramped.

101 \[n+\scriptstyle n + \scriptscriptstyle n\]
102 \[n+{\scriptstyle n + {\scriptscriptstyle n}}\]

Note: In the current version of Tralics, it is unclear what happens when you put style commands randomly in a math formula; in any case, Tralics uses the correct math style, but has difficulties in inserting <mstyle> elements: what is the scope? Currently such an element is added to the current math list, if it contains a style change command; in the example above, we have two such commands, thus three different styles, and a single list (first example) or three lists (second example). Translation of the first example is wrong, translation of the second example is correct because the style command is the first token of the list(note: ➳).

103 \[a_0+{1\over\displaystyle a_1 +
104   {\strut 1\over\displaystyle a_2+
105     {\strut 1\over\displaystyle a_3+
106        {\strut 1\over a_4}}}}\]

Translation

Without \displaystyle

Without \strut. Since a \strut is an invisible object of the size of a parenthesis, more or less the size of the digit one, its effect is hard to see.(note: ➳)

Without both

Until version 2.9.4, commands of the form \hfill were illegal in math mode. Since then, they are allowed as first or last element in arguments of commands like \overline. They are ignored, unless the result is a fraction; we demonstrate here that \hfill placed at the end of the list produces a left alignment, in the case of \over, \genfrac, or \frac. Note that the \cfrac command can be used for continued fractions: its optional argument says whether the numerator is centered, left aligned or right aligned, under the assumption that the denominator is much larger than the numerator.(note: ➳).

107 \[ {x\atop y+2}, {n \choose k}, \text{latex},
108 \genfrac{}{}{0pt}{}{x}{y+2},\binom{n}{k} \]

The \atop construction is like \over, without fraction rule; you should not use it in LaTeX, you should use \genfrac instead. The command \atopwithdelims (see below for an example) is followed by two delimiters, say A and B, it puts A before the fraction and B after it. You can use \genfrac with A and B as arguments (third argument is line thickness, fourth argument is style). The \choose command is nothing else than \atopwithdelims(). You should not use it.

109 \[{ {n\choose k}\over 2} \text{ or } { n\choose{k\over2} }
110 \text{ or } { n\choose k/2} \text{ or } {n\choose{1\over2}k}\]

As mentioned above, a construction like A over B over C needs braces, where over is a generic name for \over and variants, including \choose. Nothing special is required for \frac or \binom, so that LaTeX variant is omitted.

111 %%$${1\over2}{n\choose k}$$; $$\displaystyle{n\choose k}\over2}$$ : TeXbook
112 $\displaystyle \frac{1}{2}\binom{n}{k}$,
113 $\displaystyle{n\choose k}\over \displaystyle 2$,
114 $\dfrac{\dbinom{n}{k}}2$.
115 {\def\P{\mathchoice{D}{T}{S}{SS}}
116 $\displaystyle{\P\choose \P}\P\over \displaystyle \P$,
117 $\dfrac{\dbinom{\P}{\P}\P}\P$.}

Exercise 17.6: ${\displaystyle \frac{1}{2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)}$ and $\frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{n}{k}\right)}}{{\displaystyle 2}}$; LaTeX version ${\displaystyle \frac{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}{2}}$; version with \P: $\frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{T}{T}\right)D}}{{\displaystyle D}}$, ${\displaystyle \frac{\left({\displaystyle \genfrac{}{}{0pt}{}{T}{T}}\right)T}{T}}$. Note that the `d´ in \dfrac or \dbinom means that the expression is in display style, hence numerator and denominator are in text style; as a consequence the \choose is same as the \dbinom; on the other hand, the \over produces a fraction in text style, with numerator and denominator in display style, while the \dfrac produces a fraction in display style, with numerator and denominator in text style. In the dvi file, the distance between D and the fraction rule is the same as the width of the rule, and the distance between the T and the rule is approximatively one third of the height of the T; my HTML FM browser uses larger values, making the difference more obvious.

Exercise 17.7: ${\displaystyle \left(\genfrac{}{}{0pt}{}{p}{2}\right)x^2{y}^{p-2}-\frac{1}{1-x}\frac{1}{1-x^2}}$. Fine space added for legibility.

118 \[{\displaystyle{a\over b}\above3pt\displaystyle{c\over d}}
119 \text{~~and~~} \genfrac{}{}{3pt}{}{\dfrac{a}{b}}{\dfrac{c}{d}}\]

There are six operators that behave alike: \over, \atop and \above, and the same with delimiters. The `above´ commands read a dimension, the thickness of the fraction rule. In the current version of Tralics, this has better to be explicit (if you want the value of \parskip, you must use \genfrac).

120 ${\displaystyle \sum x_n}, {\textstyle \sum x_n},
121 {\scriptstyle \sum x_n}, {\scriptscriptstyle \sum x_n}$

Different sums ${\displaystyle \sum x_n},\sum x_n,\sum x_n,\sum x_n$. The position of the index is unaffected by the style.

122 \[ {\displaystyle \int_{-\infty}^{\infty} \sum_{n=1}^m},
123 {\textstyle \int_{-\infty}^{\infty} \sum_{n=1}^m},
124 {\displaystyle\int\limits_0^{\frac{\pi}{2}} \sum\nolimits_{n=1}^m} \]

This example demonstrates that the position of scripts depends on the style if the kernel is a math operator, and the presence of key words like `limits´, `nolimits´ or `displaylimits´. Scripts can be added to a MathML object using m/sub/sup, and for an operator using m/under/over. In this last case, the attribute movablelimits corresponds to the `displaylimits´ keyword. If set, `under´ means `sub´ if the style is not display. Tralics uses `under´ or `sub´, depending on where the indices should be placed, and never sets the attribute. On the other hand, when TeX typesets a `under´ or `sub´, it sometimes uses a wrong strategy. The last expression (display style sum without limits) uses a \msubsup element; in this case, there is an implicit \nolimits, we cannot make it explicit, because such a command has to follow an operator, and there is no easy way to check that the first child of the element is an operator (it is easy to see that it is a <mo> element, a bit more complicated to see that it contains a Unicode character, and quite impossible to say that its translation leaves TeX in state where \nolimits is allowed). Note also that we could convert the argument to an operator, but this is not always a good idea. In this example the HTML version is correct, the Pdf version is wrong(note: ➳).

125 \[\lim_ax+\lim\nolimits_l x+\lim\limits_ax+\lim\displaylimits_ax\]
126 \[\textstyle \lim_lx+\lim\nolimits_l x+\lim\limits_ax+\lim\displaylimits_lx\]
127 \[\sin_lx+\sin\nolimits_l x+\sin\limits_lx+\sin\displaylimits_ax\]
128 \[\textstyle \sin_lx+\sin\nolimits_l x+\sin\limits_lx+\sin\displaylimits_lx\]

More about limits. Knuth says that \nolimits\limits produces limits; so that \sin\limits produces limits; a special feature of amsmath is that the token that follows the \sin is ignored if it is a \limits token. According to amsmath, an index A should be below the operator, an index L is a normal subscript; the example here shows that Tralics behaves more like TeX.

129 \[\operatorname*{sin}_a \operatornamewithlimits{sin}_a \qopname\relax{n}{sin}_a
130 \operatorname{sin}_l  \qopname\relax{o}{sin}_l
131 \mathop{\rm sin}_a\mathop{\rm sin}\limits_a\]
132 \[\textstyle\operatorname*{sin}_l \operatornamewithlimits{sin}_l
133  \qopname\relax{n}{sin}_l\operatorname{sin}_l  \qopname\relax{o}{sin}_l
134 \mathop{\rm sin}_l\mathop{\rm sin}\limits_a\]

These are LaTeX commands that can define operators like sin or lim:

135 \[\sum_{\scriptstyle0\le i\le m\atop\scriptstyle0<j<n}P(i,j)\qquad
136 \sum_{\stackrel{0\le i\le m}{0<j<n}}P(i,j)\]

In the first expression given here, and in exercise 17.9 that follows, scripts use an explicit `script style´ command, hence should be typeset in scriptstyle size. This is not the case in the HTML version: we have an atop in an atop, and the style of the inner one is wrong. The second formula uses a LaTeX command, this is not the right command for stacking indices because the first argument uses a smaller style than the second. It seems to me that the top line of first formula, bottom line of the second formula have the right size. Other lines are too big (in Firefox) or much too small (in Amaya).

Exercise 17.8: ${\displaystyle \sum _{i=1}^p\sum _{j=1}^q\sum _{k=1}^r{a}_{ij}{b}_{jk}{c}_{ki}}$.

Exercise 17.9: ${\displaystyle \sum _{\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{1\le i\le n}{1\le j\le q}}{1\le k\le r}}{a}_{ij}{b}_{jk}{c}_{ki}}$ or ${\displaystyle \sum _{\genfrac{}{}{0pt}{}{1\le i\le n}{\genfrac{}{}{0pt}{}{1\le j\le q}{1\le k\le r}}}{a}_{ij}{b}_{jk}{c}_{ki}}$.

137 \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}}}\]

138 \[ \left(\left[\left\lbrack a\left\{\left\lbrace\left\lfloor b
139 \left\lceil\left\langle\left/ c\left|\left\|\left\uparrow d
140 \left\downarrow\left\updownarrow\frac 12 \right\Updownarrow
141 \right\downarrow t
142 \right\Uparrow\right\Vert\right\vert x\right\backslash\right\rangle
143 \right\rceil y \right\rfloor\right\rbrace\right\} z\right\rbrack\right]\right)\]

We would expect all delimiters to be of the same size; but not all characters can be indefinitely extensible. On Firefox, we see a large slash, reverse slash and angle brackets; all other characters have maximum size. On Amaya, slash and reverse slash have a normal size, as well as double vertical bars; on AM some delimiters are replaced by a question sign, namely floor, ceiling, angle brackets, reverse slash and arrows (on following formulas, arrows are however visible). AF uses floor instead of ceiling and vice-versa. On FM the rendering of up-down-array is wrong.

There are problems with slash, backslash, double vertical bar on Tralics 2.9.4. Concerning the double vertical bar, the following expression contains two different characters: $\Vert =\parallel$. The first character is Unicode U+2016 (entity &Vert; in the file mmlalias.ent), obtained by \Vert in Tralics 2.9.4, accessible as \@Vert in following versions, it is not shwon by Amaya; the second character is Unicode U+2225 (entity &DoubleVerticalBar;, &parallel; and &shortparallel; in the file mmlalias.ent), obtained by \parallel in Tralics 2.9.4, and also by \Vert in following versions. We changed this because the first character has fixed size as the following demonstrates $\Vert \frac{1}{2}\parallel$ (the effect is only visible in HTML with FM).

Translation of backslash changed: it is now Unicode U+2216, you can use the backslash character (Unicode U+5C) by saying \char`\^^5c. Example: old is $X\backslash Y$, new is $X\setminus Y$, note the different spacing.

Example of \bigl and \bigr, followed by the same code without these operations; currently there is no difference in the HTML version; in the Pdf there is more space in the equations on the LHS between closing brace and opening brace (same for bracket).

Example of big

144 \[ \big(\big[\big\lbrack \big\{\big\lbrace\big\lfloor
145 \big\lceil\big\langle\big/ \big|\big\|\big\uparrow
146 \big\downarrow\big\updownarrow\frac 12 \big\Updownarrow
147 \big\downarrow \big\Uparrow\big\Vert\big\vert \big\backslash\big\rangle
148 \big\rceil\big\rfloor\big\rbrace\big\}\big\rbrack\big]\big)\]

I see the following: every up to the slash, and starting with the backslash is big; remaining items are small in the Pdf version, big in Firefox, sometimes small, sometimes large in Amaya.

149 \[ \big(\bigl[\Big\lbrack \Bigl\{\bigg\lbrace\biggl\lfloor
150 \Bigg\lceil\Biggl\langle /|\|\uparrow \downarrow\updownarrow\frac 12
151 \Updownarrow\downarrow\Uparrow\Vert\vert\backslash
152 \Biggr\rangle\Bigg\rceil\biggr\rfloor\bigg\rbrace\Bigr\}\Big\rbrack\bigr]\big)\]

A formula with all variants

Up to version 2.9.4, there was a bug in handling these big things. Currently, Tralics inserts \left and \right delimiters wherever possible. In the Pdf version, this formula is identical to the previous one. In firefox, operators between slash and backslash (that have no big) are small before the fraction, large after that. This is strange. In Amaya, everything between the slash and the backslash is small, except the \vert.

Exercise 17.10: ${\displaystyle \left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}\right){\left|\varphi (x+iy)\right|}^2=0}$.

153 Phi is $\phi,^^^^03c6$, ^^^^03c6  and varphi is $\varphi,^^^^03d5$, ^^^^03d5,

There is a problem in the rendering of the letter phi; the straight phi character is Unicode U+03D5. As the example given here shows, the rendering of phi and varphi in math mode, at least on my machine: Phi is $\phi ,\phi$, φ and varphi is $\varphi ,\varphi$, ϕ, epsilon is $ϵ$, varepsilon is $\epsilon$. We changed the translation, so that the Firefox and Pdf versions show, in math mode, the desired result. Note that AM shows the same characeter for phoi ad varphi in math mode, and has trouble with epsilon(shown as a square).

154 $\bigl (x\in A(n)\bigm|x\in B(n)\bigr)$,
155 $\bigcup_n X_n\bigm\|\bigcap_n Y_n$,
156 $\displaystyle{ {a+1\over b}\bigg/{c+1\over d}}$

Other big $\left(x\in A\left(n\right)|x\in B(n)\right)$ and ${\bigcup }_n{X}_n\parallel {\bigcap }_n{Y}_n$ and ${\displaystyle \frac{a+1}{b}/\frac{c+1}{d}}$.

Exercise 17.12: $\left(x+f\left(x\right)\right)/\left(x-f\left(x\right)\right)$.

157 \[1+\left(\frac{1}{1-x^2}\right)^3,\qquad \pi(n)=\sum_{k=2}^n
158 \left\lfloor\frac{\phi(k)}{k-1}\right\rfloor, \qquad
159 \left|\left|x\right|-|\left|y\right|\right|, \qquad
160 \left(\sum_{k=1}^n A_k\right),\qquad  \biggl(\sum_{k=1}^n A_k\biggr)\]

Exercise 17.13: Comparison between \left and \bigl; translation is the same in Tralics.

Exercise 17.14: was wrong in Tralics 2.9.4.

TeX sets \nulldelimiterspace, unused by Tralics.

The \vcenter command is not implemented. Any box can be put into a formula by simply saying \hbox or \vbox or \vtop or \box or \copy; this is not implemented in Tralics, you can insert text in math mode via \text or \hbox.

Exercise 17.15: This shows the use of \mathchoice.

161 \def\puzzle{\mathchoice{D}{T}{S}{SS}}
162 \[\puzzle{\puzzle \over\puzzle^{\puzzle^\puzzle}} \qquad
163 \frac{\puzzle}{\puzzle^{\puzzle^\puzzle}} \]

Exercise 17.16: The \square command is built-in: $\square$.

The \mathpalette command is introduced in Tralics 2.9.5, not tested yet.

164 \def\legendre{\overwithdelims()}\def\Legendre{\genfrac(){}{}}
165 \def\euler{\atopwithdelims<>}\def\Euler{\genfrac<>{0pt}{}}
166 \def\grimm{\abovewithdelims][4pt}\def\Grimm{\genfrac][{4pt}{}}
167 \[{a\legendre b}, \Legendre{a}{b}, {n \euler k}, \Euler{n}{k},
168 {\dfrac ab \grimm \dfrac cd}\Grimm{x}{y} \]

Exercise 17.17: Knuth says: the size of the surrounding delimiters depends only on the size, not on the size of the fractions, this is false for Tralics:

If $f$ is the value of \delimiterfactor and $\delta$ the value of \delimitershortfall, and we have a formula that extents $y_1$ units above the axis, and $y_2$ units below, if $y=2max(y_1,y_2)$, then the delimiter size is at least $y·f/1000$, and at least $y-\delta$. This does not apply to Tralics.

The remainder of the chapter discusses some low-level commands that are not fully implemented in Tralics. The commands \textfont, \scriptfont and \scriptscriptfont take a small number as argument, and provide a reference to a font. The following code

169 \font\tenrm=somefont at 12pt
170 \font\Helvetica=someotherfont scaled 1013
171 \textfont0=\tenrm
172 \scriptscriptfont2=\scriptfont3
173 \the\fontdimen3\scriptscriptfont15
174 $\textfont0=\tenrm 9\hbox{$9\textfont0=\Helvetica$}$%
175 ${\textfont0=\Helvetica 9}$ % exercise 17.18

gives: 0.0pt$99$$9$. The dimension should be zero, and the math formula contains three identical digits. Exercise 17.18 says that the fonts used by TeX in the last case is not Helvetica.

You cannot use more than 16 different math fonts in a formula. Assume that you need symbol A from font a; you can say that \textfont6 is your font a, and your symbol is character 137 in that font. What if you want symbol B from font b? you can use font number 7, but this is obviously not the solution. We explain here the amsmath strategy by telling what happens if you use \tt in a math formula. First of all, in a formula, this evaluates to \mathtt, and the real command is \@mathtt. This is a self-modifying command; after first use it is equivalent to \fam9 (this means that normal characters in text size use the \textfont9). The first use allocates the number 9, and defines the font, moreover the font information is remembered (the font \textfont9 will be defined for all subsequent math formulas). Note that the value of \textfont9 can be cmtt12, but this is recomputed if the current font size changes. For most other commands, the number is allocated when the font is declared, this means that a slot is allocated even when the font is never used.

Four slots are preallocated, with the names operators, letters, symbols and largesymbols. If you load packages amsmath, amscd, amssymb, and bm, four other slots are used, and there are only 8 slots remaining. In the case of this document, we test all 14 MathML fonts. How is this possible? In fact some characters are typeset outside math mode (sans-serif characters for instance) and in some cases poor-man-bold is used. Moreover, we had to remove some font families (these are commented out in the file raweb-uni.sty).

Exercise 17.19: Math code are not implemented in Tralics, \oplus is character U+2295 and \bullet is character U+2022.

176 \mathcode`<="2203 \mathcode`*="313C
177 \mathcode`a="8000 {\catcode`a=13 \gdef a{A}}
178 $a<b*c$
179 %\mathchardef\@M=10000

In TeX, the previous code is equivalent to $A*b<c$, in Tralics, it gives: $A<b*c$. Since version 2.9.10, a capital letter A is produced. The last line is commented out: the assignment is done at bootstrap. You should use \@M as the constant 10000, not as control-P in family 7 (binary operation).

Following code is unlikely to work

180 \def\sum\{\mathchar"1350}
181 \mathchardef\sum"1350
182 \def\n@space{\nulldelimiterspace\z@ \m@th}
183 \newdimen\p@ \p@=1pt
184 \def\bigl#{\mathopen{\hbox{$\left#1\vbox to8.5\p@{}\right.\n@space$}}}
185 \delcode `x="123456
186 \def\langle{\delimiter"426830A }
187 \bigl\delimiter"426830A
188 \def\sqrt{\radical"270370 }
189 \def\idehat{\mathaccent"362 }

2.3. Fine points of Mathematics Typing, TB 18

2.3.1. Punctuation

Say: If $x<0$, we have shown that

Do not say: for $x=a,b$, or $c$, but: for $x=a$, $b$, $c$, or use a tie: or $c$.

Exercise 18.1: $R(n,t)=O\left(t^{n/2}\right)$, as $t\to 0^{+}$. Adding braces improves the Html version: $R(n,t)=O\left(t^{n/2}\right)$.

2.3.2. Non-italics letters in formulas

TeX has 32 predefined operators, some of them behave like sums according to the placement of limits.

The last two operators are defined by amsmath. Variants shown later. More formulas (there should be small space around these operators, that is invisibile on my browser).

Exercise 18.2:

Example of formulas using \rm: $\sqrt{\mathrm{Var}\left(X\right)}$, $x_{\max }-x_{\min }$, $\mathrm{LL}\left(k\right)\Rightarrow \mathrm{LR}\left(k\right)$, $exp(x+\mathrm{constant})$, and $x^3+\mathrm{lower}\mathrm{order}\mathrm{terms}$.

Some formulas using \hbox for roman font: $\sqrt{\text{Var}\left(X\right)}$, $\text{LL}\left(k\right)\Rightarrow \text{LR}\left(k\right)$, $exp(x+\text{constant})$, and $x^3+\text{lower}\text{order}\text{terms}$.

The same formulas using \text for roman font: $\sqrt{\text{Var}\left(X\right)}$, $\text{LL}\left(k\right)\Rightarrow \text{LR}\left(k\right)$, $exp(x+\text{constant})$, and $x^3+\text{lower}\text{order}\text{terms}$.

190 \def\Varliminf{\mathop{\underline{\mathmo{lim}}}}
191 \def\Varlimsup{\mathop{\overline{\mathmo{lim}}}}
192 \[\lim_{n\to\infty}x_n\text{ exists}\iff
193 \Varlimsup_{n\to\infty}x_n=\Varliminf_{n\to\infty}x_n.\]

Exercise 18.3:

The code above is wrong, because `lim´ is a known operator, with movable limits, hence, in non-display mode, both the underline and the real subscript will move. The real definitions adds movablelimits = `false´ to the operator.

194 $\gcd(m,n)=\gcd(n,m\bmod n)$, or $x\equiv y+1\pmod{m^2}$
195 $x\equiv0(\pmod y^n)$.

Modulo: $gcd(m,n)=gcd(n,mmodn)$, or $x\equiv y+1(mod{m}^2)$

Exercise 18.4: B.L. User got the unexpected formula $x\equiv 0\left({(mody)}^n\right)$.

Exercise 18.5: In this example, there are braces around n/p, but not k/p; this can affect the size of the slash operator (HTML version only).

Example of bold face $\mathbf{a}+\mathbf{b}={\Phi }_{\mathbf{m}}$; normal Phi is U+03A6, bold Phi is U+1D6BD, italic Phi is U+1D6F7, bold italic Phi is U+1D731, sans serif bold Phi is U+1D76B, sans serif bold italic Phi is U+1D7A5. In the current version of Tralics, font changes apply only to ASCII letters. Commands like \cal are robust, so that $\cal Ab\Phi$ produces $\mathcal{Ab}\Phi$; command \mit is not implemented. Note that in some cases, digits and lower case letters do not exist in the font, and may be replaced by random glyphs.

Exercise 18.6: If you want ${\overline{\mathbf{x}}}^{\mathrm{T}}\mathbf{Mx}=0\iff \mathbf{x}=\mathbf{0}$ in Tralics 2.9.4, you have to explicitly say that you want a bold face zero, and an upright T, see below. In version 2.9.5, nothing special needed. Note how we use colon-equal here.

196 $\bf\bar x^{\mathmo{T}}Mx={\rm0}\iff x=\mathmn[mathvariant][bold]{0}$
197 Compare $This\ is\ math\ italics$ with {\it This is text italics}.
198 Compare $different$ and $\it different$. We have also
199 $\it last\mathmo{:=}first$, $\it x\_coord(point\_2)$

Compare $Thisismathitalics$ with This is text italics. Compare $different$ and $\mathit{different}$. We have also $\mathit{last}:=\mathit{first}$, $\mathit{x}\_\mathit{coord}(\mathit{point}\_\mathit{2})$.

Exercise 18.8: We use \mathit instead of \it here.

Exercise 18.9: The following code was obtaining by taking the answer from the TeXbook, with the following modifications: the \sfcode of the semi colon not changed; math formulas put outside scope of \bf, because my browser shows letters (not digits) in bold face otherwise. Note that the XML to HTML processor converted the whole environment (seven paragraphs) into a single paragraph with <br> as separator (inter-paragraph width too big), and a huge left margin (for fun). We show here the start of the code:

200 \begin{xmlelement+}{pseudocode}\XMLaddatt{leftskip}{5cm}
201 \obeylines
202 \def\coleq{\mathmo{:=}}
203 \textbf{for} $j\coleq 2$ \textbf{step} $1$ \textbf{until} $n$ \textbf{do}
204 ...
205 \end{xmlelement+}

for $j:=2$ step $1$ until $n$ do
   begin $\mathit{accum}:=A\left[j\right]$; $k:=j-1$; $A\left[0\right]:=\mathit{accum}$;
   while $A\left[k\right]>\mathit{accum}$ do
      begin $A[k+1]:=A\left[k\right]$; $k:=k-1$;
      end;
   $A[k+1]:=\mathit{accum}$;
   end.

2.3.3. Spacing between formulas

Compare

with (lot of white space in the formula)

and (no unnecessary white space in the formula)

Normally, the spacing should be the same in the last two formulas. In fact, Knuth says that \quad is the same as \hskip 1em\relax. The translation of a \quad in math mode is <mspace width="1.em">, but the \hskip command reads a dimension, and converts one em into ten points(note: ➳).

Exercise 18.10: Three versions:

Let $H$ be a Hilbert space, $C$ a closed bounded convex subset of $H$, $T$ a non-expansive self map of $C$. Suppose that as $n\to \infty$, $a_{n,k}\to 0$ for each $k$, and ${\gamma }_n={\sum }_{k=0}^{\infty }{(a_{n,k+1}-a_{n,k})}^{+}\to 0$. Then for each $x$ in $C$, $A_{n}x={\sum }_{k=0}^{\infty }{a}_{n,k}{T}^{k}x$ converges weakly to a fixed point of $T$.

Let $H$ be a Hilbert space,  $C$ a closed bounded convex subset of $H$,  $T$ a non-expansive self map of $C$. Suppose that as $n\to \infty$,  $a_{n,k}\to 0$ for each $k$, and ${\gamma }_n={\sum }_{k=0}^{\infty }{(a_{n,k+1}-a_{n,k})}^{+}\to 0$. Then for each $x$ in $C$,  $A_{n}x={\sum }_{k=0}^{\infty }{a}_{n,k}{T}^{k}x$ converges weakly to a fixed point of $T$.

Let $C$ be a closed, bounded, convex subset of a Hilbert space $H$, and let $T$ be a non-expansive self map of $C$. Suppose that as $n\to \infty$, we have $a_{n,k}\to 0$ for each $k$, and ${\gamma }_n={\sum }_{k=0}^{\infty }{(a_{n,k+1}-a_{n,k})}^{+}\to 0$. Then for each $x$ in $C$, the infinite sum $A_{n}x={\sum }_{k=0}^{\infty }{a}_{n,k}{T}^{k}x$ converges weakly to a fixed point of $T$.

Comments: the translation of the three characters: space, backslash, space is formed of two spaces. Such a construct is used four times in the first version. This produces a nice dvi file, but HTML interprets the double space the same as a single space. In the second version, we have used a double tilde character. This inhibits line breaks. (My browser does not seem to honor this). No special characters appear in the last example.

2.3.4. Spacing within formulas

206 $a\,b\>c\;d\!e\quad f\qquad g\ h$~j
207 %\the\thinmuskip\the\medmuskip\the\thickmuskip

The math spacing commands: $abcdefghj$. In TeX, the value of the glue inserted depends on three registers: \thinmuskip (3mu), \medmuskip (4mu plus 2mu minus 4mu) and \thickmuskip (5mu plus 5mu). You can try to execute the line above that is commented out: all dimensions come out as 0mu in Tralics, but the math formula contains 0.166667em, 0.222222em, 0.277778em, etc, this is the same as 3mu, 4mu and 5mu, because 1em=18mu. In Tralics, \mskip 18mu is equivalent to \hskip1em, and, as said above, this is the same as \hskip10pt. The same is true for \mkern18mu; the translation is an empty <mspace> element, with attribute value = `10.0pt´. The amount of space given by backslash-space changed from 6pt to 4pt (this is the same amount of space used by the \text command).

208 $\int_0^\infty f(x)\,dx$, $y\,dx-x\,dy$, $dx\,dy=r\,dr\,d\theta$, $x\,dy/dx$

Translation ${\int }_0^{\infty }f\left(x\right)dx$, $ydx-xdy$, $dxdy=rdrd\theta$, $xdy/dx$.

Exercise 18.11:

209 $55\,\mathrm{mi/hr}$, $g=9.8\,\mathrm{m/sec}^2$,
210 $1\mathrm{ml}=1.000028\,\mathrm{cc}$

Units: $55\mathrm{mi}/\mathrm{hr}$, $g=9.8\mathrm{m}/{\sec }^2$, $1\mathrm{ml}=1.000028\mathrm{cc}$. Note that digits are outside the scope of \mathrm.

Exercise 18.12: Inline math, displaystyle: ${\displaystyle \hslash =1.054\times {10}^{-27}\mathrm{erg}\sec .}$

211 $(2n)!/\bigl(n!\,(n+1)!\bigr)$, $\sqrt2\,x$,
212 $\sqrt{\,\log x}$, $O\bigl(1/\sqrt n\,\bigr)$,
213 $[\,0,1)$, $\log n\,(\log\log n)^2$, $x^2\!/2$, $n/\!\log n$,
214 $\Gamma_{\!2}+\Delta^{\!2}$, $R_i{}^j{}_{\!kl}$, $\int_0^x\!\int_0^y dF(u,v)$
215  \[\frac{52!}{13!\,13!\,26!}\qquad \int\!\!\!\int_D dx\,dy\]

Test: $\left(2n\right)!/\left(n!(n+1)!\right)$, $\sqrt{2}x$, $\sqrt{logx}$, $O\left(1/\sqrt{n}\right)$, $[0,1)$, $logn{(loglogn)}^2$, $x^2/2$, $n/logn$, ${\Gamma }_2+{\Delta }^2$, $R_i{}^j{}_{kl}$, ${\int }_0^x{\int }_0^{y}dF(u,v)$

Same formulas, without thin spaces

Test: $\left(2n\right)!/\left(n!(n+1)!\right)$, $\sqrt{2}x$, $\sqrt{logx}$, $O\left(1/\sqrt{n}\right)$, $[0,1)$, $logn{(loglogn)}^2$, $x^2/2$, $n/logn$, ${\Gamma }_2+{\Delta }^2$, $R_i{}^j{}_{kl}$, ${\int }_0^x{\int }_0^{y}dF(u,v)$

Amaya shows a strange behavior here. Without the space the first integral is larger than the second. In thee case of negative space, the absolute value is used.

Exercise 18.14: intervals with left/right: $\left]-\infty ,T\right[\times \left]-\infty ,T\right[$; with mathopen, mathclose $]-\infty ,T[\times ]-\infty ,T[$; with nothing $]-\infty ,T[\times ]-\infty ,T[$. And with left/right: $\left]-\infty ,\frac{3}{4}\right[\times \left]-\infty ,\frac{3}{4}\right[$; with mathopen, mathclose $]-\infty ,\frac{3}{4}[\times ]-\infty ,\frac{3}{4}[$; with nothing $]-\infty ,\frac{3}{4}[\times ]-\infty ,\frac{3}{4}[$.

Exercise 18.15: The spacing in $x++1$ differs between TeX and MathML.

2.3.5. Ellipses

216 $x_1+\cdots+x_n$, $x_1=\cdots=x_n=0$, $A_1\times\cdots\times A_n$,
217 $f(x_1,\ldots,x_n)$, $x_1x_2\ldots x_n$, $(1-x)(1-x^2)\ldots(1-x^n)$,
218 $n(n-1)\ldots(1)$.

Translation of these formulas $x_1+\cdots +x_n$, $x_1=\cdots =x_n=0$, $A_1\times \cdots \times A_n$, $f(x_1,...,x_n)$, $x_1{x}_2...x_n$, $(1-x)(1-x^2)...(1-x^n)$, $n(n-1)...\left(1\right)$.

Exercise 18.16: Answer $x_1+x_1{x}_2+\cdots +x_1{x}_2...x_n$, $(x_1,...,x_n)·(y_1,...,y_n)=x_1{y}_1+\cdots +x_n{y}_n$. Prove that ${(1-x)}^{-1}=1+x+x^2+\cdots$. Clearly $a_i<b_i$, for $i=1$, 2, $...$, $n$. The coefficients $c_0$, $c_1$, ..., $c_n$ are positive. With braces surrounding the LHS: ${(1-x)}^{-1}=1+x+x^2+\cdots$.

Exercise 18.17: Clearly $a_i<b_i$, for $i=1,2,...,n$. Note that the `2´ here is a math digit, while it is text digit in the sentence above.

Exercise 18.18: Knuth used \dots.

2.3.6. Line breaking

The TeX book explains how TeX can break a math formula. This does not apply to Tralics. Commands \nobreak and \allowbreak do nothing in math mode.

2.3.7. Braces

You should use braces only for grouping. You can use a brace character via \{ in text or math mode. This character is a delimiter (it can be preceded by \left or \big).

219 $\{a,b,c\}$, $\{1,2,\ldots,n\}$, $\{\mathrm{red,white,blue}\}$
220 $\{\,x\mid x>5\,\}$, $\{\,x:x>5\,\}$,
221 $\bigl\{\,\bigl(x,f(x)\bigr)\bigm|x\in D\,\big\}$
222 $\bigl\{\,\bigl(x,f(x)\bigr)\bigm\mid x \in D\,\big\}$
223 $\bigl\{\,\bigl(x,f(x)\bigr)\mid x\in D\,\big\}$

Translation $\{a,b,c\}$, $\{1,2,...,n\}$, $\{\mathrm{red},\mathrm{white},\mathrm{blue}\}$, $\{x\mid x>5\}$, $\{x:x>5\}$, $\left\{\left(x,f\left(x\right)\right)|x\in D\right\}$ $\left\{\left(x,f\left(x\right)\right)|x\in D\right\}$. Note that \mid gives better spacing than a single bar $\left\{\left(x,f\left(x\right)\right)\mid x\in D\right\}$, $\left\{\left(x,f\left(x\right)\right)\mid x\in D\right\}$.

Exercise 18.21: $\left\{x^3\mid h\left(x\right)\in \{-1,0,+1\}\right\}$

Exercise 18.22: $\{p\mid p$ and $p+2$ are prime$\}$.

We show here the use of the cases enviroment, this is a two-column cmatrix, with a left brace delimiter on the left.

Exercise 18.23: The \cases command is not implemented, but there is a environment. You cannot use \noalign. After a double backslash you can put a dimension in brackets, but this is currently ignored.

224 \[\overbrace{x+\cdots+x}^{k\; \textrm{times}} \qquad \underbrace{x+y+z}_{>\,0}\]

We show here braces can stretch horizontally, when used as over-accent or under-accent.

2.3.8. Matrices

The plain TeX command \matrix should not be used. The `array´ environment can be used in math mode, and you must specify for each column the alignment method. The `matrix´ environment can be used, cells are centered. He we use `pmatrix´, because it adds automatically parentheses.

225 \[A=\begin{pmatrix}x-\lambda&1&0\\0&x-\lambda&1\\0&0&x-\lambda\end{pmatrix}\]
226 \[\begin{pmatrix}a&b&c\\d&e&f\end{pmatrix}
227 \begin{pmatrix}u&x\\v&y\\w&z\end{pmatrix}\]

Exercise 18.24: This looks bad in the HTML version, but it OK is Pdf. Delimiters are \lgroup and \rgroup.

Exercise 18.25:

Border matrix(note: ➳)

Small matrices $\left(\genfrac{}{}{0pt}{}{11}{00}\right)$ and $\left(\genfrac{}{}{0.0pt}{}{a}{l}\genfrac{}{}{0.0pt}{}{b}{m}\genfrac{}{}{0.0pt}{}{c}{n}\right)$. Note that horizontal alignment is only approximative.

2.3.9. Vertical Spacing

228 \def\Limsup{\mathop{\smash\limsup\vphantom\liminf}}
229 $\Limsup\limits_3=\limsup\limits_3$

Compare: $\underset{3}{lim\; sup\phantom{lim\; inf}}=\underset{3}{lim\; sup}$! The second index should be lower than the first.

Commands \raise and \lower not yet implemented. Commands \llap and \rlap not yet implemented.(note: ➳)

230 \def\undertext#1{$\underline{\hbox{#1}}$}
231 \undertext{This} \undertext{does} \undertext{not} \undertext{always}
232 \undertext{work} \undertext{right}.
233 \def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
234 \undertext{This} \undertext{does} \undertext{not} \undertext{always}
235 \undertext{work} \undertext{right}.

Exercise 18.26: A sentence where each word is an underlined math formula (text only): $\underline{\text{This}}$ $\underline{\text{does}}$ $\underline{\text{not}}$ $\underline{\text{always}}$ $\underline{\text{work}}$ $\underline{\text{right}}$. The same, where the \smash is used to hide the depth of the text (all lines vertically aligned): $\underline{\text{This}}$ $\underline{\text{does}}$ $\underline{\text{not}}$ $\underline{\text{always}}$ $\underline{\text{work}}$ $\underline{\text{right}}$.

236 $\rm Fe_2^{+2}Cr_2^{\vphantom{+2}}O_4^{\vphantom{+2}}$

Use of phantoms: ${\mathrm{Fe}}_2^{+2}{\mathrm{Cr}}_2^{\phantom{+2}}{\mathrm{O}}_4^{\phantom{+2}}$

2.3.10. Special features for math hackers

Commands \nonscript, \everymath and \everydisplay are OK.

237 {\everydisplay{a}\everymath{b} \def\X{x\nonscript\qquad y}
238 \[u = \text{ v $w$ }\frac{\X}{\textstyle\X} \]}

2.3.11. Summary

Exercise 18.27: $n^{\mathrm{th}}$ from the TeXbook, $n^{\mathrm{th}}$ is LaTeX, and $n$^th is textsuperscript.

Exercise 18.28: ${\mathbf{S}}^{-1}\mathbf{TS}=\mathbf{dg}({\omega }_1,...,{\omega }_n)\Lambda$, uses \mathnormal, no bf lambda?

Exercise 18.29: $Pr(m=n\mid m+n=3)$.

Exercise 18.30: $sin{18}^{\circ }=\frac{1}{4}(\sqrt{5}-1)$.

Exercise 18.31: $k=1.38\times {10}^{-16}\mathrm{erg}{/}^{\circ }\mathrm{K}$.

Exercise 18.32: $\overline{\Phi }\subset N{L}_1^{*}/N={\overline{L}}_1^{*}\subseteq \cdots \subseteq N{L}_n^{*}/N={\overline{L}}_n^{*}$.

Exercise 18.33: $I\left(\lambda \right)=\int {\int }_{D}g(x,y)e^{i\lambda h(x,y)}dxdy$.

Exercise 18.34: ${\int }_0^1\cdots {\int }_0^{1}f(x_1,...,x_n)d{x}_1...d{x}_n$. Note: Firefox shows small integral signs, and larger ones in the previous exercise.

Exercise 18.35: Using cases environment

Exercise 18.36: with a \frac

Exercise 18.37:

Exercise 18.38: Using \frac and \binom

Exercise 18.39: Using \genfrac

Exercise 18.40:

239 \def\X{\char`'}
240 \[{
241 \{\underbrace{\overbrace{\mathstrut a,\ldots,a}^{k\;a\X\mathrm{s}},
242   \overbrace{\mathstrut
243     b,\ldots,b}^{l\;b\X\mathrm{s}}}_{k+l\;\mathrm{elements}}\}
244 }\quad\text{vs}\qquad {
245 {\{\,}\underbrace{\overbrace{\mathstrut a,\ldots,a}^{k\;a'\mathrm{s}},
246   \overbrace{\mathstrut
247     b,\ldots,b}^{l\;b'\mathrm{s}}}_{k+l\;\mathrm{elements}}{\}\,}.}\]

Exercise 18.41: In this formula, we want the denote the plural of the token $a$ by as apostrophe followed by the letter s. The first attempt is not good. The second is a bit better, because it is a-prime followed by s. The first expression is formed of open brace, underbraced formula, closing brace; and this produces large braces in the HTML version; small braces are obtaing by replacing the first token by a list, containing the brace and little bit space.