# Top Tips for New LaTeX Users

This article is aimed at relatively new $\LaTeX$ users. It is written particularly for my own students, with the aim of helping them to avoid making common errors. The article exists in two forms: this WordPress blog post and a PDF file generated by $\LaTeX$, both produced from the same Emacs Org file. Since WordPress does not handle $\LaTeX$ very well I recommend reading the PDF version.

## 1. New Paragraphs

In $\LaTeX$ a new paragraph is started by leaving a blank line.

Do not start a new paragraph by using \\ (it merely terminates a line). Indeed you should almost never type \\, except within environments such as array, tabular, and so on.

## 2. Math Mode

Always type mathematics in math mode (as $..$ or $$..$$), to produce “ $y = f(x)$” instead of “y = f(x)”, and “the dimension $n$” instead of “the dimension n”. For displayed equations use $$, $..$, or one of the display environments (see Section 7). Punctuation should appear outside math mode, for inline equations, otherwise the spacing will be incorrect. Here is an example. • Correct: The variables x, y, and z satisfy x^2 + y^2 = z^2. • Incorrect: The variables x, y, and z satisfy x^2 + y^2 = z^2. For displayed equations, punctuation should appear as part of the display. All equations must be punctuated, as they are part of a sentence. ## 3. Mathematical Functions in Roman Mathematical functions should be typeset in roman font. This is done automatically for the many standard mathematical functions that $\LaTeX$ supports, such as \sin, \tan, \exp, \max, etc. If the function you need is not built into $\LaTeX$, create your own. The easiest way to do this is to use the amsmath package and type, for example, \usepackage{amsmath} ... % In the preamble. \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\inert}{Inertia}  Alternatively, if you are not using the amsmath package you can type \def\diag{\mathop{\mathrm{diag}}}  ## 4. Maths Expressions Ellipses (dots) are never explicitly typed as “…”. Instead they are typed as \dots for baseline dots, as in x_1,x_2,\dots,x_n (giving $x_1,x_2,\dots,x_n$) or as \cdots for vertically centered dots, as in x_1 + x_2 + \cdots + x_n (giving $x_1 + x_2 + \cdots + x_n$). Type ith instead of i'th or i^{th}. (For some subtle aspects of the use of ellipses, see How To Typeset an Ellipsis in a Mathematical Expression.) Avoid using \frac to produce stacked fractions in the text. Write $n^3/3$ flops instead of $\frac{n^3}{3}$ flops. For “much less than”, type \ll, giving $\ll$, not <<, which gives $<<$. Similarly, “much greater than” is typed as \gg, giving $\gg$. If you are using angle brackets to denote an inner product use \langle and \rangle: • incorrect: <x,y>, typed as <x,y>. • correct: $\langle x,y \rangle$, typed as \langle x,y \rangle ## 5. Text in Displayed Equations When a displayed equation contains text such as “subject to $x \ge 0$”, instead of putting the text in \mathrm put the text in an \mbox, as in \mbox{subject to x \ge 0}. Note that \mbox switches out of math mode, and this has the advantage of ensuring the correct spacing between words. If you are using the amsmath package you can use the \text command instead of \mbox. ### Example $$
\min\{\, \|A-X\|_F: \mbox{$X$ is a correlation matrix} \,\}.
$$ ## 6. BibTeX Produce your bibliographies using BibTeX, creating your own bib file. Note three important points. • “Export citation” options on journal websites rarely produce perfect bib entries. More often than not the entry has an improperly cased title, an incomplete or incorrectly accented author name, improperly typeset maths in the title, or some other error, so always check and improve the entry. • If you wish to cite one of my papers download the latest version of njhigham.bib (along with strings.bib supplied with it) and include it in your \bibliography command. • Decide on a consistent format for your bib entry keys and stick to it. In the format used in the Numerical Linear Algebra group at Manchester a 2010 paper by Smith and Jones has key smjo10, a 1974 book by Aho, Hopcroft, and Ullman has key ahu74, while a 1990 book by Smith has key smit90. ## 7. Spelling Errors and $\LaTeX$ Errors There is no excuse for your writing to contain spelling errors, given the wide availability of spell checkers. You’ll need a spell checker that understands $\LaTeX$ syntax. There are also tools for checking $\LaTeX$ syntax. One that comes with TeX Live is lacheck, which describes itself as “a consistency checker for LaTeX documents”. Such a tool can point out possible syntax errors, or semantic errors such as unmatched parentheses, and warn of common mistakes. ## 8. Quotation Marks $\LaTeX$ has a left quotation mark, denoted here \lq, and a right quotation mark, denoted here \rq, typed as the single left and right quotes on the keyboard, respectively. A left or right double quotation mark is produced by typing two single quotes of the appropriate type. The double quotation mark always itself produces the same as two right quotation marks. Example: $hello''$ is typed as \lq\lq hello \rq\rq. ## 9. Captions Captions go above tables but below figures. So put the caption command at the start of a table environment but at the end of a figure environment. The \label statement should go after the \caption statement (or it can be put inside it), otherwise references to that label will refer to the subsection in which the label appears rather than the figure or table. ## 10. Tables $\LaTeX$ makes it easy to put many rules, some of them double, in and around a table, using \cline, \hline, and the | column formatting symbol. However, it is good style to minimize the number of rules. A common task for journal copy editors is to remove rules from tables in submitted manuscripts. ## 11. Source Code $\LaTeX$ source code should be laid out so that it is readable, in order to aid editing and debugging, to help you to understand the code when you return to it after a break, and to aid collaborative writing. Readability means that logical structure should be apparent, in the same way as when indentation is used in writing a computer program. In particular, it is is a good idea to start new sentences on new lines, which makes it easier to cut and paste them during editing, and also makes a diff of two versions of the file more readable. ### Example: Good: $$
U(\zbar) = U(-z) =
\begin{cases}
-U(z),   & z\in D, \\
-U(z)-1, & \mbox{otherwise}.
\end{cases}
$$ Bad: $$U(\zbar) = U(-z) =
\begin{cases}-U(z),   & z\in D, \\
-U(z)-1, & \mbox{otherwise}.
\end{cases} ## 12. Multiline Displayed Equations For displayed equations occupying more than one line it is best to use the environments provided by the amsmath package. Of these, align (and align* if equation numbers are not wanted) is the one I use almost all the time. Example: \begin{align*} \cos(A) &= I - \frac{A^2}{2!} + \frac{A^4}{4!} + \cdots,\\ \sin(A) &= A - \frac{A^3}{3!} + \frac{A^5}{5!} - \cdots, \end{align*}  Others, such as gather and aligned, are occasionally needed. Avoid using the standard $\LaTeX$ environment eqnarray, because it doesn’t produce as good results as the amsmath environments, nor is it as versatile. For more details see the article Avoid Eqnarray. ## 13. Synonyms This final category concerns synonyms and is a matter of personal preference. I prefer \ge and \le to the equivalent \geq \leq\ (why type the extra characters?). I also prefer to use .. for math mode instead of $$..$$ and .. for display math mode instead of $..$. My preferences are the original $\TeX$ syntax, while the alternatives were introduced by $\LaTeX$. The slashed forms are obviously easier to parse, but this is one case where I prefer to stick with tradition. If dollar signs are good enough for Don Knuth, they are good enough for me!

I don’t think many people use $\LaTeX$‘s verbose

\begin{math}..\end{math}


or

\begin{displaymath}..\end{displaymath}


Also note that \begin{equation*}..\end{equation*} (for unnumbered equations) exists in the amsmath package but not in in $\LaTeX$ itself.

# Programming Languages: An Applied Mathematics View A lot of applied mathematics relies on computation, whether symbolic or numeric, so many applied mathematicians need to program as part of their work. It was therefore natural to include an article on programming languages in The Princeton Companion to Applied Mathematics.

The article, which I wrote, has two main purposes. The first is to give a history of those programming languages relevant to applied mathematics. The first such language, and indeed the first high-level programming language, was Fortran (1957). The language was standardized in 1966 and it is still going strong, with the most recent standard published in 2008. Developments in programming languages show no sign of abating, with the introduction in recent years of new languages such as Scala (2003), Clojure (2007, a dialect of Lisp), and Julia (2012), as well as new standards for older languages such as C (2011) and C++ (2011).

The second purpose of the article is to discuss mathematical aspects of programming, including

• notation (infix, prefix, reverse-Polish)
• implementation of complex arithmetic
• floating-point semantics
• high-precision computations
• types
• complexity analysis of codes
• structured programming
• literate programming
• domain-specific languages

One issue that I discuss is the mutually beneficial influences that mathematics and programming languages have had on each other. For example, the notation for the ceiling and floor functions that map a real number to the next larger or smaller integer, respectively, illustrated by $\lceil 4.3 \rceil = 5$ and $\lfloor 4.3 \rfloor = 4$, was first introduced in the programming language APL. The colon notation for array subscripting, A(i:j,r:s), used in Algol 68 and MATLAB, is now routinely used in linear algebra, in both equations and in pseudocode.

On the other hand, mathematics has influenced, or anticipated, syntax in programming languages. In the 1800s Cayley proposed two different notations to distinguish between the products $AB^{-1}$ and $B^{-1}A$ in the context of groups, but both were ungainly and difficult to typeset. In 1928, Hensel suggested the notation $A/B$ and $B\backslash A$. Although his suggestion appears to have attracted little or no attention, it was was reinvented by Cleve Moler for MATLAB and is now a notation familiar to anyone who works in numerical linear algebra.

At the start of the article I include a figure containing the first program written for a stored-program computer, namely the highest factor routine that ran on the Manchester “Baby” on June 21, 1948. The program was by Tom Kilburn and is taken from Geoff Tootill‘s notebook. Tootill is still alive (aged 93), and he kindly gave permission for me to reproduce the image.

The article, which has the same title as this post, can be downloaded in pre-publication form as an EPrint.

# Wobbly Tables

In The Princeton Companion to Applied Mathematics (page 50) I mention that a four-legged table provides an example of an ill-posed problem. If we take a table having four legs of equal length lying on a flat surface and shorten one leg by an arbitrarily small amount then the weight supported by that leg will jump from one quarter of the total weight to zero. A table with one leg shorter than the others wobbles, as may one sitting on an uneven floor, and how to cure wobbly tables has been the subject of a number of papers over the years. The tongue-in cheek article

Hanspeter Kraft, The Wobbly Garden Table, Journal of Biological Physics and Chemistry 1, 95-96, 2001

describes how an engineer, a physicist, and a mathematician would go about solving the problem. The engineer would invent an adjustable leg. The physicist would submit a research proposal to tackle the more general problem of “the stability of multiply-legged objects on rough surfaces”. The mathematician would construct an argument based on the intermediate value theorem to show that stability can be restored with a suitable rotation of no more than 90 degrees. This argument has been discussed by several authors, but turning it into a mathematically precise statement with appropriate assumptions on the table and the ground on which it rests is not easy.

The two most recent contributions to this topic that I am aware of are

A. Martin, On the Stability of Four-Legged Tables, Physics Letters A, 360, 495-500, 2007

Bill Baritompa, Rainer Löwen, Burkard Polster, and Marty Ross, Mathematical Table Turning Revisited, arXiv:math/0511490v1, 17 pp., 2008

In the latter paper it is shown that if the ground on which a rectangular table rests slopes by less than 35.36 degrees and the legs of the table are at least half as long as its diagonals then the rotation trick works.

For more insight into this problem you may like to watch the video below in which Matthias Kreck explains the problem with the aid of some excellent animations.

# Mathematics in Color Color is a fascinating subject. Important early contributions to our understanding of it came from physicists and mathematicians such as Newton, Young, Grassmann, Maxwell, and Helmholtz. Today, the science of color measurement and description is well established and we rely on it in our daily lives, from when we view images on a computer screen to when we order paint, wallpaper, or a car, of a specified color.

For practical purposes color space, as perceived by humans, is three-dimensional, because our retinas have three different types of cones, which have peak sensitivities at wavelengths corresponding roughly to red, green, and blue. It’s therefore possible to use linear algebra in three dimensions to analyze various aspects of color.

## Metamerism

A good example of the use of linear algebra is to understand metamerism, which is the phenomenon whereby two objects can appear to have the same color but are actually giving off light having different spectral decompositions. This is something we are usually unaware of, but it is welcome in that color output systems (such as televisions and computer monitors) rely on it.

Mathematically, the response of the cones on the retina to light can be modeled as a matrix-vector product $Af$, where $A$ is a 3-by- $n$ matrix and $f$ is an $n$-vector that contains samples of the spectral distribution of the light hitting the retina. The parameter $n$ is a discretization parameter that is typically about 80 in practice. Metamerism corresponds to the fact that $Af_1 = Af_2$ is possible for different vectors $f_1$ and $f_2$. This equation is equivalent to saying that $Ag = 0$ for a nonzero vector $g = f_1-f_2$, or, in other words, that a matrix with fewer rows than columns has a nontrivial null space.

Metamerism is not always welcome. If you have ever printed your photographs on an inkjet printer you may have observed that a print that looked fine when viewed indoors under tungsten lighting can have a color cast when viewed in daylight.

## LAB Space: Separating Color from Luminosity

In digital imaging the term channel refers to the grayscale image representing the values of the pixels in one of the coordinates, most often R, G, or B (for red, green, and blue) in an RGB image. It is sometimes said that an image has ten channels. The number ten is arrived at by combining coordinates from the representation of an image in three different color spaces. RGB supplies three channels, a space called LAB (pronounced “ell-A-B”) provides another three channels, and the last four channels are from CMYK (cyan, magenta, yellow, black), the color space in which all printing is done.

LAB is a rather esoteric color space that separates luminosity (or lightness, the L coordinate) from color (the A and B coordinates). In recent years photographers have realized that LAB can be very useful for image manipulations, allowing certain things to be done much more easily than in RGB. This usage is an example of a technique used all the time by mathematicians: if we can’t solve a problem in a given form then we transform it into another representation of the problem that we can solve.

As an example of the power of LAB space, consider this image of aeroplanes at Schiphol airport.

Suppose that KLM are considering changing their livery from blue to pink. How can the image be edited to illustrate how the new livery would look? “Painting in” the new color over the old using the brush tool in image editing software would be a painstaking task (note the many windows to paint around and the darker blue in the shadow area under the tail). The next image was produced in just a few seconds.

How was it done? The image was converted from RGB to LAB space (which is a nonlinear transformation) and then the coordinates of the A channel were replaced by their negatives. Why did this work? The A channel represents color on a green–magenta axis (and the B channel on a blue–yellow axis). Apart from the blue fuselage, most pixels have a small A component, so reversing the sign of this component doesn’t make much difference to them. But for the blue, which has a negative A component, this flipping of the A channel adds just enough magenta to make the planes pink.

You may recall from earlier this year the infamous photo of a dress that generated a huge amount of interest on the web because some viewers perceived the dress as being blue and black while others saw it as white and gold. A recent paper What Can We Learn from a Dress with Ambiguous Colors? analyzes both the photo and the original dress using LAB coordinates. One reason for using LAB in this context is its device independence, which contrasts with RGB, for which the coordinates have no universally agreed meaning.

## The Princeton Companion to Applied Mathematics

My article Color Spaces and Digital Imaging in The Princeton Companion to Applied Mathematics gives an introduction to the mathematics of color and the representation and manipulation of digital images. In particular, it emphasizes the role of linear algebra in modeling color and gives more detail on LAB space.

I have one update to the article. Since the book went to press a second edition of the book that I cite by Dan Margulis, Photoshop LAB Color: The Canyon Conundrum And Other Adventures In The Most Powerful Colorspace, has appeared (amazon.com and amazon.co.uk). I do not yet have the book but it appears to have a number of improvements on the excellent first edition.

Finally, in the article I mention the problem of finding good color maps and the problems with the commonly used rainbow color map. For a nicely illustrated talk on this topic see Perceptual Color Maps in matplotlib for Oceanography given at SciPy 2015 by Kristen Thyng. The author speaking about rainbow color maps at a UoM School of Mathematics Alumni Event at The Royal Institution, London. Photo: Chris Mann Photography.

# Knuth on Knowing Your Audience

Donald Knuth has a great ability to summarize things in pithy, quotable nuggets. A good example is the following sentence from his 2001 book Things a Computer Scientist Rarely Talks About:

The amount of terror that lives in a speaker’s stomach when giving a lecture is proportional to the square of the amount he doesn’t know about his audience.

Knuth’s point is about preparation, and it brings to mind the words of Benjamin Franklin, “By failing to prepare, you are preparing to fail”.

It’s essential to find out as much as possible about your audience, not just so that you feel more confident, but also so that what you deliver is appropriate for that audience.

As academics we are used to giving seminars and conference talks for which we know that the audience will be made up of peers, and we usually just need to ascertain where to aim the talk on the axes general researcher–specialist and graduate student–experienced researcher.

For any other talk it is important to go to some effort to find out who will be in the audience, perhaps asking for a list of attendees if the event requires registration. For an after-dinner talk you may want to know whether certain key people who you are thinking of mentioning will be in the audience. For a talk to a general audience you will want to assess the base level of technical knowledge that can be assumed.

Keep these thoughts in mind when that sought-after invitation to give a “TED talk” arrives in your mailbox.