Month: July 2022

Creativity Workshop on Numerical Linear Algebra

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Sixteen members of the Numerical Linear Algebra group at the University of Manchester recently attended a two-day creativity workshop in order to generate ideas for our research and other activities. The workshop was facilitated by Dennis Sherwood, who is an expert in creativity and has run many such workshops. Dennis and I have previously collaborated on workshops for the Manchester Numerical Analysis group (2013), the EPSRC NA-HPC Network (2014), and the SIAM leadership (2018).

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A creativity workshop brings together a group of people to tackle questions using a structured approach, in which people share what they know about the question, ask “how might this be different” about the aspects identified, and then discuss the resulting possibilities. A key feature of these workshops is that every idea is captured—on flip charts, coloured cards, and post-it notes—and ideas are not evaluated until they have all been generated. This approach contrasts will the all-too-common situation where excellent ideas generated in a discussion are instantly dismissed because of a “that will never work” reaction.

At our workshop a number of topics were addressed, covering strategic plans for the group and plans for future research projects and grant proposals, including “Mixed precision algorithms”, “Being a magnet for talent”, and “Conferences”. Many ideas were generated and assessed and the group is now planning the next steps with the help of the detailed 70-page written report produced by Dennis.

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One idea has already been implemented: we have a new logo; see A New Logo for the Numerical Linear Algebra Group.

For more on the creativity process mentioned here, as well as details of creativity workshops, including sample briefs that can be used at them, see the new book by Dennis and me: How to Be Creative: A Practical Guide for the Mathematical Sciences (SIAM, 2022).

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How to Space Displayed Mathematical Equations

In a displayed mathematical equation with more than one component, how much space should be placed between the components?

Here are the guidelines I use, with examples in LaTeX. Recall that a \quad is approximately the width of a capital M and \qquad is twice the width of a \quad.

Case 1. Equation with Qualifying Expression

An equation or other mathematical construct is separated from a qualifying expression by a \quad. Examples:

\notag |a_{ii}| \ge \displaystyle\sum_{j\ne i} |a_{ij}|, \quad i=1\colon n.

\notag fl(x\mathbin{\mathrm{op}}y) = (x\mathbin{\mathrm{op}} y)(1+\delta), \quad |\delta|\le u, \quad \mathbin{\mathrm{op}} =+,-,*,/.

\notag y' = t^2+y^2, \quad 0\le t\le 1, \quad y(0)=0.

When the qualifying expression is a prepositional phrase it is given standard sentence spacing. Examples:

\notag \min_x c^Tx \quad \mathrm{subject~to~} Ax=b,~ x\ge 0.

\notag \|J(v)-J(w)\| \le \theta_L \|v-w\| \quad \mathrm{for~all~} v,w \in \mathbb{R}^n.

The first example was typed as (using the equation* environment provided by the amsmath package)

\begin{equation*}
\min_x c^Tx \quad \text{subject to $Ax=b$, $x\ge 0$}.
\end{equation*}

Here, the qualifying phrase is placed inside a \text command, which jumps out of math mode and formats its argument as regular text, with the usual interword spacing in effect, and we re-enter math mode for the conditions. This is better than writing

\min_x c^Tx \quad \text{subject to} ~Ax=b, ~x\ge 0.

with hard spaces. Note that \text is a command from the amsmath package, and it is similar to the LaTeX command \mbox and the TeX command \hbox, both of which work equally well here.

Case 2. Equation with Conjunction

When an equation contains a conjunction such as and or or, the conjunction has a \quad on each side. Examples:

\notag x = 1 \quad \mathrm{or} \quad x = 2.

\notag a = \displaystyle\sum_{j=1}^n c_j v_j \quad \mathrm{where} \quad c_j = \langle a,~ u_j\rangle~\mathrm{for}~j=1,2,\dots,n.

In the second example, one might argue for a \quad before the qualifying “for”, on the basis of case 1, but it I prefer the word spacing. This example was typed as

\begin{equation*}
     a = \sum_{j=1}^n c_j v_j \quad \text{where} \quad
     \text{$c_j = \langle a, u_j\rangle$ for $j=1,2,\dots,n$}.
\end{equation*}

Case 3. Multiple Equations

Two or more equations are separated by a \qquad. Examples:

\notag A = e_1^{}e_3^T, \qquad B = e_1^{}e_4^T, \qquad C = e_2^{}e_3^T, \qquad D = e_2^{}e_4^T

\notag \begin{aligned} AXA &= A, \qquad & XAX &= X,\\ (AX)^* &= AX, \qquad & (XA)^* &= XA. \end{aligned}

Limitations

It is important to emphasize that one might diverge from following these (or any other) guidelines, for a variety of reasons. With a complicated display, or if a narrow text width is in use (as with a two-column format), horizontal space may be at a premium so one may need to reduce the spacing. And the guidelines do not cover every possible situation.

Notes

My guidelines are the same ones that were used in typesetting the Princeton Companion to Applied Mathematics, and I am grateful to Sam Clark (T&T Productions), copy editor and typesetter of the Companion, for discussions about them. Cases 1 and 3 are recommended in my Handbook of Writing for the Mathematical Sciences (2017).

The SIAM Style Guide (link to PDF) prefers a \qquad in Case 1 and \quad in Case 3 with three or more equations. The AMS Style Guide (link to PDF) has the same guidelines as SIAM. Both SIAM and the AMS allow an author to use just a \quad between an equation an a qualifying expression.

In the TeXbook (1986, p. 166), Knuth advocates using a \qquad between an equation and a qualifying expression.