Dear Dirichlet, Issue 07

Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Send your problems to deardirichlet@chalkdustmagazine.com.

Dear Dirichlet,

I’ve recently had the good fortune of winning three pigs at the village fete. However, I’m not sure whether my triangular garden is big enough for them as well as my collection of metal, wooden and other deckchairs. The pigs are of substantial size and my tape measure is not long enough to measure the longest side of the garden. I’ve also heard that pigs are very intelligent and would like to hear suggestions for entertaining them.

— Pearl among swine, Lower Brailes

Prize crossnumber, Issue 07

Our original prize crossnumber is featured on pages 52 and 53 of Issue 07.

Clarification: Added “non-zero” to clues 10A, 19D.
Clarification: For 20D, 0 is not a factor of any number, so the number contains no 0s.

Rules

• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
• One randomly selected correct answer will win a £100 Maths Gear goody bag. Three randomly selected runners up will win a Chalkdust t-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide, with free UK shipping. Find out more at mathsgear.co.uk
• To enter, submit the sum of the across clues via this form by 1 August 2018. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 19 August 2018.

Top Ten: Mathematical celebration days

This issue, Top Ten features the top ten mathematical celebration days! Then vote here on the top ten units for Issue 08!

Reentering the top ten, it’s the day of monthly MathsJam: the second last Tuesday of the month.

At 9 this issue, the favorite day of fans of the American date format and decimals: Pi day (14 March).

Conference bingo

How to play

Next time you’re at a conference, you and a friend/coworker each pick one of the grids below. When you see something happen cross it off. The first player to cross off five boxes in a row wins. The grids below appeared in Issue 07 of Chalkdust. To generate new random grids, click here.

Grid A

 A question that’s actually a comment Someone cites your paper Run out of coffee Slides projected in the wrong aspect ratio A proof is left as an exercise Someone forgot their Mac adaptor A talk overruns into the break Your free pen runs out Laser pointer gets shined in your eye Someone reading Chalkdust Spill ketchup on your last clean shirt Ugly Beamer slides Ugly PowerPoint slides Fire drill You fall asleep in a talk Someone uses confusing notation You understand a talk A video doesn’t work You run out of tea bags in your room Drunk Fields medallist Someone uses chalk The roof is leaking Board pen runs out Someone else playing conference bingo One of everything for breakfast

Grid 1

 PDF slides not full-screened Slide clicker batteries run out You sleep through first talk Permanent marker on a whiteboard No clock in the room No milk for tea You lose your name tag Only bad biscuits left Skip talks to go sightseeing You smash your conference mug You lose your programme Your supervisor is drunk You can’t connect to eduroam Person in front of you playing Minesweeper Awkward forced socialising You’re the only one using the hashtag You get on the wrong German train Someone gets your name wrong You get someone’s name wrong You forgot toothpaste Someone’s phone goes off in a talk Windows needs to install updates Projector overheats Too hot in room Too cold in room

How to make: a catastrophe machine

• cardboard
• scissors
• pins
• rubber bands

Instructions

1. Cut out a card circle whose diameter is the length of an unstretched rubber
band. Use pins to attach everything together, as shown above.

2. Drag the lower rubber band around to investigate catastrophe theory.

Page 3 model: Frictional unemployment

If I had a pound for every time someone assumed I studied maths because I wanted to be an economist without writing essays, I’d have enough to make it worth following the stock market. However, once the indignation fades, I can see the attraction—there are a lot of interesting uses of mathematics in economics. One of the most basic, yet most important, is modelling unemployment.

Unemployment might be caused by too few jobs in an area. Or, it may also be due to a lack of information being provided to employers or potential workers: there may be perfectly good jobs available that qualified workers simply don’t know about. This sort of unemployment is called frictional unemployment.

We split the labour force $L$ into two separate populations: employed ($N$) and unemployed ($U$). We then define $s$ and $f$ to be the rates at which people gain and lose employment:

The rate of change in unemployment is:
\begin{align*}
\frac{\text{d} U}{\text{d}t}&=\text{number becoming unemployed} -\text{number entering work}\\
&=sN(t)-fU(t)
\end{align*}
If we assume that the total size of the labour force is constant, then this leads us to:
$$\frac{\text{d}u}{\text{d}t}+(s+f)u=s,$$
where $u$ is the proportion of the labour force that is unemployed. A lovely first order ODE, which can be solved using the integrating factor method (an exercise left for the reader). Simple enough that even an economist would understand!