Sunday 18 March:
The Royal Oak pub. This is where we鈥檙e all supposed to meet, and by early
evening people are beginning to trickle in. In all, there will be more than 80
mathematicians, and 14 representatives from eight companies, keen to pick their
brains. It鈥檚 a neat idea鈥攖he companies bring along things that they鈥檙e
having trouble with, and the academics spend the week working on the problems
for free.
There are advantages for both sides. For instance, even though I am an
applied mathematician working on real-world problems, my research tends to be
long-term, and I鈥檓 unlikely to see the practical results. Occasionally, some of
the problems can even lead to further research projects and
publications鈥擨鈥檓 still collaborating on work that originated in a similar
workshop four years ago. But for the most part, this week is the only time I鈥檒l
look at these problems. It is a welcome chance to forget my own research and
work on something completely different that could have immediate practical
benefits.
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Monday 19 March:
Main lecture hall. One by one, the industrialists take the stand to present
their problems. This year, there are eight problems to choose from, ranging from
modelling the effects of council housing policy on homeless populations for
Shelter to what鈥檚 wrong with mobile phones.
Some mathematicians like to stay with the same problem all week, to really
get their teeth into it, while others like me tend to wander around, helping out
with a problem for a few hours, then moving on to the next. There are plenty of
questions in these opening sessions as the academics try to strip away the
sometimes baffling company jargon and get to the heart of the problems.
I like the look of two problems in particular. The first involves tiny but
annoying pinholes that are messing up a screen-printing process. Apparently, the
printing involves forcing paint through a fine wire mesh, some of whose holes
are masked to make the desired pattern. The process is supposed to leave a nice
uniform layer of paint on the material below, but pinholes sometimes form where
the wires in the mesh cross. The company is looking for ways to fix the problem
by changing the properties of the paint鈥攅asier than changing the whole
printing process.
The second problem I plan to work on involves cutting jumbo rolls of paper to
the individual widths demanded by different customers. For each new order, the
blades on the cutting machine are arranged into a set of patterns, optimised by
a computer program to minimise the amount of waste paper at the edge of each
roll. Since each new pattern means that the blades on the cutting machine must
be repositioned鈥攚hich is time-consuming鈥攖he question is whether
different patterns can be combined.
The company, Graycon, has already worked out general ways to turn two
patterns into one, and three into two. Could we find efficient four to three and
five to four reductions? Could we find a more general way of taking any number
of patterns and reducing them by one鈥攁n n to n鈭�1
reduction? Would this catch all possible reductions? To cap it all, we have to
find a way to compute the answers to these questions by running a program for
less than a minute鈥攔oughly the time the factory operator is willing to
wait鈥攐n a simple Pentium processor.
We retire to the safety of the bar to plan our attack.
Tuesday 20 March:
Work on the problems begins in earnest. Each problem is allocated its own
room and participants wander in and out as they wish.
I decide to start on the pinholing. The first task is physics, rather than
maths. We need to work out how the pinholes form before we can build a
mathematical model for the process and check the model鈥檚 predictions against any
experimental results the company can give us.
Alistair Fitt from the University of Southampton stands at the board, and the
rest of us sit on tables and chairs, offering suggestions. This brainstorming
process is what I love most about study groups. Nobody teaches you this at
university, but you can learn a lot by simply thinking out loud with a bunch of
other mathematicians. Students tend to be more cautious about putting forward an
idea in case everyone else thinks it鈥檚 ridiculous. But it鈥檚 soon clear that
everybody has their fair share of foolish ideas, and the newcomers start to
relax.
We need to work out not only why the pinholes form, but also what holds them
open long enough for the paint to dry. Initially, when the mesh is removed, the
paint film will be uneven, and forces such as gravity and surface tension will
make it spread out into a uniform film. Of these forces, we reckon that surface
tension will be most important in trying to close up pinholes.
The next step is to work out what could resist the action of the surface
tension. The prime candidate is the viscous resistance鈥攖he force that
makes it difficult to stir treacle. Balancing these two forces, we work out that
typical pinholes should close in about a second. Unfortunately, paint takes much
longer than this to dry and pinholes stay open in wet paint for several
minutes.
As the way forward becomes muddier, the group splinters into smaller groups,
each discussing aspects of the problem and taking turns to quiz the
industrialists. As answers go in and out of focus, groups merge and splinter
again.
We discover that the paint contains small particles about 5 micrometres in
diameter. This is why it will behave like a solid when under a small amount of
stress, and only begin to flow when the stress reaches a critical value. It鈥檚 a
phenomenon known as a yield stress.
This gives us our first stab at working out why the defects in the paint
surface last so long: perhaps yield stress is holding the pinholes open while
the paint dries. Unfortunately, holding open the tiny pinholes that are seen in
real life would take ten times the experimentally measured yield stress. As the
group begins to put together a model for the printing process to see whether the
way pinholes form sheds some light on their unusual longevity, I decide to check
out what is happening elsewhere.
In the paper cutting room, two groups are beavering away. At the back of the
room one group is using high-level mathematics to work on the problem. I鈥檓 not
familiar with this branch of mathematics, so I join the other group, which has
managed to formulate the problem, but unfortunately its statement turns out to
be too cumbersome to be useful.
On the plus side, they have solved one question already: by trial and error,
the group has come up with an example of three patterns which could be reduced
to one, but not to two. In other words, a general nto n鈭�1
solution would not pick up all possible reductions.
As the day goes on, we get a feel for the problem by working through some
simple examples applying Graycon鈥檚 two-to-one and three-to-two reductions. Each
pattern is a collection of customer widths into which the jumbo roll will be
divided. We decide that a general approach will have to work one step at a time
by switching widths between patterns. For example, say the jumbo roll is 300
centimetres long, and we have two patterns, one with widths of 100 cm, 100 cm
and 80 cm and one with 90 cm, 90 cm and 110 cm. In this case, we can switch a
width of 100 cm in the first pattern for one of 90 cm in the second pattern to
get two new patterns with widths of 100 cm, 90 cm and 80 cm, and 100 cm, 90 cm
and 110 cm. These patterns are more similar than the original two, since they
contain two out of three widths in common.
If we can switch the remaining width of 80 cm in the first case with one of
110 cm in another pattern then we will have two identical patterns, and so will
have reduced the total number of patterns by one. We work through some
three-to-two reductions by switching one roll at a time and try to identify what
made a good switch鈥攐ne that increases the similarity between patterns.
By the end of the day, we have developed some simple rules which we hope will
form the basis of an algorithm. Discussions continue over dinner and well into
the night in the bar.
Wednesday 21 March:
Yesterday evening, the paper cutting seemed in good shape, so I begin back in
the pinholing room. Sadly, there is still no sign of a mechanism to explain why
pinholes stay open. Several ideas are suggested, checked out and chucked out
because they end up producing a totally unrealistic pinhole shape.
Meanwhile, back in paper cutting, the group has developed an algorithm. To
function as a computer program, the rules have to be exact and comprehensive
enough to cope with any situation without the need for humans to tinker with it.
After working through more examples by hand and ironing out some of the
wrinkles, Mark Shephard, a graduate student from Oxford, volunteers to turn it
into computer code.
The high-level maths group reports back. They have been able to show that the
simplest problem involving only three cuts per jumbo roll belongs to a class of
mathematical problems known as NP hard (鈥淗ard maths, no problem鈥�, New
杏吧原创, 28 October, 1995 p 40). For these problems, the time it takes to
solve them increases exponentially with the size of the problem. In other words,
finding the absolute minimum number of patterns would be very difficult and time
consuming.
The rest of us are not too disappointed鈥攖his means we have not missed
something simple, and our back of the envelope algorithm is probably the best we
can hope for in a week.
Thursday 22 March:
Back in the pinholing room, we find a possible clue: there are more pinholes
at higher printing pressures, and in paints containing larger particles. John
Hinch, a reader in applied maths at Cambridge, promptly comes up with a theory
to account for the longevity of pinholes. The yield stress is caused by the
solid particles becoming locked against each other, and depends on their
concentration. If there were more particles concentrated near the pinholes, this
could boost the yield stress enough to balance the surface tension.
But what could increase the particle concentration? Sam Howison, maths
lecturer at Oxford, has an idea. When the mesh is pushed down, the particles are
squeezed out and shoot sideways. When the mesh is lifted, they should all
return, but if it is lifted quickly, an air bubble may form between the mesh and
the paint. The upshot could be a pinhole, surrounded by a higher concentration
of paint particles. It has taken until Thursday afternoon, but we finally have a
plausible mechanism.
Back in paper cutting, the algorithm has been turned into computer code. We
test it on a few examples and it turns six patterns into three and ten into
seven. This last example takes 2 seconds on an old 286 processor鈥攚ell
within our allowed computing budget.
A leader is chosen for each problem to collect all the ideas and work out how
to present them to the whole gathering the next day. Being a leader means
spending the evening writing presentations for the morning.
For the rest of us, the week鈥檚 work is over, and we can relax in the bar.
Friday 23 March:
Back in the main lecture theatre, the academics say their piece. I am always
amazed at the process by which the groups鈥� tangled ideas turn into a coherent
presentation by Friday morning. Most problems have gone well. The industrialists
seem happy.
The Graycon people are delighted with the initial performance of the
algorithm, but will need to test it thoroughly. And experiments will be needed
to check whether we are right about the mechanism for pinhole formation. If it
bears up to closer examination, we can think about devising a cure.
Perhaps next year.