ERLIN — When the euro was introduced in 12 countries
last Jan. 1, politicians called it the herald of a new era in
European integration — the advent of a common currency that
would help break down political and cultural barriers as well
as economic ones.
But for mathematicians, the euro has turned those 12
countries (plus the three tiny states of the Vatican, Monaco
and San Marino) into an almost ideal laboratory, providing a
euros-and-cents way to measure the extent of European
integration — and perhaps to understand other phenomena as
well, from the course of epidemics to the movement of stock
prices.
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While the euro is worth the same in every country that uses
it, each one mints its own euro coins, with a distinctive
design on the reverse. Every time a euro coin from Finland
appears in Greece, for example, it provides a tiny but precise
data point about the relationship between the countries.
So to find out more about their continent, thousands of
Europeans are digging through their pockets and keeping track
of their coins, and mathematicians are tabulating the results
in order to figure out just how fast the coins are
spreading.
"This is a historically unique opportunity," said Dr.
Dietrich Stoyan, a statistics professor at the University of
Freiberg in Germany, who is in charge of one of the
coin-counting projects. But the goal is not just to learn more
about Europe. "I hope that studying this process will help
people studying epidemics," Dr. Stoyan said. What makes this
special is that the precise launching date of the coins is
known. "We know when this `epidemic' broke out," he noted.
Moreover, the total amount minted was roughly proportional
to each country's economy compared with the overall European
economy. Germany, an economic powerhouse, minted 32.6 percent
of all coins, France about 15 percent, and tiny Luxembourg
(which used the face of its Grand Duke Henri) just two-tenths
of 1 percent. (All of the euro notes look the same.) Each
country introduced its own coins exclusively within its
borders.
On New Year's Day, the coins, all of them valid across the
entire euro zone, began to spread across national borders.
Now, statisticians are looking forward to the summer travel
season, when German tourists can simply pay for their espresso
in Paris with the coins they received in change that morning
at the baker's, back home in Munich.
But how does one model this mingling? Is it simple
diffusion, like opening 12 cans of gas in a giant room and
watching the molecules mix? Or does the relationship between
each country depend on a complex set of equations between each
country that considers the distance between the countries and
the number of commuters, travelers and bank trucks going back
and forth?
Dr. Stoyan has gone the complex route. His model is
composed of 144 interdependent differential equations that
take as many of the known variables into account as possible,
including traveler volume data from travel bureau
associations.
Too many variables, says a group from the University of
Amsterdam, which has chosen simply to take a high-level look
at the coin flow. Its model, based on a branch of probability
theory called Markov chains, assumes that a relatively
constant percentage of Dutch coins will leave the Netherlands
each month, and that a different, smaller percentage of Dutch
coins that have already left the country will return. "It's
just a guess," said Misja Nuyens, a doctoral candidate in
probability theory at the University of Amsterdam, who is a
member of the group. "We'll see if we were right, or sadly
mistaken."
Each experiment has a Web site for participants to post the
contents of their wallets. Data are tabulated and posted as
the experiments go on.
So far, Dr. Stoyan and the Amsterdam group have been
surprised to learn that large-denomination coins — for one and
two euros — move much faster than smaller ones. Neither knows
exactly why that is. Dr. Stoyan hypothesizes that people tend
to dump their small coins out of their pockets at the end of
the day and are therefore less likely to take them traveling.
Mr. Nuyens believes that the coins are used much more
often.
And when will the coins reach statistical equilibrium? The
models have yielded different results. The Dutch group
believes that half of all coins in Holland will be of foreign
origin a year from now and that statistical equilibrium across
Europe will be reached in five to seven years.
"I don't think it will go that fast," said Dr. Stoyan,
whose differential equation model predicts it will take
several decades.
Dr. Ger Koole, a mathematics professor at the Free
University in Amsterdam who is involved with the Dutch
project, hopes the findings will help expand understanding of
applied mathematics.
Phenomena like the "on hold" times at call centers and
stock price movements, he said, are similar to the movements
of euro coins: systems whose development depends largely or
entirely on the previous state the system was in.
For example, to predict the number of calls that will be on
hold, you need to know how many calls were on hold in the
previous minute and then calculate the probability that calls
will have been added (new calls coming in) or have been
removed (calls answered or hung up). These simple
calculations, repeated for every time period in question,
quickly can predict the development of complex systems.
Dr. Koole says it is too early to know what will be learned
from the euro experiments. The only way to improve
understanding of these phenomena is "the constant application
of theory to see what you end up with," he said.
The groups agree that the experiment has so many unknowns
that it is difficult to gain scientific clarity, at least at
this point. "I have the distinct impression that a real mania
has broken out among coin collectors," Dr. Stoyan said, which
would take rare coins disproportionately more often out of
circulation. In addition, the self-selected group of
participants who log their coins each month is not necessarily
a statistically representative sample.
The best data sources have turned out to be classrooms. The
two groups report that science and math teachers have latched
onto the projects as a way to illustrate their subjects. They
either ask all of the students to examine the change in their
pockets or buy rolls of coins at the bank and count them in
class.
Each team has a different model for the effect of the
summer travel season. The Dutch expect that July and August,
the two busiest months, will each count as two or three
off-season months. Dr. Stoyan's model integrates the increased
number of travelers. The two teams are hoping for a spike in
interest in the projects, which will get even more coins
counted and improve the results.
Dr. Stoyan, though, is already dreaming of one change that
would really make a difference. "Now if I could just get them
to send me the actual money," Dr. Stoyan said with a
laugh.