96 Normat 59:2, 96 (2011)
Summary in English
Ulf Persson Mikael Passare 1959-2011
(Swedish)
An obituary of a Swedish mathemati-
cian.
Lars Holst, Om Eulers “, och
fi
2
/6 ur sannoikhetsteoretisk synpunkt.
(Swedish).
The purpose of the article is to give
derivations of classical identities involv-
ing Eulers -function from the point
of view of probability theory. Specifi-
cally it is shown by a simple probabil-
ity argument that if X
i
are independent
stochastic variables with exponential
distributions with normalized expecta-
tion 1 then the associated s tochastic
variables max(X
1
,...X
n
) and
q
i
X
i
/i
have the same distributions. This is ex-
ploited to show that the integral
⁄
n
0
y
t
(1 ≠ y/n)
n≠1
dy
can be expressed as
e
≠t(
q
n
k=1
1/k≠n ln n)
n
Ÿ
k=1
e
t/k
1+t/k
from which follows in the limit as
n æŒthe product formula for (1+ t)
given by
lim
næŒ
n!n
t
(t +1)...(t + n)
= e
≠“t
Œ
Ÿ
k=1
e
t/k
1+t/k
Finally by using generating functions he
is able to derive the product formula for
the sine function out of the -function.
Pernille H. Petersen What is the
most important aspect of Smales horse-
shoe?
The author concludes that the question
is not well-posed. It depends on the con-
text. She considers an example in celes-
tial mechanics, due to Moser involving
three bodies, one of neglible mass, and
two others that instead pertain to the
behavior of the Henón m ap. In those ex-
amples the horseshoe mapping is used
as a tool and hence the fact that it is
conjugate to a shift operator turns out
to be the important thing in view of its
application. Smale himself, however, de-
nies this, and instead claims that it was
its structural stability that was most
important to him when he discovered
while struggling to interpret into geom-
etry a counterexample provided by Nor-
man Levinson. However later on Smale
has instead emphasized its chaotic na-
ture as being the thing of interest, which
further confirms her thesis on the de-
pendence upon context in judging the
importance of a result or a concept.
Ulf Persson Bisecting segments of con-
vex sets
In this article segments bisecting an
area of a convex region are studied. It
turns out that in the middle of a convex
figure which is not invariant under re-
flection through a suitable point, there
appears a small region through every
point of which multiple such segments
can be drawn. The region is bounded by
a ramification curve with many cusps,
because every direction only appe ars
once as a tangent.
