48 Normat 58:1, 48 (2010)

Summary in English

G.Almkvist and A.Meurman, Je-

sus Guilleras formler för 1/π

2

och su-

perkongruenser(Swedish).

Striking inﬁnite series for

1

π

have

been given by Ramanujam and later

the brothers Chudnovsky (who applied

theirs to compute π with a billion dec-

imals). In 2002 Guillera provided a se-

ries for

1

π

2

, which is, along with varia-

tions concerning congruences of partial

sums, discussed at length. Intriguingly

an old theorem by the Swedish math-

ematician Fritz Carlson, to the eﬀect

that any entire function vanishing on

the non-negative integers and satisfying

a bound Ke

c|z|

(with crucially c < π) on

the right halfplane must vanish, comes

into play. The authors also show how

commercial packages such as Maple can

be used eﬀectively in the investigations.

In particular they come up with their

own examples and congruences, such as

∞

X

n=0



2n

n



4



3n

n



74n

3

+ 135n

2

+ 69n + 6

(n + 1)

3

1

2

12n

=

64

π

2

The article ends with a series of exer-

cises for the readers, some of them con-

cerning Bernouilli numbers and poly-

nomials, and an intriguing variation

thereof introduced by Zagier.

Tom Britton, Smittsamma sjukdo-

mars matematik (Swedish).

The article deals with the mathemat-

ics of infectuous diseases. First two sim-

ple deterministic models for infectuous

spreads are considered. The ﬁrst one

concerns epidemies, which are of short

dramatic duration and concern a ﬁxed

homogenuous population. The second

deals with the endemic case, which lasts

for a very long time, and in which the

population is continually renewed. Both

models involve a system of non-linear

diﬀerential equations, from which one

may observe some qualitative aspects,

as well as derive the basis for quantita-

tive simulations. In addition stochastic

models, which are more realistic if more

complicated, are introduced, along with

a study of graphs to model social con-

nectivity (particularly relevant for the

spread of HIV). Finally the author dis-

cusses practical applications.

Ulf Persson, Oktahedergruppen och

dess generaliseringar II (Swedish).

This is a second instalment in a planned

series of three. This one considers the

hypercube (the tesseract) and its dual in

R

4

, and their common group of symme-

tries. In particular it makes a thorough

study of the group, including its conju-

gacy classes and gives geometric inter-

pretations and presents various projec-

tions.

Olav Gebhardt og Marius Over-

holt, Et kombinatorisk kuriosum (Nor-

wegian).

Because of some misprints, this arti-

cle originally found in Normat 2009:4 is

reprinted in this issue.

Given n strands of black and white

pearls, each of length k. Assume that

each pair of strands agree in at least m

more positions than they disagree, and

that at each position the excess num-

ber of pearls of one color over the other

across the strands is at most d. The au-

thors ﬁnd a necessary condition on n, k,

m and d, and prove it both combinato-

rially and by linear algebra.