48 Normat 60:1, 48 (2012)

48 Normat 59:1, 48 (2011)

Summary in English

Ulf Persson, Torsten Ekedahl 1955-

2011 (Swedish)

An obituary of an untimely deceased

Swedish mathematician of renown.

Hiroshi Okumura, Are those Archi-

medean triples circles really triplts?

Dilations and the arbelos. (English).

The arbelos is a classical geometric ﬁ-

gure formed by three mutually tangent

circular arcs whose three intersections

points are collinear. Special circles as-

sociated to the arbelos was already stu-

died by Archimedes, and in these two

articles, further variations on the theme

are presented.

Ola Christensen, Kristian Borg

Thomsen, Mads Paulsen Gibbs’ fæ-

nomen swinger i 2D (Danish)

The point-wise convergence of the par-

tial sums of a Fourier series to a func-

tion can be established for piece-wise

diﬀerentiable functions, with jump dis-

continuities allowed. The convergence is

not uniform at the points of disconti-

nuity, and its deviation from uniform

continuity can be made very precise as

about 9% of the jump. This is known as

the Gibbs’ phenomenon. The introduc-

tory part of the article contains a de-

rivation of the phenomenon, while the

main part discusses what appears for

functions of two variables. The situa-

tion is then restricted to functions of

type f(x, y)=f

1

(x)f

2

(y) which is simp-

le enough to allow explicit calculations

but general enough to allow a glimpse of

the complexities that arises. It turns out

that the ’overshooting’ is unpredictable

from knowing the jump of the disconti-

nuity.

Jorge Jimenéz Urroz, A note on se-

ries of positive numbers (English)

There is no ultimate test for conver-

gence of a series, thus it is always of

interest to ﬁnd weaker and weaker cri-

teria. In this note the author revives

a criterion due to de la Vallée Pous-

sin. More speciﬁcally, let a

n

Ø 0 and

q

n

a

n

divergent, meaning that the par-

tial sums s

N

are monotonically increas-

ing to Œ.Thenletf(x) be a decreasing

function such that lim

x‘æŒ

f(x)=0.

Then there are natural conditions in

terms of f for the series

q

n

f(s

n

)a

n

and

q

n

f(s

n≠1

)a

n

to be convergent re-

spective divergent. Variations and corol-

laries of these criteria are presented.

Ulf Persson, Hopningspunkter(Swedish)

An elementary discussion of accu-

mulation points, especially strong such,

with the conclusion that any closed

countable set can be reduced to the em-

pty set by recursively removing all iso-

lated points.