192 Normat 57:4, 192 (2009)

Summary in English

Johan Hoﬀman and Claes John-

son, The Mathematical secret of Flight.

Are you afraid of ﬂying? If so there may

be little comfort in being assured by

NASA that all the popular explanations

of ﬂight are wrong and furthermore that

no authoritative explanation is availab-

le. The purpose of this article is to gi-

ve a new mathematical (and physical)

explanation of the phenomenon of ﬂight

intending to plug up this disconcerting

gap. Whether it will actually assuage

your fears is quite another matter.

The explanation is based on com-

putational solutions to the standard

mathematical model of ﬂuid mechanics

- the Navier-Stokes/Euler equations, so-

lutions which have recently led to new

discoveries of the dynamics of turbulent

air-ﬂows around a wing. It is shown that

the large lift/drag quotient required is

eﬀected by a fortunate combination of

certain features of slightly viscous in-

compressible ﬂow including a crucial in-

stability mechanism.

The lift L is caused by the down-wash

of air engineered by the wing, while the

drag D is counteracted by leading ed-

ge suction. The great mystery is how a

small amount of drag can generate so

much lift. In fact L/D needs to be lar-

ge (10 or more) to make ﬂight feasib-

le. Elementary considerations by New-

ton, later reﬁned by d’Alembert showed

that ﬂight was mathematically impos-

sible. This is known as d’Alemberts pa-

radox. The feat of the brothers Wright

further rubbed in theoretical embarrass-

ment. The Russian Zhukovsky saved

ﬂuid-dynamics from complete collapse

by explaining how lift was generated

through the perturbation by a circu-

lar ﬂow, supplemented by an explana-

tion of drag in terms of a viscous boun-

dary layer later given by the physi-

cist Prandtl. Those were, however, not

enough to transform the art of an air-

craft engineer into a science, and are in

fact shown to be incorrect. The classical

fallacy was to concentrate on the poten-

tial ﬂow, a highly unstable mathemati-

cal solution with no physical relevance.

Instead one should consider the far more

stable solution involving a turbulent ﬂu-

ctuating layer. Lift and drag are insepa-

rable, generated by the same mechanism

of counter-rotating low-pressure rolls of

the turbulently swirling ﬂow.

Olav Gebhardt og Marius Over-

holt, Et kombinatorisk kuriosum (Nor-

wegian).

Given n strands of black and white pe-

arls, 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 number of

pearls of one color over the other across

the strands is at most d. The authors

ﬁnd a necessary condition on n,k,m and

d, and prove it both combinatorially

and by linear algebra.

Tord Sjödin, Gram-Schmid’s algoritm

för en allmän vinkel(Swedish).

The classical Gram-Schmid algorithm

allows you to exhibit an orthogonal ba-

sis with respect to a positive deﬁnite

form (the normalization that each basis

element has length one is of course tri-

vial). In this paper the question is posed

whether it also works when orthogona-

lity is replaced with some other angle θ.

The author shows that it is possible if

and only if −

1

n−1

< cos θ < 1 where n

is the dimension of the space.