3,259 research outputs found
Desarguesian spreads and field reduction for elements of the semilinear group
The goal of this note is to create a sound framework for the interplay
between field reduction for finite projective spaces, the general semilinear
groups acting on the defining vector spaces and the projective semilinear
groups. This approach makes it possible to reprove a result of Dye on the
stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend
it to P{\Gamma}L(n, q). Moreover a result of Drudge [5] relating Singer cycles
with Desarguesian spreads, as well as a result on subspreads (by Sheekey,
Rottey and Van de Voorde [19]) are reproven in a similar elementary way.
Finally, we try to use this approach to shed a light on Condition (A) of
Csajbok and Zanella, introduced in the study of linear sets [4]
Hennebique’s journal 'Le Béton armé': a close reading of the genesis of concrete construction in Belgium
In June 1898, François Hennebique issued the monthly journal Le Béton Armé. Published until 1939, with 378 issues in all, this platform on the interface between information and propaganda serves as a perfect means to obtain a comprehensive overview of Hennebique’s legacy. Giving an insight into the increasing sphere of action, the growing number of applications and the hierarchic structure and policy of the firm, the journal is a work of reference, essential to document the unremitting development of concrete construction. By means of a close reading (based on the collections preserved at Ghent University and the Centre d’archives du XXe siècle de l’ifa in Paris), the content, meaning, and changing discourse of Le Béton Armé will be critically analyzed. Fitting within the scope of a PhD on the history of concrete construction in Belgium (www.architecture.ugent.be/concrete), particular attention will be given to the application of ‘le système Hennebique’ in Belgium
On sets without tangents and exterior sets of a conic
A set without tangents in \PG(2,q) is a set of points S such that no line
meets S in exactly one point. An exterior set of a conic is a set
of points \E such that all secant lines of \E are external lines of
. In this paper, we first recall some known examples of sets
without tangents and describe them in terms of determined directions of an
affine pointset. We show that the smallest sets without tangents in \PG(2,5)
are (up to projective equivalence) of two different types. We generalise the
non-trivial type by giving an explicit construction of a set without tangents
in \PG(2,q), , prime, of size , for all
. After that, a different description of the same set in
\PG(2,5), using exterior sets of a conic, is given and we investigate in
which ways a set of exterior points on an external line of a conic in
\PG(2,q) can be extended with an extra point to a larger exterior set of
. It turns out that if mod 4, has to lie on , whereas
if mod 4, there is a unique point not on
A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear
In this paper, we show that a small minimal blocking set with exponent e in
PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear
set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal
blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional
space, are F_p-linear, hence confirming the linearity conjecture for blocking
sets in this particular case
A linear set view on KM-arcs
In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in
PG(2, q) such that every line contains 0, 2 or t of its points. We use field
reduction to give a different point of view on the class of translation arcs.
Starting from a particular F2-linear set, called an i-club, we reconstruct the
projective triads, the translation hyperovals as well as the translation arcs
constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show
the KM-arcs of type q/4 recently constructed by Vandendriessche are translation
arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q/4. We show that this family, apart from new examples that are not translation
KM-arcs, contains all translation KM-arcs of type q/4
Pseudo-ovals in even characteristic and ovoidal Laguerre planes
Pseudo-arcs are the higher dimensional analogues of arcs in a projective
plane: a pseudo-arc is a set of -spaces in
such that any three span the whole space. Pseudo-arcs of
size are called pseudo-ovals, while pseudo-arcs of size are
called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from
applying field reduction to an arc in .
We explain the connection between dual pseudo-ovals and elation Laguerre
planes and show that an elation Laguerre plane is ovoidal if and only if it
arises from an elementary dual pseudo-oval. The main theorem of this paper
shows that a pseudo-(hyper)oval in , where is even and
is prime, such that every element induces a Desarguesian spread, is
elementary. As a corollary, we give a characterisation of certain ovoidal
Laguerre planes in terms of the derived affine planes
Characterisations of elementary pseudo-caps and good eggs
In this note, we use the theory of Desarguesian spreads to investigate good
eggs. Thas showed that an egg in , odd, with two good
elements is elementary. By a short combinatorial argument, we show that a
similar statement holds for large pseudo-caps, in odd and even characteristic.
As a corollary, this improves and extends the result of Thas, Thas and Van
Maldeghem (2006) where one needs at least 4 good elements of an egg in even
characteristic to obtain the same conclusion. We rephrase this corollary to
obtain a characterisation of the generalised quadrangle of
Tits.
Lavrauw (2005) characterises elementary eggs in odd characteristic as those
good eggs containing a space that contains at least 5 elements of the egg, but
not the good element. We provide an adaptation of this characterisation for
weak eggs in odd and even characteristic. As a corollary, we obtain a direct
geometric proof for the theorem of Lavrauw
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