177 research outputs found
Subgeometries in the Andr\'e/Bruck-Bose representation
We consider the Andr\'e/Bruck-Bose representation of the projective plane
in . We investigate the representation
of -sublines and -subplanes of
, extending the results for of \cite{BarJack2} and
correcting the general result of \cite{BarJack1}. We characterise the
representation of -sublines tangent to or contained in the
line at infinity, -sublines external to the line at infinity,
-subplanes tangent to and -subplanes secant to
the line at infinity
Effective calculation of LEED intensities using symmetry-adapted functions
The calculation of LEED intensities in a spherical-wave representation can be substantially simplified by symmetry relations. The wave field around each atom is expanded in symmetry-adapted functions where the local point symmetry of the atomic site applies. For overlayer systems with more than one atom per unit cell symmetry-adapted functions can be used when the division of the crystal into monoatomic subplanes is replaced by division into subplanes containing all symmetrically equivalent atomic positions
Combinatorial problems in finite geometry and lacunary polynomials
We describe some combinatorial problems in finite projective planes and
indicate how R\'edei's theory of lacunary polynomials can be applied to them
Collineation groups of translation planes of small dimension
A subgroup of the linear translation complement of a translation plane is
geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for geometrically primitive. If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup of order w2a+b where wa divides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane
The tangent splash in \PG(6,q)
Let B be a subplane of PG(2,q^3) of order q that is tangent to .
Then the tangent splash of B is defined to be the set of q^2+1 points of
that lie on a line of B. In the Bruck-Bose representation of
PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface
corresponding to B and the planes corresponding to the tangent splash of B. We
then give a geometric construction of the unique order--subplane determined
by a given tangent splash and a fixed order--subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
Inherited Groups and Kernels of Derived Translation Planes
When an affine plane is converted to another plane by derivation, the point permutations which act as collineations of both planes form the inherited group. The full group can be larger than the inherited group. For finite translation planes in which some of the Baer subplanes involved are not vector spaces over the kernel of the original plane then the full collineation group of the derived plane is the inherited group provided the order of the plane is greater than 16
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