16 research outputs found
Flag-transitive L_h.L*-geometries
The classification of finite flag-transitive linear spaces, obtained by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl [20] at the end of the eighties, gave new impulse to the program of classifying various classes of locally finite flag-transitive geometries belonging to diagrams obtained from a Coxeter diagram by putting a label L or L ∗ on some (possibly, all) of the singlebond strokes for projective planes
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems
in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October
5, 2012, Brussels, Belgiu
Harmonic Equiangular Tight Frames Comprised of Regular Simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies
Harmonic Equiangular Tight Frames Comprised of Regular Simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies
Harmonic Equiangular Tight Frames Comprised of Regular Simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies