Dense near octagons with four points on each line, III

This is the third paper dealing with the classification of the dense near octagons of order (3, t). Using the partial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3, t) with a big hex, we get an incomplete classification for the dense near octagons of order (3, t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved

An alternative definition of the notion valuation in the theory of near polygons.

Valuations of dense near polygons were introduced in \cite{DB-Va:1}. A valuation of a dense near polygon $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathrm{I})$ is a map $from the point-set$\mathcal{P}$of$\mathcal{S}$to the set$\N$of nonnegative integers satisfying very nice properties with respect to the set of convex subspaces of$\mathcal{S}$. In the present paper, we give an alternative definition of the notion valuation and prove that both definitions are equivalent. In the case of dual polar spaces and many other known dense near polygons, this alternative definition can be significantly simplified

On the valuations of the near polygon ℍn

We characterize the valuations of the near polygon that are induced by classical valuations of the dual polar space DW (2n - 1, 2) into which it is isometrically embeddable. An application to near 2n-gons that contain as a full subgeometry is given

Localization of the 17q breakpoint of a constitutional 1;17 translocation in a patient with neuroblastoma within a 25-kb segment located between the ACCN1 and TLK2 genes and near the distal breakpoints of two microdeletions in neurofibromatosis type I patients.

We have constructed a 1.4-Mb P1 artificial chromosome/bacterial artificial chromosome (PAC/BAC) contig spanning the 17q breakpoint of a constitutional translocation t(1;17)(p36.2;q11.2) in a patient with neuroblastoma. Three 17q breakpoint-overlapping cosmids were identified and sequenced. No coding sequences were found in the immediate proximity of the 17q breakpoint. The PAC/BAC contig covers the region between the proximally located ACCN1 gene and the distally located TLK2 gene and SCYA chemokine gene cluster. The observation that the 17q breakpoint region could not be detected in any of the screened yeast artificial chromosome libraries and the localization of the 17q breakpoint in the vicinity of the distal breakpoints of two microdeletions in patients with neurofibromatosis type 1 suggest that this chromosomal region is genetically unstable and prone to rearrangement

The valuations of the near polygon n

We show that every valuation of the near 2n-gon G(n), n >= 2, is induced by a unique classical valuation of the dual polar space DH(2n - 1, 4)into which G(n) is isometrically embeddable

On generalized hexagons of order (3, t) and (4, t) containing a subhexagon

We prove that there are no semi-finite generalized hexagons with $q + 1$ points on each line containing the known generalized hexagons of order $q$ as full subgeometries when $q$ is equal to $3$ or $4$, thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when $q$ is equal to $2$ was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order $4$ we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.Comment: 13 pages, minor revisions based on referee reports, to appear in European Journal of Combinatoric

On geometric SDPS-sets of elliptic dual polar spaces

AbstractLet n∈N∖{0,1} and let K and K′ be fields such that K′ is a quadratic Galois extension of K. Let Q−(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q+(2n+1,K′) of Witt index n+1 in PG(2n+1,K′). For even n, we define a class of SDPS-sets of the dual polar space DQ−(2n+1,K) associated to Q−(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ−(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ−(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ−(2n+1,K) into one of the half-spin geometries for Q+(2n+1,K′)