69,709 research outputs found
Maximal partial spreads and the modular n-queen problem III
AbstractMaximal partial spreads in PG(3,q)q=pk,p odd prime and q⩾7, are constructed for any integer n in the interval (q2+1)/2+6⩽n⩽(5q2+4q−1)/8 in the case q+1≡0,±2,±4,±6,±10,12(mod24). In all these cases, maximal partial spreads of the size (q2+1)/2+n have also been constructed for some small values of the integer n. These values depend on q and are mainly n=3 and n=4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3,q),q=pk where p is an odd prime and q⩾7, of size n for any integer n in the interval (q2+1)/2+6⩽n⩽q2−q+2
Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
It is shown that the maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ;
in Finite Geometry terms, the maximum number of planes in
mutually intersecting in at most a point is .
Optimal binary subspace codes are classified into
isomorphism types, and a computer-free construction of one isomorphism type is
provided. The construction uses both geometry and finite fields theory and
generalizes to any , yielding a new family of -ary
subspace codes
Abstract hyperovals, partial geometries, and transitive hyperovals
Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries
On the Veldkamp Space of GQ(4, 2)
The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized
quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply
giving several examples of Veldkamp lines (V-lines) having two or even three
Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size
five, one also finds V-lines of cardinality three and two. There, however,
exists a subspace of the Veldkamp space isomorphic to PG(3, 4) having 45 perps
and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four
distinct types. A V-line of the first type consists of five perps on a common
line (altogether 27 of them), the second type features three perps and two
ovoids sharing a tricentric triad (240 members), whilst the third and fourth
type each comprises a perp and four ovoids in the rosette centered at the
(common) center of the perp (90). It is also pointed out that 160 non-plane
ovoids (tripods) fall into two distinct orbits -- of sizes 40 and 120 -- with
respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the
first/second orbit sharing with the GQ(2, 2) a tricentric/unicentric triad,
respectively. Finally, three remarkable subconfigurations of V-lines
represented by fans of ovoids through a fixed ovoid are examined in some
detail.Comment: 6 pages, 7 figures; v2 - slightly polished, subsection on fans of
ovoids and three figures adde
An extreme, blueshifted iron line profile in the Narrow Line Seyfert 1 PG 1402+261; an edge-on accretion disk or highly ionized absorption?
We report on a short XMM-Newton observation of the radio-quiet Narrow Line
Seyfert 1 PG 1402+261. The EPIC X-ray spectrum of PG 1402+261 shows a strong
excess of counts between 6-9 keV in the rest frame. This feature can be modeled
by an unusually strong (equivalent width 2 keV) and very broad (FWHM velocity
of 110000 km/s) iron K-shell emission line. The line centroid energy at 7.3 keV
appears blue-shifted with respect to the iron Kalpha emission band between
6.4-6.97 keV, while the blue-wing of the line extends to 9 keV in the quasar
rest frame. The line profile can be fitted by reflection from the inner
accretion disk, but an inclination angle of >60 deg is required to model the
extreme blue-wing of the line. Furthermore the extreme strength of the line
requires a geometry whereby the hard X-ray emission from PG 1402+261 above 2
keV is dominated by the pure-reflection component from the disk, while little
or none of the direct hard power-law is observed. Alternatively the spectrum
above 2 keV may instead be explained by an ionized absorber, if the column
density is sufficiently high (N_H > 3 x 10^23 cm^-2) and if the matter is
ionized enough to produce a deep (tau~1) iron K-shell absorption edge at 9 keV.
This absorber could originate in a large column density, high velocity outflow,
perhaps similar to those which appear to be observed in several other high
accretion rate AGN. Further observations, especially at higher spectral
resolution, are required to distinguish between the accretion disk reflection
or outflow scenarios.Comment: Accepted for publication in ApJ (18 pages, 5 figures, 1 table
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