241 research outputs found
On upper bounds on the smallest size of a saturating set in a projective plane
In a projective plane (not necessarily Desarguesian) of order
a point subset is saturating (or dense) if any point of is collinear with two points in. Using probabilistic methods, the
following upper bound on the smallest size of a saturating set in
is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln
(q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any
constant a random point set of size in with is a saturating
set with probability greater than Our probabilistic
approach is also applied to multiple saturating sets. A point set is -saturating if for every point of the number of secants of through is at least , counted with
multiplicity. The multiplicity of a secant is computed as
The following upper bound on the smallest
size of a -saturating set in is proved:
\begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim
2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*}
By using inductive constructions, upper bounds on the smallest size of a
saturating set (as well as on a -saturating set) in the projective
space are obtained.
All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and
some references are adde
Rational points on K3 surfaces and derived equivalence
We study K3 surfaces over non-closed fields and relate the notion of derived
equivalence to arithmetic problems.Comment: 30 page
On Saturating Sets in Small Projective Geometries
AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)
The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
We review the bootstrap method for constructing six- and seven-particle
amplitudes in planar super Yang-Mills theory, by exploiting
their analytic structure. We focus on two recently discovered properties which
greatly simplify this construction at symbol and function level, respectively:
the extended Steinmann relations, or equivalently cluster adjacency, and the
coaction principle. We then demonstrate their power in determining the
six-particle amplitude through six and seven loops in the NMHV and MHV sectors
respectively, as well as the symbol of the NMHV seven-particle amplitude to
four loops.Comment: 36 pages, 4 figures, 5 tables, 1 ancillary file. Contribution to the
proceedings of the Corfu Summer Institute 2019 "School and Workshops on
Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25
September 2019, Corfu, Greec
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