122 research outputs found
Equivalent definitions for (degree one) Cameron-Liebler classes of generators in finite classical polar spaces
In this article, we study degree one Cameron-Liebler sets of generators in
all finite classical polar spaces, which is a particular type of a
Cameron-Liebler set of generators in this polar space, [9]. These degree one
Cameron-Liebler sets are defined similar to the Boolean degree one functions,
[15]. We summarize the equivalent definitions for these sets and give a
classification result for the degree one Cameron-Liebler sets in the polar
spaces W(5,q) and Q(6,q)
The frequency map for billiards inside ellipsoids
The billiard motion inside an ellipsoid Q \subset \Rset^{n+1} is completely
integrable. Its phase space is a symplectic manifold of dimension , which
is mostly foliated with Liouville tori of dimension . The motion on each
Liouville torus becomes just a parallel translation with some frequency
that varies with the torus. Besides, any billiard trajectory inside
is tangent to caustics , so the
caustic parameters are integrals of the
billiard map. The frequency map is a key tool to
understand the structure of periodic billiard trajectories. In principle, it is
well-defined only for nonsingular values of the caustic parameters. We present
four conjectures, fully supported by numerical experiments. The last one gives
rise to some lower bounds on the periods. These bounds only depend on the type
of the caustics. We describe the geometric meaning, domain, and range of
. The map can be continuously extended to singular values of
the caustic parameters, although it becomes "exponentially sharp" at some of
them. Finally, we study triaxial ellipsoids of \Rset^3. We compute
numerically the bifurcation curves in the parameter space on which the
Liouville tori with a fixed frequency disappear. We determine which ellipsoids
have more periodic trajectories. We check that the previous lower bounds on the
periods are optimal, by displaying periodic trajectories with periods four,
five, and six whose caustics have the right types. We also give some new
insights for ellipses of \Rset^2.Comment: 50 pages, 13 figure
Tight sets in finite classical polar spaces
We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q)
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