31,906 research outputs found
Counting Steiner triple systems with classical parameters and prescribed rank
By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of
a Steiner triple system on points is at least , and equality
holds only for the classical point-line design in the projective geometry
. It follows from results of Assmus \cite{A} that, given any integer
with , there is a code containing
representatives of all isomorphism classes of STS with 2-rank at most
. Using a mixture of coding theoretic, geometric, design
theoretic and combinatorial arguments, we prove a general formula for the
number of distinct STS with 2-rank at most contained
in this code. This generalizes the only previously known cases, , proved
by Tonchev \cite{T01} in 2001, , proved by V. Zinoviev and D. Zinoviev
\cite{ZZ12} in 2012, and (V. Zinoviev and D. Zinoviev \cite{ZZ13},
\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and
simplifying the proofs. This enumeration result allows us to prove lower and
upper bounds for the number of isomorphism classes of STS with 2-rank
exactly (or at most) . Finally, using our recent systematic
study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain
analogous results for the ternary case, that is, for STS with 3-rank at
most (or exactly) . We note that this work provides the first
two infinite families of 2-designs for which one has non-trivial lower and
upper bounds for the number of non-isomorphic examples with a prescribed
-rank in almost the entire range of possible ranks.Comment: 27 page
Notice of proposal to amend regulation S-X
Notice is hereby given that the Securities and Exchange Commission, pursuant to authority conferred upon it by the Securities Act of 1933, particularly Sections 6, 7, 8, 10 and 19 (a) thereof, the Securities Exchange Act of 1934, particularly Sections 12, 13, 15 (d) and 23 (a) thereof, and the Investment Company Act of 1940, particularly Sections 8, 30, 31 (c) and 38 (a) thereof, has under consideration a proposal to revise Articles 1, 2, 3, 4, 5 and 11 of Regulation S-X
Minimal triangulations of sphere bundles over the circle
For integers and or 1, let
denote the sphere product if and the
twisted bundle over if . The main results of
this paper are: (a) if (mod 2) then has a unique minimal triangulation using vertices, and
(b) if (mod 2) then has
minimal triangulations (not unique) using vertices. The second result
confirms a recent conjecture of Lutz. The first result provides the first known
infinite family of closed manifolds (other than spheres) for which the minimal
triangulation is unique. Actually, we show that while
has at most one -vertex triangulation (one if
(mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic -vertex triangulations of these -manifolds grows exponentially with
for either choice of . The result in (a), as well as the minimality
part in (b), is a consequence of the following result: (c) for ,
there is a unique -vertex simplicial complex which triangulates a
non-simply connected closed manifold of dimension . This amazing simplicial
complex was first constructed by K\"{u}hnel in 1986. Generalizing a 1987 result
of Brehm and K\"{u}hnel, we prove that (d) any triangulation of a non-simply
connected closed -manifold requires at least vertices. The result
(c) completely describes the case of equality in (d). The proofs rest on the
Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version
of Alexander duality.Comment: 15 pages, Revised, To appear in `Journal of Combinatorial Theory,
Ser. A
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