94 research outputs found
On Nyman, Beurling and Baez-Duarte's Hilbert space reformulation of the Riemann hypothesis
There has been a surge of interest of late in an old result of Nyman and
Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many
authors have contributed to this circle of ideas, culminating in a beautiful
refinement due to Baez-Duarte. The purpose of this little survey is to
dis-entangle the resulting web of complications, and reveal the essential
simplicity of the main results.Comment: 10 page
On -stellated and -stacked spheres
We introduce the class of -stellated (combinatorial) spheres
of dimension () and compare and contrast it with the
class () of -stacked homology -spheres.
We have , and for . However, for each there are
-stacked spheres which are not -stellated. The existence of -stellated
spheres which are not -stacked remains an open question.
We also consider the class (and ) of
simplicial complexes all whose vertex-links belong to
(respectively, ). Thus, for , while . Let
denote the class of -dimensional complexes all whose
vertex-links are -stacked balls. We show that for , there is a
natural bijection from onto which is the inverse to the boundary map .Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note:
substantial text overlap with arXiv:1102.085
Non-existence of 6-dimensional pseudomanifolds with complementarity
In a previous paper the second author showed that if is a pseudomanifold
with complementarity other than the 6-vertex real projective plane and the
9-vertex complex projective plane, then must have dimension , and -
in case of equality - must have exactly 12 vertices. In this paper we prove
that such a 6-dimensional pseudomanifold does not exist. On the way to proving
our main result we also prove that all combinatorial triangulations of the
4-sphere with at most 10 vertices are combinatorial 4-spheres.Comment: 11 pages. To appear in Advances in Geometr
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