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 $k$-stellated and $k$-stacked spheres

We introduce the class $\Sigma_k(d)$ of $k$-stellated (combinatorial) spheres
of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the
class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres.
We have $\Sigma_1(d) = {\cal S}_1(d)$, and $\Sigma_k(d) \subseteq {\cal
S}_k(d)$ for $d \geq 2k - 1$. However, for each $k \geq 2$ there are
$k$-stacked spheres which are not $k$-stellated. The existence of $k$-stellated
spheres which are not $k$-stacked remains an open question.
We also consider the class ${\cal W}_k(d)$ (and ${\cal K}_k(d)$) of
simplicial complexes all whose vertex-links belong to $\Sigma_k(d - 1)$
(respectively, ${\cal S}_k(d - 1)$). Thus, ${\cal W}_k(d) \subseteq {\cal
K}_k(d)$ for $d \geq 2k$, while ${\cal W}_1(d) = {\cal K}_1(d)$. Let
$\bar{{\cal K}}_k(d)$ denote the class of $d$-dimensional complexes all whose
vertex-links are $k$-stacked balls. We show that for $d\geq 2k + 2$, there is a
natural bijection $M \mapsto \bar{M}$ from ${\cal K}_k(d)$ onto $\bar{{\cal
K}}_k(d + 1)$ which is the inverse to the boundary map $\partial \colon
\bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d)$.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 $M$ is a pseudomanifold
with complementarity other than the 6-vertex real projective plane and the
9-vertex complex projective plane, then $M$ must have dimension $\geq 6$, and -
in case of equality - $M$ 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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