8,128 research outputs found
Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences
Let d and k be integers with 1 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K.
We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d > =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even
A generalization of Voronoi's reduction theory and its application
We consider Voronoi's reduction theory of positive definite quadratic forms
which is based on Delone subdivision. We extend it to forms and Delone
subdivisions having a prescribed symmetry group. Even more general, the theory
is developed for forms which are restricted to a linear subspace in the space
of quadratic forms. We apply the new theory to complete the classification of
totally real thin algebraic number fields which was recently initiated by
Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known
sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke
Math.
The logic of causally closed spacetime subsets
The causal structure of space-time offers a natural notion of an opposite or
orthogonal in the logical sense, where the opposite of a set is formed by all
points non time-like related with it. We show that for a general space-time the
algebra of subsets that arises from this negation operation is a complete
orthomodular lattice, and thus has several of the properties characterizing the
algebra physical propositions in quantum mechanics. We think this fact could be
used to investigate causal structure in an algebraic context. As a first step
in this direction we show that the causal lattice is in addition atomic, find
its atoms, and give necesary and sufficient conditions for ireducibility.Comment: 17 pages, 8 figure
- …