37,656 research outputs found
On densities of lattice arrangements intersecting every i-dimensional affine subspace
In 1978, Makai Jr. established a remarkable connection between the
volume-product of a convex body, its maximal lattice packing density and the
minimal density of a lattice arrangement of its polar body intersecting every
affine hyperplane. Consequently, he formulated a conjecture that can be seen as
a dual analog of Minkowski's fundamental theorem, and which is strongly linked
to the well-known Mahler-conjecture.
Based on the covering minima of Kannan & Lov\'asz and a problem posed by
Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and
investigate densities of lattice arrangements of convex bodies intersecting
every i-dimensional affine subspace. Then it becomes natural also to formulate
and study a dual analog to Minkowski's second fundamental theorem. As our main
results, we derive meaningful asymptotic lower bounds for the densities of such
arrangements, and furthermore, we solve the problems exactly for the special,
yet important, class of unconditional convex bodies.Comment: 19 page
Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a
bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound
the distance between an arbitrary parameter and the spectrum
in terms of the boundary -norm of a normalized trial solution of the
Helmholtz equation . We also bound the -norm of the
error of this trial solution from an eigenfunction. Both of these results are
sharp up to constants, hold for all greater than a small constant, and
improve upon the best-known bounds of Moler--Payne by a factor of the
wavenumber . One application is to the solution of eigenvalue
problems at high frequency, via, for example, the method of particular
solutions. In the case of planar, strictly star-shaped domains we give an
inclusion bound where the constant is also sharp. We give explicit constants in
the theorems, and show a numerical example where an eigenvalue around the
2500th is computed to 14 digits of relative accuracy. The proof makes use of a
new quasi-orthogonality property of the boundary normal derivatives of the
eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure
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