22,518 research outputs found
Generalized offsetting of planar structures using skeletons
We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams
Discrete space-time geometry and skeleton conception of particle dynamics
It is shown that properties of a discrete space-time geometry distinguish
from properties of the Riemannian space-time geometry. The discrete geometry is
a physical geometry, which is described completely by the world function. The
discrete geometry is nonaxiomatizable and multivariant. The equivalence
relation is intransitive in the discrete geometry. The particles are described
by world chains (broken lines with finite length of links), because in the
discrete space-time geometry there are no infinitesimal lengths. Motion of
particles is stochastic, and statistical description of them leads to the
Schr\"{o}dinger equation, if the elementary length of the discrete geometry
depends on the quantum constant in a proper way.Comment: 22 pages, 0 figure
Addicted to Bryozoans
Most people are familiar with the shells of large molluscs: clams, scallops, mussels, oysters, snails, päua. It may seem surprising that almost every group of marine creatures produces at least somekind of skeletal structure – biomineralisation is everywhere in the sea. Vertebrates like us make our skeletons from what we eat, but invertebrates like urchins, barnacles, worms and corals maketheir skeletons straight out of seawater. Despite that common origin, there is huge variation in their shapes and sizes and textures. When they die, shells break up into fragments, littering the seafloor with evidence of the past. I decided to come to New Zealand to work with Prof Cam Nelson, a world authority on temperate-latitude shells and the sediments that come from them, to try to understand more about calcareous algae. Why would a plant that needs the sun make itself a coat of armour?
The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case
We consider the spectral statistics of large random band matrices on
mesoscopic energy scales. We show that the two-point correlation function of
the local eigenvalue density exhibits a universal power law behaviour that
differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in
the physics literature by Altshuler and Shklovskii [4]; it describes the
correlations of the eigenvalue density in general metallic samples with weak
disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas
for band matrices. In two dimensions, where the leading term vanishes owing to
an algebraic cancellation, we identify the first non-vanishing term and show
that it differs substantially from the prediction of Kravtsov and Lerner [33]
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