27,669 research outputs found
On a property of 2-dimensional integral Euclidean lattices
Let be any integral lattice in the 2-dimensional Euclidean space.
Generalizing the earlier works of Hiroshi Maehara and others, we prove that for
every integer , there is a circle in the plane that
passes through exactly points of .Comment: 9 page
Dr TIM: Ray-tracer TIM, with additional specialist scientific capabilities
We describe several extensions to TIM, a raytracing program for ray-optics
research. These include relativistic raytracing; simulation of the external
appearance of Eaton lenses, Luneburg lenses and generalized focusing
gradient-index (GGRIN) lenses, which are types of perfect imaging devices;
raytracing through interfaces between spaces with different optical metrics;
and refraction with generalised confocal lenslet arrays, which are particularly
versatile METATOYs.Comment: 12 pages, 16 figure
Flows of constant mean curvature tori in the 3-sphere: The equivariant case
We present a deformation for constant mean curvature tori in the 3-sphere. We
show that the moduli space of equivariant constant mean curvature tori in the
3-sphere is connected, and we classify the minimal, the embedded, and the
Alexandrov embedded tori therein. We conclude with an instability result.Comment: v2: 33 pages, 9 figures. Instability result adde
Hyperuniformity of Quasicrystals
Hyperuniform systems, which include crystals, quasicrystals and special
disordered systems, have attracted considerable recent attention, but rigorous
analyses of the hyperuniformity of quasicrystals have been lacking because the
support of the spectral intensity is dense and discontinuous. We employ the
integrated spectral intensity, , to quantitatively characterize the
hyperuniformity of quasicrystalline point sets generated by projection methods.
The scaling of as tends to zero is computed for one-dimensional
quasicrystals and shown to be consistent with independent calculations of the
variance, , in the number of points contained in an interval of
length . We find that one-dimensional quasicrystals produced by projection
from a two-dimensional lattice onto a line of slope fall into distinct
classes determined by the width of the projection window. For a countable dense
set of widths, ; for all others, . This
distinction suggests that measures of hyperuniformity define new classes of
quasicrystals in higher dimensions as well.Comment: 12 pages, 14 figure
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