267,862 research outputs found
Distance k-Sectors Exist
The bisector of two nonempty sets P and Q in a metric space is the set of all
points with equal distance to P and to Q. A distance k-sector of P and Q, where
k is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the
bisector of C_{i-1} and C_{i+1} for every i = 1, 2, ..., k-1, where C_0 = P and
C_k = Q. This notion, for the case where P and Q are points in Euclidean plane,
was introduced by Asano, Matousek, and Tokuyama, motivated by a question of
Murata in VLSI design. They established the existence and uniqueness of the
distance trisector in this special case. We prove the existence of a distance
k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in
Euclidean spaces of any (finite) dimension, or more generally, in proper
geodesic spaces (uniqueness remains open). The core of the proof is a new
notion of k-gradation for P and Q, whose existence (even in an arbitrary metric
space) is proved using the Knaster-Tarski fixed point theorem, by a method
introduced by Reem and Reich for a slightly different purpose.Comment: 10 pages, 5 figure
Vacuum Structure, Lorentz Symmetry and Superluminal Particles
If textbook Lorentz invariance is actually a property of the equations
describing a sector of the excitations of vacuum above some critical distance
scale, several sectors of matter with different critical speeds in vacuum can
coexist and an absolute rest frame (the vacuum rest frame) may exist without
contradicting the apparent Lorentz invariance felt by "ordinary" particles
(particles with critical speed in vacuum equal to , the speed of light).
The sectorial Lorentz symmetry may be only a low-energy limit, in the same way
as the relation (frequency) = (speed of sound) (wave
vector) holds for low-energy phonons in a crystal. We study the consequences of
such a scenario, using an ansatz inspired by the Bravais lattice as a model for
some vacuum properties. It then turns out that: a) the Greisen-Zatsepin-Kuzmin
cutoff on high-energy cosmic protons and nuclei does no longer apply; b)
high-momentum unstable particles have longer lifetimes than expected with exact
Lorentz invariance, and may even become stable at the highest observed cosmic
ray energies or slightly above. Some cosmological implications of superluminal
particles are also discussed.Comment: 18 pages, LaTe
High-Energy Nuclear Physics with Lorentz Symmetry Violation
If textbook Lorentz invariance is actually a property of the equations
describing a sector of the excitations of vacuum above some critical distance
scale, several sectors of matter with different critical speeds in vacuum can
coexist and an absolute rest frame (the vacuum rest frame) may exist without
contradicting the apparent Lorentz invariance felt by "ordinary" particles
(particles with critical speed in vacuum equal to , the speed of light).
Sectorial Lorentz invariance, reflected by the fact that all particles of a
given dynamical sector have the same critical speed in vacuum, will then be an
expression of a fundamental sectorial symmetry (e.g. preonic grand unification
or extended supersymmetry) protecting a parameter of the equations of motion.
Furthermore, the sectorial Lorentz symmetry may be only a low-energy limit, in
the same way as the relation (frequency) = (speed of sound)
(wave vector) holds for low-energy phonons in a crystal. In this context,
phenomena such as the absence of Greisen-Zatsepin-Kuzmin cutoff for protons and
nuclei and the stability of unstable particles (e.g. neutron, several
nuclei...) at very high energy are basic properties of a wide class of
noncausal models where local Lorentz invariance is broken introducing a
fundamental length. Observable phenomena are expected at very short wavelength
scales, even if Lorentz symmetry violation remains invisible to standard
low-energy tests. We present a detailed discussion of the implications of
Lorentz symmetry violation for very high-energy nuclear physics.Comment: Contributed Paper 435 to the EPS-HEP97 Conference, Jerusalem August
19 - 26, 1997 ; 16 pages, LaTe
On Vertex- and Empty-Ply Proximity Drawings
We initiate the study of the vertex-ply of straight-line drawings, as a
relaxation of the recently introduced ply number. Consider the disks centered
at each vertex with radius equal to half the length of the longest edge
incident to the vertex. The vertex-ply of a drawing is determined by the vertex
covered by the maximum number of disks. The main motivation for considering
this relaxation is to relate the concept of ply to proximity drawings. In fact,
if we interpret the set of disks as proximity regions, a drawing with
vertex-ply number 1 can be seen as a weak proximity drawing, which we call
empty-ply drawing. We show non-trivial relationships between the ply number and
the vertex-ply number. Then, we focus on empty-ply drawings, proving some
properties and studying what classes of graphs admit such drawings. Finally, we
prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Dark costs, missing data: shedding some light on services trade
A structural gravity model is used to estimate barriers to services trade across many sectors, countries
and time. Since the disaggregated output data needed to flexibly infer border barriers are often missing
for services, we derive a novel methodology for projecting output data. The empirical implementation
sheds light on the role of institutions, geography, size and digital infrastructure as determinants of
border barriers. We find that border barriers have generally fallen over time but there are differences
across sectors and countries. Notably, border effects for the smallest economies have remained stable,
giving rise to a divergent pattern across countries
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