1,736 research outputs found
Bounding the size of an almost-equidistant set in Euclidean space
A set of points in d-dimensional Euclidean space is almost equidistant if among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in Rd has cardinality O(d4/3)
Spacelike distance from discrete causal order
Any discrete approach to quantum gravity must provide some prescription as to
how to deduce continuum properties from the discrete substructure. In the
causal set approach it is straightforward to deduce timelike distances, but
surprisingly difficult to extract spacelike distances, because of the unique
combination of discreteness with local Lorentz invariance in that approach. We
propose a number of methods to overcome this difficulty, one of which
reproduces the spatial distance between two points in a finite region of
Minkowski space. We provide numerical evidence that this definition can be used
to define a `spatial nearest neighbor' relation on a causal set, and conjecture
that this can be exploited to define the length of `continuous curves' in
causal sets which are approximated by curved spacetime. This provides evidence
in support of the ``Hauptvermutung'' of causal sets.Comment: 32 pages, 16 figures, revtex4; journal versio
Modulation Diversity in Fading Channels with Quantized Receiver
In this paper, we address the design of codes which achieve modulation
diversity in block fading single-input single-output (SISO) channels with
signal quantization at receiver and low-complexity decoding. With an
unquantized receiver, coding based on algebraic rotations is known to achieve
modulation coding diversity. On the other hand, with a quantized receiver,
algebraic rotations may not guarantee diversity. Through analysis, we propose
specific rotations which result in the codewords having equidistant
component-wise projections. We show that the proposed coding scheme achieves
maximum modulation diversity with a low-complexity minimum distance decoder and
perfect channel knowledge. Relaxing the perfect channel knowledge assumption we
propose a novel training/estimation and receiver control technique to estimate
the channel. We show that our coding/training/estimation scheme and minimum
distance decoding achieve an error probability performance similar to that
achieved with perfect channel knowledge
A CENTER OF A POLYTOPE: AN EXPOSITORY REVIEW AND A PARALLEL IMPLEMENTATION
ABSTRACT. The solution space of the rectangular linear system Az b, subject to x> 0, is called a polytope. An attempt is made to provide a deeper geometric insight, with numerical examples, into the condensed paper by Lord, et al. [1], that presents an algorithm to compute a center of a polytope. The algorithm is readily adopted for either sequential or parallel computer implementation. The computed center provides an initial feasible solution (interior point) of a linear programming problem. KEY WORDS AND PHRASES. Center of a polytope, consistency check, Euclidean distance, initial feasible solution, linear programming, Moore-Penrose inverse, nonnegative solution
Quantum gravity without vacuum dispersion
A generic prediction of quantum gravity is the vacuum dispersion of light,
and hence that a photon's speed depends on its energy. We present further
numerical evidence for a scale dependent speed of light in the causal dynamical
triangulation (CDT) approach to quantum gravity. We show that the observed
scale dependent speed of light in CDT can be accounted for by a scale dependent
transformation of geodesic distance, whose specific functional form implies a
discrete equidistant area spectrum. We make two non-trivial tests of the
proposed scale transformation: a comparison with the leading order quantum
correction to the gravitational potential and a comparison with the generalised
uncertainty principle. In both cases, we obtain the same functional form.
However, contrary to the widespread prediction of vacuum dispersion in quantum
gravity, numerous experiments have now definitively ruled out linear vacuum
dispersion beyond Planckian energy scales, and have now even constrained
quadratic dispersion. Motivated by these experimental constraints we seek to
reconcile quantum gravity with the absence of vacuum dispersion. We point out
that given a scale dependent geodesic distance, a scale dependent time interval
becomes essential to maintaining an invariant speed of light. We show how a
particular scale dependent time interval allows a photon's speed to remain
independent of its energy.Comment: Version published in International Journal of Modern Physics D. 13
pages, 3 figure
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