3,617 research outputs found
Einstein equations in the null quasi-spherical gauge III: numerical algorithms
We describe numerical techniques used in the construction of our 4th order
evolution for the full Einstein equations, and assess the accuracy of
representative solutions. The code is based on a null gauge with a
quasi-spherical radial coordinate, and simulates the interaction of a single
black hole with gravitational radiation. Techniques used include spherical
harmonic representations, convolution spline interpolation and filtering, and
an RK4 "method of lines" evolution. For sample initial data of "intermediate"
size (gravitational field with 19% of the black hole mass), the code is
accurate to 1 part in 10^5, until null time z=55 when the coordinate condition
breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed
Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction
A method for concatenating quantum error-correcting codes is presented. The
method is applicable to a wide class of quantum error-correcting codes known as
Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate
in the Shannon theoretic sense and that are decodable in polynomial time are
presented. The rate is the highest among those known to be achievable by CSS
codes. Moreover, the best known lower bound on the greatest minimum distance of
codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of
the AE of the journal, the present version has become a combination of
(thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195.
Problem formulations of polynomial complexity are strictly followed. An
erroneous instance of a lower bound on minimum distance was remove
Geometric representation of interval exchange maps over algebraic number fields
We consider the restriction of interval exchange transformations to algebraic
number fields, which leads to maps on lattices. We characterize
renormalizability arithmetically, and study its relationships with a
geometrical quantity that we call the drift vector. We exhibit some examples of
renormalizable interval exchange maps with zero and non-zero drift vector, and
carry out some investigations of their properties. In particular, we look for
evidence of the finite decomposition property: each lattice is the union of
finitely many orbits.Comment: 34 pages, 8 postscript figure
Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces
We investigate the asymptotic density of error-correcting codes with good
distance properties and prescribed linearity degree, including sublinear and
nonlinear codes. We focus on the general setting of finite
translation-invariant metric spaces, and then specialize our results to the
Hamming metric, to the rank metric, and to the sum-rank metric. Our results
show that the asymptotic density of codes heavily depends on the imposed
linearity degree and the chosen metric
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