3,088 research outputs found
Maximum number of -rational points on nonsingular threefolds in
We determine the maximum number of -rational points that a
nonsingular threefold of degree in a projective space of dimension
defined over may contain. This settles a conjecture by Homma and
Kim concerning the maximum number of points on a hypersurface in a projective
space of even dimension in this particular case.Comment: 8 page
Numerical evolution of axisymmetric, isolated systems in General Relativity
We describe in this article a new code for evolving axisymmetric isolated
systems in general relativity. Such systems are described by asymptotically
flat space-times which have the property that they admit a conformal extension.
We are working directly in the extended `conformal' manifold and solve
numerically Friedrich's conformal field equations, which state that Einstein's
equations hold in the physical space-time. Because of the compactness of the
conformal space-time the entire space-time can be calculated on a finite
numerical grid. We describe in detail the numerical scheme, especially the
treatment of the axisymmetry and the boundary.Comment: 10 pages, 8 figures, uses revtex4, replaced with revised versio
Matroid connectivity and singularities of configuration hypersurfaces
Consider a linear realization of a matroid over a field. One associates with
it a configuration polynomial and a symmetric bilinear form with linear
homogeneous coefficients. The corresponding configuration hypersurface and its
non-smooth locus support the respective first and second degeneracy scheme of
the bilinear form. We show that these schemes are reduced and describe the
effect of matroid connectivity: for (2-)connected matroids, the configuration
hypersurface is integral, and the second degeneracy scheme is reduced
Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the
second degeneracy scheme is integral. In the process, we describe the behavior
of configuration polynomials, forms and schemes with respect to various matroid
constructions.Comment: 64 pages, 4 figure
Self-force with (3+1) codes: a primer for numerical relativists
Prescriptions for numerical self-force calculations have traditionally been
designed for frequency-domain or (1+1) time-domain codes which employ a mode
decomposition to facilitate in carrying out a delicate regularization scheme.
This has prevented self-force analyses from benefiting from the powerful suite
of tools developed and used by numerical relativists for simulations of the
evolution of comparable-mass black hole binaries. In this work, we revisit a
previously-introduced (3+1) method for self-force calculations, and demonstrate
its viability by applying it to the test case of a scalar charge moving in a
circular orbit around a Schwarzschild black hole. Two (3+1) codes originally
developed for numerical relativity applications were independently employed,
and in each we were able to compute the two independent components of the
self-force and the energy flux correctly to within . We also demonstrate
consistency between -component of the self-force and the scalar energy flux.
Our results constitute the first successful calculation of a self-force in a
(3+1) framework, and thus open opportunities for the numerical relativity
community in self-force analyses and the perturbative modeling of
extreme-mass-ratio inspirals.Comment: 23 pages, 13 figure
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