9,638 research outputs found
Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant
We use Fisher Matrix analysis techniques to forecast the cosmological impact
of astrophysical tests of the stability of the fine-structure constant to be
carried out by the forthcoming ESPRESSO spectrograph at the VLT (due for
commissioning in late 2017), as well by the planned high-resolution
spectrograph (currently in Phase A) for the European Extremely Large Telescope.
Assuming a fiducial model without variations, we show that ESPRESSO
can improve current bounds on the E\"{o}tv\"{o}s parameter---which quantifies
Weak Equivalence Principle violations---by up to two orders of magnitude,
leading to stronger bounds than those expected from the ongoing tests with the
MICROSCOPE satellite, while constraints from the E-ELT should be competitive
with those of the proposed STEP satellite. Should an variation be
detected, these measurements will further constrain cosmological parameters,
being particularly sensitive to the dynamics of dark energy.Comment: Phys. Lett. B (in press
The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
One of the most widely used methods for eigenvalue computation is the
iteration with Wilkinson's shift: here the shift is the eigenvalue of the
bottom principal minor closest to the corner entry. It has been a
long-standing conjecture that the rate of convergence of the algorithm is
cubic. In contrast, we show that there exist matrices for which the rate of
convergence is strictly quadratic. More precisely, let be the matrix having only two nonzero entries and let
be the set of real, symmetric tridiagonal matrices with the same spectrum
as . There exists a neighborhood of which is
invariant under Wilkinson's shift strategy with the following properties. For
, the sequence of iterates exhibits either strictly
quadratic or strictly cubic convergence to zero of the entry . In
fact, quadratic convergence occurs exactly when . Let be
the union of such quadratically convergent sequences : the set has
Hausdorff dimension 1 and is a union of disjoint arcs meeting at
, where ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit
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