75,630 research outputs found
On the spectrum of the magnetohydrodynamic mean-field alpha^2-dynamo operator
The existence of magnetohydrodynamic mean-field alpha^2-dynamos with
spherically symmetric, isotropic helical turbulence function alpha is related
to a non-self-adjoint spectral problem for a coupled system of two singular
second order ordinary differential equations. We establish global estimates for
the eigenvalues of this system in terms of the turbulence function alpha and
its derivative alpha'. They allow us to formulate an anti-dynamo theorem and a
non-oscillation theorem. The conditions of these theorems, which again involve
alpha and alpha', must be violated in order to reach supercritical or
oscillatory regimes.Comment: 35 pages, 4 figures, to be published in SIAM J. Math. Anal
Yan’s oscillation theorem revisited
AbstractYan’s contribution [J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98 (1986) 276–282] was an important breakthrough in the development of the Theory of Oscillation. This frequently cited paper has stimulated extensive investigations in the field. During the last decade, an integral oscillation technique has been developed to such an extent as to allow us to revisit Yan’s fundamental oscillation theorem and remove one of the conditions, leaving the other assumptions and the conclusion intact, thus enhancing this keystone result
Supermassive black holes or boson stars? Hair counting with gravitational wave detectors
The evidence for supermassive Kerr black holes in galactic centers is strong
and growing, but only the detection of gravitational waves will convincingly
rule out other possibilities to explain the observations. The Kerr spacetime is
completely specified by the first two multipole moments: mass and angular
momentum. This is usually referred to as the ``no-hair theorem'', but it is
really a ``two-hair'' theorem. If general relativity is the correct theory of
gravity, the most plausible alternative to a supermassive Kerr black hole is a
rotating boson star. Numerical calculations indicate that the spacetime of
rotating boson stars is determined by the first three multipole moments
(``three-hair theorem''). LISA could accurately measure the oscillation
frequencies of these supermassive objects. We propose to use these measurements
to ``count their hair'', unambiguously determining their nature and properties.Comment: 8 pages. This essay received an honorable mention in the Gravity
Research Foundation Essay Competition, 200
REMARK ON MEDIAN OSCILLATION DECOMPOSITION AND DYADIC POINT WISE DOMINATION
In this note, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures. In the light of the analogy between median and mean oscillation, our extension can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO. As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John-Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of zero complexity. Furthermore, by using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity.Peer reviewe
Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences
This paper deals with the problem of quantifying the approximation a
probability measure by means of an empirical (in a wide sense) random
probability measure, depending on the first n terms of a sequence of random
elements. In Section 2, one studies the range of oscillation near zero of the
Wasserstein distance
^{(p)}_{\pms} between \pfrak_0 and
\hat{\pfrak}_n, assuming that the \xitil_i's are i.i.d. with \pfrak_0 as
common law. Theorem 2.3 deals with the case in which \pfrak_0 is fixed as a
generic element of the space of all probability measures on (\rd,
\mathscr{B}(\rd)) and \hat{\pfrak}_n coincides with the empirical measure.
In Theorem 2.4 (Theorem 2.5, respectively) \pfrak_0 is a d-dimensional Gaussian
distribution (an element of a distinguished type of statistical exponential
family, respectively) and \hat{\pfrak}_n is another -dimensional Gaussian
distribution with estimated mean and covariance matrix (another element of the
same family with an estimated parameter, respectively). These new results
improve on allied recent works (see, e.g., [31]) since they also provide
uniform bounds with respect to , meaning that the finiteness of the p-moment
of the random variable \sup_{n \geq 1} b_n
^{(p)}_{\pms}(\pfrak_0,
\hat{\pfrak}_n) is proved for some suitable diverging sequence b_n of positive
numbers. In Section 3, under the hypothesis that the \xitil_i's are
exchangeable, one studies the range of the random oscillation near zero of the
Wasserstein distance between the conditional distribution--also called
posterior--of the directing measure of the sequence, given \xitil_1, \dots,
\xitil_n, and the point mass at \hat{\pfrak}_n. In a similar vein, a bound
for the approximation of predictive distributions is given. Finally, Theorems
from 3.3 to 3.5 reconsider Theorems from 2.3 to 2.5, respectively, according to
a Bayesian perspective
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