71,066 research outputs found
Complexity in complex analysis
We show that the classical kernel and domain functions associated to an
n-connected domain in the plane are all given by rational combinations of three
or fewer holomorphic functions of one complex variable. We characterize those
domains for which the classical functions are given by rational combinations of
only two or fewer functions of one complex variable. Such domains turn out to
have the property that their classical domain functions all extend to be
meromorphic functions on a compact Riemann surface, and this condition will be
shown to be equivalent to the condition that an Ahlfors map and its derivative
are algebraically dependent. We also show how many of these results can be
generalized to finite Riemann surfaces.Comment: 30 pages, to appear in Advances in Mat
Coronal Holes
Coronal holes are the darkest and least active regions of the Sun, as
observed both on the solar disk and above the solar limb. Coronal holes are
associated with rapidly expanding open magnetic fields and the acceleration of
the high-speed solar wind. This paper reviews measurements of the plasma
properties in coronal holes and how these measurements are used to reveal
details about the physical processes that heat the solar corona and accelerate
the solar wind. It is still unknown to what extent the solar wind is fed by
flux tubes that remain open (and are energized by footpoint-driven wave-like
fluctuations), and to what extent much of the mass and energy is input
intermittently from closed loops into the open-field regions. Evidence for both
paradigms is summarized in this paper. Special emphasis is also given to
spectroscopic and coronagraphic measurements that allow the highly dynamic
non-equilibrium evolution of the plasma to be followed as the asymptotic
conditions in interplanetary space are established in the extended corona. For
example, the importance of kinetic plasma physics and turbulence in coronal
holes has been affirmed by surprising measurements from UVCS that heavy ions
are heated to hundreds of times the temperatures of protons and electrons.
These observations point to specific kinds of collisionless Alfven wave damping
(i.e., ion cyclotron resonance), but complete models do not yet exist. Despite
our incomplete knowledge of the complex multi-scale plasma physics, however,
much progress has been made toward the goal of understanding the mechanisms
responsible for producing the observed properties of coronal holes.Comment: 61 pages, 12 figures. Accepted by the online journal "Living Reviews
in Solar Physics." The abstract has been abbreviated slightly, and some
figures are degraded in quality from the official version, which will be
available at http://solarphysics.livingreviews.org
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