73,023 research outputs found
Riesz external field problems on the hypersphere and optimal point separation
We consider the minimal energy problem on the unit sphere in
the Euclidean space in the presence of an external field
, where the energy arises from the Riesz potential (where is the
Euclidean distance and is the Riesz parameter) or the logarithmic potential
. Characterization theorems of Frostman-type for the associated
extremal measure, previously obtained by the last two authors, are extended to
the range The proof uses a maximum principle for measures
supported on . When is the Riesz -potential of a signed
measure and , our results lead to explicit point-separation
estimates for -Fekete points, which are -point configurations
minimizing the Riesz -energy on with external field . In
the hyper-singular case , the short-range pair-interaction enforces
well-separation even in the presence of more general external fields. As a
further application, we determine the extremal and signed equilibria when the
external field is due to a negative point charge outside a positively charged
isolated sphere. Moreover, we provide a rigorous analysis of the three point
external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure
On the singularity type of full mass currents in big cohomology classes
Let be a compact K\"ahler manifold and be a big cohomology
class. We prove several results about the singularity type of full mass
currents, answering a number of open questions in the field. First, we show
that the Lelong numbers and multiplier ideal sheaves of
-plurisubharmonic functions with full mass are the same as those of the
current with minimal singularities. Second, given another big and nef class
, we show the inclusion Third, we characterize big classes whose full
mass currents are "additive". Our techniques make use of a characterization of
full mass currents in terms of the envelope of their singularity type. As an
essential ingredient we also develop the theory of weak geodesics in big
cohomology classes. Numerous applications of our results to complex geometry
are also given.Comment: v2. Theorem 1.1 updated to include statement about multiplier ideal
sheaves. Several typos fixed. v3. we make our arguments independent of the
regularity results of Berman-Demaill
Stabilizing the Dilaton in Superstring Cosmology
We address the important issue of stabilizing the dilaton in the context of
superstring cosmology. Scalar potentials which arise out of gaugino condensates
in string models are generally exponential in nature. In a cosmological setting
this allows for the existence of quasi scaling solutions, in which the energy
density of the scalar field can, for a period, become a fixed fraction of the
background density, due to the friction of the background expansion. Eventually
the field can be trapped in the minimum of its potential as it leaves the
scaling regime. We investigate this possibility in various gaugino condensation
models and show that stable solutions for the dilaton are far more common than
one would have naively thought.Comment: 13 pages, LaTex, uses psfig.sty with 3 figure
Cosmology
In these lectures we first concentrate on the cosmological problems which,
hopefully, have to do with the new physics to be probed at the LHC: the nature
and origin of dark matter and generation of matter-antimatter asymmetry. We
give several examples showing the LHC cosmological potential. These are WIMPs
as cold dark matter, gravitinos as warm dark matter, and electroweak
baryogenesis as a mechanism for generating matter-antimatter asymmetry. In the
remaining part of the lectures we discuss the cosmological perturbations as a
tool for studying the epoch preceeding the conventional hot stage of the
cosmological evolution.Comment: 47 pages, set of lectures given at the 2011 European School of
High-Energy Physics, Cheile Gradistei, Romania, 7-20 Sep 2011, edited by C.
Grojean, M. Mulder
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