488 research outputs found
Geodesic-Einstein metrics and nonlinear stabilities
In this paper, we introduce notions of nonlinear stabilities for a relative
ample line bundle over a holomorphic fibration and define the notion of a
geodesic-Einstein metric on this line bundle, which generalize the classical
stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We
introduce a Donaldson type functional and show that this functional attains its
absolute minimum at geodesic-Einstein metrics, and we also discuss the
relations between the existence of geodesic-Einstein metrics and the nonlinear
stabilities of the line bundle. As an application, we will prove that a
holomorphic vector bundle admits a Finsler-Einstein metric if and only if it
admits a Hermitian-Einstein metric, which answers a problem posed by S.
Kobayashi.Comment: 21 pages, the final version, to appear in Transactions of the
American Mathematical Societ
Geometry of logarithmic forms and deformations of complex structures
We present a new method to solve certain -equations for
logarithmic differential forms by using harmonic integral theory for currents
on Kahler manifolds. The result can be considered as a -lemma
for logarithmic forms. As applications, we generalize the result of Deligne
about closedness of logarithmic forms, give geometric and simpler proofs of
Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral
sequences at -level, as well as certain injectivity theorem on compact
Kahler manifolds.
Furthermore, for a family of logarithmic deformations of complex structures
on Kahler manifolds, we construct the extension for any logarithmic
-form on the central fiber and thus deduce the local stability of log
Calabi-Yau structure by extending an iteration method to the logarithmic forms.
Finally we prove the unobstructedness of the deformations of a log Calabi-Yau
pair and a pair on a Calabi-Yau manifold by differential geometric method.Comment: Several typos have been fixed. Final version to appear in Journal of
Algebraic Geometr
Resource Destroying Maps
Resource theory is a widely-applicable framework for analyzing the physical
resources required for given tasks, such as computation, communication, and
energy extraction. In this paper, we propose a general scheme for analyzing
resource theories based on resource destroying maps, which leave resource-free
states unchanged but erase the resource stored in all other states. We
introduce a group of general conditions that determine whether a quantum
operation exhibits typical resource-free properties in relation to a given
resource destroying map. Our theory reveals fundamental connections among basic
elements of resource theories, in particular, free states, free operations, and
resource measures. In particular, we define a class of simple resource measures
that can be calculated without optimization, and that are monotone
nonincreasing under operations that commute with the resource destroying map.
We apply our theory to the resources of coherence and quantum correlations
(e.g., discord), two prominent features of nonclassicality.Comment: 12 pages including Supplemental Material, published versio
A Bayesian approach to parameter estimation for kernel density estimation via transformations
In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibit a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel choice. In the current literature, there have been some developments in the area of estimating densities based on transformed data, but bandwidth selection depends on pre-determined transformation parameters. Moreover, in the bivariate situation, each dimension is considered separately and the correlation between the two dimensions is largely ignored. We extend the Bayesian sampling algorithm proposed by Zhang, King and Hyndman (2006) and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is well captured through the bivariate density estimator based on transformed data.Bandwidth parameter; kernel density estimator; Markov chain Monte Carlo; Metropolis-Hastings algorithm; power transformation; transformation parameter.
Cooling and work extraction under memory-assisted Markovian thermal processes
We investigate the limits on cooling and work extraction via Markovian
thermal processes assisted by a finite-dimensional memory. Here the memory is a
-dimensional quantum system with trivial Hamiltonian and initially in a
maximally mixed state. For cooling a qubit system, we consider two paradigms:
cooling under coherent control and cooling under incoherent control. For both
paradigms, we derive the optimal ground-state populations under the set of
general thermal processes (TP) and the set of Markovian thermal processes
(MTP), and we further propose memory-assisted protocols, which bridge the gap
between the performances of TP and MTP. For the task of work extraction, we
prove that when the target system is a qubit in the excited state the minimum
extraction error achieved by TP can be approximated by Markovian thermal
processes assisted by a large enough memory. Our results can bridge the
performances of TP and MTP in thermodynamic tasks including cooling and work
extraction.Comment: Published versio
Necessary and sufficient conditions for local creation of quantum correlation
Quantum correlation can be created by a local operation from some initially
classical states. We prove that the necessary and sufficient condition for a
local trace-preserving channel to create quantum correlation is that it is not
a commutativity-preserving channel. This condition is valid for arbitrary
finite dimension systems. We also derive the explicit form of
commutativity-preserving channels. For a qubit, a commutativity-preserving
channel is either a completely decohering channel or a mixing channel. For a
three-dimensional system (qutrit), a commutativity-preserving channel is either
a completely decohering channel or an isotropic channel.Comment: Theorem 2 has been modifie
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