16,966 research outputs found
Effectiveness of Demailly's strong openness conjecture and related problems
In this article, stimulated by the effectiveness in Berndtsson's solution of
the openness conjecture and continuing our solution of Demailly's strong
openness conjecture, we discuss conditions to guarantee the effectiveness of
the conjecture and establish such an effectiveness result. We explicitly point
out a lower semicontinuity property of plurisubharmonic functions with a
multiplier, which is implicitly contained in our paper arXiv:1401.7158. We also
obtain optimal effectiveness of the conjectures of Demailly-Koll\'{a}r and
Jonsson-Mustat\u{a} respectively.Comment: 31 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:1401.715
Characterization of multiplier ideal sheaves with weights of Lelong number one
In this article, we characterize plurisubharmonic functions of Lelong number
one at the origin, such that the germ of the associated multiplier ideal sheaf
is nontrivial: in arbitrary complex dimension, their singularity must be the
sum of a germ of smooth divisor and of a plurisubharmonic function with zero
Lelong number. We also present a new proof of the related well known
integrability criterion due to Skoda.Comment: 14 pages, 0 figures. Revised versio
Fast model-fitting of Bayesian variable selection regression using the iterative complex factorization algorithm
Bayesian variable selection regression (BVSR) is able to jointly analyze
genome-wide genetic datasets, but the slow computation via Markov chain Monte
Carlo (MCMC) hampered its wide-spread usage. Here we present a novel iterative
method to solve a special class of linear systems, which can increase the speed
of the BVSR model-fitting tenfold. The iterative method hinges on the complex
factorization of the sum of two matrices and the solution path resides in the
complex domain (instead of the real domain). Compared to the Gauss-Seidel
method, the complex factorization converges almost instantaneously and its
error is several magnitude smaller than that of the Gauss-Seidel method. More
importantly, the error is always within the pre-specified precision while the
Gauss-Seidel method is not. For large problems with thousands of covariates,
the complex factorization is 10 -- 100 times faster than either the
Gauss-Seidel method or the direct method via the Cholesky decomposition. In
BVSR, one needs to repetitively solve large penalized regression systems whose
design matrices only change slightly between adjacent MCMC steps. This slight
change in design matrix enables the adaptation of the iterative complex
factorization method. The computational innovation will facilitate the
wide-spread use of BVSR in reanalyzing genome-wide association datasets.Comment: Accepted versio
Lelong numbers, complex singularity exponents, and Siu's semicontinuity theorem
In this note, we present a relationship between Lelong numbers and complex
singularity exponents. As an application, we obtain a new proof of Siu's
semicontinuity theorem for Lelong numbers.Comment: 5 pages, revised versio
Frequency Detection and Change Point Estimation for Time Series of Complex Oscillation
We consider detecting the evolutionary oscillatory pattern of a signal when
it is contaminated by non-stationary noises with complexly time-varying data
generating mechanism. A high-dimensional dense progressive periodogram test is
proposed to accurately detect all oscillatory frequencies. A further
phase-adjusted local change point detection algorithm is applied in the
frequency domain to detect the locations at which the oscillatory pattern
changes. Our method is shown to be able to detect all oscillatory frequencies
and the corresponding change points within an accurate range with a prescribed
probability asymptotically. This study is motivated by oscillatory frequency
estimation and change point detection problems encountered in physiological
time series analysis. An application to spindle detection and estimation in
sleep EEG data is used to illustrate the usefulness of the proposed
methodology. A Gaussian approximation scheme and an overlapping-block
multiplier bootstrap methodology for sums of complex-valued high dimensional
non-stationary time series without variance lower bounds are established, which
could be of independent interest
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