ISSN 1439-8516 CN 11-2039/O1
Let L be a second order positive, elliptic differential operator that is self-adjoint with respect to some C∞ density dx on a compact connected manifold M. We proved that if 0 < α < 1, α/2 < s < α and f ∈ Hs(M) then the fractional Schrödinger propagator eitLα/2 on M satisfies eitLα/2 f(x) -f(x)=o(ts/α-ε) almost everywhere as t → 0+, for any ε > 0.
We characterize the complex differential equations of the form
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where aj(x) are meromorphic functions in the variable x for j=0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.
Let X be a C1+ vector field on a compact Riemannian manifold M with dimension d ≥ 3. Let Λ be a transitive singular hypebolic set with positive volume. We show that Λ=M and Λ is a uniformly hyperbolic set without singularities.
In the high-dimensional setting, this article considers a canonical testing problem in multivariate analysis, namely testing coefficients in linear regression models. Several tests for high-dimensional regression coefficients have been proposed in the recent literature. However, these tests are based on the sum of squares type statistics, that perform well under the dense alternatives and suffer from low power under the sparse alternatives. In order to attack this issue, we introduce a new test statistic which is based on the maximum type statistic and magnifies the sparse signals. The limiting null distribution of the test statistic is shown to be the extreme value distribution of type I and the power of the test is analysed. In particular, it is shown theoretically and numerically that the test is powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature.
Let Dr:={z=x + iy ∈ C:|z|< r}, r ≤ 1. For a normalized analytic function f in the unit disk D:=D1, estimating the Dirichlet integral Δ(r, f)=∫ ∫Dr|f'(z)|2 dxdy, z=x + iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of Dr under f counting multiplicities. In this paper, our main objective is to estimate areas of images of Dr under non-vanishing analytic functions of the form (z/f)μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in D.
In this paper, we study superlinear elliptic equations with mixed boundary value conditions in annular domains. It is assumed that the nonlinearities depend on the derivative terms. Some results about existence of solutions are established by using the Nehari manifold technique, as well as iterative technique.
In this paper, we give an equitable presentation for the multiparameter quantum group associated to a symmetrizable Kac-Moody Lie algebra, which can be regarded as a natural generalization of the Terwilliger's equitable presentation for the one-parameter quantum group.
It is a common issue to compare treatment-specific survival and the weighted log-rank test is the most popular method for group comparison. However, in observational studies, treatments and censoring times are usually not independent, which invalidates the weighted log-rank tests. In this paper, we propose adjusted weighted log-rank tests in the presence of non-random treatment assignment and dependent censoring. A double-inverse weighted technique is developed to adjust the weighted log-rank tests. Specifically, inverse probabilities of treatment and censoring weighting are involved to balance the baseline treatment assignment and to overcome dependent censoring, respectively. We derive the asymptotic distribution of the proposed adjusted tests under the null hypothesis, and propose a method to obtain the critical values. Simulation studies show that the adjusted log-rank tests have correct sizes whereas the traditional weighted log-rank tests may fail in the presence of non-random treatment assignment and dependent censoring. An application to oropharyngeal carcinoma data from the Radiation Therapy Oncology Group is provided for illustration.
The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F. Defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept called "castling". Three types of castlings are essential to grasp the arithmetics. The divisor function τ on Thompson's monoid S satisfies τ(uv) ≤ τ(u)τ(v) for any u, v ∈ S. Then the limit τ0(u)=limn→∞(τ(un))1/n exists. The quantity Ç(S)=sup1≠u∈S τ0(u)/τ(u) describes the complexity for castlings in S. We show that Ç(S)=1. Moreover, the Möbius function on S is calculated. And the Liouville function Ω on S is studied.
In this paper, we consider composite quantile regression for partial functional linear regression model with polynomial spline approximation. Under some mild conditions, the convergence rates of the estimators and mean squared prediction error, and asymptotic normality of parameter vector are obtained. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least-squares based method when there are outliers in the dataset or the random error follows heavy-tailed distributions. Finally, we apply the proposed methodology to a spectroscopic data sets to illustrate its usefulness in practice.
