Showing papers in "Science China-mathematics in 2017"

A general extension theorem for cohomology classes on non reduced analytic subspaces

Abstract: The main purpose of this paper is to generalize the celebrated L² extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is Kahler and holomorphically convex, but not necessarily compact.

Existence of positive solutions to some nonlinear equations on locally finite graphs

Abstract: Let G = (V, E) be a locally finite graph, whose measure μ(x) has positive lower bound, and Δ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz (1973), we establish existence results for some nonlinear equations, namely Δu + hu = f(x, u), x ∈ V. In particular, we prove that if h and f satisfy certain assumptions, then the above-mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation Δu + hu = f(x, u) + ϵg. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.

Weighted variation inequalities for differential operators and singular integrals in higher dimensions

Abstract: We prove weighted q-variation inequalities with 2 < q < ∞ for sharp truncations of singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given. Parallel results are proven for differential operators.

Regularity of discrete multisublinear fractional maximal functions

Abstract: We investigate the regularity properties of discrete multisublinear fractional maximal operators, both in the centered and uncentered versions. We prove that these operators are bounded and continuous from l1(ℤd) × l1(ℤd) × · · · × l1(ℤd) to BV(ℤd), where BV(ℤd) is the set of functions of bounded variation defined on ℤd. Moreover, two pointwise estimates for the partial derivatives of discrete multisublinear fractional maximal functions are also given. As applications, we present the regularity properties for discrete fractional maximal operator, which are new even in the linear case.

Gradient estimates for parabolic systems from composite material

Abstract: In this paper, we derive W 1,∞ and piecewise C 1,α estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise Holder continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56: 892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.

Existence of solutions for a critical fractional Kirchhoff type problem in ℝ N

Abstract: This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

Abstract: The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurers and the reinsurers surplus processes. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.

Optimality conditions for sparse nonlinear programming

Abstract: The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable function subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Fr${\rm\acute{e}}$chet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn-Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

Spreading and vanishing in a West Nile virus model with expanding fronts

Abstract: We study a simplified version of a West Nile virus (WNv) model discussed by Lewis et al. (2006), which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number~$R_0$ for the non-spatial epidemic model is defined and a threshold parameter $R_0 ^D$ for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with a coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by an ODE. The risk index~$R_0 ^F (t)$ associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to the system when the spreading occurs is considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.

Homogeneous wavelets and framelets with the refinable structure

Abstract: Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Abstract: Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$ be a pair of expansive dilations and φ: ℝ n ×ℝ m ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$ -function characterization, the $$\vec g_\lambda ^*$$ -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ is equivalent to the standard infinite atomic decomposition norm As an application, we show that, for a given admissible triplet ( $$\left( {\varphi ,q,\vec s} \right)$$ ), if T is a sublinear operator and maps all ( $$\left( {\varphi ,q,\vec s} \right)$$ )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to B Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to L φ (R n × R m ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to itself, whose kernels are adapted to the action of $$\vec A$$ The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ n × ℝ m and are new even for classical product Orlicz-Hardy spaces

Further study of a weighted elliptic equation

Abstract: A Liouville type result is established for non-negative entire solutions of a weighted elliptic equation. This provides a positive answer to a problem left open by Du and Guo (2015) and Phan and Souplet (2012) (see (CJ) by Du and Guo (2015) and Conjecture B by Phan and Souplet (2012)). Meanwhile, some regularity results are also obtained. The main results in this paper imply that the number ps is the critical value of the Dirichlet problems of the related equation, even though there are still some open problems left. Our results also apply for the equation with a Hardy potential.

Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces

Abstract: We introduce the weak Hardy-Morrey spaces in this paper. We also obtain the atomic decompositions of the weak Hardy-Morrey spaces. By using these decompositions, we establish the Hardy inequalities on the weak Hardy-Morrey spaces.

Some results on the regularization of LSQR for large-scale discrete ill-posed problems

Abstract: LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems (CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition (TSVD) method. We establish bounds for the distance between the $k$-dimensional Krylov subspace and the $k$-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank $k$ approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.

Limit theorems for a supercritical branching process with immigration in a random environment

Abstract: Let (Zn) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size Wn converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Zn.

Geometry and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss

Abstract: It is known that the space of homogeneous type introduced by Coifman and Weiss (1971) provides a very natural setting for establishing a theory of Hardy spaces This paper concentrates on how the geometrical conditions of the space of homogeneous type play a crucial role in building a theory of Hardy spaces via the Littlewood-Paley functions

Isoparametric hypersurfaces in Funk spaces

Abstract: Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.

Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel

Abstract: This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of f′(0) = 0, we can no longer use the common method which mainly depends on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, so we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.

Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

Abstract: For a commutative ring $R$ and a faithfully flat $R$-algebra $S$ we prove, under mild extra assumptions, that an $R$-module $M$ is Gorenstein flat if and only if the left $S$-module $\tp{S}{M}$ is Gorenstein flat, and that an $R$-module $N$ is Gorenstein injective if and only if it is cotorsion and the left $S$-module $\Hom{S}{N}$ is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.

Symmetry of Solutions for a Fractional System

Abstract: We consider a pseudo-differential system involving different fractional orders. Through an iteration method, we obtain the key ingredients—the maximum principles—of the method of moving planes. Then we derive symmetry on non-negative solutions without any decay assumption at infinity.

A conservative local discontinuous Galerkin method for the solution of nonlinear Schrödinger equation in two dimensions

Abstract: In this study, we present a conservative local discontinuous Galerkin (LDG) method for numerically solving the two-dimensional nonlinear Schr¨odinger (NLS) equation The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux We will propose two kinds of time discretization methods for the semi-discrete formulation One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation The other one is Krylov implicit integration factor (IIF) method which demands much less computational effort Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon

A Stein deficit for the logarithmic Sobolev inequality

Abstract: We provide some lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality in terms of the so-called Stein characterization of the Gaussian distribution. The techniques are based on the representation of the relative Fisher information along the Ornstein-Uhlenbeck semigroup by the Minimum Mean-Square Error from information theory.

Traveling waves for a diffusive SEIR epidemic model with standard incidences

Abstract: This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered (SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.

Traces of weighted function spaces: Dyadic norms and Whitney extensions

Abstract: The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950’s. In this paper, we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney extension operator.

Stochastic calculus with respect to G-Brownian motion viewed through rough paths

Abstract: We study rough path properties of stochastic integrals of Ito’s type and Stratonovich’s type with respect to G-Brownian motion. The roughness of G-Brownian motion is estimated and then the pathwise Norris lemma in G-framework is obtained.

Strong openness of multiplier ideal sheaves and optimal L 2 extension

Abstract: In this paper, we reveal that our solution of Demaillys strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollar and Jonsson-Mustata implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtssons positivity of vector bundles associated to holomorphic fibrations over a unit disc.

Finite sample breakdown point of Tukey’s halfspace median

Abstract: Tukey’s halfspace median (HM), servicing as the multivariate counterpart of the univariate median, has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property (the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point (FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one—the Tukey’s halfspace median. This paper presents a precise result on FSBP for Tukey’s halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.

Curvature identities on almost Hermitian manifolds and applications

Abstract: We systematically derive the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection. As applications of the curvature identities, we obtain some results about the integrability of quasi Kahler manifolds and nearly Kahler manifolds.

Markov selection and W -strong Feller for 3D stochastic primitive equations

Abstract: This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.

Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition

Abstract: We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To do this we introduce molecules with variable growth condition. Our results are new even for ℝ n case.