Topics on the Geometry of Rational Homogeneous Spaces

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary and mainly reflects the interests of the author.     

Long time dynamics of the 3D radial NLS with the combined terms

In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation.     

Weighted Oscillation and Variation Inequalities for Singular Integrals and Commutators Satisfying Hrmander Type Conditions

This paper is devoted to investigating the weighted L~p-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type(p, p) estimates for 1 p ∞ and the weighted weak type(1,1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hormander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini's conditions as model examples. Meanwhile, we also obtain the weighted L~p-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(R~d)-functions.     

Molecular Characterization of Anisotropic Musielak–Orlicz Hardy Spaces and Their Applications

Let A be an expansive dilation on R~n and φ:R~n× [0,∞)→[0,∞) an anisotropic Musielak–Orlicz function.Let H_A~φ(R~n) be the anisotropic Hardy space of Musielak–Orlicz type defined via the grand maximal function.In this article,the authors establish its molecular characterization via the atomic characterization of H_A~φ(R~n).The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case(namely,A:=2I_(n×n)) coincides with the range of well-known classical molecules and,moreover,even for the isotropic Hardy space H~p(R~n)with p∈(0,1](in this case,A:=2I_(n×n),φ(x,t) :=t~p for all x∈R~n and t∈[0,∞)),this molecular characterization is also new.As an application,the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from H_A~φ(R~n) to L~φ(R~n) or from H_A~φ(R~n) to itself.     

Fractional Square Functions and Potential Spaces, Ⅱ

In this paper, we study fractional square functions associated with the Poisson semigroup for Schrdinger operators. We characterize the potential spaces in the Schrdinger setting by using vertical, area and g*λ fractional square functions.     

On Nonuniqueness of Geodesics and Geodesic Disks in the Universal Asymptotic Teichmüller Space

The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmüller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in [[μ]] with h_ζ~*(μ) h~*(μ) for some point ζ∈ ?D, then there exist infinitely many geodesic segments joining [[0]]and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [[μ]] in AT(D).     

Submanifolds in Cauchy Riemann Geometry

In this paper I would like to make a report on the results about hypersurfaces in the Heisenberg group and invariant curves and surfaces in CR geometry. The results are contained in the papers [8, 9, 16] and [14]. Besides, I would also report on the results about the strong maximum principle for a class of mean curvature type operators in [10].     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

Sewing homeomorphism and conformal invariants

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sufficient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Hlder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.     

A Complete Characterization of Multivariate Normal Stable Tweedie Models through a Monge-Ampère Property

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.     

广义n-赋范空间中的Vogt定理

本文推广Vogt定理到广义n-赋范空间，即证明了两个广义n-赋范空间之间的保持ρ-诱导距离映射是仿射的.     

δ-BiHom-Jordan-李超代数的表示

本文研究δ-BiHom-Jordan-李超代数的表示.特别是详细地研究δ-BiHom-Jordan-李超代数的伴随表示、平凡表示、形变.作为应用，还讨论δ-BiHom-Jordan-李代数的导子.     

点星网与度量空间的映像

拓扑空间X的覆盖列{P_i}_(i∈N)被称为空间X的点星网,若x∈X,则{st(x,P_i)}_(i∈N)是x在X中的网.本文刻画具有cs有限cs覆盖列的点星网的空间,并将其表示为度量空间在确定映射下的像.在假设集族性质β满足适当的条件下,证明对拓扑空间X下述条件相互等价:(1) X具有β且cs覆盖列的点星网.(2)X具有β且sn覆盖列的点星网.(3)X是Cauchy sn对称空间且具有σ-β的cs网.(4) X是Cauchy sn对称空间且具有σ-β的sn网.(5) X是度量空间的序列覆盖、π且σ-β映像.(6) X是度量空间的1序列覆盖、紧且σ-β映像.这些工作以局部有限集族与点有限集族为特例,拓展了从基到cs网的研究,丰富了映射与空间的相互分类思想.     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Linear arboricity of regular digraphs

A linear directed forest is a directed graph in which every component is a directed path.The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and P′eroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs,regular digraphs with high directed girth and random regular digraphs and we improve some wellknown results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.     

高新技术企业R&D绩效测度系统的开发

九十年代以来 ,就如何测度R&D绩效 ,已有许多学者提出一些具体的测度系统 (方法 ) ,但这些系统本身还存在某些不足 ,另外 ,学者们没有对测度系统的实施和更新进行研究。针对这种情况 ,本文从设计、实施、更新的角度对高新技术企业R&D绩效测度系统进行了有效的开发。