A specialissue dedicated to Professor LU QiKeng on the occasion of his 80th birthday

The Guest EditorS of the ISSUe afC Siu Yum一Tongy乞u StePhen 5.T. Liu Kefeng Zhou XiangyuA specialissue dedicated to Professor LU QiKeng on the occasion of his 80th birthday~~     

New series of odd non-congruent numbers Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and Sn of the elliptic curve En: y2 = x(x -n)(x - 2n) and its dual curve En: y2 = x3 6nx2 n2x have the smallest size: Sn = {1}, Sn = {1,2,n,2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.     

Julia sets of permutable Holomorphic maps Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.     

Remarks on the rank properties of formal CR maps Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.     

Coefficient multipliers of mixed norm space in the ball Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0相似文献   

Very weak solutions to the two-dimensional Monge-Ampére equation Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β 1/5. Our approach is based on combining the approach of Lewicka and Pakzad(2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al.(2018) for the isometric immersion problem.     

The Teichmuller distance on the space of spherical CR structures Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We define the Teichmuller pseudodistance on the space of spherical CR structures on a fixed compact manifold by using quasiconformal mappings between spherical CR manifolds. The pseudodistance is shown to be a complete distance.     

Stochastic Hamiltonian flows with singular coefficients Dedicated to the 60th Birthday of Professor Michael R¨ockner

In this paper, we study the following stochastic Hamiltonian system in R~(2d)(a second order stochastic differential equation):dX_t = b(X_t,X_t)dt + σ(X_t,X_t)dW_t,(X_0,X_0) =(x, v) ∈ R~(2d),where b(x, v) : R~(2d)→ R~d and σ(x, v) : R~(2d)→ R~d ? R~d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H_p~(2/3,0) and ?σ∈ L~p for some p 2(2 d + 1), where H_p~(α,β)is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that(x, v) → Z_t(x, v) :=(Xt,Xt)(x, v) forms a stochastic homeomorphism flow,and(x, v) → Z_t(x, v) is weakly differentiable with ess.sup_(x,v)E(sup_(t∈[0,T])|?Z_t(x, v)|~q) ∞ for all q ≥ 1 and T≥ 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli(2008) and Trevisan(2016).     

Clifford algebra approach to pointwise convergence of Fourier series on spheres Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.     

The Roper-Suffridge extension operator and classes of biholomorphic mappings Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent.     

Invariant holomorphic extension in several complex variables Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Two fundamental problems on the invariant holomorphic extensions have been posed, which are naturally arose from our solution of the extended future tube conjecture and closely and deeply related to the general theory of Stein manifolds due to Cartan-Serre. In this paper, the relationship is presented between the two problems, the motivation of considering the problems, and the methods to approach the problems. We have also posed some questions and conjectures related to this two problems.     

Some Penrose transforms in complex differential geometry Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this article, we review a construction in the complex geometry often known as the Penrose transform. We then present two new applications of this transform. One concerns the construction of symmetries of the massless field equations from mathematical physics. The other concerns obstructions to the embedding of CR structures on the three-sphere.     

Characterization of isolated homogeneous hypersurface singularities in C~4 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let V be a hypersurface with an isolated singularity at the origin in Cn 1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Isolated complex singularities and their CR links Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.     

On a non-abelian invariant on complex surfaces of general type Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S. Cheung, Bun Wong and Stephen S. T. Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.     

Some remarks and problems on complex homogeneous domains Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We discuss some problems on rigidity of canonical realizations of complex homogeneous domains at Cn     

Hodge decomposition theorem on strongly Kahler Finsler manifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (?)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M,F) is a compact strongly Kahler Finsler manifold, we define a (?)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in Kahler manifolds is still true in the more general compact strongly Kahler Finsler manifolds.     

On quasi-convex mappings of orderαin the unit ball of a complex Banach space Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order a in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order a defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.