Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

无穷维复空间的拟凸Riemann区域(英)

本文利用多极集的测度研究了凸空间上的Riemann区域的拟凸性。     

Finite Degree Holomorphic Covers of Compact Riemann Surfaces

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.     

Riemann Surfaces of Some Static Dispersion Models and Projective Spaces

We show that the analytic continuation of the S-matrix elements, which are meromorphic functions of the energy in the complex plane with the cuts (-,-1] and [+1,+), from the physical sheet to nonphysical ones results in a system of nonlinear difference equations. A global analysis of this system is performed in the projective spaces P_N and P _N+1. We discuss the connection between the spaces P _N and P _N+1 and obtain some particular solutions of the initial system.     

有限维张量积空间上的强可分算子

设F是一个域,a∈F~nF~m.若存在h∈F~m,k∈F~m,使得a=hk,则称a是可分的.空间F~nF~m上的线性算子A称为是强可分的,是指x∈F~nF~m,x可分Ax可分.本文证明了F~nF~n上的线性算子A是强可分的当且仅当存在F~n上的线性双射A_1与A_2,使得A=A_1A_2或A=A_1~T A_2;证明了F~nF~m(n≠m)上线性算子A是强可分的当且仅当存在F~n与F~m上的线性双射A_1与A_2,使得A=A_1A_2.最后,给出了可分算子、强可分算子和秩1保持映射之间的关系.     

Beauville Surfaces,Moduli Spaces and Finite Groups

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.     

Surfaces minimales du type de Riemann deE 4 et groupes de transformations

Résumé On étudie, en relation avec un certain sous-groupe du groupe général des affinités de l'espace euclidien à quatre dimensions, la possibilité de déformation isométrique des surfaces minimales du type de Riemann de ce dernier espace. Entrata in Redazione il giorno 8 marzo 1972.     

Discrete Welding on Riemann Surfaces

The process of conformal welding had been investigated for nearly 50 years. In recent years welding has drawn increased interest with applications ranging from image recognition to string theory. We describe a discrete form of conformal welding for Riemann surfaces using circle packings and show how it can be used to approximate a continuous welding.     

Statistical Cluster Points of Sequences in Finite Dimensional Spaces

In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of -statistical convergence. A sequence xis -statistically convergent to a set Cif Cis a minimal closed set such that for every > 0 the set has density zero. It is shown that every statistically bounded sequence is -statistically convergent. Moreover if a sequence is -statistically convergent then the limit set is a set of statistical cluster points.     

Fredholm Composition Operators on Riemann Surfaces

It is proved that the invertibility of a composition operator on the differential form space for a Riemann surface is equivalent to its Fredholmness. In addition, the Fredholmness of weighted composition operators is discussed.     

Linear Independence of Gabor Systems in Finite Dimensional Vector Spaces

We discuss the linear independence of systems ofmvectors in n-dimensional complex vector spaces where the m vectors are time-frequency shifts of one generating vector. Such systems are called Gabor systems. When n is prime, we show that there exists an open, dense subset with full-measure of such generating vectors with the property that any subset of n vectors in the corresponding full Gabor system of n² vectors is linearly independent. We derive consequences relevant to coding, operator identification and time-frequency analysis in general.     

黎曼曲面上Riemann边值问题的一点注记

本文给出了紧黎曼曲面上关于Ｒｉｅｍａｎｎ边值问题的Ａｂｅｌ定理，由此定理可得经典Ａｂｅｌ定理，并且解决了非紧黎曼曲面上关于Ｒｉｅｍａｎｎ边值问题的ＣｏｕｓｉｎＩ，ＩＩ问题。     

Inverse Conductivity Problem on Riemann Surfaces

An electrical potential U on a bordered Riemann surface X with conductivity function σ>0 satisfies equation d(σ d ^c U)=0. The problem of effective reconstruction of σ from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: U| _bX ↦ σ d ^c U| _bX is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. Novikov (Funkc. Anal. Ego Priloz. 22:11–22, 2008) for simply connected X. We apply for this new kernels for on the affine algebraic Riemann surfaces constructed in Henkin (, 2008).      

Super-Liouville Equations on Closed Riemann Surfaces

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler–Lagrange equations, culminating in a blow up analysis.     

Energy and Foliations on Riemann Surfaces

We define the energy of foliations on Riemann surfaces. We prove that meromorphic vector fields are critical points and we compute their energies using the Green’s function. We then generalize the results to principal circle bundles over Riemann surfaces.Mathematics Subject Classification (2000): 53C12, 53C15.     

Vekua Integral Operators on Riemann Surfaces

On an arbitrary (in general, noncompact) Riemann surface R, we study integral operators T and ∏ analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy--Riemann equation $\bar \partial f = F$ in the class of functions f exhibiting Λ₀-behavior in the vicinity of the ideal boundary of R.