Characterization of convex domains in C~2 with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in Cn. Suppose there exists {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in C~2 with C~2 boundary.Suppose that there is a sequence {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

Domains in complex surfaces with non-compact automorphism groups

Let X be an arbitrary complex surface and D a domain in X that has a non-compact group of holomorphic automorphisms. A characterization of those domains D that admit a smooth, weakly pseudoconvex, finite type boundary orbit accumulation point is obtained.     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Unbounded domains in ℂ2 with non-compact automorphisms group

Let D be a domain in ?² such that the orbit of an internal point under the group Aut(D) of the holomorphic automorphisms of D accumulates to a smooth boundary point p. It is well known that if p is a strictly pseudoconvex point, such a local information forces D to be biholomorphic to the ball. We prove that if p is only weakly pseudoconvex of finite type, the domain D need not to be biholomorphic to a natural standard model. On the other end our results show that, under some convexity assumptions, D may be realized as a bounded domain with real analytic boundary except at most at one point.     

Some Minkowski planes with 3-dimensional automorphism group

A new rather large family of 2-dimensional locally compact topological Minkowski planes with an at least 3-dimensional automorphism group is introduced here. Isomorphism classes and automorphisms of these planes are determined.     

Some new designs with prescribed automorphism groups

We establish the existence of simple designs with parameters 2‐(55, 10, 4), 3‐(20, 5, 4), 3‐(21, 7, 30), 4‐(15, 5, 2), 4‐(16, 8, 45), 5‐(16, 7, 10), and 5‐(17, 8, 40), which have previously been unknown. For the corresponding t, v, and k, we study the set of all λ for which simple t‐ designs exist.     

A moduli space of non-compact curves on a complex surface

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by finitely many holomorphic functions.     

The holomorphic automorphism group of the complex disk

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.     

A collection of results on Hamiltonian cycle systems with a nice automorphism group

We collect some old and new results on Hamiltonian cycle systems of the complete graph (or the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties.     

Bergman-Hartogs型域的全纯自同构群

我们考虑一类以有界对称域D为底的Bergman-Hartogs型域Ω={(wm(1),...,w(r),z)∈C1×···×Cmr×D:∥w(1)∥2p1+···+∥w(r)∥2prKD(z,z)-q},其中KD(z,z)是D上的Bergman核函数,r 1且为正整数,参数p1,...,pr1和q0为实数.我们给出它的全纯自同构群,并且证明当r=1时此自同构群为最大全纯自同构群;当r1时,若Ω的全纯自同构变换F将(0,z)∈{0}×D映到(0,z*)∈{0}×D,则F在我们给出的全纯自同构群中.     

Graphs with symmetric automorphism group

We investigate the properties of graphs whose automorphism group is the symmetric group. In particular, we characterize graphs on less than 2n points with group S_n, and construct all graphs on n + 3 points with group S_n. Graphs with 2n or more points and group S_n are discussed briefly.     

The t-designs with prescribed automorphism group,new simple 6-designs

We introduce an algorithm for the construction of a complete system of representatives of t-designs with given parameters t ? (v, k, λ) and prescribed full automorphism group A. It is based on the following observation published by Kramer and Mesner in 1976: The t ? (v, k, λ) designs admitting automorphism group A are exactly the 0-1-solutions $ \overrightarrow x $ of the following system of linear equations M are incidence matrices, which we compute by means of double cosets. Representing the set of all solutions of the above system of equations implicitly by a graph gives us the possibility either to extract the solutions explicitly or to compute their precise numbers, which often are very big. We use the lattice of overgroups of A in the full symmetric group S_v for the construction or enumeration of the isomorphism types of the t-designs with full automorphism group A from these solutions. To the best of our knowledge our approach for the first time allows one to compute the precise number of isomorphism types or even these designs themselves for substantial numbers. We determined the (number of) isomorphism types for many known parameter sets and found new simple 6-designs with parameters and full automorphism group PΓL₂(27). We constructed all isomorphism types of these designs; their precise numbers are 3,367,21 743,38 277, respectively. © 1993 John Wiley & Sons, Inc.     

The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd^∗+d^∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d^∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d^∗ is determined and an expression for d^∗ is obtained. The operator dd^∗+d^∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.