Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.      

Reducts of the random partial order

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that there are precisely five such structures. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure.     

The automorphism group of a certain unbounded non-hyperbolic domain

In this paper we determine the automorphism group of the Fock–Bargmann–Hartogs domain _Dn,m$D_{n,m}$ in Cⁿ×C^m${\mathbb{C}}^n\times {\mathbb{C}}^m$ which is defined by the inequality ‖ζ‖²<e^{−μ‖z‖2}${\Vert \zeta \Vert }^2<e^{-\mu {\Vert z\Vert }^2}$.     

On orthogonal resolutions of the classical Steiner quadruple system SQS(16)

A Steiner quadruple system SQS(16) is a pair where V is a 16-set of objects and is a collection of 4-subsets of V, called blocks, so that every 3-subset of V is contained in exactly one block. By classical is meant the boolean quadruple system, also known as the affine geometry AG(4,2). A parallel class is a collection of four blocks which partition V. The system possesses a resolution or parallelism, since can be partitioned into 35 parallel classes. Two resolutions are called orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. We prove that there are at most nine further resolutions which, together with the classical one, are pairwise orthogonal.      

Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

For each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three‐dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three‐dimensional case, and Kowalski and Nikv?cevi?'s results [6, Theorems 3.1 and 4.1] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integral point sets in higher dimensional affine spaces over finite fields

We consider point sets in the m-dimensional affine space where each squared Euclidean distance of two points is a square in Fq. It turns out that the situation in is rather similar to the one of integral distances in Euclidean spaces. Therefore we expect the results over finite fields to be useful for the Euclidean case.We completely determine the automorphism group of these spaces which preserves integral distances. For some small parameters m and q we determine the maximum cardinality I(m,q) of integral point sets in . We provide upper bounds and lower bounds on I(m,q). If we map integral distances to edges in a graph, we can define a graph Gm,q with vertex set . It turns out that Gm,q is strongly regular for some cases.     

A Note on the Wong-Rosay Theorem in Complex Manifolds

The authors prove a version, in utmost generality, of the Bun Wong-Rosay theorem on a complex manifold M. The essence of the result is that a domain Ω?M with non-compact automorphism group and boundary orbit accumulation point that is strongly pseudoconvex must be biholomorphic to the unit ball in C ⁿ .