The structure of hemigroups

A hemigroup is a continuous binary operation on a spaceM which satisfies (xy)(zy)=xz. The structure of these and their relationship with semigroups is described.     

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their sets of homogeneous geodesics through a point. Dedicated to the memory of Professor Aldo Cossu Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce.     

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

Affine Vogan diagrams and symmetric pairs

We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the associated and symplectic symmetric pairs.     

Reductive homogeneous pseudo-Riemannian manifolds

A classification of homogeneous pseudo-Riemannian structures and a characterization of each primitive class are obtained. Several examples are also given.     

Double structures and jets

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.     

Affine and projective transformations of Berwald connections

We discuss the infinitesimal affine transformations of the Berwald connection of a spray, and the relation between the projective transformations of a spray and the affine transformations of its Berwald-Thomas-Whitehead connection.     

Contact and Conformal Maps in

When the action of the conformal group O(1, n+1) on may be characterized in simple differential geometric terms, even locally: a theorem of Liouville states that a C⁴ map between domains and in whose differential is a (variable) multiple of a (variable) isometry at each point of is the restriction to of a transformation x g·x, for some g in O(1,n+1). In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G/P, where P is a minimal parabolic subgroup.     

Loops on spheres having a compact-free inner mapping group

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable 1-dimensional compact loops explicitly using the theory of Fourier series. Authors’ addresses: ágota Figula, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany and Institute of Mathematics, University of Debrecen, P.O.B. 12, H-4010 Debrecen, Hungary; Karl Strambach, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany     

Constrained Willmore surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint. C. Bohle, G. P. Peters and U. Pinkall are partially supported by DFG SPP 1154.     

Parallel permit grammars and some graph representations of Petri nets

The graph of a set grammar is introduced in such a way that each set rule of the grammar is represented by a cartesian subgraph of it. The correspondence between cartesian subgraphs and transitions of Petri nets (which satisfy the axiom of extensionality) is established. The set grammars with input (initial) and output (terminal) elements are studied in an analogy to Chomsky's string grammars and their strong equivalence. Permit rules and parallel permit rules are introduced in such a way that parallel permit grammars are more general tools than Petri nets themselves, because the equivalence between homogeneous parallel permit grammars and set grammars (and Petri nets) is proved.     

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

De Branges Spaces of Entire Functions Symmetric About the Origin

We define and investigate the class of symmetric and the class of semibounded de Branges spaces of entire functions. A construction is made which assigns to each symmetric de Branges space a semibounded one. By employing operator theoretic tools it is shown that every semibounded de Branges space can be obtained in this way, and which symmetric spaces give rise to the same semibounded space. Those subclasses of Hermite-Biehler functions are determined which correspond to symmetric or semibounded, respectively, nondegenerated de Branges spaces. The above assignment is determined in terms of the respective generating Hermite-Biehler functions.     

Symmetries of distributions and quadrature of ordinary differential equations

We present a geometric exposition of S. Lie's and E. Cartan's theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination.     

An upper bound for the number of Eulerian orientations of a regular graph

We establish a new upper bound for the number of Eulerian orientations of a regular graph with even degrees.C.N.R.S., Paris with partial support of P.R.C. Mathématiques-Informatique.     

Forests and score vectors

Thescore vector of a labeled digraph is the vector of out-degrees of its vertices. LetG be a finite labeled undirected graph without loops, and let σ(G) be the set of distinct score vectors arising from all possible orientations ofG. Let ϕ(G) be the set of subgraphs ofG which are forests of labeled trees. We display a bijection between σ(G) and ϕ(G). Supported in part by ONR Contract N00014-76-C-0366.     

The asymptotic number of acyclic digraphs. I

We obtain an asymptotic formula forA _n,q , the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forA _n,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatA _n,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula. Research supported by NSF under grant MCS-8300414. Research supported by NSERC under grant A4067. Research supported by NSF under grant MCS-8302282. Research supported by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowship Scheme, while this author was at the University of Newcastle, Australia.