Deletion constructions of symmetric 4-configurations. Part I.

By deletion constructions we mean several methods of generation of new geometric configurations by the judicious deletion of certain points and lines, and introduction of other lines or points. A number of such procedures have recently been developed in a systematic way. We present here one family of such constructions, and will describe other families in the following parts

Higher order selfdual toric varieties

The notion of higher order dual varieties of a projective variety, introduced in \cite{P83}, is a natural generalization of the classical notion of projective duality. In this paper we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley-Bacharach questions and with Cayley configurations.Comment: 21 page

Forbidden Subgraphs in Connected Graphs

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed

On shared indefinite expressions in coordinative structures

The paper shows that shared indefinite expressions in coordinative constructions may differ with respect to their referential properties. This is due to their being either in a focused or in a nonfocused shared constituent. Their different information-structural status follows from Rooth's theory on focus interpretation. Thus it follows that focused shared constitutents must be beyond the actual coordination and that coordinative constructions with unfocused shared constituents can be represented as ellipsis. In a focused shared constituent indefinite expressions may have a specific and an non specific unique reading as well as an non specific distributive one. For the latter we outline the idea that subjects and objects in the actual coordination form a pair of sets to which a distributing operator is attached. The set formation is further supported by plural pronouns referring to the respective set and by plural verb agreement in subsequent expressions