Let R denote a 2-fir. The notions of F -independence and algebraic subsets of R are defined. The decomposition of an algebraic subset into similarity classes gives a simple way of translating the F -independence in terms of dimension of some vector spaces. In particular to each element a ∈ R is attached a certain algebraic set of atoms and the above decomposition gives a lower bound of the leng...

A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is neither a graph nor a linear space is called proper. The aim of this paper is to c...

Given a Banach space X and a subspace Y , the pair (X,Y ) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. The main result is an easy to apply dual formulation of this property. Applications are gi...

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

This paper presents a method called Rank Inclusion in Criteria Hierarchies (RICH) for the analysis of incomplete preference information in hierarchical weighting models. In RICH, the decision maker is allowed to specify subsets of attributes which contain the most important attribute or, more generally, to associate a set of rankings with a given set of attributes. Such preference statements le...

It is surmised that the algebra of the Pauli operators on the Hilbert space of N -qubits is embodied in the geometry of the symplectic polar space of rank N and order two, W2N−1(2). The operators (discarding the identity) answer to the points of W2N−1(2), their partitionings into maximally commuting subsets correspond to spreads of the space, a maximally commuting subset has its representative ...

Let 9 be an automorphism of finite order of a free group X. We characterise the action of 9 on X by showing that X has a free basis which is the disjoint union of finite subsets Sj, where if S,, ~ {uq, «,,..., uk) then u¡9 ~ ul+x (0 < 1 < k) and uk9 = AjUqBj for some A}, B¡ in X and e = ±1. As an application of this result, we obtain a list of the conjugacy classes of periodic automorphisms of ...

Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if th...

1. Preliminaries. A functor from a category of combinatorial geometries, or equivalently a category of geometric lattices, to a category of commutative algebras will be described, and some properties of this functor will be investigated. In particular, a cohomology will be associated to each point of a geometry and will be derived from the associated algebra. If (G, S) is a geometry on a set S ...

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional. Theorem 2, the main result announced in this note, gives an ex...