A disjoint NP-pair is a pair (A,B) of nonempty, disjoint sets A and B such that both A and B belong to the complexity class NP.1 We let DisjNP denote the collection of all disjoint NP-pairs. A separator of a disjoint NP-pair (A,B) is a set S such that A ⊆ S and B ⊆ S (Figure 1). A fundamental question is whether (A,B) has a separator belonging to P. In this case the pair is P-separable; otherwi...

Two games are inseparable by semivalues if both games obtain the same allocation whatever semivalue is considered. The problem of separability by semivalues reduces to separability from the null game. For four or more players, the vector subspace of games inseparable from the null game by semivalues contains games different to zero-game. Now, for five or more players, the consideration of a pri...

We do not say that the four temperaments and astrology are inseparable, but rather that they are a feature of the rest of astrology. Furthermore, we do not talk lightly about the use of the four temperaments as being playthings. It is serious business when one uses an extrabiblical source to understand the nature of man, why he is and how he is to change. Myers then continues by charting the as...

The notion of immune sets is extended to closed sets and Π 1 classes in particular. We construct aΠ 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A,B) of separating sets for A and B is immune. We show that every perfect thin Π 1 class is immune. We define the stronger notion of prompt immunity and construct an exampl...

Non Destructive Evaluation (NDE) has become an inseparable part of modern society. Be it the field of engineering, technology, or healthcare, NDE is being used right from cradle to tomb to optimize processes, manage quality, predict the life and limit liability avoiding accidents. While NDE science and technology was qualitative during the first half of 20 century, the liberalisation of economi...

A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.

We determine the function fields of varieties of dimension ≥ 2 defined over the algebraic closure of Fp, modulo purely inseparable extensions, from the quotient by the second term in the lower central series of their pro-` Galois groups.

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (non-intersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+ 1)st Baxter number. We also show that...

An ordered partition of a set of n points in the d dimensional Euclidean space is called a separable partition if the convex hulls of the parts are pairwise disjoint. For each fixed p and d we determine the maximum possible number rp,d(n) of separable partitions into p parts of n points in real d-space up to a constant factor. Of particular interest are the values rp,d(n) = Θ(nd( p 2)) for ever...

We show that if a separable Banach space Z contains isometric copies of every strictly convex separable Banach space, then Z actually contains an isometric copy of every separable Banach space. We prove that if Y is any separable Banach space of dimension at least 2, then the collection of separable Banach spaces which contain an isometric copy of Y is analytic non Borel.