Cutting a cake, dividing up the property in an estate, determining the borders in an international dispute - such problems of fair division are ubiquitous. Fair Division treats all these problems and many more through a rigorous analysis of a variety of procedures for allocating goods, or deciding who wins on what issues, when there are disputes. Starting with an analysis of the well-known cake-cutting procedure, 'I cut, you choose', the authors show how it has been adapted in a number (...) of fields and then analyze fair-division procedures applicable to situations in which there are more than two parties, or there is more than one good to be divided. In particular they focus on procedures which provide 'envy-free' allocations, in which everybody thinks he or she has received the largest portion and hence does not envy anybody else. They also discuss the fairness of different auction and election procedures. (shrink)

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq (...) \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated. (shrink)

Let AC and AI denote the collections of graphs with vertex set ω and which have, respectively, no infinite independent subgraph, and no infinite complete subgraph. Both AC and AI are coanalytic sets of reals that are not analytic, and they are disjoint by Ramsey’s theorem. We prove that there exists a Borel set separating AC and AI, and we discuss the sense in which this is an infinite analogue of a weak version of.

It appears that both English and French contain a mixture of ergative and transitive verbal constructions. How useful is reference to a dual-structure model of syntax in determining the bias of a particular language? Certain verbal constructions in English and French could be generated from either transitive or ergative deep structures; we must therefore turn to other criteria - syntactic cohesion and predominant constituent structures in 'surface structure' - for a non-arbitrary - and adequate - decision.

To his credit, Chomsky puts them all on his website, whether it’s The New Yorker describing him as “the devil’s accountant†and “one of the greatest minds of the 20th centuryâ€, or The Nation, which lampooned him as “a very familiar kind of academic hack†whose career has been “the product of a combination of self-promotion, abuse of detractors, and the fudging of his findingsâ€. He stands accused of asserting that every US President since Franklin D Roosevelt should have been (...) impeached as war criminals; of supporting the murderous Pol Pot regime in Cambodia; and of comparing Israel to the Third Reich. (shrink)

Reviews of Peter Dear's Discipline and Experience: The Mathematical Way in the Scientific Revolution.