We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, min}(E)$ and ort}(E)$. show that these are not isolated examples; in any given system of at least one will occur as a subprojection. This characterizes those which have is the first general structural result for such systems.

This paper concerns a duality between conditionally convergent real series and permutations of their indices. It is widely known that if one is given a series E ^ which is conditionally convergent but not absolutely convergent, then there is a permutation n such that £«< # £#*(*)• Various authors have studied extensions of this result of Riemann (for example, Smith [8], Steinitz [9], Threlfall ...

A class is an e ectively closed set of reals We study properties of these classes determined by cardinality measure and category as well as by the complexity of the members of a class P Given an e ective enumeration fPe e g of the classes the index set I for a certain property such as having positive measure is the set of indices e such that Pe has the property For example the index set of bina...

If a weakly compact convex set K in a real Banach space X is strongly exposed by a dense set of functionals in X', it is proved that the functionals which expose K form a residual set in A". If v. & -* X is a measure, it follows that the set of exposing functionals of its range is a residual Gs in A". This, in turn, is found to be equivalent to a theorem of B. Walsh on the residuality of functi...

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $f$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.

Association mining may often derive an undesirably large set of frequent itemsets and association rules. Recent studies have proposed an interesting alternative: mining frequent closed itemsets and their corresponding rules, which has the same power as association mining but substantially reduces the number of rules to be presented. In this paper, we propose an e cient algorithm, CLOSET, for mi...