In this paper, we construct from any given good (3,1)-dimensional manifold pair finitely many almost identical imitations of it whose exteriors are mutative hyperbolic 3-manifolds. The equivariant versions with the mutative reduction property on the isometry group are also established. As a corollary, we have finitely many hyperbolic 3-manifolds with the same volume and the same isometry group.

Hyperidentities were introduced by Taylor, and their properties have been studied by Taylor, Bergman, Penner, Graczynska, Schweigert, Wismath, and others. In particular, Penner has produced bases for the hyperidentities of various types satisfied by the variety of semilattices.

In this paper, we look at bases for the hyperidentities satisfied by some other varieties of semigroups. We first investigate solid and hyper-associative varieties of semigroups, and use a result of Graczynska's to form a basis for the hyperidentities of type (2) for varieties with these properties. We then produce bases for the hyperidentities satisfied by the variety R B of rectangular bands, and prove that the collection of all hyperidentities satisfied by R B is not finitely based. Finally, we use the R B results to give hyperidentity bases for some varieties of nilpotent semigroups.

We are concerned with existence results for nonlinear scalar Neumann boundary value problems u″ + g(x, u) = 0, u′(0) = u′(π) = 0 where g(x, u) satisfies Carathéodory conditions and is (possibly) unbounded. On the one hand we only assume that the function (sgn u)g(x, u) is bounded either from above or from below in some function space, and we impose conditions which relate the asymptotic behavior of the function (for¦u¦large) with the first two eigenvalues of the corresponding linear problem (here G(x, u) = is the potential generated by g). On the other hand we consider cases where the function (sgn u)g(x, u) is unbounded. The potential G(x, u) is not necessarily required to satisfy a convexity condition. Our method of proof is variational, we make use of the Saddle Point Theorem.

Examples of hyperbolic knots in S3 are given such that their complements contain quasi-Fuchsian non-Fuchsian surfaces. In particular, this proves that there are hyperbolic knots that are not toroidally alternating.

We give a simple necessary and sufficient condition for the inclusion map of a subpolyhedron into a compact 3-manifold with non-empty boundary to be a homotopy equivalence.

We establish a characterization of certain trees of polygons similar to that of n-gon-trees given by Chao and Li.

Let = {A1, …, An} be a union-closed set. This note establishes a property which must be possessed by any smallest counterexample to the Union-Closed Sets Conjecture. Specifically, a counterexample to the conjecture with minimal n has at least three distinct elements, each of which appears in exactly (n − 1)/2 of the .

The paper gives a rather simple description of all maximal abelian subgroups H of the symmetric group SM acting on an arbitrary set M. In the case of finite M this result is used to determine the maximal cardinality of such an H and the maximal number of permutations without fixed points contained in an abelian subgroup of SM.