We study the representations of an element of a free group as a commutator. For a given element g of a free group F, we are interested in the set of all pairs (x, y) of elements of F such that

(1)

where [x, y] = xyx-1 y-1. If g = 1, the problem is trivial. We assume henceforth that g ≠ 1.

By an ortholattice we mean a lattice with 0 and 1 and a complementation operation which is an involutorial antiautomorphism. The free modular ortholattice on two generators has 96 elements—cf. J. Kotas [8].

It is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.

It is proved that a polynomial on several complex variables, whose coefficients depend analytically on a parameter ε, admits a factorization which is irreducible for every value of the parameter, with the possible exception of an analytic set of points. Moreover, the coefficients of the irreducible factors can be chosen to depend analytically on ε in a neighborhood of every point not belonging to this analytic set.

Our aim is to compute for all n > 2, ψ(n, h), the number of components of a certain quotient of the fixed point set of an involution in the "mod-n" Teichmuller space. This answers part of a question raised by Earle [2] and corrects and extends the answer due to Zarrow (See Theorem 2 of [6]).

It is shown that there exists a sequence of integers a1 < a2 <... such that each positive integer is a difference of elements of the sequence in exactly one way, and such that ak does not exceed a constant times k3. In fact we construct such a sequence with each ak in [C(k -1)3, Ck3), where C is an absolute constant.

In an article P. E. Himelwright and J. E. Williamson [3] proved a theorem on 1-factorability of Cartesian product of two graphs. With a very short proof we prove a more general theorem which immediately implies their theorem as a corollary. We will follow the notations and definitions of [1], [2] and [3].

Mason, [1, Theorem 2.6], proved that for any near-ring R, there are no non-trivial injective R-modules. In his proof he embedded R into a simple R -module G of arbitrarily high cardinality.

We generalize a result of E. E. Floyd on unicoherent Peano continua with involution to unicoherent locally connected regular hereditarily Lindelöf spaces. The result has an application in the theory of connectivity functions.

Let X be a closed convex subset of a Banach space and let T: X → X be a nonexpansive mapping, i.e.

In this note we prove:

Theorem. Let R be a right primitive ring with pair-wise non-isomorphic faithful irreducible modules Ml, M2,…, Mk. Let Di = EndR Mi. For each i, let be elements of Mi linearly independent over Di. For each i, let be a set of elements of Mi. Then there exists an element r of R such that uij = vijr, for i = 1, 2, …, k and j = 1, 2, …, ni.

Let N and K be finite groups with K acting on N, and let G be the semidirect product NK It was shown by Zassenhaus that if (|N|, |K|) — 1 and either N or K is solvable (an assumption later rendered redundant by the Feit-Thompson theorem), then all complements of N in G are conjugate to K.

The theorem below is one version of the Phragmén-Lindelöf principle [4], which extends the maximum modulus theorem. The theorem has many applications, including the proof of a better-known but less general result [3], which is sometimes attributed to Phragmén and Lindelöf.

For τ a piecewise C2 transformation, we present a method for obtaining an upper bound for the number of independent absolutely continuous measures invariant under τ.

Let τ = [0,1] and let τ:I→ J be a piecewise C2 transformation with infI1 |dτ/dx| > 1, where I1 = I-P and P denotes the points of discontinuity of τ and τ′