In [2] G. Baumslag presents a list of twenty-three unsolved problems on one-relator groups. We give counterexamples to two of them.

Problem 5 asks whether a maximal locally free subgroup of a one-relator group always has finite “rank” (G has “rank” k if each finitely generated subgroup of G is contained in a k-generator subgroup of G).

Critical r, λ-systems are introduced such that any nontrivial r, λ-system must be an extension of some critical system. It is shown that parametric values for which critical systems can exist are restricted to λ(v-1) ≤ r(r-1) and, further, to λ(v-1)r(r-1) if the critical system is extendible.

Let {sm} be a discrete set of points in ℂN and λm any sequence of points in ℂ. We shall be interested in finding an entire function F(z) such that F(sm)=λm. This is of course easy if no restriction is placed on F, but we shall be interested in finding an F which in addition satisfies certain growth conditions.

We shall denote the variable z=(z1,..., zN), zj=xj+iyj,

In this note the Erdös-Renyi law of large numbers is extended to stationary Gaussian sequences.

Soit M une variété réelle de classe C∞ (connexe, paracompacte) et ω une un-forme complètement intégrable sur M (ω ∧ dω = 0). On admet qu’en certains points, dits singuliers, la un-forme peut s’annuler. Désignons par Jω(A) la matrice jacobienne de ω en un tel point A. Un point singulier A de ω est non dégénéré si le rang de Jω(A) est maximum. II est clair que ces singularités sont toujours isolées dans M.

Let R be a prime ring and T be a nontrivial automorphism of R. If xxT—xTx is in the center of the ring for every x in R, then R is a commutative integral domain.

If Y is a subspace of a Banach space X, either Y contains an isomorphic copy of ℓ1 or each weak Cauchy sequence in X/Y has a subsequence that is the image under the quotient mapping of a weak Cauchy sequence in X. If X is weakly sequentially complete and Y is reflexive, X/Y is weakly sequentially complete. Related structural results are given.

Locally quasi-uniform spaces are studied, and it is shown that a topological space (X, t) admits exactly one compatible locally quasi-uniform structure if and only if t is finite.

The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.

We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))bf(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=nef(e)n-1 for all e ∈ E.

In this paper, we introduce the concept of topological left substantial subsemigroup of a locally compact semigroup S and prove that if T is such a subsemigroup in S and M(T) is the measure algebra of T, then M(T)* is topological left amenable iff M(S)* is topological left amenable, an extension of a similar result for discrete semigroups.

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

Let Γn = {z = (z1, z2,…, zn): and Re(zi)>0, ∀i}. For a multisequence Mj, = (j1, j2,…,jn) and 0 < M(j)≤∞, let , q(r) = sup|j||{r|j|Mj} and .

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

In this note we present a natural “weak” form of a certain convexity estimate for evolution inequalities as given in Agmon-Nirenberg’s paper [1], p. 139 (see also A. Friedman [2], Theorem 4.2 and 4.3). Our proof will follow that given in [1] and [2] with the natural modifications due to the enlargement of the class of solutions which are taken into account.

Matrix iterative methods of solving systems of linear algebraic equations for a class of matrices which includes strictly and irreducibly diagonally dominant matrices are considered and a convergence theorem proved.

In 1956 R. H. Fox had occasion, while investigating fundamental groups of topological surfaces, to believe that the group <a, b | ab 2=b 3 a, ba 2=a 2 b> was trivial. Using the Todd-Coxeter coset enumeration algorithm a proof was obtained, see [3], and this algorithmic proof was used to produce an algebraic proof, see [2]. In [1] Benson and Mendelsohn, using a similar method to that of [2] showed that <a, b | ab n =b n+1 a, ba n =a n+1 b> is trivial. In this note we give a direct proof for the more general problem of describing the structure of the group <a, b | ab n =b ℓ a, ba n =a ℓ b>.