We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

A noetherian ring R satisfies the descending chain condition on two-sided ideals (“is bi-artinian”) if and only if, for each prime P ∈ spec(R), R/P has a unique minimal ideal (necessarily idempotent and left-right essential in R/P). The analogous statement for merely right noetherian rings is false, although our proof does not use the full noetherian condition on both sides, requiring only that two-sided ideals be finitely generated on both sides and that R/Q be right Goldie for each Q ∈ spec(R). Examples exist, for each n∈ℕ and in all characteristics, of bi-artinian noetherian domains Dn with composition series of length 2n and with a unique maximal ideal of height n. Noetherian rings which satisfy the related E-restricted bi-d.c.c. do not, in general, satisfy the second layer condition (on either side), but do satisfy the Jacobson conjecture.

We give two new sufficient conditions for unbounded Hilbert space operators to be subnormal. The first assumes that the sequence //Tnf//2 on a suitable subset of the domain is completely monotonic, the second is similar to the one given by Lambert in [3] for bounded operators and involves the sequence of binomial expansion of the real part of the operator.

We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions.

In this paper, we study geometric structures on 2-dimensional simplicial complexes. In particular, we consider hyperbolic structures and measured foliations on these simplicial complexes. We describe the spaces of such structures and we relate the two resulting spaces in a manner which is analogous to Thurston's compactification of the Teichmüller space of a surface.

Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product of the pseudovariety of bands with a pseudovariety V of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in the case in which terms involving only multiplication and weak inversion are considered. It is also shown that, if V is a hyperdecidable (respectively canonically reducible) pseudovariety of groups, then so is W.

In this paper the authors consider the class of locally nilpotent groups that have the maximum condition on non-nilpotent subgroups.

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that

TkBTk =Tk and TkB=BTk.

Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1).

We give an example of an element r of a free group F, and an element s of minimal length in the normal closure of r in F, such that s is not conjugate to r±1 or to [r±1f], for any element f of F.

A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited set in E is relatively compact. Banach spaces with weak * sequentially compact dual balls (W*SCDB for short) are GP-spaces and l1 is a GP-space without W*SCDB. Disproving a conjecture of Rosenthal and inspired by James tree space, Hagler and Odell constructed a class of Banach spaces ([HO]-spaces) without both W*SCDB and subspaces isomorphic to l1. Schlumprecht has shown that there is a subclass of the [HO]-spaces which are also GP-spaces. It is not clear however if any [HO]-construction yields a GP-space—in fact it is not even clear that W*SCDB[lrarr ]GP-space is false in general for the class of Banach spaces containing no subspace isomorphic to l1. In this note the example of Hagler and Odell is modified to yield a GP-space without W*SCDB and without an isomorphic copy of l1.

Bai [1] gave an intrinsic integral inequality for compact minimal submanifolds of constant curvature Riemannian manifolds. In this paper, we extend Bai's result to the case of pseudo-umbilical submanifolds.

The projective Schur group of a commutative ring was introduced by Lorenz and Opolka. It was revived by Nelis and Van Oystaeyen, and later by Aljadeff and Sonn. In this paper we study the intriguing question that there seems to be no adequate version of the crossed product theorem for the projective Schur group. We present a radical group R(k) (k a field) situated between the Schur group and the projective Schur group, and we prove the crossed product theorem for R(k).

We investigate cyclic bi-embeddings in an orientable surface of Steiner triple systems on 31 points. Up to isomorphism, we show that there are precisely 2408 such embeddings. The relationship of these to solutions of Heffter's first difference problem is discussed and a procedure described which, under certain conditions, transforms one bi-embedding to another.