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.

The investigation of general F-abundant semigroups is initiated. After obtaining some properties of such semigroups, the structure of a class of F-abundant semigroups is established. In addition, a problem raised in [2] is positively answered.

It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k).

All the Gabriel topologies on a Dubrovin valuation ring are classified in terms of its prime ideals. Furthermore, these Gabriel topologies are cogenerated by two kinds of indecomposable injective modules.

Since its elaboration in 1983 by Weiss, Tabor and Carnevale, the method to explicitly build the Bäcklund transformation of a partial differential equation (PDE) from singularity analysis only has been improved in several complementary directions, and at the present time it succeeds for practically all PDEs in 1+1-dimensions. The current state of the art is presented, and the emphasis is put on understanding the method. There are two important stages: first, the definition (identified with a Darboux transformation) of a resummation variable to make the Laurent series a finite one as requested by the definition of the word integrability; second, the link (identified with a linearizing formula to be taken from the classification of Painlevé and Gambier) between this resummation variable and the Lax pair to be found.

Methods of calculating discriminant and image Milnor numbers in terms of the Milnor numbers of multiple point spaces are described for cases f:{ℂ}^3 \to {ℂ}^4 and F:{ℂ}^3 \to {ℂ}^3.

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 prove a natural factorization of supersolvable groups and then we give another characterization of them in connection with the Fitting subgroup. Applying these theorems we describe the structure of some subclasses of supersolvable groups.

We derive a finite normal series for the group {\rm Aut}\nolimits _*(X\;\!) of self-homotopy equivalences which induce identity automorphisms of homology groups, where X is a countable, simply-connected and finite-dimensional CW-complex.

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.

It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem.

In a recent paper we presented a truncation-type method of deriving Bäcklund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painlevé equations. For the third Painlevé equation we are able to obtain all fundamental Bäcklund transformations for the case where the parameters satisfy \gamma \delta \neq 0. For the fifth Painlevé equation our approach yields what appears to be all known Bäcklund transformations.

A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.

By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.

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.

Soliton equations whose solutions are expressed by Pfaffians are briefly discussed. Included are a discrete-time Toda equation of BKP type, a modified Toda equation of BKP type, a coupled modified KdV equation and a coupled modified KdV equation of derivative type.

We introduce and study the type I-, II-, and III-Λ-complete continuity property of Banach spaces, where Λ is a subset of the dual group of a compact metrizable abelian group G.

Methods from fibrewise homology theory are illustrated by computations of cohomology rings of certain mapping spaces arising in the geometry of loop groups, specifically the spaces of maps from S^1 to the classifying space BSO(n) of SO(n) and maps from S{\hskip1}^2 to BSU(n).

Two coupled bilinear equations are considered, and then two new coupled differential-difference systems are found. Also two special reductions of these two systems are studied. By using Hirota's method, Bäcklund transformation and superposition formulae, soliton solutions to these equations are presented.

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 T be a dominant operator that is a quasi-affine transform of an M-hyponormal operator. In this paper we show that if f is a function analytic on a neighborhood of the spectrum of T, then Weyl's theorem holds for f(T{\hskip1}).