A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.

We study non-commutative convolution measure algebras satisfying the condition in the title and having an involution with a non-degenerate finite dimensional *-representation. We show first that the group algebra L1(G) of a locally compact group G satisfies these conditions. Then we show that to a given algebra A with the above conditions there corresponds a locally compact group G such that A is a * and L-subalgebra of M(G) and such that the enveloping C*-algebra of A is *isomorphic to C*(G). Finally we show for certain groups that L1(G) is the only example of such algebras, thus giving a characterisation of L1(G).

Precise decay estimates as t → ∞ are derived for a class of nonlinear second order ordinary differential equations of the form

where h, g and f are functions like

With α > −1, β > −1 and γ > −1.

Let X = [a, b] be a closed bounded real interval. Let B be the closed linear space of all bounded real valued functions defined on X, and let M ⊆ B be the closed convex cone consisting of all monotone non-decreasing functions on X. For f, g ∈ B and a fixed positive w ∈ B, we define the so-called best L∞-simultaneous approximant of f and g to be an element h* ∈ M satisfying

for all h ∈ M, where

We establish a duality result involving the value of d in terms of f, g and w only.

If in addition f, g and w are continuous, then some characterisation results are obtained.

It is shown that Witt's basic dimension formula and a more recent result of Klyachko imply each other. Then Klyachko's identities between certain idempotents in the group ring of Sn are supplemented by identities involving Wever's classical idempotent. This leads to a direct proof of Klyachko's theorem (and hence Witt's formula), avoiding any commutator collecting process. Furthermore, this approach explains why the Witt dimensions are numbers which otherwise occur when “counting necklaces”.

A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

The purpose of this note is to describe some of the standard data types of computer science in the language of distributive categories. We believe that in this way we have achieved a simplification and a formal clarification of the specification of these data types.

In this paper, we shall show that if m is a natural number and for every 0 ≦ n ≦ m, and 2ω ≦ ωm are assumed, then connected, locally Lindelöf, submetaLindelöf, normal spaces of character ≦ 2ω are Lindelöf. Furthermore, we shall show that if and only if connected, locally Lindelöf, submetaLindelöf, normal spaces of character ≦ 2ω and tightness ≦ ω are Lindelöf.

We present an example showing that a classical result due to Glaeser about the closedness of composition subalgebras of infinitely differentiable functions cannot be extended to the case of weakiy uniformly differentiable functions on Banach spaces.

This paper contains two main types of results. Firstly, we discuss what sort of critical points are obtained in various direct and dual minimax principles. The techniques we apply are widely applicable. Secondly, we obtain “best possible” results on which critical points of a function on Rn are removable.

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

A group G is torsion-free, finitely generated, and nilpotent if and only if G is a supersolvable R-group. An ordered polycylic group G is nilpotent if and only if there exists an order on G with respect to which the number of convex subgroups is one more than the length of G. If the factors of the upper central series of a torsion-free nilpotent group G are locally cyclic, then consecutive terms of the series are jumps, and the terms are absolutely convex subgroups.

We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Bouudedness Principle which is valid without any completeness or barrelledness assumptions.

A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T1-neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net convergence is defined in neighbourhood lattices as a consequence of replacing ‘point’ by ‘set’ in topological convergence, then the limits of convergent nets are unique. The relationship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete T1-neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If P(X) denotes the power set of X, then the observation that a filter in a topological space (X, T) is a net in P(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Fréchet filters and to the filter generated by the limit elements of a net. Further, if the ‘filter’ convergence of a filter F in a topological space (X, T) is given by , if η(x) ⊆ F, then the relationship between ‘filter’ convergence and the net convergence of a filter in P(X) is established. Finally, it is proved that, in the neighbourhood system ‘lifted’ from a topological space to P(P(X)), the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit.

The concept of kth Hölderian functions on an interval [a, b] which generalises the notion of Hölderian (Lipschitzian) functions of positive order on [a, b] is introduced. The relationship of such functions to functions of bounded kth variation and absolutely kth continuous functions is examined. Properties induced by higher order derivatives in this new class of functions are investigated.

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.