Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

Suppose that f, g ∈ L∞[0,1] have discontinuities of the first kind only. Using the measure, max{‖f − h‖p, ‖g − h‖p}, of simultaneous Lp approximation, we show that the best simultaneous approximations, hp, to f and g by nondecreasing functions converge uniformly as p → 1. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. We discuss the inheritance of properties of f and g by hp, and of hp by h1.

Let F be a multifunction from [a, b] × ℝn × ℝn into ℝn, with non-empty closed convex values. In this paper we prove that, under suitable assumptions, the multivalued boundary value problem

has at least one solution u ∈ W2, p([a, b], ℝn). Next we point out some particular cases.

For each even dimension greater than or equal to 8, an infinite family of 3-step nilpotent Lie algebras over ℂ is constructed. In dimension m, the family contains isomorphism classes parameterised locally by approximately m3/48 essential parameters.

One particular case is studied further to get some global information about the variety of all nilpotent Lie algebras of dimension 8. Using the results obtained here, and some known facts, it is shown that there is a component consisting of algebras not having minimal possible central dimensions.

In a Minkowski plane with unit circle E, the product of the positive circumference of a plane convex body K and that of its polar dual is greater than or equal to the square of the Euclidean length of the polar dual of E. Equality holds if and only if K is a Euclidean unit circle.

We show that a C1-flow on a compact Riemannian manifold is structurally stable and topologically stable if and only if it satisfies Axiom A and the strong transversality condition. This improves Smale's conjecture for flows.

We study the existence of positive solutions of the periodic, Neumann or Dirichlet problem for the semilinear equation u″ + f(t, u) = 0, 0 ≤ t ≤ T, where f is a Carathéodory function. Our assumptions in each case are such that the problem possesses a lower solution or an upper solution.

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

The solution of the linear boundary layer problem is presented as a sum of the reduced solution and the layer function, which is approximated by a truncated orthogonal series. The layer subinterval is determined through the numerical layer length, which depends on the perturbation parameter and the degree of the spectral approximation. Modified Hermite polynomials are used as an orthogonal basis. Systematic methods are presented to obtain the recurrence relations for the coefficients in modified Hermite series solution of linear differential equations with polynomial coefficients. The upper bound for the error function is constructed and the numerical example is included.

A ring R is said to be an AE-ring if every endomorphism of its additive group R+ is a ring endomorphism. Clearly, the zero ring on any abelian group is an AE-ring. In a recent article, Birkenmeier and Heatherly characterised the so-called standard AE-lings, that is, the non-trivial AE-rings whose maximal 2-subgroup is a direct summand. The present article demonstrates the existence of non-standard AE-rings. Four questions posed by Birkenmeier and Heatherly are answered in the negative.

In this paper we obtain a sufficient and necessary condition for an analytic function f on D with Hadamard gaps, that is, for satisfying nk+1/nk ≥ λ 1 for all k, to belong to a kind of space consisting of analytic functions on D. The special cases of these spaces are BMOA and VMOA. In view of our result we can answer the open question given recently by Stroethoff.

An example of a Banach space is given which does not contain isomorphically l1 and does not admit the Kadec-Klee property.

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

Let W be an n−dimensional vector space over a field F. It is shown that the expected dimension of a vector subspace of W is n/2. If F is infinite, the expected dimension of a sum of a pair of subspaces of W is (n + 1)/2 if n > 1; and 3/4 if n = 1. If F is finite, with q elements, the expected dimension of a sum of subspaces of W depends on q and n. For fixed n, the limiting value of this expectation as q → ∞ is n if n is even; and n − 1/4 if n is odd. Moreover, if F is finite and n > 1, the expected dimension of a sum of three (not necessarily distinct) subspaces of W has limit n as q → ∞.

Sufficient conditions are obtained for the existence of a globally attractive positive periodic solution of the periodic diffusive delay logistic system

in which τ and K are positive periodic functions of period τ, n is a positive integer and ö is a nonnegative number; sufficient conditions are also obtained for all positive solutions to be oscillatory about the periodic solution.

We study some smoothness properties of a Banach space X that are related to the weak* asymptotic-norming properties of the dual space X*. These properties imply that X is an Asplund space and are related to the duality mapping of X.

Certain elliptic equations of higher order are studied and a sufficient condition is given that every solution is oscillatory in an exterior domain. The principal tool is an averaging technique which enables one to reduce the n–dimensional problem to a one-dimensional problem.

In this work we are concerned with the problem of the existence of an exponential dichotomy for the linear singularly perturbed system εx′ = A(t)x, where the matrix A(t) is piecewise uniformly continuous, that is, A(t) admits points of discontinuity but is uniformly continuous in any interval where it is continuous. We shall prove that the classical result regarding the existence of an exponential dichotomy extends to this case, when there is a constant γ > 0 such that |Reλ(t)| ≥ γ > 0 for any eigenvalue λ(t) of A(t). The proofs are obtained by means of the quasidiagonalisation of a non-constant matrix: For A(t), a piecewise uniformly continuous matrix and σ > 0 there exists a bounded, piecewise constant function L(t): J → ℂn×n, and a bounded matrix Δ(t, σ) such that L-1(t)A(t)L(t) = Λ(t) + Δ(t, σ), |Δ(t, σ)| ≤ σ, where Λ(t) is the diagonal matrix consisting of eigenvalues of A(t).

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

In this paper we introduce the concept of ℙ-regular, normal subset of a ℙ-regular semigroup, give an alternate characterisation of the maximum idempotent separating congruence in a ℙ-regular semigroup and finally describe the lattice of the idempotent separating congruences in a ℙ-regular semigroup.