Let be a bounded strongly pseudoconvex domain with C2 boundary . In this paper we prove that for a Bloch function in the existance of radial limits at almost all implies the existence of admissible limits almost everywhere on .

We consider two notions of well posedness for problems of the type and give conditions under which the majority (in Baire category sense) of bounded functions f defined in X × Y give rise to problems which are well posed. As a corollary we get that the problem is well posed for the majority of bounded lsc real valued functions f if, and only if, X contains a dense completely metrisable subset.

We characterise the relative Chebyshev centre of a compact subset F of a real Banach space in terms of the Gateaux derivative of the distance to farthest points. We present a relative-Chebyshev-centre characterisation of Hilbert space. In Hilbert space we show that the relative Chebyshev centre is in the closed convex hull of the metric projection of F, and we estimate the relative Chebyshev radius of F.

Let φ denote the normalised surface measure on the unit sphere Sn−1. We shall be interested in the weighted Lp estimate for Stein's maximal function Mφf, namely

where w is an Ap weight, especially for 1 < p ≤ 2. Using the Mellin transformation approach, we prove that the estimate holds for every weight wδ where w ∈ Ap and 0 ≤ δ < (p(n − 1) − n)/(n(p − 1)), for n ≥ 3 and n/(n − 1) < p ≤ 2.

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

In his work on a proof of Grushko's theorem by groupoid methods, Philip Higgins proved the following subgroup theorem.

Theorem. Suppose that θ:G → B is a surjective map of groups, where G and B are free products and , and where θ(Gλ) = Bλ for each λ. Let H be a subgroup of G with the property that the restriction of θ to H is surjective. Then there is a free decomposition of H with θ(Hλ) = Bλ for each λ.

As Higgins has noted, there is a common strategy in the proff of this theorem and in the groupoid proofs of Kurosh and Neilsen-Schreier subgroup theorems. Higgins conjectured the existence of a common generalisation of his theorem and the Kurosh theorem, proposed a plausible statement for such a result, and hinted that the common strategy of these proofs might be extended to give a proof of the conjecture. Examined at more detailed level, however, the common strategy used in the proofs of the Kurosh theorem and Higgin's theorem is seen to diverge into two strads. In this paper, it is shown that the construction entailed in these divergent approaches are in genera; incompatible. Thus any proof of Higgin's conjecture must require substantially different techniques. In particular, the proof of a theorem by Ordman that purports to affirm the conjecture is incorrect, and the approach used in his argument cannot yield a valid proof.

Certain Toeplitz type operators acting on the Bergman space A1 of the unit ball are considered and pluriharmonic symbols of commuting Toeplitz type operators are characterised by using M-harmonic function theory.

The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above by

where [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

In this paper, we tire concerned with duality of a multiobjective fractional program. Two different dual problems are introduced with respect to the primal multiobjective fractional program. Under a mild assumption, we prove a weak duality theorem and a strong duality theorem for each type of duality. Finally, we explore some relations between these two types of duality.

We improve the intuition and some ideas of G. D. Faulkner and M. C. Vipera presented in the article “Remainders of compactifications and their relation to a quotient lattice of the topology” [Proc. Amer. Math. Soc. 122 (1994)] in connection with the question about internal conditions for locally compact spaces X and Y under which βX/X ≅ βY/Y or, more generally, under which the remainders of compactifications of X belong to the collection of the remainders of compactifications of Y. We point out the reason why the quotient lattice of the topology considered by Faulkner and Vipera cannot lead to a satisfactory answer to the above question. We replace their lattice by the qoutient lattice of a new equivalence relation on a Wallman base in order to describe a method of constructing a Wallman-type compactification which allows us to deduce more complete solutions to the problems investigated by Faulkner and Vipera.

In the present note the magnitude of constants in Telyakovskii-type theorems is investigated. Our general approach to construct the linear operators yielding good constants is the one via Boolean sums. Explicit values for the constants in question are given for general convolution-type operators; the classical Fejér-Korovkin kernel is then used as an example for which one obtains rather small values. Furthermore, also an asymptotic assertion is derived which indicates the room left for improvement of the main results. This leads to a natural conjecture concluding this article.

We prove under quite general assumptions the existence of a positive solution to the equation Δu + f(x, u) + g(x)x.∇u = 0 in exterior domains of Rn (n ≥ 3).

We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l1.

Several interesting criteria for constructing triangulations associated with a given set of points in a plane have been introduced. In order to obtain optimal triangula-tion with respect to the min-max-angle criterion, it is essential to study the nature of neutral cases with respect to the criterion. Our aim in this paper is to establish precise equations for neutral set curves with respect to the min-max-angle criterion and to develop an algorithm to obtain a locally optimal triangulation with respect to the criterion.