Explicit formulas for Cotes numbers of the Gaussian Hermite quadrature formula based on the zeros of the nth Chebyshev polynomial of the second kind and their asymptotic behaviour as n → ∞ are given. This provides an answer to an analogue of Problem 26 of Tur´n.

In this paper we construct dual pairs of problems, of both the Wolfe and Mond-Weir types, in which the objective contains a a support function and is therefore not differentiable. A special case which appears repeatedly in the literature is that in which the support function is the square root of a positive semidefinite quadratic form. This and other special cases can be readily generated from our result.

The main result: Let R be a 2-torson free semiprime ring and let D: R → R be a derivation. Suppose that [[D(x), x], x] = 0 holds for all x ∈ R. In this case [D(x), x] = 0 holds for all x ∈ R.

We characterise some star sign pattern matrices and linear tree sign pattern matrices that allow a nilpotent matrix.

Let A be a hyperplane arrangement in an arbitrary finite dimensional vector space V and let G ≤ GL(V) be an automorphism group of A. If λ is a complex representation of G such that (λ,1)GH=0 for all pointwise isotropy groups GH (H ∈ A), then we prove the “local-global” result that λ does not appear in the representation of G on the Orlik-Solomon algebra of A. The result is applied to complex reflection groups and to finite orthogonal groups. It may also be viewed as a combinatorial result concerning the homology of the lattice of intersections of A. A more general version of the main result is also discussed.

We establish a common fixed point principle for a commutative family of self-maps on an abstract set. This principle easily yields the Markoff-Kakutani theorem for affine maps, Kirk's theorem for nonexpansive maps and Cano's theorem for maps on the unit interval. As another application we obtain a new common fixed point theorem for a commutative family of maps on an arbitrary interval, which generalises an earlier result of Mitchell.

Let M be a Cartan-Hadamard manifold of dimension n (n ≥ 2). Suppose that M satisfies for every x > M outside a compact set an inequality:

where b, A are positive constants and A > 4. Then M admits a wealth of bounded harmonic functions, more precisely, the Dirichlet problem of the Laplacian of M at infinity can be solved for any continuous boundary data on Sn−1(∞).

In this paper, a second-order characterisation of η-convex C1, 1 functions is derived in a Hilbert space using a generalised second-order directional derivative. Using this result it is then shown that every C1, 1 function is locally weakly convex, that is, every C1, 1 real-valued function f can be represented as f (x) = h (x) − η‖x‖2 on a neighbourhood of x where h is a convex function and η > 0. Moreover, a characterisation of the generalised second-order directional derivative for max functions is given.

One might think that if the minimal surface equation had a solution on a smooth domain D ⊂ Rn with boundary values φ, it would have a solution with boundary values tφ for all 0 ≤ t ≤ 1. We give a counterexample in R2.

Sufficient conditions are obtained for the existence and global asymptotic stability of a periodic solution in Volterra's population system of integrodifferential equations with periodic coefficients. It is shown that if (i) the intraspecific negative feedbacks are instantaneous and dominate the interspecific effects (ii) the minimum possible growth rates are stronger than the maximum interspecific effects weighted with the respective sizes of all species, when they are near their potential maximum sizes, then the system of integrodifferential equations has a unique componentwise periodic solution which is globally asymptotically stable.

Metanilpotent Lockett classes are Hall closed. There is an example of a supersoluble, not Hall closed Fitting class.

This paper investigates a certain type of numerical range introduced by Stampfli. In particular, we investigate the convexity of this set of elements of operators on Hilbert spaces and its relationship to the algebra numerical range implemented by elements of a W*-algebra.

The aim of this paper is three-fold: to fill the gap between different deformation lemmas, to obtain a unifying minimax result where multiple linking situations can occur, and to locate the critical points as solutions of minimisation problems.

Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.

For any topological near-ring (which is not a ring) whose additive group is the additive group of real numbers, we investigate the near-ring of all continuous functions, under the pointwise operations, from a compact Hausdorff space into that near-ring. Specifically, we determine all the homomorphisms from one such near-ring of functions to another and we show that within a rather extensive class of spaces, the endomorphism semigroup of the near-ring of functions completely determines the topological structure of the space.

A continuous analogue is derived for Radó's comparison formulæ. The analogue is then employed to provide a result which continues Radó's result and interpolates an inequality of Pittenger.

The value function is presented by minimisation of a cost functional over admissible controls. The associated first order Bellman equations with varying control are treated. It turns out that the value function is a viscosity solution of the Bellman equation and the comparison principle holds, which is an essential tool in obtaining the uniqueness of the viscosity solutions.

We characterise Opial's condition, the non-strict Opial condition, and the uniform Opial condition for a Banach space X in terms of properties of the duality mapping from X into X*.

Existence theorems for pre-vector variational inequalities are established under different conditions on the operator T and the function η. As an application, we establish the existence of a weak minimum of an optimisation problem on η-invex functions.