We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p–type modulus of convexity implies order p strong unicity of the simultaneous approximant.

In this paper, we consider the simultaneous representation of pairs of positive integers. We show that every pair of large positive even integers can be represented in the form of a pair of linear equations in four prime variables and k powers of two. Here, k=63 in general and k=31 under the generalised Riemann hypothesis.

In 1958, Gagliardo showed that if u is a locally integrable function on a domain Ω satisfying the cone condition, with all weak derivatives belonging to the Lebesgue space Lp (Ω) (1 ≤ p < ∞), then u belongs to Lp (Ω) also. We extend this result to Orlicz spaces, and use it to extend a result of Marcus and Mizel on Nemitsky operators between Sobolev spaces to Orlicz-Sobolev spaces.

Two characterisations are obtained for a gamma mixture distribution. The first is a generalisation of a result of Engel, Zijlstra and Philips [4] and the second based on Gumbel's bivariate exponential distribution. The two characterisatio are of direct relevance to some practical problems.

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

Let C(X) denote the complete boolean algebra of Borel sets modulo first category sets of the space X. Given an isomorphism τ between C(X) and C(Y), where X and Y are complete metric spaces, it is shown that there exists a homeomorphism T, between residual subsets A of X and B of Y, that induces τ. When X = Y one can make A = B. An analogous result is stated when τ is a complete isomorphism onto a subalgebra.

Sufficient conditions are obtained for all solutions of a class of second order nonlinear functional differential equations to be nonoscillatory.

Varieties with enough injectives satisfy the congruence extension property (CEP). We investigate the CEP in modular varieties by using the techniques developed in the first part of the present paper. As corollaries, we obtain the results of B. Davey and J. Kollár for the congruence distributive case as well as the description of all varieties of groups and rings with CEP, given by B. Biró, E.W. Kiss and P.P. Pálfy.

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed modulo arbitrary polynomials.

In this paper, we exhibit the necessary and sufficient conditions for a ring R to have only the trivial torsion parts with respect to any (hereditary) radical on the category of left R-modules.

It is shown that a finite group G is a normal subgroup of the group of units of the group ring of G over the ring of integers modulo n if and only if G is abelian or n = 2 and G is isomorphic to the symmetric group on 3 letters.

In this paper, composition operators acting on Bergman-Orlicz spaces

are studied, where ψ is a non-constant, non-decreasing convex function defined on (-∞, ∞) which satisfies the growth condition . In fact, under a mild condition on ∞, we show that every holomorphic-self map ∞ of induces a bounded composition operator on and C∞ is compact on if and only if it is compact on .

We give sufficient conditions involving f, g and ω in order that systems of differential equations of the form y″ = f(x, y, y′), x in [0, 1] with fully nonlinear boundary conditions of the form g((y(0), y(1)), (y′(0), y′(1))) = 0 have solutions y with (x, y) in . We use Schauder degree theory in a novel space. Well known existence results for the Picard, the periodic and the Neumann boundary conditions follow as special cases of our results.

In this paper, we first prove an existence theorem for the integrodifferential equation (*)where f,k,x are functions with values in a Banach space E and the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the set S of all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where .

This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.

Let a,b,c be relatively prime positive integers such that a2+b2=c2. Half a century ago, Jeśmanowicz [‘Several remarks on Pythagorean numbers’, Wiadom. Mat.1 (1955/56), 196–202] conjectured that for any given positive integer n the only solution of (an)x+(bn)y=(cn)z in positive integers is (x,y,z)=(2,2,2). In this paper, we show that (8n)x+(15n)y=(17n)z has no solution in positive integers other than (x,y,z)=(2,2,2).

Wu [‘An order characterization of commutativity for $C^{\ast }$ -algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$ -algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$ -algebras in terms of the order.

We show that the left and the right uniformities on a locally connected topological group G coincide if and only if every left uniformly continuous real-valued function on G is right uniformly continuous.