We study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.

The velocity profiles in volumes of either passive resistance or active transport are calculated for both linear and non-linear resistive models. These include fluid flows containing particles in a channel or pipe subject to either a steady or oscillatory pressure gradient, cilia induced transport, and laminar flow in a non-linear resistive shear layer. Resistive elements tend to substantially reduce the inertial phase lag component in oscillatory flows. Only small concentrations of particles are needed to reduce the flow field to the Darcy approximation. An ‘active porous media’ model for a cilia sublayer predicts accurate voleoity profiles for dense concentrations of cilia (for example, on Opaline, and in the lung).

Maximal sum-free sets in elementary abelian 3-groups and groups G = Z3 ⊕ Z3 ⊕ Zp where p is a prime congruent to 1 modulo 3 are completely characterized.

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H in the sense of Y. Billig and A. Pianzola for involutive complex Lie algebras L of arbitrary dimension. If L has such a decomposition and is infinite dimensional and simple, we show it is *-isomorphic to a direct limit of classical finite dimensional simple involutive Lie algebras of the same type A, B, C, or D.

We give transcendence measures for $p$-adic numbers $\unicode[STIX]{x1D709}$, having good rational (respectively, integer) approximations, that force them to be either $p$-adic $S$-numbers or $p$-adic $T$-numbers.

Given a positive Borel measure ${\it\mu}$ on the $n$-dimensional Euclidean space $\mathbb{C}^{n}$, we characterise the boundedness (and compactness) of Toeplitz operators $T_{{\it\mu}}$ between Fock spaces $F^{\infty }({\it\varphi})$ and $F^{p}({\it\varphi})$ with $0<p\leq \infty$ in terms of $t$-Berezin transforms and averaging functions of ${\it\mu}$. Our result extends recent work of Mengestie [‘On Toeplitz operators between Fock spaces’, Integral Equations Operator Theory78 (2014), 213–224] and others.

Maximal sum-free sets in groups Zn, where n is any positive integer such, that every prime divisor of n is congruent to 1 modulo 3, are completely characterized.

Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.

In this paper, we study the mean square asymptotic stability of a generalized half-linear neutral stochastic differential equation with variable delays applying fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are given to illustrate our results.

For $n\in \mathbb{Z} $ and $A\subseteq \mathbb{Z} $, let ${r}_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} + {a}_{2} , {a}_{1} \leq {a}_{2} \} $ and ${\delta }_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} - {a}_{2} \} $. We call $A$ a unique representation bi-basis if ${r}_{A} (n)= 1$ for all $n\in \mathbb{Z} $ and ${\delta }_{A} (n)= 1$ for all $n\in \mathbb{Z} \setminus \{ 0\} $. In this paper, we construct a unique representation bi-basis of $ \mathbb{Z} $ whose growth is logarithmic.

We introduce and study two semigroups of operators u+ and u_, defined in terms of unconditionally converging series. We prove a lifting result for unconditionally converging series that allows us to show examples of operators in u+. We obtain perturbative characterisations for these semigroups and, as a consequence, we derive characterisations for some classes of Banach spaces in terms of the semigroups. If u+(X, Y) is non-empty and every copy of c0 in Y is complemented, then the same is true in X. We solve the perturbation class problem for the semigroup u_, and we show that a Banach space X contains no copies of ℓ∞ if and only if for every equivalent norm |·| on X, the semiembeddings of (X, |·|) belong to u+.

This article develops survey lectures for general mathematical audiences which the author delivered at the 27th Annual Meeting of the Australian Mathematical Society at the University of Queensland, 1983, and at the 10th Austrian Congress of Mathematicians in Innsbruck, 1981. The central theme of these lectures was the use of probabilistic methods in the study of linear elliptic-parabolic differential equations of second order.

The starting point will be an orientative discussion of the role of Brownian motion in classical potential theory. It will then be discussed that, given an elliptic-parabolic differential operator L of a certain type, there exists a uniquely determined diffusion process which is linked with L formally in the same way in which Brownian motion is linked with the Laplace operator. The fundamental results of K. Itô, D.W.Stroock and S.R.S. Varadhan will be in the centre of this part of the paper. We will then proceed to the discussion of more refined problems of the same type for differentiable manifolds. A glimpse at stochastic Riemannian geometry will then close our tour d'horizon.

Let G be a finitely generated soluble group. The main result of this note is to prove that G is nilpotent-by-finite if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and two positive integers m = m (x,y), n = n (x, y) satisfying [x, nym] = 1. We prove also that if G is infinite and if m is a positive integer, then G is nilpotent-by-(finite of exponent dividing m) if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and a positive integer n = n (x,y) satisfying [x,nym] = 1.

Just over 60 years ago, G. Grünwald and J. Marcinkiewicz discovered a divergence phenomenon pertaining to Lagrange interpolation polynomials based on the Chebyshev nodes of the first kind. The main result of the present paper is an extension of their now classical theorem. In particular, we shall show that this divergence phenomenon occurs for odd higher order Hermite–Fejér interpolation polynomials of which Lagrange interpolation polynomials form one special case.

We prove some new theorems and reprove some old ones about finitely generated soluble groups and Lie algebras by a uniform method. Among the applications are Gruenberg's Theorem on Engel groups, for which we obtain a very short proof; and the Milnor and Wolf polynomial growth theorem. It is shown that a finitely generated soluble group with all 2-generator subgroups polycyclic is itself polycyclic, and that a finitely generated soluble Lie algebra, all of whose inner derivations are algebraic, is finite-dimensional. This last result enables us to give a partial answer to a question of Jacobson.

We prove that semi-symmetric ball-homogeneous spaces are locally symmetric and we use this result to prove that a semi-symmetric Riemannian manifold such that the volume of each sufficiently small geodesic ball is the same as in a Euclidean space, is locally flat.

It is shown that the category of fixed points of a left exact idempotent functor on a topos is again a topos. As well as a direct proof, a bicategorical proof is given which shows that the result only depends on certain bicategorical exactness properties.

In a Banach lattice or the hermitian part of a C*-algebra, every element a admits a decomposition a = a+ − a− such that and N(−a) = ‖a−‖, where N is the canonical half-norm of the positive cones. In general ordered Banach spaces, this property is related to the order structure of the duality map and the metric projectability of the positive cones, and it turns out to be equivalent to an “orthogonal” decomposability.

In this paper, we characterise the quasi-strong endomorphisms of the join of split graphs. We give conditions under which the quasi-strong endomorphisms of the join of split graphs form a monoid.

In this short paper we show that if (X, ∥ · ∥) is a Banach space, Γ a norming set for X and C is a nonempty, bounded and Γ sequentially compact subset of X, then in C the Γ-Opial condition for nets is equivalent to the Γ-Opial condition.