Approximate methods are developed for modelling the growth and collapse of clouds of cavitation bubbles near an infinite and semi-infinite rigid boundary, a cylinder, between two flat plates and in corners and near edges formed by planar boundaries. Where appropriate, comparisons are made between this approximate method and the more accurate boundary integral methods used in earlier calculations. It is found that the influence of nearby bubbles can be more important than the presence of boundaries. In confined geometries, such as a cylinder, or a cloud of bubbles, the effect of the volume change due to growth or collapse of the bubble can be important at much larger distances. The method provides valuable insight into bubble cloud phenomena.

In this paper we use the Dubovitski–Milyutin formalism to establish necessary and sufficient conditions for optimality in a nonlinear, distributed parameter control system, with convex cost criterion and initial condition not given a priori (that is it is not a known function but instead it belongs to a specified set). Our result extends a recent theorem of Lions. Finally a concrete example is worked out in detail.

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

A quasilinear singularly perturbed boundary value problem whose solution has a shock layer is investigated. Estimates of the derivatives of the solution are derived. Based on these estimates, a new independent variable is introduced. Then the transformed problem is solved numerically using finite - difference schemes. The transformation corresponds to solving the original problem on a mesh which is dense in the layer. The linear convergence uniform in the perturbation parameter is proved in the discrete L1 norm. Numerical results show uniform pointwise convergence too.

We characterise the relatively congruence distributive subquasivarieties of a modular variety using the modular commutator. Our characterisation allows us to extend the results of Dziobiak concerning relatively congruence distributive quasivarieties of nonassociative R-algebras.

A set of blocks which is a subset of a unique t – (v, k, λt) design is said to be a defining set of that design. We examine the properties of such a set, and show that its automorphism group is related to that of the whole design. Smallest defining sets are found for 2-designs and 3-designs on seven or eight varieties with block size three or four, revealing interesting combinatorial structures.

We prove that the crossed group algebra A of the infinite dihedral group over the real field defined by the generators a and b, relations b−1 ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ is a principal left ideal ring. This corrects a result of Buzási and provides the missing step towards the classification of finitely generated torsion-free RG-modules for groups G which contain an infinite cyclic subgroup of finite index.

In terms of a new notion, called prebalancedness, those extensions of a Butler group by a Butler group which are again Butler groups are characterised. It is shown that the prebalanced extensions of torsion-free abelian groups form a divisible subgroup in the group of extensions.

A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) = Σk〈Lkx, x〉uk with respect to a Hilbert basis . It is shown that with the norm | ‖Q‖ | = (Σk ‖Lk‖2)½ and inner-product 〈〈〈Q, P〉〉〉 = Σk 〈〈Lk, Mk〉〉, together with a multiplication defined componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators.

A slant immersion is defined as an isometric immersion from a Riemannnian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this article we give some fundamental results concerning slant immersions. Several results on slant surfaces in ℂ2 are also proved.

Let K be a function field in one variable over a finite field of characteristic three. If a ∈ K is not a cube, we show that the equation y2 = x3 + a has only finitely many solutions x, y ∈ K.

The infinitary divisors of a natural number n are the products of its divisors of the , where py is an exact prime-power divisor of n and (where yα = 0 or 1) is the binary representation of y. Infinitary harmonic numbers are those for which the infinitary divisors have integer harmonic mean. One of the results in this paper is that the number of infinitary harmonic numbers not exceeding x is less than 2.2 x1/2 2(1+ε)log x/log log x for any ε > 0 and x > n0(ε). A corollary is that the set of infinitary perfect numbers (numbers n whose proper infinitary divisors sum to n) has density zero.