It is shown that if a nonconstant entire function f(z) and its derivative f(k)(z), share two finite values (counting multiplicities), then f(z) = f(k)(z), where k is a positive integer.

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.

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:

(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.

(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.

(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.

For these three classes of semigroups, type-II is equal to type-II construct.

A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.

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.

Global wellposedness theorems are established for a class of abstract Cauchy initial value problems and a class of abstract Volterra equations which have a linear semigroup as a convolution kernel. These existence theorems are used to show that a class of nonlinear operators and a class of perturbed linear operators are m-accretive. The m-accretiveness results are used in turn to represent solutions to the differential and integral equations.

Under the assumption that Ca = C([−r, 0], Sn−1(a)) is positively invariant for a > 0, two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in Ca. Moreover, a feasible sufficient condition is given, which is better for n = 2 than that given by Dos Reis and Baroni.

Let R be an associative ring with unity 1, N(R) the set of nilpotents, J(R) the Jacobson radical of R and n > 1 be a fixed integer. We prove that R is commutative if and only if it satisfies (xy)n = ynxn for all x, y ∈ R \ N(R) and commutators in R are n(n + 1)-torsion free. Moreover, we extend the same result in the case when x, y ∈ R\J(R).

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.

An asymptotic relation for the expected number of excursions above a boundary g(n) by a random walk Sn, n = 1,2, ‥, N is given in terms of an integral involving g. An integral test is given to determine whether the total excursion time has finite expectation. If some moment assumptions hold then the expectation of the total excursions is finite if and only if .

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.

The problem of obtaining explicit solutions to coupled linear reaction-diffusion partial differential equations is generally recognised as technically very difficult. Frequently it is possible to deduce seemingly simple expressions for Laplace or Fourier transforms of the solution but such transforms tend not to be amenable to simple inversion and usually involve, for example, square roots within a square root. Fortunately however, a general uncoupling procedure has previously been established which provides explicit integral expressions in terms of classical heat functions. Such expressions are especially useful for problems with zero boundary data but non-zero initial data. The purpose of this paper is to provide the formal details necessary to deduce corresponding uncoupling transformations for two dependent variables, which preserve zero initial data and constant boundary data. For the case of one dependent variable such a transformation is known as Dankwerts' transformation. For coupled systems the existence of a Dankwerts' transformation means that together with the existing uncoupling transformation, solutions of boundary value problems involving constant boundary data, can be decomposed, just as for the single heat equation, into a contribution from the initial condition and zero boundary data and a contribution from non-zero boundary data and zero initial condition. The problem considered is highly non-trivial and the final expressions obtained are correspondingly complicated. Nevertheless the end results are explicit and together with standard integration routines constitute a powerful solution procedure.

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarke's non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

In axisymmetric irrotational flows of a perfect fluid under gravity there are three basic conserved quantities; axial momentum, energy and a circulation based, radial moment of momentum. This paper adapts these conservation principles to describe cavity collapse adjacent to a rigid boundary in a semi-infinite perfect fluid. They afford a global model accounting for volume change, migration and jet formation; physically the most significant features of bubble collapse close to a rigid boundary.

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 give a formula for the norm on a polynomial ring modulo an ideal in terms of the zero-set of the ideal. We hint at the relation to resultants.

The existence of maximal arcs of a certain type in symmetric designs is shown to yield semiregular group divisible designs whose duals are also semiregular group divisible. Two infinite families of such group divisible designs are constructed. The group divisible designs in these families are, in general, not symmetric.

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 this paper some new results concerning computational analysis are established. The fundamental concepts of interval analysis and fixed point theorems suitable for computational purposes are developed and applied to concrete examples illustrating this new method of proving mathematical statements.