We use a direct, geometric approach to study the free surface boundary conditions for stationary flows of viscous liquids. The free surface problem is characterised by a mapping on smooth variations of a given configuration; this mapping has a simple structure, which we determine by computing its differential, and studying it in terms of the space dimension and the surface tension coefficient. Applications are given to problems of existence, uniqueness and regularity in free surface flows.

This paper considers the Cauchy problem for the isentropic equations of gas dynamics in Euler coordinates ρt + (ρu)x = 0, (ρu)t + (ρu)2 + P(ρ))x=0 and gives the Hölder-continuous solution by applying the method of vanishing viscosity.

In this paper we introduce a new tool in geometric measure theory and then we apply it to study the rank properties of the derivatives of vector functions with bounded variation.

The Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

In this paper we obtain new results which relate the number of conjugacy classes of л-elements of a finite group and an arbitrary subgroup, which are analogous to some results about normal subgroups. We also prove some new results which show the relationship between class numbers and splitting theorems. Our proofs only involve elementary techniques.

In this paper we characterise left orders in strongly regular rings, both for classical left orders and left orders in the sense of Fountain and Gould [2] where the ring of quotients need not have an identity.

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.

A simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

Asymptotic behaviour of solutions of polyharmonic equations

on exterior domains in Rn is considered under suitable conditions on f. It is shown that finite energysolutions u have the asymptotic property

This partially extends results of Egnell [6] for m = 1.

Let In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in

Any solution to the problem

is shown to decay to zero in the strong topology of the energy space as t→ ∞. The function β is allowed to be nonmonotone and d is a nonnegative function strictly positive on a nonvoid subset of (0, л).

The Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

For a Hasse domain R in a global field F, the distribution of genera of R-lattices with specified invariants among the nonisometric quadratic spaces over F which contain them is studied. It is shown that such genera are equally distributed for spaces of dimension 2, but that this is not generally the case for spaces of dimension exceeding 2. Theresult in the binary case yields an extension of a dimension exceeding 2. The result in the binary case yields an extension of a theorem of Gauss.

It is shown that for each v ∊ {0, 1, 2}, the class of v-primitive near-rings is not axiomatisable.