Using techniques from probability theory, it has been established that if μ is a probability measure on a separable, locally compact group, then the space of μ-harmonic functions on the group can be identified with C(X) for some compact, Hausdorff space X. The space X is known as the Poisson space of μ. We generalise this result in the context of a measure μ on a locally compact semigroup S, in particular establishing the existence of a Poisson space for non-separable groups. The proof is non-probabilistic, and depends on properties of projections on C(K)(K compact Hausdorff). We then show that if S is compact and the support of μ generates S, then the Poisson space associated with μ, is X, where X×G×Y is the Rees product representing the kernel of S.

We continue with the work of earlier papers concerning the use of partial dilferential equations to prove the uniform convergence of the eigenfunction expansion associated witha left definite two-parameter system of ordinary differential equations of the second order.

This paper treats the global qualitative behaviour of all bifurcating configurations of whirling nonlinearly elastic strings with ends fixed on the axis of rotation.

If von Kármán's substitution is made in the Navier-Stokes equations, and boundary conditions corresponding to a flow in all of space with constant angular velocities at infinity are imposed, a boundary value problem analgous to those for flow above a rotating disk and between rotating disks is obtained. It is shown here that this problem has no solution.

The following system of conservation laws is considered:

where σ: ℝ→ℝ is a smooth function monotonically increasing except in an interval. Two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other. This contrasts with the corresponding situation for strictly hyperbolic systems (σ'(u)>0 for all u), for which the two criteria are equivalent.

Let X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠

It is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.

In this note we prove the following

Theorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.

Integral transformations T of the formal form

where G is a Meijer-G function are studied on the spaces ℐμ,r of functions f such that

Results are obtained on the boundedness, representation and range of the transformations, and on when they have an inverse of the same formal form.

By suitably coupling convexity and weight function methods, we prove uniqueness and continuous dependence theorems in linear elastodynamics in unbounded domains without definiteness conditions on the elasticities. The class of solutions considered allows the “growth” at large spatial distances.

The operators Δhf ≡ f(x) on function spaces and Δxn ≡ xn+1–xn on sequence spaces replace derivatives to yield analogues of the Kolmogorov inequality. Estimates for best constants are given for many spaces and for a few the best constants are actually given.

The relations v1 and v2 defined on the lattice ℒ of varieties of inverse semigroups by v1 if and only if and v2 if and only if , where denotes tie variety of groups, are both congruences on ℒ the class v1, is simply the lattice of varieties of grcups and is therefore known to have cardinality .

The class v2 is precisely the sublattice of ℒ consisting of those varieties containing . Each v1-class contains preciselyone element of v2. The main result of this paper establishes that the sublattice v2 of ℒ has breadth . From this it follows that the lattice ℒ/v1 also has breadth . Some consequences concerning varieties generated by fundamental inverse semigroups are also considered.

In this paper energy estimates for solutions of the Dirichlet problem for the biharmonicequation, expressing Saint-Venant's principle in elasticity, are proved. From these integral inequalities, estimates for the maximum modulus of solutions and the gradient of solutions with homogeneous Diriehlet's boundary conditions in a neighbourhood of an irregular boundary point or in a neighbourhood of infinity are derived. These estimates characterize the continuity of solutions and their gradients at these points.

Let V be a vector space and End (V) the semigroup of endomorphisms of V. An affine mapping of V is a map A: V → V given by xA = xα + a, where a belongs to End (V) and a is some element of V. Let (V) be the semigroup of affine mappings of V.

Let E' denote the non-injective idempotents of End (V) and let ℰ denote the idempotents of (V). In this paper 〈ℰ〉 is determined in terms of 〈E′〉 when End (V) consists of all endomorphisms of V and when End (V) only contains the continuous endomorphisms (in which case we restrict V to being an inner product space).

The authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.