The following conjecture of I. D. Macdonald is confirmed in this paper. If G is a finite ρ-group generated by elements of breadth at most n, then the nilpotency class of G is bounded in terms of nonly.

We give a new perturbation theorem for symmetric differential expressions (relatively bounded perturbations, with relative bound 1) and prove with this theorem a new limit-point criterion generalizing earlier results of Schultze. We also obtain some new results in the fourth-order case.

Liouville theorems are obtained for general elliptic PDE-systems ℒU(x): = Σlɑl≦21Aα(x) əαU(x)=0 essentially under the assumption that ℒ satisfies a coerciveness estimate over .

A spin factor is a JW-factor of type I2. It is shown that certain automorphisms of finite dimensional spin factors extend to extremal positive linear maps on complex matrix algebras which are not decomposable, and hence, do not preserve extreme rays of the positive cone.

In three recent papers by Cavaretta et al., progress has been made in understanding the structure of bi-infinite totally positive matrices which have a block Toeplitz structure. The motivation for these papers came from certain problems of infinite spline interpolation where total positivity played an important role.

In this paper, we re-examine a class of infinite spline interpolation problems. We derive new results concerning the associated infinite matrices (periodic B-spline collocation matrices) which go beyond consequences of the general theory. Among other things, we identify the dimension of the null space of these matrices as the width of the largest band of strictly positive elements.

Small cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.

Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:

For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).

For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

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.

Duchon (1978) considered interpolation in ℝn by “Dm-splines”, which are interpolating functions having, in a sense, minimum energy. The purpose of this paper is to consider the analogous interpolation at the lattice of points in ℝn with integer co-ordinates, generalising aspects of Schoenberg's (1973) theory of cardinal spline interpolation. Following Schoenberg, we prove that higher order “basic” splines can be written as convolutions of lower order ones, using a new notion of convolution due to Jones (1982).

The problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.

Linearized local disturbances on a vortex sheet are known to develop singularities after a finite term in some cases but not in others. A simple test for the appearance of such singularities is given in terms of the Fourier transform of the initial disturbance. Such singularities are a consequence of the artificiality of the vortex sheet model and should not be regarded as physically meaningful.

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {x∊H: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, where

and ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.

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 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.

We discuss smooth changes of eigenvalues under perturbation of the boundary value problems given in the title. The simple eigenvalue criterion is developed in the setting of Banach spaces, so very general perturbations of both the differential equation and the boundary conditions are allowed. Further, we need no assumptions about self-adjointness of the original or perturbed problems. The discussion is concluded with the application of the simple eigenvalue criterion to two examples.

The asymptotic behaviour of L2-solutions of one-body Schrödinger equations (–δ+V–E)ψ = 0 in ΩR = {x ∊ Rn||x|>R} is investigated. We show, for example, that if V tends to zero in a certain sense for |x|→∞, then either |x|γ exp for some γ>0 or ψ has compact support. Related results are given for potentials tending to infinity for |x|→∞.

This paper shows that if S is a sphere in a Banach space and f: S → S is an α-contraction, then f has a fixed point. The paper generalizes a result of R. D. Nussbaum which holds for α-contractions only. The proof uses the Browder nonrepulsive fixed point theorem and is motivated by recent work of M. Martelli and G. D. Cooperman.

The existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.

In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

In this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.