We discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

The class of quadratic systems having a parabola composed of integral curves is examined. Canonical forms are found for the members of this class, and conditions are obtained, using the Bendixson's Criterion and the Poincaré–Bendixson Theorem, for the existence or non-existence of limit cycles, in the case where there is a limit cycle “inside” the parabola (that is, in the convex component of its compliment).

The existence of a set of real numbers σ, depending only on ∧0, is established such that if λ ∉ σ, the system Y'=(i∧0 + R)Y can be transformed into a system Z' = (i∧ + S)Z of the Levinson form. Here ∧0 and ∧ are real diagonal matrices, with ∧0 constant, and R(x) = ξ(x)T(λx). The scalar factor ξ(x) is o(l) (x →∞) and T belongs to a certain class of periodic matrices. The consequences for non-resonance, and the necessity of this class, are discussed.

We consider a generalisation of the liquid drop problem, introduced in [1, Part II], by allowing the upper and lower surfaces to have different surface tension coefficients γv and γu. We study the existence, uniqueness and regularity of this problem. In addition, we show that as γv/γu →0, the solution of this problem converges to the solution of the “plasma problem”.

We investigate algebras associated with a (discrete) Clifford semigroup S =∪ {Ge: e ∈ E{. We show that the representation theory for S is determined by an enveloping Clifford semigroup UC(S) =∪ {Gx: x ∈ X} where X is the filter completion of the semilattice E. We describe the representation theory in terms of both disintegration theory and sheaf theory.

We show that the property of linear operators to be in the surjective hull (injective hull) of the ideal of strictly singular (strictly cosingular) operators between Banach spaces is an interpolation property with respect to the real interpolation method with parameters 0 < ủ < 1 and < p < ℞.

Suppose that M is a closed, connected and smooth manifold of dimension n = 8k + 5, with k ≧1. Let η be an n-plane bundle over M. Under suitable conditions on M, we derive necessary and sufficient conditions for the span of η to be ≧j, j = 5 or 6. We then apply the results to the tangent bundle of M. In particular, we prove a conjecture of E. Thomas, namely, if M is 3-connected mod 2, then span M ≧ 5 if, and only if, χ2(M) = 0. We prove that if also w8k(M) = 0, then span M≧6. We also derive some immersion theorems for M.

An Lp inequality for l < p < 2 is established and applications to fixed points of uniformly Lipschitz mappings and strongly unique best approximations are given.

For an H-space with a generating subspace, we construct a space whose K-cohomology is a direct sum of a truncated polynomial algebra and an ideal, which enables technical restrictions to be removed from several known results in the homotopy theory of H-spaces.

We establish that if Ω is an open square and if P is an arbitrary point of Ω, then the solutions of the wave equation with Dirichlet boundary condition utt − Δu = 0 in R × Ω, u = 0 on R × Γ can remain strictly positive during an arbitrarily large time. In fact stronger results are proved, considering several points of Ω simultaneously and prescribing different, more general, behaviour of the solution at these points.

When a symmetric rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. These equations at the boundary contain a time derivative and thus are of a dynamic nature. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity and pressure fields of the fluid motion and the angular velocity of the rigid body.

In this paper we formulate the physical problem for the case of rotation about an axis of symmetry as an abstract ordinary differential equation in two Banach spaces in which the velocity field is the only unknown. To achieve this, a method for the elimination of the pressure field, which also occurs in the boundary condition, is developed. Existence and uniqueness results for the abstract equation are derived with the aid of the theory of B-evolutions and the associated theory of fractional powers of a closed pair of operators.

We prove that for primitive positive definite binary quadratic forms having fundamental discriminant the concepts of genus and of ℚ-equivalence coincide.

Lp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.

A new proof is given of the partial regularity of minimisers under the principal assumptions of uniform strict quasiconvexity and polynomial growth. The proof is based on a Caccioppoli inequality and an “indirect” blow-up argument; this avoids the technical complications of a “direct” argument.

In this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.

Some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised. Different approaches are compared with one another, using topological and variational methods and the theory of weighted eigenvalue problems.