It is shown that, among all the L2 normalized spherical harmonics of a given degree, the L4 norm is locally maximized by the ‘highest weight’ function .

A linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

In 1973, V.I.Lomonosov introduced a new technique for finding invariant and hyperinvariant subspaces for certain classes of (continuous, linear) operators on complex Banach spaces. Recall that a closed subspace M of the Banach space X is called hyperinvariant for the operator T if S(M) ⊂ M for every operator S which commutes with T.

An intrinsic pseudodistance dM is associated to the underlying causal structure of a Lorentzian manifold (M, g). An infinitesimal form for dM is defined on the bundle of null vectors over M and is used to prove that, under certain physically reasonable energy conditions, M is conformally hyperbolic, i.e. that dM is a true distance. As a corollary we obtain sufficient conditions for the conformal hyperbolicity of a globally hyperbolic space-time. Some exact solutions of the Einstein field equations are discussed and possible applications to the study of singularities are outlined.

This paper is a continuation of (8), and is concerned with geometrical properties of special null hypersurfaces. In particular, on a one-parameter family of null hypersurfaces in four-dimensional Lorentzian manifold V4, we consider the relation between their normal and the Debever vectors, especially repeated ones. Throughout this paper, the same notations as those in (8) are used.

We obtain here estimates for the expectation of a restricted quadratic variation of L1 bounded (weak) martingales and positive submartingales, and for related functionals. These inequalities are then used to give a quantitative version of a recent result of D. J. Aldous and D. H. Fremlin concerning the L1 norms of expansions with respect to uniformly integrable L1-normalized martingale difference sequences. Another application relates precisely the degree of norm divergence of an L1-bounded martingale to the degree of non-uniform integrability of that martingale. A partial analogue of this result is proved for certain vector valued martingales with values in L1 which are related to the Radon–Nikodým property.

Asymptotic formulae for the capacitances of two oppositely charged, identical, circular, coaxial discs and two identical, infinite parallel strips separated by dielectric slabs are derived from the dual integral equations approach. The formulation in terms of dual integral equations using transforms gives rise to relatively simple Green's functions in the transformed space and renders the derivation of the asymptotic formulae relatively easy. The solution near the edge of the plates, a two-dimensional problem previously thought not solvable by the Wiener-Hopf technique, is solved indirectly using the method. Solution of the Helmholtz wave equation is first sought, and the solution to Laplace's equation is obtained by letting the wavenumber go to zero. The solution away from the edge is obtained by solving the dual integral equations approximately. The total charge on the plate is obtained by matching the solution near the edge and away from the edge giving the capacitance.

It is shown that if, in an infinite homogeneous unbounded elastic heat-conducting solid, we are given the thermal stresses which arise when any arbitrarily shaped subregion of the upper half-space z > 0 is emitting any quantity of heat into the surrounding, then on introducing a different heat-conducting material into the half-space z < 0, the new thermal stresses in z >0 and in z < 0 are explicitly expressible in an invariant form in terms of the known thermal stresses in the homogeneous infinite solid. Specialization of the derived dependence shows that the physically interesting interfacial discontinuities inherent in the problem admit the representation

while the traction transmitted across the interface takes the form

where αi and βi are constants. An application is then made to the case when the region emitting heat into the surrounding is ellipsoidal, since this is a fundamental shape of practical importance. Applications can however be made to any shape of the region emitting heat, provided that the corresponding harmonic and biharmonic potentials are available.

The paper may be considered to have its origin in the half-space theorem of Lorentz in hydrodynamics.

The existence of solutions is considered for equations of the form

for x ∈ H (a Hilbert space), P a compact linear operator on H; Q(x) a bounded linear operator on H and continuous in x and uniformly bounded; g(x) a continuous uniformly bounded map with range in H. Two situations are considered: Q(x) lies in a weakly compact set of operators for which (a) (I − PQ(x)) is invertible (non-resonance case) or (b) (I − P(Q(x) + λI)) is invertible for 0 < λ ≤ α (resonance case).

Consider an nth-order linear ordinary differential equation

Suppose the αj(x) are holomorphic for x ∈ S, 0 < x0 ≤ |x| < ∞, where S is an open sector of the complex plane with vertex at the origin and positive central angle not exceeding π. Suppose as x → ∞ in each closed subsector of S. The problem of finding a basis for the solutions of (1) can be reduced by a sequence of transformations (see, for example, Gilbert (1)) to the problem of finding a fundamental matrix for a system of the form

where q is a non-negative integer, A(x) is an m by m matrix which is holomorphic for x ∈ S, x0 < ≤ |x| < ∞, and as x → ∞ in each closed subsector of S. If m ≥ 2, A0 is an m by m matrix with elements all zero except for l's on the upper off-diagonal, and the elements of Ar for r ≥ 1 are all zero except possibly in the last row. (There is one such problem (2) for each root, μ of the equation with multiplicity m ≥ 2. Recall that the coefficient αn−r, 0 the first term in the asymptotic expansion of the coefficint αn−r(x) of equation (1).) We denote the elements of the last row of Ar by 1 ≤ k ≤ m. The system (2) has an irregular singular point at infinity.

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

The Green's functions for semi-infinite and adjacent semi-infinite media are considered in one-speed transport theory. Chandrasekhar's Principles of Invariance are used to formulate integral equations that describe the emergent and interface angular distributions. It is shown that mathematically these are equivalent to subtracting from the transport equation known solutions to related, simpler problems. Duplication of effort is avoided in this way. For adjacent semi-infinite media, a difficulty encountered in the solutions of the integral equations necessitates the application of a Laplace transform for its resolution. Some virtues of using the transform are explored, e.g. in the determination of the internal distribution where it yields concise expressions for the Green's function.