The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

This paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

The symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

We develop various facets of the theory of quadratic forms on a Hilbert space suggested by a criterion of Kato which characterizes closed forms in terms of lower semicontinuity.

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.

The groups of the title are characterized arithmetically.

Formulae are derived for the numbers of completely 0-simple and completely simple semigroups with m ℒ-classes, n ℜ-classes and underlying finite group G.

It is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

Given the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

In a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

Conditions are obtained for certain elliptic balls in ℂn+1 to have empty intersection with the Nyquist set of a vector polynomial G(z). Such conditions are shown to yield explicit criteria for the existence of periodic solutions of non-autonomous scalar differential equations of the form A* G(D)y = p.

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.