If A and B are C*-algebras and X is an operator A, B-bimodule, then points of X can be separated from closed A, B-absolutely convex subsets of X by completely bounded A, B-bimodule homomorphisms from X into B(K), where K is a Hilbert space and the A, B-bimodule structure on B(K) is induced by a pair of representations π : A → B(K) and σ : B → B(K). If A and B are von Neumann algebras and X is a normal (not necessarily dual) operator A, B-bimodule, those A, B-absolutely convex subsets of X are characterized which can be separated from points of X as above, but with the additional requirement that the two representations π and σ are normal. This requires a new topology on X, which has appeared also in connection with some other questions concerning operator modules.

The results obtained may be summarised in the following table comparing the filial and fraternal coefficients for autosomal and sex-linked variation. When dominance is complete the values given are referred to equal frequency of the allelomorphic genes, i.e.p = ½ = q.

Briefly, then, the contribution of sex-linked genes to the observed variance will tend to lower the correlation between father and son and between brother and sister. It will raise the correlation between father and daughter or between sister and sister conspicuously. These considerations emphasise the interest attaching to a separate investigation of fraternal correlation for sibs of all three sex classes in connection with studies on twins. If the correlation for dizygotic twins of like sex is higher than for dizygotic twins of unlike sex, the difference might be due to the greater similarity of environment for two individuals of the same sex, or it might be due to the contribution of sex-linked genes to the observed variance. In the latter case the correlation for boy pairs should be significantly lower than the correlation for girl pairs. The wide discrepancies in recorded data for fraternal correlations with respect to intelligence quotients might be re-examined from this standpoint. If an appreciable proportion of the total variance is due to sex-linked genes, it is of more importance that a boy should have a clever mother than a clever father.

We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

We have, during the last two years, made a series of observations upon specimens of Sagartia troglodytes which are at least fifty years old, and have thought it worth while to give a somewhat detailed account of these, as, so far as we can ascertain, there is only one other recorded case of longevity in Coelenterates (see p. 302), and very few in the whole of the Invertebrata.

Two theorems on the spectrum of operator pencils generated by the three-dimensional system of anisotropic elasticity in the complement of a plane angle are proved.

We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .

In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

With certain initial and boundary conditions the solution u* to the semilinear heat equation ∂u*/∂t = ∂u* + λ * f(u*), where f is a positive superlinear function and λ is the supremum of the open spectrum for the steady state problem Δw + λf(w) = 0, is found to exist for all time and to be unbounded. Moreover u* approaches w* a singular steady state, as / tends to infinity.

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,

where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

The aim of this paper is to study Hamiltonian elliptic system of the form 0.1

$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$

0.2

$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$

The hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.

Sharp weak type (1,1) and Lp estimates in dimension one are obtained for uncentred maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions, uncentred maximal functions fail to satisfy such estimates. Analogous results for centred maximal functions are given in all dimensions.

In this paper we prove refinements, extensions and counterparts of known integral inequalities for concave functions. Furthermore, we apply our results to find new inequalities for some special functions and mean values.

We combine the sub- and supersolution method and perturbation arguments to obtain positive solutions of singular quasi-monotone (p, q)-Laplacian systems.

Let f : Mn×n → R ∪ {∞} be a function on the space Mn×n of n by n matrices that is invariant with respect to the left and right multiplication by proper orthogonal tensors. It is shown that f(A) ≤ f(Ā) if f is convex and the partial sums of the singular values of A, Ā ∈ Mn×n satisfy certain ordering inequalities. The same holds if f is rank 1 convex and the partial products of the singular values satisfy analogous inequalities. The proofs emphasize the roles of the ordered-forces inequalities and the Baker-Ericksen inequalities for invariant convex and rank 1 convex functions. As an application, the evaluation of the convex and lamination convex hulls of fully rotationally invariant sets by Dacorogna and Tanteri is simplified and similar results are given for sets invariant only with respect to the proper orthogonal group.

This paper is concerned with homogenisation processes for parametrised families of transport equations in ℝn symmetric hyperbolic systems in several space dimensions and anisotropic wave equations. The main tools for carrying out the homogenisation are the Young measures and Radon transform combined with the integral representation of holomorphic functions of Nevanlinna-Pick's type.

The bivariate series defines a partial theta function. For fixed q, θ(q, ·) is an entire function. We show that for ∣q∣ ≤ 0.108 the function θ(q, ·) has no multiple zeros.

We study a class of BGK approximations of parabolic systems in one space dimension. We prove stability and existence of global solutions for this model. Moreover, under certain conditions, we prove a rigorous result of convergence toward the formal limit, by using compensated compactness techniques.

The main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations

0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditions

r = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].