In this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′).

We consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

The semilinear damped wave equations

subject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

Periodic solutions of certain one-dimensional non-autonomous differential equations are investigated (equation (1.4)); the independent variable is complex. The motivation, which is explained in the introductory section, is the connection with certain polynomial two-dimensional systems. Several classes of coefficients are considered; in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. In particular, we need to know when there is a full neighbourhood of periodic solutions. We give a number of sufficient conditions and investigate the implications for the corresponding two-dimensional systems.

For Q ∊ L2(0, 1) we investigate the set Γ ∊ ℝ2 of pairs (α β,) for which the problem

has a nontrivial solution which has exactly one zero in (0,1) and is positive near x = 0. We show that Γ is stable in a certain sense under small perturbations of Q. The dependence of Γ upon Q is illustrated by an example.

In this paper we establish continuity of the interface of the weak solution to an elliptic-parabolic problem. The physical background is the theory of partially saturated fluid flows in porous media. Our method is based on the maximum principle for parabolic equations. An essential assumption is that the flow is one-dimensional.

We consider nonlinear singular Volterra integral equations of the second kind. We generalise the transformation method introduced in Part I of this paper [6] to cope with both the nonlinearity and slightly more general singular kernels. We also consider a particular class of nonlinear equation for which the solution behaviour is known. Using this a priori knowledge, we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties. Applying the general theory of Part I of this paper, we prove convergence of this scheme.

We prove that a free discrete transformation group on a compact connected space X is completely determined by the way C(X) lies inside the corresponding crossed product C*-algebra.

The best constants in Landau's inequality for the difference operator in the classical sequence spaces lp are known explicitly only for p = 1, 2, ∞. This is true in both the infinite N = (0, 1, 2, …) and biinfinite Z= (… − 1, 0, 1, …) cases. It is known that there are no extremals when p = 2 in both the infinite and biinfinite cases. Also, it is known that there are extremals when p = ∞ in the biinfinite case. Here we prove that there are no extremals in the other three cases where the best constants are known explicitly. The proofs for these three cases are quite different from each other.

On donne dans cet article un théorème d'existence de solutions lipschitziennes pour des problèmes du type:

où Ω est un ouvert convexe borné de ℝn, n ≧ 2, p ≧ 2, aucune hypothèse de convexité n'est faite sur g ou f. On étend de la sorte des ŕesultats d'existence obtenus en dimension 1.

where Ω is a bounded open convex subset of ℝn, n ≧ 2, p ≧ 2; we suppose no assumption of convexity on g or f. In this way we extend existence results proved in dimension 1.)

We consider the boundary value problem

where Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

When the points of a projective space [9] map ternary cubics the maps of perfect cubes are the points of a del Pezzo surface F. Several manifolds linked with F are listed and the cubics mapped on these are noted. A new approach leads to the first of Aronhold's two invariants.

A simple method is given for the construction of real symmetric differential expressions that are not in the limit-point case but have the real half-line as essential spectrum.

We consider the second order, linear, differential equation

where q is real-valued. In the case we calculate the Titchmarsh-Weyl m-function associated with (*).

An Archimedean unital f-algebra A is called a U-algebra if, for every a∊A, there exists an invertible element u∊A such that a = u |a|. Characterisations of a U-algebra are established. As an application, an extension theorem of Hahn–Banach type on modules over a U-algebra and over the complexification of a Dedekind complete unital f-algebra is given.

The deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

In this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

Consider the Abel integral operator

where 0 < α < 1. Suppose u is in H1(0, 1) of H1-norm ≦E, and f is an element of L2(0, 1) such that ∥Au – f∥L−2 < ε. We give a regularised approximate solution uβ(f) of the equation

which satisfies

and can be computed simply by performing some integrations. The preceding error estimate can be sharpened by strengthening regularity conditions on u