Localized travelling waves to reaction-diffusion systems on the real line are investigated. The issue addressed in this work is the transition to instability which arises when the essential spectrum crosses the imaginary-axis. In the first part of this work, it has been shown that large modulated pulses bifurcate near the onset of instability; they are a superposition of the primary pulse with spatially periodic Turing patterns of small amplitude. The bifurcating modulated pulses can be parametrized by the wavelength of the Turing patterns. Furthermore, they are time periodic in a moving frame. In the second part, spectral stability of the bifurcating modulated pulses is addressed. It is shown that the modulated pulses are spectrally stable if and only if the small Turing patterns are spectrally stable, that is, if their continuous spectrum only touches the imaginary-axis at zero. This requires an investigation of the period map associated with the time-periodic modulated pulses.

In this paper we give a local classification of the integral curves of implicit differential equations where F is a smooth function and p = dy/dx, at points where Fp = 0, Fpp ≠ 0 and where the discriminant {(x, y) : F = Fp = 0} has a Morse singularity. We also produce models for generic bifurcations of such equations and apply the results to the differential geometry of smooth surfaces. This completes the local classification of generic one-parameter families of binary differential equations (BDEs).

We study the 1-relator relative presentation 〈H, x|xaxbx−1c〉 where H is a group, a, b, c ∈ H, x ∉ H and b, c ≠ 1. We give necessary and sufficient conditions for this presentation to be aspherical apart from two outstanding special cases which remain open.

Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

We say that a Banach space E is a continuation space for a given parabolic problem if the E-norm of any non-global solution has to become unbounded. We will prove that for large classes of parabolic systems of two equations, the space E = Lr1 × Lr2 can be a continuation space even though the problem is not locally well posed in E. This stands in contrast with classical results for analogous scalar equations.

In this paper we prove that the domain of hyperbolicity of the polynomial xn + λ2nn−2+λ3xn−3+ … + λn,λiϵR intersected by the half-space λ2 ≧ – 1, has the property of Whitney, i.e., every two points of this set can be connected by a piecewise-smooth curve belonging to it, whose length is ≦C times greater than the euclidian distance between the points, where the constant C does not depend on the choice of the points. Parallel with this, we show that the values x1≦x2≦…≦xn of a random variable are uniquely determined by the corresponding probabilities and by thefirst n moments.

We consider some definitions of tangent space to a Radon measure μ on ℝn that have been given in the literature. In particular, we focus our attention on a recent distributional notion of tangent vector field to a measure and we compare it to other definitions coming from ‘geometric measure theory’, based on the idea of blow-up. After showing some classes of examples, we prove an estimate from above for the dimension of the tangent spaces and a rectifiability theorem which also includes the case of measures supported on sets of variable dimension.

The present paper describes the topological and ergodic structure of the set of bounded trajectories of the flow defined by a scalar convex differential equation. We characterize the minimal subsets, the ergodic measures concentrated on them, and study the longtime behaviour of the bounded trajectories in terms of the Lyapunov exponents of the linearized equations. In particular, we obtain conditions that guarantee the existence of almost-periodic, almost-automorphic and recurrent solutions.

Each sigma-finite subalgebra from the sigma-algebra of a measure space induces a conditional expectation operator which acts on L2 as well as the set of almost everywhere nonnegative measurable functions. The concept of localising set is introduced and shown to be closely related to certain functional equations involving . Localising sets are shown to arise naturally in the study of weighted point transformations f→ϕ. f°T, where ϕ is a measurable function and T is a measurable self-map of the state space. A complete characterisation of localising sets related to such transformations is given when the underlying measure space is completely atomic.

The asymptotic behaviour of L2-solutions of one-body Schrödinger equations (–δ+V–E)ψ = 0 in ΩR = {x ∊ Rn||x|>R} is investigated. We show, for example, that if V tends to zero in a certain sense for |x|→∞, then either |x|γ exp for some γ>0 or ψ has compact support. Related results are given for potentials tending to infinity for |x|→∞.

For a non-negative function ū(x), we study the long-time behaviour of solutions of the heat equation

with the Dirichlet or Neumann boundary conditions at x = 0. We find a critical parameter λD > 0 such that the solution subjected to the Dirichlet boundary condition tends to a spatially localized wave travelling to infinity in the space variable. On the other hand, there exists a λN > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dynamics is considerably influenced by the choice of boundary conditions.

We consider weak solutions of parabolic systems of the type

where the structure function b is differentiable with respect to x and satisfies standard ellipticity and growth properties with polynomial growth rate p ∊ (2n/(n + 2), 2). We investigate regularity properties of the solution, including the existence of second-order spatial derivatives, the existence of the time derivative and the higher integrability of the spatial gradient. As an application, we derive dimension estimates for the singular set of solutions of homogeneous parabolic systems. More precisely, we establish the bound

provided the structure function depends Höolder continuously on the space variable with Höolder exponent β ∊(0, 1].

On any spacelike surface in a light cone of four-dimensional Lorentz–Minkowski space, a distinguished smooth function is considered. We show how both extrinsic and intrinsic geometry of such a surface are codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann-type result. Two remarkable families of examples of spacelike surfaces in a light cone are explicitly constructed. Finally, several results that involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a light cone are obtained.

The asymptotic behaviour for large-time values of solutions of the initial-boundary-value problem for the Benjamin–Bona–Mahony–Burgers (BBMB) equation is studied. The solution of the initial-boundary-value problem is proved to converge to the standing wave as t → ∞ uniformly with respect to x ∈ R1. The estimate obtained of the time-decay rate of the remainder appears to depend on the rate of decay at infinity of the deviation of initial data from the standing wave. Due to the known expansion formulae with respect to the eigenfunctions associated to the stationary Schrodinger operator with the standing wave as a potential, it is possible to find the second term of the large-time asymptotic expansion of the solution to the initial-boundary-value problem for the BBMB equation.

In a paper read nearly three years ago to this Society, I have attempted

1. (1) To review the existing state of statistics ;

2. (2) To define the nature of the subject, and its relation to history and the sciences;

3. (3) Broadly to group and co-ordinate the whole body of existing and possible statistics, in relation to the respective statistical sciences ; and

4. (4) In accordance with the preliminary sciences to frame a classification embracing all existing and possible sociological statistics. Moreover,

5. (5) This was shown to involve, or rather actually to constitute, an aspect of the pressing problem of the systematisation of the literature of economics, of which

6. (6) The existing schools were briefly criticised ;

7. (7) The relation of the conceptions of scientific economics to practical economics was outlined ;

8. (8) As also their relation to ethics.

The Cauchy problem for semi-linear heat equations with singular initial data is studied, where N > 2, p > (N + 2)/N, and l > 0 is a parameter. We establish the existence and multiplicity of positive self-similar solutions for the problem by applying the ordinary differential equation shooting method to the corresponding spatial profile problem.

We consider the positive solutions to the semilinear equation:

where Ω denotes a smooth bounded region in ℝN(N > 1) and ℷ 0. Here f :[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such that f(0)<0 (semipositone). Assuming that f′(∞)≡limt→∞f(t)> 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] where f′(∞)≦0.

The semilinear parabolic system ut + A(x, D)u = g(u) in (0, ∞) × Ω, Ω⊂ℝn bounded, u ∈ ℝN, with homogeneous boundary conditions B(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearity g is assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solution u is relatively compact in .

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

In this paper we prove a non-existence result for nonlinear parabolic problems with zero lower-order terms whose model is

where Δp=div(|∇u|p−2∇u) is the usual p-laplace operator, λ is measure concentrated on a set of zero parabolic r-capacity (1<p<r) and q is large enough.