We consider a purely quasi-incompressible elasticity model. We rigorously establish asymptotic expansions of near- and far-field measurements of the transient elastic wave induced by a small elastic anomaly. Our proof uses layer potential techniques for the modified Stokes system. Based on these formulae, we design asymptotic imaging methods leading to a quantitative estimation of elastic and geometrical parameters of the anomaly.

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

We consider the equation

where

Under these conditions, (1) is correctly solvable in L1(ℝ), i.e.

(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);

(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),

In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimate

where c2 is some absolutely positive constant.

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterized.

We study Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3. The close connections of these algebras with Lie algebras graded over the non-reduced root system BC1, with J-ternary algebras and with Freudenthal–Kantor triple systems are explored.

We use the theory of semiperfect rings to derive decomposition theorems for H- and co-H-spaces, generalizing the results of Wilkerson. The results are then used to prove reducibility of self-homotopy equivalences for arbitrary p-local H- and co-H-spaces.

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

We provide a general procedure for computing the algebraic K-theory of finitely generated virtually free groups. The procedure describes these groups in terms of the algebraic K-theory of various finite subgroups and various Farrell Nil groups. We illustrate this process by carrying out the computation for several interesting classes of examples. The first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature. The last two classes of examples yield new computations.

We consider the regularity of solutions to a multidimensional moving boundary problem modelling the growth of non-necrotic solid tumours. The model equations include two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure within the tumour, respectively, and a first-order partial differential equation governing the evolution of the moving boundary on which surface tension effects counteract the internal pressure. On account of the moving boundary and surface tension effects, this problem is a nonlinear problem involving non-local terms. By employing the functional analytic method and the theory of maximal regularity, we prove that the moving boundary is real analytic in temporal and spatial variables, even if the given initial data admit less regularity.