We consider composite media made of homogeneous inclusions with C1,α,boundaries. Our goal is to compare the potential uε in a perfectly periodic composite with the potential uε,d, of a perturbed periodic medium, where the periodicity defects consist of misplaced inclusions. We give an asymptotic expansion of the difference uε,d − uε away from the defects and show that, to first order, a misplaced inclusion manifests itself via a polarization tensor, which is characterized.

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations to a wide class of Banach spaces.

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.

We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

For a large class of functions f, we consider the nonlinear biharmonic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. The discussion is based on the degree theory for C2 proper Fredholm maps developed by Fitzpatrick, Pejsachowicz and Rabier.

To understand the heterogeneous spatial effect on predator–prey models, we study the behaviour of the positive steady states of a predator–prey model as certain parameters are small or large. We compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy. Our results show that the effect of the degeneracy can be clearly observed in one limiting case, but not in the others.

We prove the uniqueness of positive solutions for a class of second-order nonlinear elliptic boundary-value problems with sublinear nonlinearities.

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .

In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.

We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.

Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

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.

The continuous wavelet transform is studied on certain Gel'fand–Shilov spaces of type S. It is shown that, for wavelets belonging to the one type of S-space defined on R, the wavelet transform is a continuous linear map of the other type of the S-space into a space of the same type (latter type) defined on R × R+. The wavelet transforms of certain ultradifferentiable functions are also investigated.

We study the heat flow projected on a manifold M ⊂ L2(Ω). This manifold is defined by the condition that the integrals ∫Ωuk (t,x) dx, k = 1,…,N, are constants of motion. We show that solutions to this problem converge to a steady state as time tends to +∞. We use in an essential way a variant of the Łojasiewicz inequality.

We discuss the spectral properties of a class of sequences of what we call ‘spectral’ type. We introduce an effective method to calculate the zeta invariants for this type of sequence. Such invariants are given in terms of some new and old special functions, and we consider a number of examples in which we study the properties of these special functions.