We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

The aim of this paper is to study the asymptotic behaviour of the solutions of the linearized elasticity system, posed on thin reticulated structures involving several small parameters. We show that this behaviour depends on the relative size of the parameters. In each case, we obtain a limit system where the microstructure and macrostructure appear simultaneously. From it, we get a suitable approximation in L2 of the displacements and the linearized strain tensor.

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove that for an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.

Path formulation can be used to classify and structure efficiently multiparameter bifurcation problems around fundamental singularities: the cores. The non-degenerate umbilic singularities are the generic cores for four situations in corank 2: the general or gradient problems and the Z2-equivariant (general or gradient) problems. Those categories determine an interesting ‘Russian doll’ type of structure in the universal unfoldings of the umbilic singularities.

One advantage of our approach is that we can handle one, two or more parameters using the same framework (even considering some special parameter structure, for instance, some internal hierarchy). We classify the generic bifurcations that occur in those cases with one or two parameters.

We establish sharpness of embedding theorems for Bessel-potential spaces modelled upon Lorentz–Karamata (LK) spaces and we prove the non-compactness of such embeddings. Target spaces in our embeddings are LK spaces. As a consequence of our results, we get growth envelopes of Bessel-potential spaces modelled upon LK spaces.

We consider the generalized Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equations with dissipative term. We establish the condition under which the solutions uα,β,γ converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed by Hwang and Tzavaras. First, we obtain the approximate transport equation for the given BBM–Burgers equations. Then, using the averaging lemma, we obtain the convergence.

Given a function υ ∈ BV (Ω; Rm), we introduce the notion of a minimal lifting of Dυ. We prove that every υ ∈ BV (Ω; Rm) has a unique minimal lifting, and we show that if υk → υ strictly in BV, then the minimal liftings of υk converge weakly as measures to the minimal lifting of υ. As an application, we deduce a result about weak continuity of the distributional determinant Det D2u with respect to strict convergence.

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.

Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.

In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

This paper deals with existence and non-existence of global solutions of certain fast–slow diffusion systems with nonlinear boundary conditions. Necessary and sufficient conditions for global existence of positive solutions are obtained in terms of various parameters which appear explicitly in the definition of the systems.

It is shown that, for integrals of the type with Ω RN open, bounded and f: Ω × Rm × Rd → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences un → u in measure, υn → υ in Lp, Aυn → 0 in W−1, p is the Ax-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the form with A(i) ∈ C∞ (Ω; Ml × d) ∩ W1, ∞, i = 1, …, N, satisfying the condition and Ax denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υn} in Lp satisfying the condition Aυn → 0 in W−1,p, is obtained.