When ceramics are heated inside a microwave cavity, a well-known phenomenon is the occurrence of hot spots – localised regions of high temperature. This phenomenon was modelled by Kriegsmann ((1997), IMA J. Appl. Math. 59(2), pp. 123–146; (2001), IMA J. Appl. Math. 66(1), pp. 1–32) using a non-local evolution PDE. We investigate profile and the stability of hot spots in one and two dimensions by using Kriegsmann's model with exponential non-linearity. The linearised problem associated with hot-spot-type solutions possesses two classes of eigenvalues. The first type is the large eigenvalues associated with the stability of the hot-spot profile and in this particular model there cannot be instability associated with these eigenvalues. The second type is the small eigenvalues associated with translation invariance. We show that the hot spots can become unstable due to the presence of small eigenvalues, and we characterise the instability thresholds. In particular, we show that for the material with low heat conductivity (such as ceramics), and in the presence of a variable electric field, the hot spots are typically stable inside a plate (in two dimensions) but can become unstable for a slab (in one dimension) provided that the microwave power is sufficiently large. On the other hand, for materials with high heat conductivity, the interior hot spots are unstable and move to the boundary of the domain in either one or two dimensions. For materials with moderate heat conductivity, the stability of hot spots is determined by both the geometry and the electric field inside the microwave cavity.

We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.

Szemerédi's Regularity Lemma is an important tool for analysing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

We give sufficient conditions on two dyadic systems to obtain the equivalence of corresponding Haar systems on dyadic weighted Lebesgue spaces on spaces of homogeneous type. In order to obtain these results, we prove a Fefferman–Stein weighted inequality for vector-valued dyadic Hardy–Littlewood maximal operators with dyadic weights in this general setting.

Irregular mappings that are weak solutions of the energy–momentum equations are presented. One example is discontinuous at a countable number of points while the other is C1, but not C2. These mappings are not solutions of the usual Euler–Lagrange equations.

We study (p, q)-inequalities of Ulyanov type for moduli of smoothness of fractional order in the Lp and the Lp(ℝn) setting, p ≥ 1. In particular, we obtain estimates for the modulus of smoothness of a generalized Liouville derivative of a function via the modulus of smoothness of the function itself. We give examples showing the sharpness of these inequalities.

The three-dimensional compressible magnetohydrodynamic equation in the whole space are studied in this paper. The global classical solution is established when the initial data are small perturbations of some given constant state. Moreover, the optimal decay rate of the solution is also obtained.

A logistic equation with infinite delay is considered under conditions that force its solution to approach a positive steady state at large times. It is shown that this rate of convergence depends on the initial history in some cases, and is independent of the history in others.

Let C be a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space of C to the one attached to C, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group on C with n strings does not depend on n, even if n = 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

We study the weak interaction between a pair of well-separated coherent structures in possibly non-local lattice differential equations. In particular, we prove that if a lattice differential equation in one space dimension has asymptotically stable (in the sense of a paper by Chow et al.) travelling-wave solutions whose profiles approach limiting equilibria exponentially fast, then the system admits solutions which are nearly the linear superposition of two such travelling waves moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the travelling waves are moving in opposite directions, these solutions are not shift-periodic and hence the framework of Chow et al. does not apply. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a shift-periodic piece plus another piece which, although it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.

We establish the local well-posedness for a periodic two-component Camassa–Holm equation. We then present precise blow-up scenarios. Finally, we obtain several blow-up results and the blow-up rate of strong solutions to the equation.

For a sign-changing function a(x) we consider the solutions of the following semilinear elliptic problem in ℝn with n ≥ 3:

where γ > 0 and 0 < q < 1 < p < (n + 2)/(n − 2). Under an appropriate growth assumption on a− at infinity, we show that all solutions are compactly supported. When Ω+ = {x ∈ ℝn | a(x) > 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a(x) = a(|x|), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω+.

The equation div υ = F has a solution υ in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BVm/(m−1)(ℝm) (the space of functions in Lm/(m−1)(ℝm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F(uj) converges to zero for each sequence {uj} such that the measure norms of ∇j are uniformly bounded and uj ⇀ 0 weakly in Lm/(m−1)(ℝm).

The existence, stability and uniqueness of positive solutions to a semilinear elliptic system with sublinear nonlinearities are proved. It is shown that the precise global bifurcation diagram of the positive solutions is a monotone curve with different asymptotical behaviour according to the form of the nonlinearities. Equations with Hölder continuous nonlinearities are also considered.

We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)-superlinear on ℝ+ and (p − 1)-(sub)linear on ℝ−. Using variational methods based on the critical point theory together with truncation techniques, Ekeland's variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra having a root space decomposition with respect to the abelian subalgebra , with the property for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).