We study the structure of symmetric vortices in a Ginzburg–Landau model based on Zhang's SO(5) theory of high-temperature superconductivity and antiferromagnetism. We consider both a full Ginzburg–Landau theory (with Ginzburg–Landau scaling parameter κ < ∞) and a κ → ∞ limiting model. In all cases we find that the usual superconducting vortices (with normal phase in the central core region) become unstable (not energy minimizing) when the chemical potential crosses a threshold level, giving rise to a new vortex profile with antiferromagnetic ordering in the core region. We show that this phase transition in the cores is due to a bifurcation from a simple eigenvalue of the linearized equations. In the limiting large-κ model, we prove that the antiferromagnetic core solutions are always non-degenerate local energy minimizers and prove an exact multiplicity result for physically relevant solutions.

We develop a technique for calculating the Wielandt subgroup of a semidirect product of two finite groups of coprime order. We apply this technique to calculate the Wielandt length of a supersoluble group in terms of the Wielandt lengths of its Sylow subgroups (for small Wielandt lengths) and in terms of the nilpotency classes of its Sylow subgroups.

Let be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n ≥ 2 be an integer. We determine the best possible constants a(p, α, n), A(p, α, n), b(p, n) and B(p, n) such that the inequalities and hold for all positive real numbers x1,…,xn.

In this paper we consider the Sturm–Liouville operator d2/dx2 − 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L2(a, 0) and L2(0, b) of the space L2(a, b) = L2(a, 0) ⊕ L2(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L2(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition limx→0+(f′(x)−f′(−x)) = γf(0) with γ∈C+∪R or by the Dirichlet condition f(0+) = f(0−) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L2(a, b).

We introduce a new variant to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame. We combine it with moment lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov–Poisson system, and we solve a long-standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov–Stokes system, where we prove the same result for fairly singular kernels.

In this paper we study the diffusive Nicholson's blowflies equation. Generalizing previous works, we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible. The linearized stability of the non-zero uniform steady state is studied in detail, mainly by using the principle of the argument. Global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations. Finally, we consider what bifurcations are possible from the non-zero uniform state in the case when it is unstable.

This paper is concerned with the existence of topological multivortex solutions in (2 + 1) self-dual gauge theories such as the classical abelian Higgs model or the Chern–Simons Higgs gauge theories. A general form of topological multivortex equations is presented with the inclusion of antivortices. Two kinds of solutions are considered; topological vortex solutions and topological vortex–antivortex solutions. We construct these solutions by super and subsolution methods, and derive the exponential decay of solutions at infinity and the quantized integral formula. As an application, we prove the existence of a topological multivortex solutions in a generalized Chern–Simons Higgs theory.

We construct non-trivial laminates distributed continuously along a smooth closed ‘evolute’ curve by approximating an auxiliary (‘involute’) curve by a polygon with rank-1 sides. We give an explicit example in which the evolute has no rank-1 connections. Using these techniques, given a finite set of matrices B1,…,Br and any other matrix A, we show how to construct a laminate with positive mass at each Bi, any proportion less than 1 of its support on the Bi, and centre of mass at A.

We study a Stefan problem for coarsening of many small particles of solid phase in undercooled liquid phase which includes surface tension and kinetic undercooling effects. A mean-field model is derived by homogenization of the free boundary problem. It extends the classical theory of Lifshitz et al. by taking into account effects due to kinetic undercooling.

A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.

The Cauchy problem for the time-dependent Ginzburg–Landau equations of superconductivity in Rd (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L2. When d = 3, we establish the existence result of solutions for L3 initial data and the uniqueness result for L4 initial data.

We prove a theorem on the uniqueness of positive radial solutions to a Dirichlet problem of the n-Laplacian in a finite ball of Rn. Our proofs use only elementary analysis based on an identity due to Erbe and Tang. The result can be applied to a large class of nonlinearities, including some polynomials and functions with exponential growth; in particular, the one recently studied by Adimurthi.

We establish the uniqueness of positive radial solutions of −Δu = up − u in B(R1, R2), u = 0 on ∂B(R1, R2), where B(R1, R2) is an annulus and 0 < R1 < R2 ≤ ∞, in the following cases.

(a) n ∈ {3, 4} and 1 < p ≤ n/(n − 2).

(b) n ∈ {5, 6, 7, 8} and 1 < p ≤ p0(n) for some p0(n) < n/(n − 2).

Earlier to this result, the uniqueness has been obtained by Coffman for n = 3 and 1 < p ≤ 3 and by Yadava for p ≥ (n + 2)/(n − 2) and n ≥ 3.