An affine Hecke algebra $\mathcal{H}$ contains a large abelian subalgebra $\mathcal{A}$ spanned by the Bernstein–Zelevinski–Lusztig basis elements $\theta_x$, where $x$ runs over (an extension of) the root lattice. The centre $\mathcal{Z}$ of $\mathcal{H}$ is the subalgebra of Weyl group invariant elements in $\mathcal{A}$. The natural trace (‘evaluation at the identity’) of the affine Hecke algebra can be written as integral of a certain rational $n$-form (with values in the linear dual of $\mathcal{H}$) over a cycle in the algebraic torus $T=\textrm{Spec}(\mathcal{A})$. This cycle is homologous to a union of ‘local cycles’. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum $W_0\setminus T$ of $\mathcal{Z}$. From this result we derive the Plancherel formula of the affine Hecke algebra.

AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32

This paper deals with the stabilization of the contribution of elliptic elements to the geometric side of the general twisted trace formula. We extend the results of Langlands, Kottwitz and Shelstad to all elliptic elements for the general twisted trace formula.

AMS 2000 Mathematics subject classification: Primary 11F72; 11R39; 11R34

The problem of a bubble rising due to buoyancy in a Hele–Shaw cell filled with a viscous fluid is a classical free-boundary problem first posed and solved by Saffman & Taylor [11]. In fact, due to linearity of the flow equations the problem is reduced to that of a bubble transported by uniform fluid flow. Saffman and Taylor provided explicit expressions for the bubble shape. Steady propagation of bubbles and fingers in a Hele–Shaw cell filled with a nonlinearly-viscous fluid was studied by Alexandrou & Entov [1]. In Alexandrou & Entov [1], it was shown that for a nonlinearly viscous fluid the problem of a rising bubble cannot be reduced to that of a steadily transported bubble, and should be treated separately. This note presents a solution of the problem following the general framework suggested in Alexandrou & Entov [1]. The hodograph transform is used in combination with finite-difference and collocation techniques to solve the problem. Results are presented for the cases of a Bingham and power-law fluids.

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.

Mathematical models for the growth of nutrient-rich tissue are presented and a number of properties of the resulting models outlined. The focus is on obtaining broadly applicable results for the simplest appropriate formulations by using matched-asymptotic, moving-boundary and thin-film approaches; the relevance of the results to a variety of specific biological applications will be addressed elsewhere, as will the inclusion of additional physical phenomena.

We consider a Stefan problem with a kinetic condition at the free boundary and prove the convergence of the solution as $t$ tends to infinity either to a travelling wave solution or to a self-similar solution. The key idea is to transform this problem into a problem for a single nonlocal parabolic equation which admits a comparison principle.

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

The aim of this paper is to develop a theory for the growth of multiple crystals in a polymer melt. This leads to nonlinear moving boundary problems for the temperature, with normal growth speed of the crystal boundaries determined by a nonlinear Gibbs-Thompson relation. Particular attention is paid to the effect of impingement, i.e., the event of two crystals hitting each other, which stops the growth on the contact interface. In one space dimension, the well-posedness of the growth model coupled to the heat equation is shown for an arbitrary number of crystals, both in a quasi-stationary and in an instationary situation. The resulting evolution of a fixed crystal in presence of other crystals is compared to the pure single crystal case. Finally, some basic features of the model in higher spatial dimensions and the main problems encountered in the attempt to prove a general well-posedness result are discussed.

For nonlinear hyperbolic equations of a gas dynamic type with flow acceleration and friction terms, the classification of a special class of periodic travelling waves, which are known as roll waves, is given. As an illustration, the shallow water equations for the inclined channels of an arbitrary cross-section are considered. The analysis shows that the flow patterns depend upon the sign of the second derivative of the pressure function. The roll waves in regular channels with the convex pressure have the same structure as the waves described in Dressler [6]. For a nonconvex pressure function, the multi-jump configuration of roll waves is found.

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

Based on earlier results on existence, we study the asymptotic behaviour of solutions to the coalescence-breakage equations, including the volume-scattering phenomenon and high-energy collisions. The solutions are shown to converge towards one particular equilibrium, provided the kernels satisfy a kind of reversibility. We also derive stability of these equilibria in a suitable topology.

We study a generalization of the Svarc genus of a fibre map. For an arbitrary collection ɛ of spaces and a map f : X → Y, we define a numerical invariant, the ɛ-sectional category of f, in terms of open covers of Y. We obtain several basic properties of ɛ-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple axioms which characterize the ɛ-sectional category. In the final section, we obtain inequalities for the ɛ-sectional category of a composition and inequalities relating the ɛ-sectional category to the Fadell–Husseini category of a map and the Clapp–Puppe category of a map.

It is shown that the Cesàro averaging operators Cα, Re α > −1, introduced by Stempak, are bounded on the Dirichlet space Da if and only if a > 0, while the associated operators Aα are bounded on Da if and only if −1 < a < 2. This extends results of Galanopoulos, who considered the particular case α = 0 for 0 ≤ a ≤ 1.

In this paper, we show that if b(x) ≥ b∞ > 0 in Ω̄ and there exist positive constants C, δ, R0 such thatwhere x = (y, z) ∈ RN with y ∈ Rm, z ∈ Rn, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω ⊆ Rm a bounded C1,1 domain and Ω = ω × Rn, then the Dirichlet problem −Δu + u = b(x)|u|p−1u in Ω has a solution that changes sign in Ω, in addition to a positive solution.

Let Ω be a bounded domain in Rn, n ≥ 3 and 0 ∈ Ω. It is known that the heat problem ∂u/∂t + Lλ*u = 0 in Ω × (0, ∞), u(x, 0) = u0 ≥ 0, u0 ≢ 0, where Lλ* := −Δ − λ*/|x|2, λ* := ¼(n − 2)2, does not admit any solutions for any t > 0. In this paper we consider the perturbation operator Lλ*q := −Δ − λ*q(x)/|x|2 for some suitable bounded positive weight function q and determine the border line between the existence and non-existence of positive solutions for the above heat problem with the operator Lλ*q. In dimension n = 2, we have similar phenomena for the critical Hardy–Sobolev operator L* := −Δ − (1/4|x|2)(log R/|x|)−2 for sufficiently large R.

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

The paper considers an incompressible and irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity and surface tension. It is known that the full Euler equations have travelling two-dimensional solitary-wave solutions of small amplitude for large surface tension (Bond number greater than ⅓). This paper shows that these waves are linearly unstable to three-dimensional perturbations which oscillate along the wave crest with wavenumber in a finite band. The growth rates of these unstable modes are well approximated using the linearized Kadomtsev–Petviashvili equation with positive disper

In this paper we present a simpler proof of a result of Lewis concerning the continuity of weak solutions to the two-dimensional thermistor problem in the case where the temperature can blow up in a region with non-empty interior. Some other regularity properties are also discussed.