Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure of K. For an element α ∈ \K, a chain α = α0, α1,…,αr of elements of , such that αi is of minimum degree over K with the property that ῡ(αi−1 − αi) = sup{ῡ(αi−1 − β) | [K (β) : K] < [K (αi−1) : K]} and that αr ∈ K, is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of (cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

We characterize a topological convex space $C$ in terms of the family $\mathcal{A}(C)$ of real continuous affine functions on $C$. Our main result states that two topological convex spaces $C_1$ and $C_2$ are affine-homeomorphic if and only if $\mathcal{A}(C_1)$ and $\mathcal{A}(C_2)$ are isomorphic as ordered unital vector spaces.

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

We show the $L^1$ contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem $j(v)_t-\text{div}(\nabla w+F(w))=f(t,x)$, $w=\varphi(v)$ in $(0,T)\times\varOmega\subset \mathbb{R}^+\times\mathbb{R}^N$ with inhomogeneous Dirichlet boundary datum $g\in L^2(0,T;W^{1,2}(\varOmega))$ for $w$ (the boundary datum is taken in the sense $w-g\in L^2(0,T;H^{1}_0(\varOmega))$) and initial datum $j_o\in L^1(\varOmega)$ for $j(v)$. Here $\varphi$ and $j$ are non-decreasing, and we assume that $F$ is just continuous.

Our proof consists in doubling of variables in the interior of $\varOmega$ as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of $\varOmega$. For the latter argument, the result is restricted to the linear dependence on $\nabla w$ of the diffusion term. The proof allows for a wide class of domains $\varOmega$, including, for example, weakly Lipschitz domains with Lipschitz cracks.

We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators.

In this paper a method is developed for the asymptotic expansion of some classes of integral as a parameter k → 0+. The procedure is analogous to the method of inner and outer sums for treating certain types of infinite series whose terms contain a small parameter, and can involve heavy algebra. However, this aspect of the process can be delegated to a symbolic manipulation package.

In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form

$$-\Delta u(x)+q(x)u(x)=\lambda\eta(x)^{-1}f(\eta(x)u(x))$$

and

$$-\Delta u(x)+q(x)u(x)-\eta(x)^{-1}f(\eta(x)u(x))=\lambda u(x),$$

where $u\in H^{2}(\mathbb{R}^{N})$.

Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

Algorithms are introduced that produce optimal Markovian couplings for large finite-state-space discrete-time Markov chains with sparse transition matrices; these algorithms are applied to some toy models motivated by fluid-dynamical mixing problems at high Peclét number. An alternative definition of the time-scale of a mixing process is suggested. Finally, these algorithms are applied to the problem of coupling diffusion processes in an acute-angled triangle, and some of the simplifications that occur in continuum coupling problems are discussed.

We study model-theoretic and stability-theoretic properties of the non-abelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn on ‘negligible subsets' of free groups. We point out analogies between the free group and so-called bad groups of finite Morley rank, and prove ‘non-CM-triviality' of the free group.

The Main Conjecture of Iwasawa theory for an elliptic curve $E$ over $\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ was studied in \cite{bertolini_darmon}, in the case where $p$ is a prime of ordinary reduction for $E$. Analogous results are formulated, and proved, in the case where $p$ is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.

The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups $A_{\mathrm{o}}(F)$ are identified with higher-dimensional Podleś spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of is 1-O (d−1/2) w.h.p.

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,n at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, as n varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

We continue the study of regular partitions of hypergraphs. In particular, we obtain corresponding counting lemmas for the regularity lemmas for hypergraphs from our paper ‘Regular Partitions of Hypergraphs: Regularity Lemmas’ (in this issue).

A widely studied model for generating binary sequences is to ‘evolve’ them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially ‘easier’ (in terms of the sequence length needed) than determining the true tree. The key tool is a new and near-tight Ramsey-type result for binary trees.

In 1972, Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere Sd in such a way that two vertices joined by an edge have distance more than (ie, distance more than 2π/3 on the sphere). In 1978, Larman [LAR] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed . In addition, he conjectured that the right answer would be , which is not better than the class of all graphs. Larman'sconjecture was independently proved by Rosenfeld [MR] and Rödl [VR[. In this last paper it was shown that no bound better than can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is stilltrue for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree.

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].

We define the γ invariant as the coefficient of x1 in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1 in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1 when n true twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

Layered nanocrystals consist of a core of one material surrounded by a shell of a second material. We present computation of the atomistic strain energy density in a layered nanocrystal, using an idealised model with a simple cubic lattice and harmonic interatomic potentials. These computations show that there is a critical size r*s for the shell thickness rs at which the energy density has a maximum. This critical size is roughly independent of the geometry and material parameters of the system. Interestingly, this critical size agrees with the shell thickness at which the quantum yield has a maximum, as observed in several systems and thus leads one to support the hypothesis that maximal quantum yield is strongly correlated with maximal elastic energy density.