Proceeding from a recent construction (Fraenkel 2007, Arch. Rational Mech. Anal. 183, 187–214), this paper contains three contributions to the theory of the extreme gravity wave on water (or “wave of greatest height”) initiated by Stokes in 1880. The first is a solution, of the extreme form of the integral equation due to Nekrasov, that consists of an explicit function plus a (rigorously) estimated remainder; the remainder vanishes at the crest and at the trough of the wave and in between has a maximum value less than 0.0026 times the maximum value of the explicit part. The second contribution establishes local uniqueness for a variant of the Nekrasov equation; we prove that this equation has only one solution between upper and lower bounding functions that are moderately far apart. The third contribution consists of a table and graph of strict upper and lower bounds for the interface between air and water in the physical plane.

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

We show that for any type III1 free Araki–Woods factor = (HR, Ut)″ associated with an orthogonal representation (Ut) of R on a separable real Hilbert space HR, the continuous core M = ⋊σR is a semisolid II∞ factor, i.e. for any non-zero finite projection q ∈ M, the II1 factor qM q is semisolid. If the representation (Ut) is moreover assumed to be mixing, then we prove that the core M is solid. As an application, we construct an example of a non-amenable solid II1 factor N with full fundamental group, i.e. (N) = R*+, which is not isomorphic to any interpolated free group factor L(Ft), for 1 < t ≤ = +∞.

We study a problem on edge percolation on product graphs G × K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. Every edge in G × K2 is present with probability p independent of other edges. The bunkbed conjecture states that for all G and p, the probability that (u, 0) is in the same component as (v, 0) is greater than or equal to the probability that (u, 0) is in the same component as (v, 1) for every pair of vertices u, v ∈ G.

We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs G, in particular outerplanar graphs.

This paper gives a complete description of the spherical unitary dual of split classical real and p-adic groups. The proof makes heavy use of the affine graded Hecke algebra.

Let R = (r1, . . ., rm) and C = (c1, . . ., cn) be positive integer vectors such that r1 + ⋯ + rm = c1 + ⋯ + cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (‘typical table’) Z defined as follows. We let g(x) = (x + 1)ln(x + 1) − x ln x for x ≥ 0 and let g(X) = ∑i,jg(xij) for a non-negative matrix X = (xij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative m × n matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.

The number of closed tree-like walks in a graph is closely related to the moments of the roots of the matching polynomial for the graph. Thus, by counting these walks up to a given length it is possible to find approximations for the matching polynomial. This approach has been used in two separate problems involving asymptotic enumerations of 1-factorizations of regular graphs. Nevertheless, a systematic way to count the required walks had not previously been found.

In this paper we give an algorithm to count closed tree-like walks in a regular graph up to a given length. For small m, this provides expressions for the number of m-matchings in the graph in terms of the numbers of copies of certain small subgraphs that appear in the graph. The simplest of these expressions were already known, having been rediscovered by numerous authors using ad hoc methods. We offer the first general method for producing the expressions. We also find generating functions that isolate the contribution from the simplest kind of subgraph – namely a single cycle of arbitrary length.

We introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.

The statement of Theorem 2.7.6 in the indicated paper is incorrect. For instance, if m = 0 (i.e., the base field K contains no additional derivations), it is inconsistent with Theorem 2.7.4 due to the distinction between intrinsic and extrinsic generic radii of convergence. Theorems 2.7.12 and 2.7.13 are incorrect for similar reasons.

A digraph is m-labelled if every arc is labelled by an integer in {1, . . ., m}. Motivated by wavelength assignment for multicasts in optical networks, we introduce and study n-fibre colourings of labelled digraphs. These are colourings of the arcs of D such that at each vertex v, and for each colour α, in(v, α) + out(v, α) ≤ n with in(v, α) the number of arcs coloured α entering v and out(v, α) the number of labels l such that there is at least one arc of label l leaving v and coloured with α. The problem is to find the minimum number of colours λn(D) such that the m-labelled digraph D has an n-fibre colouring. In the particular case when D is 1-labelled, λ1(D) is called the directed star arboricity of D, and is denoted by dst(D). We first show that dst(D) ≤ 2Δ−(D)+1, and conjecture that if Δ−(D) ≥ 2, then dst(D) ≤ 2Δ−(D). We also prove that for a subcubic digraph D, then dst(D) ≤ 3, and that if Δ+(D), Δ−(D) ≤ 2, then dst(D) ≤ 4. Finally, we study λn(m, k) = max{λn(D) | D is m-labelled and Δ−(D) ≤ k}. We show that if m ≥ n, then for some constant C. We conjecture that the lower bound should be the correct value of λn(m, k).

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,

where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

Some of the proofs in the above paper are incomplete.

The pressure term in the incompressible elasticity equation needs to be considered and estimated in order to pass to the limit in the homogenization process. This will require Sections 2.2 and 3.2 to be rewritten.

A corrected version will be submitted in due course.