We consider uniformly elliptic, second-order, linear partial differential equations depending on three variables in bounded domains. We obtain interior Hölder estimates for the first derivatives of the bounded solutions independent of the regularity assumptions of the differential operator.

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

We study closed Einstein–Weyl structures on compact K-contact, Sasakian and cosymplectic manifolds. In particular we prove that compact Sasakian and cosymplectic manifolds endowed with a closed Einstein–Weyl structure are η-Einstein.

Among the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

A compact contact Ricci soliton (whose potential vector field is the Reeb vector field) is Sasaki–Einstein. A compact contact homogeneous manifold with a Ricci soliton is Sasaki–Einstein.

We give a simple proof of the Lalonde–McDuff Conjecture for aspherical manifolds.

We consider the existence and multiplicity of standing-wave solutions

of nonlinear Schrödinger equations with electromagnetic fields and critical nonlinearity

Under suitable assumptions, we prove that it has at least one solution and that, for any m ∈ ℕ, it has at least m pairs of solutions.

Let A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of .

We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to , and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to .

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

Ordering constraints are formally analogous to instances of the satisfiability problem in conjunctive normal form, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering if and only if the relative ordering of its variables obeys the constraint considered.

The naturally arising satisfiability problems are NP-complete for many types of constraints. We look at random ordering constraints. Previous work of the author shows that there is a sharp unsatisfiability threshold for certain types of constraints. The value of the threshold, however, is essentially undetermined. We pursue the problem of approximating the precise value of the threshold. We show that random instances of the betweenness constraint are satisfiable with high probability if the number of randomly picked clauses is ≤0.92n, where n is the number of variables considered. This improves the previous bound, which is <0.82n random clauses. The proof is based on a binary relaxation of the betweenness constraint and involves some ideas not used before in the area of random ordering constraints.

This paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.