Let M(n, A) denote the maximum possible cardinality of a family of binary strings of length n, such that for every four distinct members of the family there is a coordinate in which exactly two of them have a 1. We prove that M(n, A) [les ] 20.78n for all sufficiently large n. Let M(n, C) denote the maximum possible cardinality of a family of binary strings of length n, such that for every four distinct members of the family there is a coordinate in which exactly one of them has a 1. We show that there is an absolute constant c < 1/2 such that M(n, C) [les ] 2cn for all sufficiently large n. Some related questions are discussed as well.

Vapnik and Chervonenkis proposed in [7] a combinatorial notion of dimension that reflects the ‘combinatorial complexity’ of families of sets. In the three decades that have passed since that paper, this notion – the Vapnik–Chervonenkis dimension (VC-dimension) – has been discovered to be of primal importance in quite a wide variety of topics in both pure mathematics and theoretical computer science.

In this paper we turn our attention to classes with infinite VC-dimension, a realm thrown into one big bag by the usual VC-dimension analysis. We identify three levels of combinatorial complexity of classes with infinite VC-dimension. We show that these levels fall under the set-theoretic definition of σ-ideals (in particular, each of them is closed under countable unions), and that they are all distinct. The first of these levels (i.e., the family of ‘small’ infinite-dimensional classes) coincides with the family of classes which are non-uniformly PAC-learnable.

Maybe the most surprising contribution of this work is the discovery of an intimate relation between the VC-dimension of a class of subsets of the natural numbers and the Lebesgue measure of the set of reals defined when these subsets are viewed as binary representations of real numbers.

As a by-product, our investigation of the VC-dimension-induced ideals over the reals yields a new proper extension of the Lebesgue measure. Another offshoot of this work is a simple result in probability theory, showing that, given any sequence of pairwise independent events, any random event is eventually independent of the members of the sequence.

Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture.

We derive improved isoperimetric inequalities for discrete product measures on the n-dimensional cube. As a consequence, a general theorem on the threshold behaviour of monotone properties is obtained. This is then applied to coding theory when we study the probability of error after decoding.

In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp–Wilson exact simulation for all Markov random fields on ℤd satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts.

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxationAssume there exists an equilibrium curve A(u), such that r(u,A(u)) = 0. Under some assumptions on σ and r, we prove the existence of global (in time) solutions of bounded variation, uε, υε, for ε > 0 fixed.

As ε → 0, we prove the convergence of a subsequence of uε, υε to some u, υ that satisfy the equilibrium equations

A parabolic system with an unknown boundary is considered. The boundary condition at the unknown boundary is concerned with its mean curvature. The physical prototype is superconductivity of type I. Existence of classical solutions is proved by the use of fixed-point theorem, uniqueness is obtained as well.

We extend and clarify some of observations due to Barron and Jensen concerning the relation between subdifferentials and superdifferentials of a function and extend the comparison principle for semicontinuous solutions of Hamilton–Jacobi equations with convex Hamiltonians to that in infinite-dimensional Hilbert spaces.

We prove the existence of a very singular solution towhen 1 < p < (N + 2)/(N + 1).

In this paper we construct new classes of stationary solutions for the Cahn–Hilliard equation by a novel approach.

One of the results is as follows. Given a positive integer K and a (not necessarily non-degenerate) local minimum point of the mean curvature of the boundary, then there are boundary K-spike solutions whose peaks all approach this point. This implies that for any smooth and boundeddomain there exist boundary K-spike solutions.

The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (lemma 3.5), where the variables are closely related to the peak locations.

This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.

A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.

Linked equationsare studied on [0,1] subject to boundary conditions of the formResults are given on existence, location, asymptotics and perturbation of the eigenvalues λj and oscillation of the eigenfunctions yi.

The creation and propagation of oscillations in a model for the dynamics of fine structure under viscoelastic dampingis studied. It is shown that oscillations in the velocity ut are lost immediately as time evolves, while oscillations in the initial strain ux cannot be created, and they persist for all time if initially present. Uniqueness of generalized solutions (Young measures) is obtained, and a characterization of these Young measures is provided in the case of periodic modulated initial data.

Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functionalover the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

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.

Farah recently proved that many Borel P-ideals. on satisfy the following requirement: any measurable homomorphism has a continuous lifting which is a homomorphism itself. Ideals having such a property were called Radon–Nikodym (RN) ideals. Answering some Farah's questions, it is proved that many non-P ideals, including, for instance, Fin ⊗ Fin, are Radon–Nikodym. To prove this result, another property of ideals called the Fubini property, is introduced, which implies RN and is stable under some important transformations of ideals.

The set of integers represented as the sum of three cubes of natural numbers is widely expected to have positive density (see Hooley [7] for a discussion of this topic). Over the past six decades or so, the pursuit of an acceptable approximation to the latter statement has spawned much of the progress achieved in the theory of the Hardy-Littlewood method, so far as its application to Waring's problem for smaller exponents is concerned. Write R(N) for the number of positive integers not exceeding N which are the sum of three cubes of natural numbers.

Let ℳ be the collection of all intersections of balls, considered as a subset of the hyperspace ℳ of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. It is proved that ℳ is uniformly very porous if and only if the space fails the Mazur intersection property.