In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

Basic methods from bifurcation theory are applied to derive a criterion that predicts when a symmetric tricritical point may occur in a transition between condensed ordered phases described by any finite number of scalar order parameters. At such a point, a change of order takes place in the phase transition, which passes from first to second order, or vice versa.

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

We start here the study of some algebraic varieties related to cluster algebras. These varieties are defined as the fibres of the projection map from the cluster variety to the affine space of coefficients. We compute the number of points over finite fields on these varieties, for all simply laced Dynkin diagrams. We also compute the cohomology with compact support in some cases.

Let 2 be the space of the holomorphic foliations on ℂℙ2 of degree 2. In this paper we study the linear action PGL(3, ℂ) × 2 → 2 given by gX = DgX ^(g−1) in the sense of the Geometric Invariant Theory. We obtain a characterisation of unstable and stable foliations according to properties of singular points and existence of invariant lines. We also prove that if X is an unstable foliation of degree 2, then X is transversal with respect to a rational fibration. Finally we prove that the geometric quotient of non-degenerate foliations without invariant lines is the moduli space of polarised del Pezzo surfaces of degree 2.

Some nonlinear eigenvalue problems related to the modelling of the steady-state deflection of an elastic membrane associated with a Micro-Electromechanical System capacitor under a constant applied voltage are analysed using formal asymptotic methods. These problems consist of certain singular perturbations of the basic membrane nonlinear eigenvalue problem Δu = λ/(1 + u)2 in Ω with u = 0 on ∂Ω, where Ω is the unit ball in 2. It is well known that the radially symmetric solution branch to this basic membrane problem has an infinite fold-point structure with λ → 4/9 as ϵ ≡ 1 − ||u||∞ → 0+. One focus of this paper is to develop a novel singular perturbation method to analytically determine the limiting asymptotic behaviour of this infinite fold-point structure in terms of two constants that must be computed numerically. This theory is then extended to certain generalisations of the basic membrane problem in the N-dimensional unit ball. The second main focus of this paper is to analyse the effect of two distinct perturbations of the basic membrane problem in the unit disk resulting from either a bending energy term of the form −δΔ2u to the operator, or inserting a concentric inner undeflected disk of radius δ. For each of these perturbed problems, it is numerically shown that the infinite fold-point structure for the basic membrane problem is destroyed when δ > 0, and that there is a maximal solution branch for which λ → 0 as ϵ ≡ 1 − ||u||∞ → 0+. For δ > 0, a novel singular perturbation analysis is used in the limit ϵ → 0+ to construct the limiting asymptotic behaviour of the maximal solution branch for the biharmonic problem in the unit slab and the unit disk, and for the annulus problem in the unit disk. The asymptotic results for the bifurcation curves are shown to compare very favourably with full numerical results.

In the paper ‘Approximate transmission conditions through a rough thin layer. The case of periodic roughness’ (Eur. J. Appl. Math. 2010; 21: 51–75), Ciuperca et al. derive transmission conditions that are equivalent to a rough thin layer for the conductivity problem. These conditions involved a pair of vector fields (A0, a0) and two constants D1 and D2 that are given implicitly by solving a partial differential equation in the infinite strip × /2π. We give here an explicit expression of a0 in terms of A0, which shows that D1 equals zero. We infer an explicit characterization of the polarization tensor as given by Capdeboscq and Vogelius (ESAIM:M2AN. 2003; 37: 159–173).

Uniform large deviation principles for positive functionals of all equivalent types of infinite-dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula, which for an infinite sequence of independent and identically distributed real Brownian motions and a Poisson random measure was shown in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes. Ann. Inst. H. Poincaré (to appear)].

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of n×n matrices with entries that are polynomials or more general analytic functions.

For an increasing monotone graph property the local resilience of a graph G with respect to is the minimal r for which there exists a subgraph H ⊆ G with all degrees at most r, such that the removal of the edges of H from G creates a graph that does not possess . This notion, which was implicitly studied for some ad hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model (n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random regular graphs of constant degree. We investigate the local resilience of the typical random d-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values of d, with high probability, the local resilience of the random d-regular graph, n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model (n, p), for every positive ϵ > 0 and large enough values of K, if p > then, with high probability, the local resilience of (n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that if d is large enough then, with high probability, a typical random d-regular graph G is such that, in the unbiased Maker–Breaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle.