How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.

For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an ‘interaction graph’. During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.

We give a concentration inequality involving a family of independent random permutations, which is useful for analysing certain randomized methods for graph colouring.

We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.

Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in a layer, driven by a nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on horizontal coordinates, external mass forces are absent, and the motion starts from rest then the free boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. In present paper, we formulate conditions for both the unique solvability of the reduced problem globally in time and the collapse of the solution in finite time. We further study qualitative properties of the solution such as its behaviour for large time (in the case of global solvability of the problem), and the asymptotics of the solution near the collapse moment in the opposite case.

The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration uctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.

We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.

The paper presents a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically finding the mesh on which the functions to be found are approximated by the best way. The power of the method to describe ‘bad functions’ well makes it possible to reproduce all the main results of asymptotic theory for the almost-highest waves (Longuet-Higgins & Fox, 1977, 1978, 1996) by a direct numerical simulation. The method is able to compute two full periods of the oscillations of wave properties for all wave height-to-length ratios. The end of the second period corresponds to the wave steepness that achieves 99.99997% of the limiting value. So, the validity of the asymptotic formulae by Longuet-Higgins & Fox is proved for the steep waves of any finite depth. The refined value of the maximum slope of the free-surfaces is found to be 30.3787°.

We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.

The paper compares computational aspects of four approaches to compute conservation laws of single Differential Equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev–Petviashvili, Karney–Sen–Chu–Verheest, Boussinesq, Tzetzeica, Benney.

The free boundary model of diffusion-induced grain boundary motion derived in Cahn et al. [3], Fife et al. [6] and Cahn & Penrose [4] is extended, in the case of thin metallic films, to account for bidirectional motion, together with the appearance of S-shapes and double seam configurations. These are often observed in the laboratory. Computer simulations based on the extended model are given to illustrate these and other features of bidirectional motion. More generally, the extension accounts for the motion of grain boundaries whose traces on the film's surface are curved. The new free boundary model is one of forced motion by curvature, the forcing term possibly changing sign due to the bidirectionality. The thin film model is derived systematically under explicit assumptions, and an adjustment for grooving is included.

We analyze the front structures evolving under the difference-differential equation ∂tCj = −Cj+C2j−1 from initial conditions 0 [les ] Cj(0) [les ] 1 such that Cj(0) → 1 as j → ∞ suffciently fast. We show that the velocity v(t) of the front converges to a constant value v* according to v(t) = v*−3/(2λ*t)+(3√π/2) Dλ*/(λ*2Dt)3/2+[Oscr](1/t2). Here v*, λ* and D are determined by the properties of the equation linearized around Cj = 1. The same asymptotic expression is valid for fronts in the nonlinear diffusion equation, where the values of the parameters λ*, v* and D are specific to the equation. The identity of methods and results for both equations is due to a common propagation mechanism of these so-called pulled fronts. This gives reasons to believe that this universal algebraic convergence actually occurs in an even larger class of equations.

We consider the Ginzburg–Landau equation in dimension two. We introduce a key notion of the vortex (interaction) energy. It is defined by minimizing the renormalized Ginzburg–Landau (free) energy functional over functions with a given set of zeros of given local indices. We find the asymptotic behaviour of the vortex energy as the inter-vortex distances grow. The leading term of the asymptotic expansion is the vortex self-energy while the next term is the classical Kirchhoff–Onsager Hamiltonian. To derive this expansion we use several novel techniques.

The behaviour of liquid crystal materials used in display devices is discussed. The underlying continuum theory developed by Frank, Ericksen and Leslie for describing this behaviour is reviewed. Particular attention is paid to the approximations and extensions relevant to existing device technology areas where mathematical analysis would aid device development. To illustrate some of the special behaviour of liquid crystals and in order to demonstrate the techniques employed, the specific case of a nematic liquid crystal held between two parallel electrical conductors is considered. It has long been known that there is a critical voltage below which the internal elastic strength of the liquid crystal exceeds the electric forces and hence the system remains undeformed from its base state. This bifurcation behaviour is called the Freedericksz transition. Conventional analytic analysis of this problem normally considers a magnetic, rather than electric, field or a near-transition voltage since in these cases the electromagnetic field structure decouples from the rest of the problem. Here we consider more practical situations where the electromagnetic field interacts with the liquid crystal deformation. Assuming strong anchoring at surfaces and a one dimensional deformation, three nondimensional parameters are identified. These relate to the applied voltage, the anisotropy of the electrical permittivity of the liquid crystal, and to the anisotropy of the elastic stiffness of the liquid crystal. The analysis uses asymptotic methods to determine the solution in a numerous of different regimes defined by physically relevant limiting cases of the parameters. In particular, results are presented showing the delicate balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.

We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state a are governed by two asymptotic equations: the first-order asymptotic equation if a is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions. It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the ‘concentration-annihilation’ phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv2 for which v2 may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.

We show that certain interpolation results for compact operators established by Cobos and co-workers cannot be extended to general closed operator ideals. We shall also characterize compactness of an embedding in terms of functions related to the classical K- and J-functionals of interpolation theory.

In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H-measures defined on the space of positions and frequencies. The present paper is devoted to the investigation of the Tartar equationwhich describes the evolution of the H-measure μt associated with a sequence of oscillatory solutions of the linear transport equationin cases when a given solenoidal velocity field v(x, t) is sufficiently smooth. Here, (t, x, y) ∈ (0, T) × Ω × S1, 0 < T < +∞, Ω is a bounded open subset of R2 and S1 is the unit circle in R2, given coefficients Yij = Yij(y) are infinitely smooth.

Assuming that v belongs to , we establish the well posedness of Cauchy problem for the Tartar equation in the same measure class as the H-measures are in. For this purpose, we develop and use an extension of the theory of Lagrange coordinates for a case of non-smooth solenoidal velocity fields.

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

We study a boundary perturbation problem for a one-dimensional Schrödinger equation in which the potential has a regular singularity near the perturbed end point. We give the asymptotic behaviour of the eigenvalues under the perturbation. This problem arose out of the author's studies of singular elliptic operators in higher dimensions and we illustrate this point with an example. The class of potentials to which this method applies is larger than that covered by standard results, which assume uniform ellipticity of the operator or a perturbative term that is analytic in the perturbation parameter.