We study semirandom k-colourable graphs made up as follows. Partition the vertex set V = {1, . . ., n} randomly into k classes V1, . . ., Vk of equal size and include each Vi–Vj-edge with probability p independently (1 ≤ i < j ≤ k) to obtain a graph G0. Then, an adversary may add further Vi–Vj-edges (i≠j) to G0, thereby completing the semirandom graph G = G*n,p,k. We show that if np ≥ max{(1 + ϵ)klnn, C0k2} for a certain constant C0>0 and an arbitrarily small but constant ϵ>0, an optimal colouring of G*n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0max{klnn, k2}, a k-colouring of G*n,p,k can be computed in polynomial expected time. Moreover, an optimal colouring of G*n,p,k can be computed in expected polynomial time if k ≤ ln1/3n and np ≥ C0klnn. By contrast, it is NP-hard to k-colour G*n,p,k With high probability if .

We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n−1 + λn−4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ϵ (a scaled version of the number of vertices in components of size greater than εn2/3) is almost constant.

In this note we give a very short proof for the description of all maximum size two-part Sperner systems.

We consider the distribution of the value of the optimal k-assignment in an m × n matrix, where the entries are independent exponential random variables with arbitrary rates. We give closed formulas for both the Laplace transform of this random variable and for its expected value under the condition that there is a zero-cost (k − 1)-assignment.

Improving upon work in [2], the precise value of the set reconstruction number is given for all cyclic groups of odd order.

This paper has two parts. In the first part we consider a simple Markov chain for d-regular graphs on n vertices, where d = d(n) may grow with n. We show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. In the second part of the paper, a related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even constant d. This is a model for a certain peer-to-peer network. We prove that the related chain has mixing time which is bounded above by a polynomial in N, the expected number of vertices, provided certain assumptions are met about the rate of arrival and departure of vertices.

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node, without leaving any trace. We consider the task of locating a black hole in a (partially) synchronous tree network, assuming an upper bound on the time of any edge traversal by an agent. The minimum number of agents capable of identifying a black hole is two. For a given tree and given starting node we are interested in the fastest-possible black hole search by two agents. For arbitrary trees we give a 5/3-approximation algorithm for this problem. We give optimal black hole search algorithms for two ‘extreme’ classes of trees: the class of lines and the class of trees in which any internal node (including the root which is the starting node) has at least two children.

Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form p∈ pc(u,∂tu). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.

We investigate the asymptotic behaviour as $k\to+\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\varOmega)$ of solutions of the equations $\Delta^2u_k=V_k\mathrm{e}^{4u_k}$ on $\varOmega$, where $\varOmega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to+\infty}V_k=1$ in $C^0_{\mathrm{loc}}(\varOmega)$. The corresponding two-dimensional problem was studied by Brézis and Merle and Li and Shafrir, who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Robert and Struwe in 2006, such a quantization does not hold in dimension $4$ for the problem in its full generality. We prove here that, under a natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension $2$.

This paper studies blowing-up properties of a unique positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight function and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property are discussed, based on the variational characterization of the unique positive principal eigenvalue. A counterexample is constructed, which shows that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true in the Neumann case.

We consider the problem $-\text{div}(p(x)\nabla u)=\lambda{u}+\alpha|u|^{r-1}u$ in $\varOmega$, $\partial u/\partial\nu=Q(x)|u|^{q-2}u$ on $\partial\varOmega$, where $\varOmega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $q=2(N-1)/(N-2)$ and $2<r<q$. Under some conditions on $\partial\varOmega$, $p$, $Q$, $\lambda$, $\alpha$ and the mean curvature at some point $x_0$, we prove the existence of solutions of the above problem. We use variational arguments, namely the concentration–compactness principle, min–max principle and the mountain-pass theorem.

In this article the linear theory of thermoviscoelastic mixtures is considered. The fundamental solution of the system of linear-coupled partial differential equations of steady oscillations (steady vibrations) of the theory of thermoviscoelastic mixtures is constructed in terms of elementary functions and basic properties are established.

The main goal of this work is to study the existence and uniqueness of a positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.

This article provides a borrower's optimal strategies to terminate a mortgage with a fixed interest rate by paying the outstanding balance all at once. The problem is modelled as a free boundary problem for the appropriate analogue of the Black-Scholes pricing equation under the assumption of the Vasicek model for the short-term rate of investment. Here the free boundary provides the optimal time at which the mortgage contract is to be terminated. A number of integral identities are derived and then used to design efficient numerical codes for computing the free boundary. For numerical simulation, parameters for the Vasicek model are estimated via the method of maximum likelihood estimation using 40 years of data from US government bonds. The asymptotic behaviour of the free boundary for the infinite horizon is fully analysed. Interpolating this infinite horizon behaviour and a known near-expiry behaviour, two simple analytical approximation formulas for the optimal exercise boundary are proposed. Numerical evidence shows that the enhanced version of the approximation formula is amazingly accurate; in general, its relative error is less than 1%, for all time before expiry.

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

To prove that certain standard 2-complexes are aspherical we explore a strategy that combines a well-known method based on a graph theoretic lemma of Stallings with a process of reversing the orientation of edges in spherical diagrams. We apply this strategy to labelled-oriented-tree complexes and, more generally, to labelled-oriented-graph (LOG) complexes and obtain classes of aspherical LOG complexes to which, traditionally (without the reversing of edges), Stallings's lemma could not be applied to prove asphericity. These classes contain examples whose asphericity, as far as we know, could not be established by any previous method.

We consider the fourth-order thin film equation (TFE)with the unstable second-order diffusion term. We show that, for the first critical exponentwhere N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the formFor the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0+ and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationstudied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0,nh), where the valuecharacterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p0, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.