We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Fix q and let Mn be an n × n matrix with entries drawn independently from the finite field Fq according to some distribution μn. It is shown that, except in certain pathological cases, the probability that Mn is nonsingular is asymptotically the same as for uniform entries; that is,

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It has been conjectured by Alspach [2] that given integers n and m1, …, mt with 3 [les ] mi [les ] n and [sum ]ti=1mi = (n2) (n odd) or [sum ]ti=1mi = (n2) − n/2 (n even), then one can pack Kn (n odd) or Kn minus a 1-factor (n even) with cycles of lengths m1, …, mt. In this paper we show that if the cycle lengths mi are bounded by some linear function of n and n is sufficiently large then this conjecture is true.

We show that connected matroids with symmetric Tutte polynomials are not necessarily self-dual; in fact, we construct matroids that are not self-dual, have arbitrarily high connectivity, and yet have symmetric Tutte polynomials.

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

In 1978, Dhar suggested a model of a lattice gas whose states are partial orders. In this context he raised the question of determining the number of partial orders with a fixed number of comparable pairs. Dhar conjectured that in order to find a good approximation to this number, it should suffice to enumerate families of layer posets. In this paper we prove this conjecture and thereby prepare the ground for a complete answer to the question.

This work studies mathematical issues associated with steady-state modelling of diffusion-reaction-conduction processes in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. The discussion is applicable to various electrodes where the rate-determining reaction occurs at the electrolyte-solid interface; molten carbonate fuel cell cathodes are used as a specific example. New modelling in terms of component potentials (linear combinations of electrochemical potentials) is shown to be consistent with tradition concentration modelling. The current density is proved to be finite, and asymptotic expressions for both current density and total current are derived for suffciently small contact angles. Finally, numerical and asymptotic examples are presented to illustrate the strengths and weaknesses of these expressions.

The purpose of this paper is to analyse the free boundary problem for the Black–Scholes equation for pricing the American call option on stocks paying a continuous dividend. Using the Fourier integral transformation method, we derive and analyse a nonlinear singular integral equation determining the shape of the free boundary. Numerical experiments based on this integral equation are also presented.

Rapid solidification fronts are studied using a phase field model. Unlike slow moving solutions which approximate the Mullins–Sekerka free boundary problem, different limiting behaviour is obtained for rapidly moving fronts. A time-dependent analysis is carried out for various cases and the leading order behaviour of solidification front solutions is derived to be one of several travelling wave problems. An analysis of these problems is conducted, leading to expressions for front speeds in certain limits. The dynamics leading to these travelling wave solutions is derived, and conclusions about stability are drawn. Finally, a discussion is made of the relationship to other solidification models.

The paper is devoted to the stability of stationary solutions of an evolution system, describing heat explosion in a two-phase medium, where a parabolic equation is coupled with an ordinary differential equation. Spectral properties of the problem linearized about a stationary solution are analyzed and used to study stability of continuous branches of solutions. For the convex nonlinearity specific to combustion problems it is shown that solutions on the first increasing branch are stable, solutions on all other branches are unstable. These results remain valid for the scalar equation and they generalize the results obtained before for heat explosion in the radially symmetric case [1].

The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.