In recent papers the present authors considered the effects of small cross-flow on the evolution of two unequal oblique waves. In these studies the relative size of the crossflow meant that a diffusion (or buffer) layer was required around the critical layer to smooth out the algebraic growth in the mean-flow distortion generated by the nonlinear critical-layer interactions. The present analysis increases the cross-flow to an order of magnitude such that the buffer and critical layers coalesce. In this instance the nonlinear critical layer contains viscous as well as nonequilibrium effects. The resulting amplitude equations are solved for perturbations initiated at a fixed station in the flow.

If K is a convex body in d and 1≤k≤d − 1, we define Pk(K) to be the Minkowski sum or Minkowski average of all the projections of K onto k-dimensional subspaces of d. The operator Pd − 1, was first introduced by Schneider, who showed that, if Pd − 1(K) = cK, then K is a ball. More recently, Spriestersbach showed that, if Pd − 1(K) = cK then K = M. In addition, she gave stability versions of this result and Schneider's. We will describe further injectivity results for the operators Pk. In particular, we will show that Pk is injective if k≤d/2 and that P2 is injective in all dimensions except d = 14, where it is not injective.

A 2×2 table of nonnegative integers is subdivided additively into n 2×2 subtables of nonnegative integers in an arbitrary way. The first case of Simpson's Paradox (SP) occurs when the determinant of the original table is less than zero, but the determinant of each of the n 2×2 subtables is greater than or equal to zero. The second case of SP occurs when the previous inequalities hold with ‘less than’ and ‘greater than or equal’ replaced by ‘greater than’ and ‘less than or equal’, respectively. For the case n=2, this paper calculates the asymptotic proportion of the subdivisions of the original 2×2 table such that SP occurs. It is shown that this asymptotic proportion is bounded above by 1/12.

We determine the asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order. One corollary of our result is the following. If a maximum random walk, constrained to use each edge at most once, is taken on Kn, then the probability that all the edges are eventually used is asymptotic to e3/4n−½. Some similar results are obtained about Eulerian circuits and spanning trees in random regular tournaments. We also give exact values for up to 21 nodes.

For an external problem in IRd (d=2, 3) such that the unknown function satisfies the wave equation outside a finite domain, we generate artificial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artificial boundary which is a circle (d=2) or a sphere (d=3): The questions of numerical implementation of the artificial conditions (that are non-local in both space and time) are considered. Special attention is paid to the reduction of necessary computational resources; in particular, a way of incorporating these conditions into numerical methods which makes the computational formulae local in time is suggested. The aspects of treating artificial boundaries of a non-spherical shape are discussed. Numerical examples of two- and three-dimensional scattering problems demonstrate the accuracy of proposed artificial boundary conditions.

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

A cocircuit C* in a matroid M is said to be non-separating if and only if M[setmn ]C*, the deletion of C* from M, is connected. A vertex-triad in a matroid is a three-element non-separating cocircuit. Non-separating cocircuits in binary matroids correspond to vertices in graphs. Let C be a circuit of a 3-connected binary matroid M such that [mid ]E(M)[mid ][ges ]4 and, for all elements x of C, the deletion of x from M is not 3-connected. We prove that C meets at least two vertex-triads of M. This gives direct binary matroid generalizations of certain graph results of Halin, Lemos, and Mader. For binary matroids, it also generalizes a result of Oxley. We also prove that a minimally 3-connected binary matroid M which has at least four elements has at least ½r*(M)+1 vertex-triads, where r*(M) is the corank of the matroid M. An immediate consequence of this result is the following result of Halin: a minimally 3-connected graph with n vertices has at least 2n+6/5 vertices of degree three. We also generalize Tutte's Triangle Lemma for general matroids.

The paper deals with the problem of estimating the distance, in radial or Hausdorff metrics, between two centred star bodies of Rd, d≤3, in terms of the distance between the corresponding intersection bodies.

Let C be a convex cone in ℛd with non-empty interior and a compact basis K. If H1 and H2 are any two parallel hyperplanes tangent to K, whose slices with C are two other compact basis K1 and K2, let D, D1 and D2 be the truncated subcones of C generated by K, K1 and K2. We prove that K is an ellipsoid if, and only if, vol (D)2 = vol (D1) vol (D2) for every such pair of hyperplanes H1, and H2.

We prove that in every finite dimensional normed space, for “most” pairs (x, y) of points in the unit ball, ║x − y║ is more than √2(1 − ε). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than √2(1 − ε).

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge TE is considered, adopting for simplicity a linearized analysis of the vortex sheets that collect the released vorticity and form the trailing wake. The main purpose of the paper is to discuss the uniqueness of the bounded flow solution and the role of the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at TE; and the physical flow solution requires the removal of the divergent part of u (the Kutta condition). This process yields a linear functional equation along the trailing edge involving both the normal vorticity ω released into the wake, and the multiplicative factor of the eigenfunction, a1. Uniqueness is then shown to depend upon the topology of the trailing edge. If δTE=[empty ], as, for example, in an annular-aerofoil configuration, both ω and a1 are uniquely determined by the Kutta condition, and the bounded flow u is unique. If δTE≠[empty ], as, for example, in a connected-wing configuration, there is an infinity of bounded flows, parametrized by a1. Numerical results of relevance for these typical configurations are presented to show the different role of the eigenfunction in the two cases.

We consider the function χ(Gk), defined to be the smallest number of colours that can colour a graph G in such a way that no vertices of distance at most k receive the same colour. In particular we shall look at how small a value this function can take in terms of the order and diameter of G. We get general bounds for this and tight bounds for the cases k=2 and k=3.

For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, [oplus ] 1}. For every n and every k[les ]n/2, we describe a probabilistic distribution on formulas in the basis {∧, [oplus ] 1} in some given set of n variables and of size at most [lscr](k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)[les ][lscr](k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then

formula here

where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in [lscr](k), they are quasi-polynomially related whenever [lscr](k)[ges ]lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

We prove that there exists n0, such that, for every n[ges ]n0 and every 2-colouring of the edges of the complete graph Kn, one can find two vertex-disjoint monochromatic cycles of different colours which cover all vertices of Kn.

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

We prove the existence of generalized weak solutions of a new model for the process of in situ vitrification. The model describes the steady process of vitrification of soil, i.e. the melting of soil by means of electrical current and subsequent solidification as a rock. This is the core of a new technology developed to treat buried low-level radioactive or toxic waste. It is a free boundary problem consisting of a degenerate elliptic equation for the electric potential coupled with a stationary Stefan problem for melting the soil, which results from the Joule heating. The degeneracy reflects the smallness of the ratio of the electrical conductivity of the soil to that of the molten rock. The existence result is proved by considering a sequence of approximate nondegenerate problems, obtaining the necessary a priori estimates and passing to the limit. We also establish a sufficient condition for the solution to exhibit a molten region.

New Lie and conditional symmetries of nonlinear diffusion equations with convection terms are constructed. Examples of new ansätze and exact solutions for some nonlinear equations (in particular, for the Murray and the Newell–Whitehead equations) are presented.