As usual in Waring's problem, we let G(k) be the least number s such that all sufficiently large natural numbers can be written as the sum of s or fewer k-th powers of positive integers. Hardy and Littlewood gave the first general upper bound, G(k)≤(k − 2)2k − 1 + 5. Later [4], they reduced this to order k2 by first constructing an auxiliary set of natural numbers below x which are sums of sk-th powers, of cardinality with for large k. The argument, which is brief and elementary (see [15, Chapter 6]), is now referred to by R. C. Vaughan's term “diminishing ranges”, since the k-th powers are taken from intervals of decreasing lengths. This idea of choosing the variables from restricted ranges was refined by Vinogradov, whose application to exponential sum estimates gave for large k (see [18]). The recent iterative method of Vaughan and Wooley [16, 19, 21], which halves Vinogradov's bound, may be viewed as an evolved diminishing ranges argument, producing an auxiliary set with .

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k2r + 2. In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r2k2+(2/(c − 2)) if r ≠ 1 and p > k2+(2/(c − 1)) if r = 1.

Recent advances in the theory of exponential sums (see, for example, [6], [7], [8], [12]) have contributed to corresponding progress in our understanding of the solubility of systems of simultaneous additive equations (see, in particular, [1], [2], [3], [4]). In a previous memoir [11] we developed a version of Vaughan's iterative method (see Vaughan [8]) suitable for the analysis of simultaneous additive equations of differing degrees, discussing in detail the solubility of simultaneous cubic and quadratic equations. The mean value estimates derived in [11] are, unfortunately, weaker than might be hoped, owing to the presence of undesirable singular solutions in certain auxiliary systems of congruences. The methods of [12] provide a flexible alternative to Vaughan's iterative method, and, as was apparent even at the time of their initial development at the opening of the present decade, such ideas provide a means of avoiding altogether the aforementioned problematic singular solutions. The systematic development of such an approach having been described recently in [15], in this paper we apply such methods to investigate the solubility of pairs of additive equations, one cubic and one quadratic, thereby improving the main conclusion of [11].

Let G be an abelian group of order n and let μ be a sequence of elements of G with length 2n−k+1 taking k distinct values. Assuming that no value occurs n−k+3 times, we prove that the sums of the n-subsequences of μ must include a non-null subgroup. As a corollary we show that if G is cyclic then μ has an n-subsequence summing to 0. This last result, conjectured by Bialostocki, reduces to the Erdos–Ginzburg–Ziv theorem for k=2.

Let f(n, k, l) be the expected length of a longest common subsequence of l sequences of length n over an alphabet of size k. It is known that there are constants γ(l)k such that f(n, k, l)→γ(l)kn as n→∞, and we show that γ(l)k=Θ(k1/l−1) as k→∞. Bounds for the corresponding constants for the expected length of a shortest common supersequence are also presented.

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.

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.

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.