We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

In a recent paper [1] I made the following

Definition. The function f: ℤ+ ∪ {0} → ℤ is a pseudo-polynomial if

for all integers n ≥ 0, k ≥ 1.

The construction of the fiber polytope ∑(P, Q) of a projection π:P→Q of polytopes is extended to flags of projections. While the faces of the fiber polytope are related to subdivisions of Q induced by the faces of P, those of an iterated fiber polytope are related to discrete homotopies between polyhedral subdivisions. In particular, in the case of projections

starting with an (n + 1)-simplex, vertices of the successive iterates correspond to, respectively, subsets, permutations and sequences of permutations of an n-set. The first iterate will always be combinatorially an n-cube, and, under certain conditions, the second will have the structure of the (n−1)-dimensional permutohedron.

New classes of pairs e, p are presented for which the Gauss sums corresponding to characters of order e over finite fields of characteristic p are pure, i.e., have a real power. Certain pure Gauss sums are explicitly evaluated.

Let Q(x1, …, xn) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality

is soluble for every ε > 0 in integers x1, …, xn, not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n. We say that a non-singular Q is of type (r, n—r) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that

Extreme value problems and particularly those arising in the combinatorial field have a peculiar interest and challenge in that an exact solution is rarely possible. We discuss here four combinatorial extreme value problems each concerned with the distribution of the largest (or smallest) of a set of mutually dependent variables. These problems, widely different in character possess three points in common. First the probability distribution functions of the variables considered cannot be obtained explicitly, nor secondly can the moments of the variable, and thirdly the probability distribution functions are difficult to evaluate even for moderate sized samples. We shall treat the situation where the variables, or functions of them, have probability distribution functions which tend to an exponential limit, analogous to the well-known limit for extreme values in the case of independent events. The p.d.f.'s for the upper tails of the distributions which we consider are found to be very closely contained within a pair of inequalities (Bonferroni Inequalities), especially in the regions of statistical significance.

When Souslin and Lusin initiated and developed the theories of the Souslin operation, of projective sets and of analytic sets, they attached great importance to the constructive nature of their definitions (see [1], [2] and [3]). When Choquet (see [4], [5] and [6]) made his very successful extension of these theories to an arbitrary Hausdorff space Ω, he defined an analytic set in Ω to be a continuous image in Ω of a Kσδ-set in an unspecified compact Hausdorff space X. Thus, a priori, the construction of the analytic sets in Ω requires the preliminary construction of all compact Hausdorff spaces X.

It is a familiar fact that the sequence of K-free numbers, i.e. those having no K-th power divisor greater than 1, has asymptotic density 1/ζ(K). Let χ(K) denote the characteristic function of this sequence, χ(2) = χ. Roth [5] proved that, if k/(n3/13(log n)4/13) → ∞, then

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

Consider two convex bodies K, K′ in Euclidean space En and paint subsets β, β′ on the boundaries of K and K′. Now assume that K′ undergoes random motion in such a way that it touches K.

Let E be a continuum and n ≥ 4 a given positive integer. A system of points z1,…, zn єE that maximizes

is called a system of Fekete points. Such a system may be not unique.

The fluid mechanical problem with which we are concerned is the behaviour of fluid occupying the half-space x > 0 above a rotating disc which is coincident with the horizontal plane x = 0 and rotating about its axis which remains fixed. Studying rotationally symmetric solutions of this problem, von Kármán [1] (see also [2; p. 93]; [3; p. 133]) reduced it to the solution of two simultaneous equations in functions f(x), g(x) which may, with suitable normalisation, be written in the form

with the boundary conditions

Throughout the paper, let m be a natural number and let F(x1,…, xn) be a form of degree k ≥ 2 with integer coefficients, n ≥ 3. We are concerned with finding solutions of the congruence

for which x is a small non-zero integer vector. For example, in the case k = 2 it was shown by Schinzel, Schlickewei and Schmidt [11] that is a solution of (1) satisfying

provided that n is odd. This is best possible for n = 3, as we shall see later. Of course we can get an exponent (1/2) + (1/(2n – 2)) trivially for even n. I do not know how to improve on this. D. R. Heath-Brown (private communication) can improve the exponent in (2) to (l/2) + ε for n ≥ 4 and prime m > C1(ε).

We consider an eigenvalue problem for a divergence-form elliptic operator Aε that has high-contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ε Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing “asymptotic one-to-one correspondence” between the eigenvalues and the eigenfunctions of the operators Aε and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of Aε to A0 but also the Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.

An incident sound field is scattered by a semi-infinite rigid screen with periodically arranged slits or circular apertures and an approximate solution is sought when the slit (or aperture) width is small and the wavelength is large compared with the separation. An integral equation formulation is used to show that the scattering properties of the screens are equivalent to those of a homogeneous compliant plate. The effective compliance is estimated and is found to be essentially uniform over the plate, with corrections close to the edges.

Let L be a lattice in euclidean n-space Rn of determinant d(L). A well-known problem in the geometry of numbers is whether

(1) if d(L) = l and there exists a sphere centred at the origin containing n linearly independent points of L on its boundary and none other than the origin inside, then any sphere in Rn of radius ½√ contains a point of L.

Let K be any field of characteristic 0 and let T and X be algebraically independent over K. For n ≥ 1 let k(n) ≥ 2 be an integer and let fn(X, T) = xk(n) + T ε K [X, T]. We shall regard T as a “parameter” and X as a “variable”. We put F1(X, T) = f1(X, T) and define, for n ≥ 1,

Let f(z) be an entire function. The definition of a Phragmén–Lindelöf indicator of f(z) requires the preliminary construction of a fairly regular comparison function V(r).

If f(z) is of order λ (0 < λ < + ∞), and of mean type, one takes

and associates with f(z) the indicator

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in ℝd such that μi(ℝd)=k for all i, there is a partition of ℝd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1 for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

Introduction. Gale diagrams, and similar techniques, have been used in several recent investigations into the combinatorial properties of convex polytopes. Here we describe other applications, namely to problems concerning sections and projections of polytopes of a given combinatorial type. For example, in §4 we shall show that every n–dimensional convex polytope with at most n + 2 vertices possesses the universal shadow boundary (usb) property: If is a subcomplex of the boundary complex ℬ (P) of P, and set is homeomorphic to an (n – 2)-sphere, then there exists a polytope P', combinatorially isomorphic to P, such that the complex corresponding to is a shadow boundary of P'. We shall also show that this is the best possible result in that polytopes with n + 3 or more vertices do not, in general, have the usb property. A second application of the method, to be described in §5, is to the construction of non-realisable complexes.