We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope 𝔭 parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_{p}$. They are unified by the theme of “breaking the $3/2$ threshold”, epitomizing the previous state of the art. This concerns two questions pivotal for the sum-product theory, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of $\mathbb{F}$, as well as the number of values of the symplectic form determined by a finite subset of $\mathbb{F}^{2}$. We establish the estimate $|R[A]|\gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios, defined by triples of elements of a set $A\subset \mathbb{F}$ (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates), pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set $P\subset \mathbb{F}^{2}$, the number of distinct values of the symplectic form $\unicode[STIX]{x1D714}$ on pairs of distinct points $u,u^{\prime }$ of $P$ is $|\unicode[STIX]{x1D714}(P)|\gtrsim |P|^{2/3+c}$, with an explicit $c$. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein. Our proofs make use of specially suited applications of new incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. The technical thrust of the paper uses additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any $\mathbb{F}$, by showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^{2}$, the quantity often arising in applications of geometric incidence estimates.

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function

For a cell compex M decomposing the closed orientable 2-manifold Pg of genus g let pi (M) and νj(M) denote the number of i-gonal cells (faces, countries) and j-valent vertices of the graph (= 1 – skeleton) of M, respectively. It will be supposed that i, j ≥ 3. From Euler's formula follows

and

is even, where e is the number of edges of M. As seen, the relation above does not impose restrictions on the numbers p4 (M), ν4(M). In an attempt to characterize the vectors {pi(M)}, {νi(M)}which partially determine the combinatorial structure of the cell-decompositions, the first step could involve answering the question: Given sequences p = (p3, p5, …), ν = (ν3, ν5, …) of non-negative integers satisfying conditions

and

does there exist a cell-decomposition M of Pg, for which pi (M) = pi, νj(M) = νj for all i, j ≠ 4? (If so, the sequences p, ν are called realizable in the sequel. M itself is a realization of p, ν. For brevity's sake we will often use the word map instead of cell-decomposition in the sequel.)

This functional equation for Dedekind's

has been established in many different ways; however, it seems that the following proof, a straightforward application of the theorem of residues, has not been observed before. Since

it suffices to prove that

Let Κ be a finite number field and let ο be the ring of algebraic integers in Κ. The algebraic integers in a finite extension field Λ of Κ form a ring . We shall be concerned here with the structure of such rings , viewed as modules over ο. It will be useful to begin with a brief discussion of a new concept of the discriminant of Λ/Κ, introduced in a preceding paper [1], which will be our principal tool (see also [2]).

Stokes's stream-function for an axisymmetrical irrotational motion of perfect incompressible liquid, bounded by a plane [1], or by a sphere [2], has been expressed in terms of the stream-function for the motion in an unbounded liquid. Corresponding results are found in this note for the slow steady motion of a viscous liquid, when the motion in unbounded liquid is irrotational.

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

The vertex-connectivity and the edge-connectivity of a graph involve minimum sets of vertices and edges, respectively, whose removal results in a disconnected graph. However, the mixed case of separating sets consisting of both vertices and edges appears to have been overlooked. Such considerations might apply to vulnerability problems, such as that of disrupting a railway network with both tracks and depots being destroyed. Depending on the relative costs, a particular combination of tracks and depots might be optimal for the purpose.

A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

Estimates involving polynomials can often naturally be given in terms of the discriminants of these polynomials or of the resultants of pairs of polynomials. Since the discriminant of a polynomial with multiple zeros vanishes as does the resultant of two polynomials with common zeros these results become trivial when applied to such polynomials or pairs of polynomials. Therefore it is often necessary to exclude polynomials with multiple zeros from a given investigation. In theoretical studies of a measure-theoretical nature this often does not affect the results; however for the purpose of constructing polynomials with specified properties it can be an advantage if it is not necessary to restrict the attention to polynomials without multiple zeros.

Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsøe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.

It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

If D is an integral domain with quotient field K, then by an overring of D we shall mean a ring D′ such that D ⊂ D′ ⊂ K. Gilmer and Ohm in [GO], Davis in [D] and Pendleton in [P] have studied the class of integral domains D having the property that each overring D′ of D is of the form Ds for some multiplicative system S of D. In [D] and [P] a domain with this property is called a Q-domain and in [GO] such a domain is said to have the QR-property. Slightly altering the terminology of [D] and [P], we shall say “QR-domain” instead of “Q-domain”. Noetherian QR-domains are precisely the Dedekind domains having torsion class group [GO; p. 97] or [D; p. 200]. Pendleton in [P; p. 500] classifies QR-domains as Prüfer domains satisfying the additional condition that the radical of each finitely generated ideal is the radical of a principal ideal. It is pointed out in [P; p. 500] that this characterization of QR-domains still leaves unresolved the question of whether the class group of a QR-domain is necessarily a torsion group. We show in §1 that a QR-domain need not have torsion class group. Our construction is direct; however, the problem can be viewed in terms of the realization of certain ordered abelian groups as divisibility groups of Prüfer domains, and we conclude §1 with a brief discussion of this approach to the problem.

Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G∣−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On,ϵ(Hn−1+δG+ϵ) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

During the past 55 years substantial progress concerning sharp constants in Poincaré-type and Steklov-type inequalities has been achieved. Original results of H. Poincaré, V. A. Steklov and his disciples are reviewed along with the main further developments in this area.

Let X be a metric space and τ a non-negative function on the subsets of X. By the well-known Carathéodory process, we generate outer measures μδ(τ), for δ > 0, and (see §3). When, for every A ⊂ X, τA = (diamA)s for s ≥ 0, μ(τ) is the Hausdorff s-dimensional measure, and, if τA = h(diam A) for a monotone continuous function h with h(0) = 0, μ(τ) is the Hausdorff h-measure. In both of these cases, μ(τ) has been extensively studied.

It is well known that if, in a continued fraction

there is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then

where qn is the denominator of the nth convergent to (1). Thus if

ξ is transcendental.

A long-standing conjecture claims that the Diophantine equation

has finitely many solutions, and, maybe, only those given by

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝd, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝd. If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W, provided that A is regular enough (e.g., polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ AC are replaced by prescribed threshold volumes of B in A and W in AC. Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B. An analogous result holds for the hit-or-miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.