Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

We define s-dimensional Hausdorff outer measure on subsets of ℝn by

Here | | denotes the diameter of a set. The Souslin sets, which include the Borel sets, are Hs-measurable for any s ≥ 0.

In the paper, we provide an effective criterion for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

The flow induced by an oscillating circular cylinder which may perform transverse, torsional and axial vibrations is considered. The steady streaming associated with purely transverse vibrations of the cylinder may be significantly modified by the presence of, and interaction with, torsional oscillations. Similarly the interaction between the transverse and axial vibrations introduces a modification to the axial flow, which results in a steady streaming motion in the axial direction.

A (convex) d-polytope is the convex hull of a finite set of points in Euclidean d–space Ed. The (Minkowski) sum of two polytopes P1 and P2 is defined by

.

Let Λ be a lattice in 3-dimensional space which provides a double covering for spheres of unit radius. By this we mean that if X is any point of space, there are at least two distinct lattice points P, Q such that XP ≤ 1, XQ ≤ 1. Let d(Λ) be the determinant of Λ. We shall prove that

where

For finite coverings in euclidean d-space Ed we introduce a parametric density function. Here the parameter controls the influence of the boundary of the covered region to the density. This definition gives a new approach to covering which is similar to the approach for packing in [BHW1], [BHW2]. In this way we obtain a unified theory for finite and infinite covering and generalize similar results, which were developed by various authors since 1950 for d=2, to all dimensions.

Kan and Thurston, in their paper [5], asked whether each smooth closed manifold other than S2 or RP2 has the same integral homology as a closed aspherical manifold. F. E. A. Johnson in [3], [4] is concerned with the answer to this question when the smooth closed manifold is an n-dimensional sphere Sn. He asked whether there exist aspherical manifolds Xг which have the homology of Sn.

In [1] I showed that for any θ belonging to (0,1) there exists a constant C(θ) such that for at least θp of the non-principal characters χ modulo the prime p we have both

and

A length function, for a group, associates to an element x a real number |x| subject to certain axioms, including a cancellation axiom which embodies certain cancellation properties for elements of a free group. Integer valued length functions were introduced by Roger Lyndon [1] where, with a more restrictive set of axioms than ours, it is shown that a length function for a group is given by a restriction of the usual length function on some free product.

A compact space K is said to have the Namioka Property, or to belong to the class *, if, for every Baire space B and every separately continuous function Ψ:B × K → ℝ, there is dense δ subset H of B such that Ψ is (jointly) continuous at all points of H × K. Although the terminology is more recent, the idea of looking at properties of this kind goes back to Namioka's paper [6] on separate and joint continuity. Talagrand [8] gave the first example of a compact space that is not in * and it is now known [4] that there are even examples of scattered compact spaces that are not in *. On the other hand, many good classes of compact spaces have been shown to be contained in *, probably the most general being the class of continuous images of Valdivia compacts [2]. The aim of this note is to prove the following stability result: a compact space which is a countable union of closed subsets with the Namioka Property does itself possess that property.

Consider a slab which is made from a homogeneous and isotropic thermoelastic material and which occupies the region 0 ≤ x ≤ a, where x, y, z are the usual rectangular cartesian coordinates. Suppose that the slab undergoes a motion in which the displacement vector is parallel to the x-axis and the displacement and the temperature are functions of the coordinate x and the time t ( ≥ 0) only. Suppose too that the faces of the slab are clamped, that the face x = 0 is maintained at a constant temperature, and that heat is supplied to unit area of the face x = a at a prescribed rate h(t).

Suppose x1, x2Є[0,1] and α = log 3/ log 4. Then

The relation is readily seen to be satisfied with equality for both of X1, x2 equal to any of the values 0, ½, 1 so that the value of α is “best possible”.

We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$-neighborhood of the unit ball, the only solutions of $\unicode[STIX]{x1D6F1}^{2}K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$-dimensional convex body, $\unicode[STIX]{x1D6F1}K$ is the projection body of $K$ and $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$

We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

A number of oscillating systems have been described in the literature which have proved useful in the study of elastico-viscous liquids. The most common type of system in use is the forced oscillating system of the coaxial-cylinder type; in this, the outer cylinder wall is made to oscillate about its axis with a prescribed frequency, and the resulting motion of the inner solid cylinder (constrained by a torsion wire) is recorded (see, for example, [1]). Another type of oscillating system— a, free oscillating system, also of the coaxial-cylinder type—was considered by the present authors in a previous paper [2]; this latter system is one which may prove to be simpler to design and to control in practice.

We shall say that the sets A, B ⊂ Rk are equivalent, if they are equidecomposable using translations; that is, if there are finite decompositions and vectors x1,…, xd∈Rk such that Bj = Aj + xj, (j = 1,…,d). We shall denote this fact by In [3], Theorem 3 we proved that if A ⊂ Rk is a bounded measurable set of positive measure then A is equivalent to a cube provided that Δ(δA)<k where δA denotes the boundary of A and Δ(E) denotes the packing dimension (or box dimension or upper entropy index) of the bounded set E. This implies, in particular, that any bounded convex set of positive measure is equivalent to a cube. C. A. Rogers asked whether or not the set

A sequence {an} of integers is said to be primitive if whenever i ≠ j. For example, if n is any positive integer, the sequence

is primitive. This is an important example in the light of the elementary result (see [2; p. 244]) that if 0 < a1 < a2 < … < ar ≤ 2n is primitive then necessarily r ≤ n; i.e. at most half of the positive integers ≤ 2n can be members of the sequence. Besicovitch [2; p. 257] has obtained the surprising result that, given ε > 0, there exists an infinite primitive sequence {ai} such that

where A(n) denotes the number of ai ≤ n.

We consider a Markoff spectrum for the set of indefinite binary quadratic forms with real coefficients which represent zero non-trivially. As was done for the classical Markoff spectrum, we show that 1/3 is the largest accumulation point of the set and explicitly determine the countably infinite number of elements greater than 1/3. Unlike the situation for the classical Markoff spectrum, there is a countably infinite number of limit points greater than 1/3.