Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.

In the present note we discuss some properties of a ‘measure of asymmetry’ of convex bodies in n-dimensional Euclidean space. Various measures of asymmetry have been treated in the literature (see, for example, (1), (6); references to most of the relevant results may be found in (4)). The measure introduced here has the somewhat surprising property that for n ≥ 3 the n-simplex is not the most asymmetric convex body in En. It seems to be the only measure of asymmetry for which this fact is known.

Let HX be the trigraded Hilbert function of a set X of reduced points in $\mathbb{P}$1 × $\mathbb{P}$1 × $\mathbb{P}$1. We show how to extract some geometric information about X from HX. This paper generalises a similar result of Giuffrida, Maggioni and Ragusa about sets of points in $\mathbb{P}$1 × $\mathbb{P}$1.

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

1. In this paper I consider various properties of the area of a surface including the relationship between Hausdorff measure and the Lebesgue area. The results here obtained will enable me, in a subsequent paper, to investigate the tangential properties of parametric surfaces. For simplicity of exposition I confine my attention to surfaces homoeomorphic to a disk, but the general case follows easily from cyclic element theory.

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

In this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case.

We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of ℤ+2 in the critical regime n1 ≍ √n2 and in the subcritical regime n1 = o(√n2). This work completes the results established in the fifties by Auluck, Nanda and Wright.

In the construction of the seismological tables it has been found that the times of several pulses at a given distance show a well-marked maximum frequency for each, about which the bulk of the observations are spread with a standard deviation of 1·5−3 sec. For others there was only a vague concentration, spread over about half a minute (Jeffreys and Bullen, 1936–40). The theoretical times of the latter type could all be calculated from those of the former by using the principle of stationary time for small variations of the path, and the observed times might be either shorter or longer than the theoretical ones. There was a complete association between the type of distribution of the residuals and the nature of the path. Whenever the time as calculated was a minimum for all small variations of the path, the distribution of residuals was of the former type; when it was stationary without being either a true minimum or a maximum it was of the latter.

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

Let α be a positive integer, and El, …, Eα Hadamard sets of positive integers. It is shown that E = E1 + … + Eα determines a Littlewood–Paley decomposition of Z.

Suppose that is a Hadamard set of positive integers such that nj+1/nj ≥ 2 for all j. Let α be a positive integer, and

We show that F(α) also determines a Littlewood-Paley decomposition of Z.

Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.

We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

In (general) elastico-viscous liquids the response to stress at any instant will depend on previous rheological history, the equations of state needed to describe the rheological properties of a typical material element at any instant t being expressible in the form of a (properly invariant†) set of integro-differential equations relating its deformation, stress and temperature histories (as defined by a metric tensor (of a convected frame of reference), a stress tensor and the temperature measured in the element as functions of previous time t'( < t)) together with the time lag (t – t') and physical constant tensors associated with the element (1). Thus in any type of oscillatory motion a rheological history will necessarily be a function of the frequency of the forcing agent, the rheological history of a number of different types of elastico-viscous liquids in some simple shearing oscillatory flows being a rather simple oscillatory history (see, for example, (2–4)). It is, therefore, to be expected that a liquid with elastic properties will behave somewhat differently from any inelastic viscous liquid when subjected to any kind of oscillatory motion, and it is for this reason that oscillatory motions have been used extensively to detect and measure the elastic properties of liquids (see, for example, (2–5)).

A Haken sphere $S$ in a 3-manifold $M$ with a Heegaard splitting $V\cup_F W$ is a separating 2-sphere in $M$ which intersects the Heegaard surface $F$ only in an essential circle in $F$. We describe how any two Haken spheres in the genus 2 Heegaard splitting of connected sum of two lens spaces are related, which extends Scharlemann–Thompson's corresponding result on Haken spheres in genus 2 Heegaard splitting for $S^3$.

It has been shown that univalent circle packings filling the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result.

We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.

We show that, for every structurally finite transcendental entire function, the Hausdorff dimension of its Julia set is two.

The Atiyah–Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic $K$-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic $K$-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.

The solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are defined as

1. In (1) M. G. Smith solved the equation

where

for R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namely

and

The examples are unconnected.

According to some published footnotes, examples are known of Serre fibrations which are not Hurewicz fibrations. However, as far as I know no such example has been published. The following example, which was stimulated by a more complicated example of G. Horrocks, seems worth mentioning, since it is extremely simple and does suggest some further questions.