The Newhouse gap lemma is generalized by finding a geometric condition which ensures that N-fold sums of compact sets, which might even have thickness zero, are intervals. A new proof is also obtained of a lower bound on the thickness of the sum of two Cantor sets.

Let G be a plane domain with ∞ ∊ G. Let E be the compact complement of G and cap E the logarithmic capacity. We shall assume that cap E = 1 and E ⊂ {|Z| ≤ R. Then R ≥ 1, with equality if and only if E is a closed disc.

In this note we shall show that the assumptions made by Murray [1] on the initial state of the gas and the initial magnetic field are inconsistent with the steady state momentum equation and the energy balance equation (20). The form of second term in equation (20) requires

For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3), (5.4), indicate that some rather subtle hypothesis is needed. The first such hypothesis, see Davenport and Lewis [1], was very far from being necessary for the solubility of the equation. It would seem that any supplementary hypothesis which (loosely) is somewhere near necessary and also (together with the NCC) somewhere near sufficient deserves separate detailed investigation before one proceeds to use it.

In [7[ a functor Ext is defined in terms of C*-extensions. It is a covariant functor from the homotopy category of compact, metrizable spaces to abelian groups. Further details are given in [7, 8, 9, 11]. From [7, 14] Ext extends to a Steenrod homology theory, Ext*, which may be identified with the one associated with unitary K-theory. Since Lie groups are fundamental to K-theory (see [2, p. 24]) one might expect Ext(G) to be of interest when G is a Lie group.

Let X be an infinite set and T(X) be the full transformation semigroup on X. In [4] and [6] Howie gives a description of the subsemigroup of T(X) generated by its idempotents. In order to do this he defines, for α in T(X),

and refers to the cardinals s(α) = |S(α)|, d(α) = |Z(α)| and |c(α) = |C(α)| as the shift, the defect, and the collapse of α respectively. Then putting

he proves that the subsemigroup of T(X) generated by its idempotents is . Furthermore, both F and Q are generated by their idempotents

We give a simple graph-theoretic proof of a classical result due to Nash-Williams on covering graphs by forests. Moreover, we derive a slight generalization of this statement where some edges are preassigned to distinct forests.

Given a Hausdorff topological vector space with dimensiongreater than one, the barycentre of simple masses can be seen as the unique associative, internal and continuous mapping defined on these masses. Moreover, if the associated dual space separates points, by extending the continuity property, one can characterize also the barycentre of masses with compact convex support.

In Mahler's classification of complex numbers [10] (see [4]), a transcendental number ξ is called a U-number if there exists a fixed integer N ≥ 1 so that for all ω > 0, there exists a polynomial so that

where the height h(f) = max {|α0|, |α1|, …, |αN|}. The number ξ is called a Um-number if the above holds for N = m but for no smaller value of N (examples and further details may be found in [9,1 and 2]). Thus the set of U1-numbers is precisely the set of Liouville numbers. In this paper we investigate the statistical behavior of the partial quotients of real U-numbers, in particular, U2-numbers. In addition, we demonstrate the existence of a U2-number with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number. Related results involving badly approximate U2-numbers are also discussed.

In this note we present a simple way of obtaining the universal field of fractions of certain free rings as a subfield of an ultrapower of a (by no means unique) skew field. This method of embedding was discovered by Amitsur in [1]; our presentation uses Cohn's specialization lemma and the embedding is constructed in terms of full matrices over the rings in question (Theorem 3.2). In particular, if k is an infinite commutative field, the universal field of fractions of a free k-algebra can be realized as a subfield of an ultrapower of any skew extension of k, with centre k, which is infinite dimensional over k. Thus many problems concerning the universal field of fractions of a free k-algebra can be settled by studying skew extensions of k of relatively simple structure. More precisely and more generally, let E be a skew field with centre k and denote by R the free E-ring on X over k. Write U for the universal field of fractions of R and assume that U embeds in an ultrapower of a skew field D. Then by Łos' theorem, U inherits the first-order properties of D which can be expressed by universal sentences.

The following theorem appears in Weyl's famous memoir [3] of 1916.

THEOREM A. Let λ1 ≤ λ2 ≤ … ≤ λn ≤ … be an increasing sequence of positive integers. Suppose that, of the numbers λ1, … λn, the first h1 are equal to each other, then the following h2 and so on, and finally that the last hm coincide. Let hj. If

the sequenceis uniformly distributed (mod 1)for almost all x, in the Lebesgue sense.

Gravity and surface-tension effects are examined for inviscid–inviscid interactions between two fluids close to a wall. The ratios of density and viscosity of the two fluids are taken to be small. A nonlinear integro–differential equation is found to govern the near-wall flow velocity, interface shape and pressure; analysis, computation and comparisons are then applied. Travelling-state solutions are of particular interest.

§1. Preliminaries. A Cauchy process in d-dimensional Euclidean space, Rd, is a stochastic process, Xt(ω), with stationary independent increments and with a continuous transition density, p(t, y − x) defined by

and

where m, the isotropic measure, is a probability measure on Sd, the unit sphere in Rd, such that when d > 1 the support of m is not contained in any d − 1 dimensional subspace. In (2) w is given by

where . It follows that for each t > 0 and y we have p(t, y) > 0 and that for each t > 0 p(t, y) is a bounded and continuous function of y. Xt(ω) can be considered as being a standard Markov process (for a full description of the definition of such a process see Chapter 1 of [1]) and in particular we can assume that the sample functions of Xt(ω) are right continuous and have left limits. We can also assume that Xt(ω) enjoys the strong Markov property. We write Px and Ex for probabilities and expectations conditional on X0(ω) = x, and we write P for P0.

In its simplest form, the theorem of Ascoli with which we are concerned is an extension of the Bolzano-Weierstrass theorem: it states that, if X and Y are bounded closed sets, of real or complex numbers, andis a sequence of equicontinuous functions mapping X into Y, thenhas a uniformly convergent subsequence. As is well known, this result has played a fundamental part in the development of several theories; it has also been widely generalized, by processes of abstraction and localization, and a very useful version of the theorem runs as follows (cf. [4], 233–234):

(A) Suppose that X is a locally compact regular space, and that Y is a Hausdorff space whose topology is determined by a uniform structure. Let YX be the space of all functions that map X into Y, with the topology of locally uniform convergence with respect to. Then a closed setin YX is compact if, at each point x of X, (i) the set(x) is relatively compact, in Y, and, (ii) is equicontinuous with respect to.

Let K be an algebraic number field of degree n = rl + 2r2 (in the usual notation) over the rationals with discriminant d. Let ZK denote the ring of integers in K. It is usual to speak of an integer Πi ∈ Zk as an almost-prime of order l, if the principal ideal (Πi) has at most l prime ideal factors, counted according to multiplicity. Let P1, …, Pn be positive real numbers with Pk = Pk+r2, k = r1 + l, …, r1 + r2 and P = P1 … Pn ≥ 1.

1. Certain geometric properties of the valuation theory were considered by O. Zariski in [7]. We have proved some related results in [1] and we consider further similar problems in this paper.

Let V be an irreducible algebraic primal situated in Sd, where d≥3. Throughout the ground field is the field K of complex numbers. For simplicity we assume that V lies in an affine space Ad with coordinates x1,…,xd. Let O be a point on V not at infinity and we take it to be the origin of Ad. Apply a monoidal transformation to V with O as the basis; We obtain thereby a (d−l)-fold V1 lying on a non-singular d-fold U1 situated in an affine space of dimension N1 Since V and V1 are birationally equivalent, we may identify their function fields and thus we denote their common function field by Σ.

In this paper it is shown that a simplicial action of Zp (p a prime) on an n-dimensional simplicial complex which is a Poincaré duality space of formal dimension n for Zp coefficients cannot have just one isolated fixed point.

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and

$$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$

${\it\gamma}(\cdot )$

$\mathbb{A}$

$X,Y\subseteq A$

$$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$

$\mathbb{A}$

${\it\gamma}(X+Y)$

$|S|$

$S$

$\mathbb{A}$

Let τ(n; k, h) denote the number of divisors of n which are congruent to h (mod k), and τ(n; k) the number of divisors of n prime to k, so that

Let

Erdős [1] proved that, if ε and η are fixed arbitrary positive numbers, then for almost all integers n ≤ x, we have

provided

Hall and Sudbery [2] showed that it is sufficient that

and apart from the ε, this upper bound for k is best possible, for it is clear that k must not exceed the normal order of τ(n). For n ≤ x, Hardy and Ramanujan [3] showed that this normal order is (log x)log 2 = 2 log log x.

A subsemigroup S of a semigroup Q is an order in Q if, for every q ∈ Q, there exist a, b, c, d ∈ S such that q = a−1b = cd−1 where a and d are contained in (maximal) subgroups of Q and a−1 and d−1 are their inverses in these subgroups. A semigroup which is a union of its subgroups is completely regular.