The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

When a displacement front meets a heterogeneity in a porous medium, its shape will be altered. The amount of distortion depends on the size and shape of the heterogeneity, the amount of variation of the heterogeneous properties, and the mobility ratio between the displaced and displacing fluids. The solution for a circular permeability discontinuity is known when the capillary pressure is uniform and the mobility ratio M is unity [4]. Here, we extend the theory to the case where there is a small change ε in capillary pressure as well as a nearly unit mobility ratio M = 1 + εγ. Corrections can then be found, in closed form, to first order in ε, to the shape of the front and the pressure field. Computations using these expressions are simpler than the full free-boundary problem, and some analytical estimates are possible in further limits. Finally, the theory is extended to the case of a front passing a number of such heterogeneous patches which are widely spaced.

Let T be a nonlinear Volterra integral operator of the form

(with I compact, a < b), whose kernel K = K(x, t, u) satisfies the following conditions:

with

The membrane theory for calculating stresses in symmetrically loaded elastic shells of revolution was introduced in 1828 by Lamé and Clapeyron [1] who assumed that a thin shell is incapable of resisting bending. In 1892 Love [2] gave the general equations of equilibrium for an element of an elastic shell taking bending into consideration and obtained expressions for the strains in terms of the displacements as well as the stress-strain relations. Since then the problem of the elastic shell has been the subject of numerous researches. The spherical shell has, however, drawn the attention of many investigators due to its importance in structural and mechanical engineering, e.g. roof and boiler constructions.

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

The results we present were motivated by the product measure problem for Baire measures. For two completely regular Hausdorff spaces X and Y, with totally finite a- additive measures μ and ν defined on the Baire σ- algebras ℬ0(X) and ℬ0(Y) respectively, under what conditions may we define a measure λ on the Baire σ-algebra ℬ0(X × Y), extending the product measure μ ⊗ ν defined on the product σ-algebra ℬ0(X) × ℬ0(Y) and satisfying a Fubini theorem?

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).

We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least

Let K0(x) be a simple transcendental extension of a field K0, υ0 be a valuation of K0 with value group G0 and residue field K0. Suppose is an inclusion of totally ordered abelian groups with [G1: G0] < ∞ such that G is the direct sum of G1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K0(x) extending υ0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k0. This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ0 to K0(x).

Given a sequence a(l), a(2), a(3), … of complex numbers such that a(n) ≤ 0(nc) for some c > 0, we define, for Im(z) > 0,

where q(λ) = exp (2πiz/λ), λ > 0 and a is a real number. Throughout this paper, for complex numbers x, w with x ≠ 0, xw = exp (w log x) and the principal branch is taken for the logarithm. Then it is easily verified that the infinite product converges absolutely and uniformly in every compact subset of the upper half plane H. Hence f(z) is holomorphic in H. The aim of this paper is to determine holomorphic functions in H defined by (1) which satisfy the special transformation formula

for some real number k under certain assumptions.

The authors regret that a misprint has occurred in the main theorem of their recent paper with the above title [Mathematika, 23 (1976), 220–226]. The congruence condition which p1 should satisfy modulo if p1 has odd order.

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.

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.

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.

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

For every integer $k\geqslant 3$ we construct a $k$-gonal curve $C$ along with a very ample divisor of degree $2g+k-1$ (where $g$ is the genus of $C$) to which the vanishing statement from the Green–Lazarsfeld gonality conjecture does not apply.

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.

A subsemigroup S of a semigroup Q is a left order in Q, and Q is a semigroup of left quotients of S, if every element of Q can be written as a−1b for some a, b∈S with a belonging to a group -class of Q. Necessary and sufficient conditions on a semigroup S are obtained in order that S be a left order in a completely 0-simple semigroup Q. The class of all completely 0-simple semigroups of left quotients of S is related to the set of certain left congruences on S. Axioms are provided for semigroups which occur in the discussion of left orders in completely 0-simple semigroups.

Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.