A topological ordered space (or pospace) is a poset (X, <) with a topology on X for which the relation < is closed in the product X × X. The topology of X is then necessarily Hausdorff. The basic theory of pospaces was developed by Nachbin in his book [5]; and others have extended it, but the resulting body of knowledge is not very geometrical. There are few concrete examples, other than the unit interval I with its natural order, and Euclidean spaces (Rn, ≤), the Hilbert cube (H, ≤) (each with the vector order), and some function spaces.

Abstract. We study the envelope of the family of planes which bisect the volume of a tetrahedron.

In Theorem 7.13 of [1], Proposition 3.1 of [5], and Theorem 1 of [10], minimal group actions on R-trees are considered. If a group G acts on a tree T, then a Lyndon length function lu is associated with each point u∈T. Abstract minimal length functions are defined in Section 2 of this paper by a simple reduction process, where lengths of elements are reduced by a fixed amount (except that any length must remain non-negative). It is shown in Theorem 2.3 that minimal length functions correspond to minimal actions by following Chiswell's construction of actions on trees from length functions, given in [4]. A parallel result to Theorem 1 of [10] is given for minimal length functions in Theorem 2.2. One outcome of these results is that to determine which length functions can arise from an action of a group on the same tree, it suffices to consider only minimal length functions. Section 1 is concerned with some preparatory properties on lengths of products of elements. These lead in Proposition 1.6 to an alternative description of the maximal trivializable subgroup associated with a length function, defined in [3].

§1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,

where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define

§1. Introduction. Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X. The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ-diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each ε>0, it is possible to write

where each set Xi, i≥1, has the property that each non-empty subset of Xi, has a non-empty relatively open subset of ρ-diameter less than ε. If is any family of subsets of X, we say that X is σ-fragmented by the metric ρ, using sets from, if, for each ε>0, the sets Xi, i ≥ 1, in (1.1) can be taken from

Let R be a commutative, Noetherian ring and let Q be the total quotient ring of R. We shall call B an intermediate ring if R ⊂ B ⊂ Q. In [S] it is proved, for an integral domain R, that if R ⊂ B ⊂ Rf where B is flat over R, then B is a finitely generated R-algebra. We observe that the result holds for any commutative, Noetherian ring where f is a non-zero divisor. Our proof [Theorem 1.1] is a little different and straight; it is given for completeness. The idea of the proof in [S] lies in finding an ideal I of R such that IB = B, and for any λ∈I, b∈B there exists m ≥ 1 such that λmb ∈ R. We shall show that even if an intermediate ring B is finitely generated R-algebra, there may not exist any ideal I of R such that IB = B, moreover, if B is not finitely generated R-algebra, we may have IB = B for some ideal I in R.

An explicit formula is given for the volume of the polar dual of a polytope. Using this formula, we prove a geometric criterion for critical (w.r.t. volume) sections of a regular simplex.

§1. Introduction. In 1985, Sárkõzy [11] proved a conjecture of Erdõs [2] by showing that the greatest square factor s(n)2 of the “middle” binomial coefficient satisfies for arbitrary ε > 0 and sufficiently large n

Where

Dörner [8] has recently proved a result for systems of additive equations over algebraic number fields, which we restate in a more specific setting.

Abstract. We show that the set of T-numbers in Mahler's classification of transcendental numbers supports a measure whose Fourier transform vanishes at infinity. A similar argument shows that U-numbers also support such a measure.