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.

Abstract. We consider the structure of the Kac modules V(Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V(Λ) are constructed and it is shown that the number of composition factors of V(Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1(m/n), and conjectured to be valid for all finite dimensional irreducible representations of sl(m/n) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).

Let K ⊂ Rd be a convex body and choose points xl, x2, …, xn randomly, independently, and uniformly from K. Then Kn = conv {x1, …, xn} is a random polytope that approximates K (as n → ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K – vol Kn when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).

Abstract. Sufficient conditions are derived for all bounded solutions of general classes of integrodifferential equations of arbitrary order with variable coefficients to be either oscillatory or convergent to zero asymptotically.

Abstract. For F a field we compute, explicitly and directly, the right Krull dimension of the algebra Qop⊗FQ for certain semisimple Artinian F-algebras Q. (Here Qop denotes the opposite ring of Q.) We use our calculation to give alternative proofs of some theorems of J. T. Stafford and A. I. Lichtman. Our methods involve a detailed study of skew polynomial rings.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.