Let x denote a primitive character to a prime-power modulus k = pα. The expected estimate

for the incomplete character sum has been established for r = 1 and 2 by D. A. Burgess and recently, he settled the case r = 3 for all primes p < 3, (cf. [2] for the proof and for references). Here, a short proof of the main inequality (Theorem 2) which leads to this result is presented; the argument being based upon my characterization in [3] of the solution-set of a related congruence.

Knopp's inequality for convex functions ϕ on an interval I = [m,M] states that

for an explicit functional H, and all integrable g: [0, 1] → I. In this paper we give results of this kind in which the integral operator, ∫, is replaced by a general isotonic linear functional.

This is a sequel to the author's paper "On the semigroup of probability measures of a locally compact semigroup." We continue to investigate the relationship between amenability of spaces of functions and functionals associated with a locally compact semigroups S and its convolution semigroup MO(S) of probability measures and fixed point properties of actions of S and MO(S) on compact convex sets.

The construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

We determine, except for one unsettled case, which combinations of Weierstrass weights can occur on irreducible rational nodal curves of arithmetic genus three. It is shown that the number of nonsingular Weierstrass points on such curves can be any integer between 0 and 6, except 1.

We show that, if X and Y are Banach spaces such that X has the Mazur's intersection property and such that there exists T, an operator from Y into X so that T* and T** are injective, then there exists on Y an equivalent norm which has the Mazur's intersection property.

We deduce from this result and from a result of M. Talagrand that there exists on the long James space J(η) an equivalent norm which has the Mazur's intersection property.

A random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

Let {x} denote the fractional part of x. We find an asymptotic formula of , where k is any positive integer and a is any real number ≥ 1, and so for the sum ∑n<xf(n), where f(n) belongs to a class of additive functions.

Hidden behind a sums of squares formula are other such formulae not obtainable by restriction. This drastically simplifies the combinatorics involved in the existence problem of sums of squares formulae, and leads to a proof that the product of two sums of 16 squares cannot be rewritten as a sum of 28 squares, if only integer coefficients are permitted. We also construct all [10, 10, 16] formulae.

Let G be a locally compact abelian group. An amalgam space (Lp ℓq)(G) (1 ≦ p,q ≦ ∞) is a Banach space of functions which belong locally to LP(G) and globally to ℓq. In this paper we present noninclusion results related to the Hausdorff-Young theorem for amalgams.

Recently, Balder proved a version of FatoiTs lemma in several dimensions which, inter alia, generalizes a version of this lemma due to Artstein. Here we show how the latter result can be used to derive the former, by using Chacon's biting lemma.

We explicitly describe an infinite collection of pairs of Azumaya algebras over the ring of integers of real quadratic number fields K which are maximal orders in the usual quaternion algebra over K, hence Brauer equivalent, but are not isomorphic. The result follows from an identification of the groups of norm one units, using the classification of Coxeter.

Let R be a prime ring with involution satisfying x/2 ∊ R whenever x ∊ R. Assume that R has two nontrivial symmetric idempotents e1, e2 whose sum is not 1, and that the subrings determined by e1, e2, 1 — (e1 + e2) are not orders in simple rings of dimension at most 4 over their centers. Then if L is a Lie derivation of the skew elements K into R there exists a subring A of R, A ⊆ , a derivation D :A → RC, the central closure of R, and a mapping T:R → C, satisfying L = D + T on K and = 0.

Let P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

We determine an algebraic condition necessary and sufficient for a group G to act freely on the nth Cartesian power of an even sphere, and characterize the abelian groups that satisfy this condition.

Let A be an H-simple commutative H-module algebra, with AH = k and dimk H ≤ dimk A < ∞. We show that this implies that A # H is isomorphic to Mn(k), a central simple algebra. We apply this to characterize certain group graded algebras, algebras acted upon by a group as automorphisms, or by a nilpotent Lie algebra as derivations.

Denote by d2 the immanant afforded by Sn and the character corresponding to the partition (2, 1n-2). If n ≥ 4, the following analog of Oppenheim's inequality is proved:

for all n-by-n positive semidefinite hermitian A and B.

A new concept of generic sign-stability is proposed, and a necessary and sufficient condition for this property is given. This result shows that the condition proposed by Quirk and Ruppert [12] is correct almost everywhere, and helps to clarify the counterexample presented by Jeffries [4].

A complex nilmanifold X is isomorphic to a product X ⋍ ℂp x N/┌, where N is a simply connected nilpotent complex Lie group and ┌ is a discrete subgroup of N not contained in a proper connected complex subgroup of N. The pair (N, ┌) is uniquely determined up to holomorphic group isomorphisms.

Let a be a constant, |a| = 1. We shall prove meromorphic (M) and bounded-holomorphic (BH) versions of the following prototype: (P) Let f and g be holomorphic in a domain D. Then, |f'| = |g'| in D if and only if there exist constant a, b with f = ag + b in D. (M) Let f and g be meromorphic in D. Then, |f'|/(1 + |f|2) = |g'|/(1 + |g|2) in D if and only if there exist a, b with |b| ≦ ∞ such that f = a(g - b)/(\ + g). (BH) Let f and g be holomorphic and bounded, |f| < 1, |g| < 1, in D. Then, |f'|/ (1 - |f|2) = |g'|/(1 - |g|2) in D if and only if there exist a, b with |b| < 1, such that f = a(g - b)/(1 - g).