Let be a = ℤ-order in A, a finite dimensional Q-algebra. K0() denotes the Grothendieck group of projective right -modules. locally isomorphic to } is a subgroup of K0() and is called the locally free classgroup of . (If = ℤΓ for some finite group Γ then as all ℤΓ projectives are locally free [12].)

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results is

provided m ≥ log n. Related questions and results are also discussed.

Abstract. In this paper is proved a formula for the number of neighbourly d-polytopes with d + 3 vertices, when d is odd.

In the study of heat transfer to fluids flowing in pipes or channels, the inversion of a Fourier transform requires consideration of the zeros of a certain function. The principal contribution here is a proof that these zeros are all real and simple, properties assumed by previous authors.

Erdoső has asked whether there exists a positive constant c < 1 such that

where pn is the n-th prime. In 1943 Selberg «3» proved, on the assumption of the Riemann Hypothesis, that

uniformly for H > 0. By an elementary extension of Selberg's method we prove the following two results free of any unproved assumptions.

Let p be an odd prime, let q be a divisor of p – 1, and let α be a primitive q-th root modulo p. For each natural number r the metacyclic group Gr is defined by

Radon's theorem [8] asserts that, if X is a finite set of s points in Rd and s ≥ d + 2, then X admits a. Radon partition, that is, a partition {X1; X2} of X into disjoint subsets X1 and X2, such that

The stability against small disturbances which do not bend the vortex lines of a circular cylindrical vortex sheet of variable radius is examined. If the jump in tangential velocity across the sheet falls off with time faster than (time)–íthe disturbances are bounded (or grow less rapidly than linearly with time) so that the sheet is effectively stable. If the tangential velocity jump falls off with time slower than (time)–í the sheet is unstable.

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.

The equations governing the evolution of the modulated amplitude of a pointcentred disturbance to a slightly supercritical flow are shown to have solutions which become infinite at a finite time and at a single point. The amplitude develops a sharp peak and the structure of this peak is found, for real and complex coefficients in the governing equations. Such a solution can only occur if the coefficients satisfy certain conditions