The Quarterly Journal of Mathematics - current issue

WHEN JORDAN SUBMODULES ARE BIMODULES

Bresar, M., Kissin, E., Shulman, V. S. — 2008-11-18

Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yA Y for all A A and y Y (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C*-algebras having Jordan ideals that are not ideals.

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Buchweitz, R.-O., Green, E. L., Snashall, N., Solberg, O. — 2008-11-18

Let = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions over Koszul algebras, Arch. Math. 85 (2005), 118–127.], and this is used to find a minimal projective resolution P of over the enveloping algebra ^e. Using these results, we show that the multiplication in the Hochschild cohomology ring of relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of is shown to be surjective onto the graded centre of the Koszul dual.

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

Choi, Y. S., Garcia, D., Maestre, M., Martin, M. — 2008-11-18

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L(µ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every -finite measure µ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely k^k/(1–k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L(µ, X) when µ is atomless, and C_w(K, X), C_w*(K, X*) when K is perfect.

ON THE SUM OF THE FIRST n PRIMES

Cilleruelo, J., Luca, F. — 2008-11-18

In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of the first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as the square root of the sum of the first n primes are well distributed modulo 1.

ON THE ERROR TERMS AND EXCEPTIONAL SETS IN CONJECTURAL SECOND MAIN THEOREMS

Levin, A., McKinnon, D., Winkelmann, J. — 2008-11-18

We study the error terms and exceptional sets in conjectural Second Main Theorems in Nevanlinna theory and Diophantine approximation (Vojta's conjecture). In particular, we give a geometric description of the exceptional set in the case of surfaces and the trivial divisor. Examples are given which show that, in general, the exceptional sets in conjectural Second Main Theorems must depend on the parameter in these conjectures. As a consequence, we obtain counterexamples to a conjecture of S. Lang on the forms of the error terms in conjectural Second Main Theorems.

ON DAVENPORT-STOTHERS INEQUALITIES AND ELLIPTIC SURFACES IN POSITIVE CHARACTERISTIC

Schutt, M., Schweizer, A. — 2008-11-18

We show that the Davenport–Stothers inequality from characteristic 0 fails in any characteristic p > 3. The proof uses elliptic surfaces over P¹ and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti and Szpiro. For characteristics 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P¹. Our ideas are also related to supersingular surfaces (in Shioda's sense).

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

Zhou, H.-S., Zhu, H. — 2008-11-18

In this paper, we consider the following semilinear elliptic problem:

where f(x, t) tends to p(x) and q(x) L(R^N), respectively, as t -> 0 and t -> +. We prove that there exist two numbers l and L with L < l such that problem (P) has at least one positive solution for (– l, –L) and has no positive solution for all [–l,–L]. The existence and non-existence of positive solutions for problem (P) at = –l and = –L are also discussed.