After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group.

This brief note is to demonstrate the existence of a new class of (exceptional) Lie algebras of type D^ The construction stems from a cyclic sixth degree extension P /$ , together with an element y of norm 1 in the unique cubic subfield F/$ of P /$ , where y(£NPf!?(P*). Each such y will determine a non-Jordan (see [l] for definition) Lie algebra 8(7), of type D4111. Two algebras of this form, 8...

Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Traditionally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which ther...

We show that for every 3-connected cubic graph G, the prism G×K2 has cycles of every even length. Furthermore, if G has a triangle, then G × K2 is pancyclic.

Expressions for the pressure-dependent apparent elastic constants of cubic crystals are derived in terms of their original secondand third-order elastic constants. New expressions are obtained for the apparent compliances and their pressure derivatives. Results are shown to be in agreement with those based on the apparent elastic moduli. ( 1998 Elsevier Science Ltd. All rights reserved.

In this paper we prove that the generalized version of the Minimal Resolution Conjecture stated in [7] holds for certain general sets of points on a smooth cubic surface X ⊂ P. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

We present some recently obtained constructions of linear quasi-perfect codes with small density arising from plane cubic curves.

The paper characterizes (2t + 1)-regular graphs with circular flow number 2 + 2 2t−1 . For t = 1 this is Tutte’s characterization of cubic graphs with flow number 4. The class of cubic graphs is the only class of odd regular graphs where a flow number separates the class 1 graphs from the class 2 graphs. We finally state some conjectures and relate them to existing flow-conjectures.

In this paper we study sequences of lattices which are, up to similarity, projections of Z onto hyperplanes v⊥, with v ∈ Z. We show a sufficient condition to construct sequences converging at rate O(1/ ‖v‖) to integer lattices and exhibit explicit constructions for some important families of lattices. The problem addressed here arises from a question of communication theory.