ELEMENTARY CLASSES AND ACCESSIBLE CATEGORIES T. BEKE∗ AND J. ROSICKÝ∗∗ Abstract. We investigate properties of accessible categories with directed colimits and their relationship with categories arising from Shelah’s Abstract Elementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks. We investigate properties of accessible c...

Let Q be a diassociative A-loop which is centrally nilpotent of class 2 and which is not a group. Then the factor over the centre cannot be an elementary abelian 2-group.

I have been asked to discuss the status of QCD. It seems to me that there are three main points to be made about the present status of QCD: • QCD is right, and we can do many beautiful things with it. • There are several important concrete problems that lie just beyond the edge of our current understanding. • There are some foundational issues in QCD, and some recent developments, that may poin...

We show that the qualitative form of the prime k-tuple conjecture implies a certain conjecture related to the least size of the interval containing exactly ` prime numbers. Also, we prove that the latter conjecture implies the prime k-tuple conjecture for some minimal admissible set. The proof is self-contained and elementary.

We give a new proof of the Minkowski-Hlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective. 1

In this article, we discuss a theorem about polygons in the plane, which involves in an intriguing manner the number 12. The statement of the theorem is completely elementary, but its proofs display a surprisingly rich variety of methods, and at least some of them suggest connections between branches of mathematics that on the surface appear to have little to do with one another. We describe fo...

Let G be an abelian group of finite order n, and let h be a positive integer. A subset A of G is called weakly h-incomplete if not every element of G can be written as the sum of h distinct elements of A; in particular, if A does not contain h distinct elements that add to zero, then A is called weakly h-zero-sum-free. We investigate the maximum size of weakly hincomplete and weakly h-zero-sum-...

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is “real” or not. We attempt to flesh out the details of Connes’ ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemann...

An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof answers this question in the affirmative.