Critical sets for families of Latin squares

In this paper I give details of new constructions for critical sets in latin squares. These latin squares, of order n, are such that they can be partitioned into four subsquares each of which is based on the addition table of the integers module n/2, an isotopism of this or a conjugate.

A history of agricultural production at Neolithic Catalhoyuk East, Turkey

A history of agricultural production is proposed for Neolithic Catalhoyuk East, central Turkey, using archaeobotanical, environmental, population and settlement studies. In the aceramic early phase of site occupation, intensive strategies developed as changes in population and environment caused stress on food supplies produced within a limited territory. Food exchange may have been part of the social means by which Catalhoyuk and nearby contemporary settlements amalgamated into the single site of the main occupation phase. Population change, inherited territories and continuing environmental impact led to the development of an extensive system of agriculture using widely dispersed dry soils, with an intensive regime applied to nearby alluvial soils. Social tensions caused by the evolution of this system contributed to the fissioning of the site by the Chalcolithic.

Critical sets in direct products of back circulant Latin squares

To date very Few families of critical sets for latin squares are known. The only previously known method for constructing critical sets involves taking a critical set which is known to satisfy certain strong initial conditions and using a doubling construction. This construction can be applied to the known critical sets in back circulant latin squares of even order. However, the doubling construction cannot be applied to critical sets in back circulant latin squares of odd order. In this paper a family of critical sets is identified for latin squares which are the product of the latin square of order 2 with a back circulant latin square of odd order. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.

On the distance between distinct group Latin squares

In an article in 1992, Drapal addressed the question of how far apart the multiplication tables of two groups can be? In this article we continue this investigation; in particular, we study the interaction between partial equalities in the multiplication tables of the two groups and their subgroup structure. (C) 1997 John Wiley & Sons, Inc.