It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

Nonoscillatory solutions of a nonlinear neutral type higher order difference equations are classified by means of their asymptotic behaviors. Existence criteria are then provided for justification of such classficiation.

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. We study the orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0. The methods are essentially number-theoretic.

1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50; secondary 12E20.

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

In previous work [2] calculations of subquadratic second order Dehn functions for various groups were carried out. In this paper we obtain estimates for upper and lower bounds of second order Dehn functions of HNN-extensions, and use these to exhibit an infinite number of different superquadratic second order Dehn functions. At the end of the paper several open questions concerning second order Dehn functions of groups are discussed.