In a crystal containing paramagnetic impurities the parameters that specify mean squares and products of the intrinsic strain components at the impurity sites can in principle be obtained from EPR linewidths provided that the spin-strain coupling tensor is known. It is shown that if the strain-inducing defects are distributed with the symmetry of the crystal lattice the number of such parameters is restricted by the relation ( . E~, . E~~) = & u 6 , , (~) J~k )where &jJi is the ith component of the rth repetition of the irreducible representation A of the crystal's point group. Some strain-broadening measurements in MgO are considered. Ruby linewidths have been measured and an expression containing four such strain parameters, a mean-square crystallite misalignment, and an expression for the linewidth arising from Cr3' interaction with A1 nuclei fitted to them to obtain meaningful strain parameters.

Algorithms for solving linear PDEs implemented in modern computer algebra systems are usually limited to equations with two independent variables. In this paper, we propose a generalization of the theory of Laplace transformations to second order partial differential operators in ‫ޒ‬ 3 (and, generally, ‫ޒ‬ n ) with a principal symbol decomposable into the product of two linear (with respect to derivatives) factors. We con sider two algorithms of generalized Laplace transformations and describe classes of operators in ‫ޒ‬ 3 to which these algorithms are applicable. We correct a mistake in [8] and show that Dini type transformations are in fact generalized Laplace transformations for operators with coefficients in a skew (noncommutative) Ore field.

For hyperbolic first-order systems of linear partial differential equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry we propose new procedures to obtain their complete closed-form non-stationary solutions. The methods used include the classical Laplace cascade method as well as its recent generalizations for systems with more than 2 equations and more than 2 independent variables. As an example we present the complete non-stationary solution (probability distribution) for Verhulst model driven by Markovian coloured dichotomous noise. Mathematics Subject Classification (2000). 60-08, 65C20, 68W30.

We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in R n . Practically all previously proposed differential transformations of LPDOs are particular cases of this transformation (intertwining Laplace transformation, ILT ). We give a complete algorithm of construction of ILT and describe the classes of operators in R n suitable for this transformation.

(the theory of Egorov curvilinear orthogonal systems of coordinates in R3), and some other cases. They also derived nonlinear superposition formulas for all these cases. In the modern theory of (1 + I)-dimensional nonlinear integrable systems of partial differential equations, the B~icklund transformations (see [16, 17]) and the corresponding nonlinear superposition formulas have been obtained for most such systems.A theory of Bacldund transformations is also .available for (2 + l)-dimensional nonlinear integrable equations (i.e., equations with 2 "space" and 1 "time" independent variables). Such problems (described by equations or systems of equations with three independent variables) also were intensively studied in the classical differential geometry at the turn of the century: the theory of arbitrary curvilinear orthogonal systems of coordinates in R3 (and also in En, which is a (2 + 1)-dimensional 38