Let X be a cadlag Markov process with separable metric state space S, governed by semigroup of transition kernels (N_t)_t≥0. Let f be a bounded, non-negative, continuous function on S², vanishing in a uniform neighbourhood of the diagonal. Define and suppose that sup{|J_tf(x)|: t>0, x εS}<∞ and that exists for each xεS. Then du for each t≥0 and each xεS.

An approach is developed for describing inhomogeneous periodic structures in a system of Fermi particles with a short-range pair potential in the formalism of a functional integral and a functional effective action. Instability of a homogeneous base state at large pressures and also instability of a weakly inhomogeneous bodycentered structure are demonstrated.

Specifying optimal economic policy using an econometric model has passed through three phases in its evolution. The first phase started with a two-stage approach in Tinbergen’s tradition: first, one estimates an econometric model by the standard techniques of statistical estimation, then one uses optimal control theory in the estimated model to determine an optimal policy. This approach failed to recognize two types of interdependence. One is that the changes in policy or optimal control would induce changes in the structural response of the model. Thus the post-control model equations would differ from the pre-control ones, thus generating some inconsistency. The second is that future errors or uncertainty conditional on the adoption of a given set of policies would be very different from the past by the same reasoning. The second phase of econometric model building for specifying an optimal economic policy explicitly allows for the interdependence of the two conditional aspects: the problem of optimal estimation given the control or policy variables and the problem of optimal regulation or control, given the estimated model using all past information upto the current date.

According toK. Strubecker ([18]–[21]) a three dimensional real affine space with the metricds²=dx²+dy² is called a simply isotropic spaceJ₃⁽¹⁾ . InJ₃⁽¹⁾ exist 41 types of bundles of linear line complexes. In this paper we study the metric theory of a bundle of type 1. Especially we investigate the congruence of axis, we give some isotropic and affine results and we study the complementary bundle.