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The purpose of this article is to show that the image of the homological boundary map attached to a filtration for an attractorrepeller pair of a smooth flow on a compact manifold is a submodule of the Alexander cohomology of certain order of the connecting set (some restrictions have to be imposed in order to have a valid argument). In particular, this gives an affirmative answer to a conjecture in Conley index theory which states that if the boundary map is not zero in two dimensions, the connecting set cannot be contractible. The Conley index. Blocks. The connecting setSuch an invariant set S is called isolated invariant if there is a compact neighborhood N of S with the property that the maximal invariant set in N is S. The neighborhood itself is called an isolating neighborhood and its existence guarantees that S is compact. Describing invariant sets based on an algebraic or a topological invariant is an old idea which goes back to the works of Poincaré and Morse. Charles Conley introduced a general approach to this problem based on the notion of index pairs. Definition 1.1. Let S be an isolated invariant set for a continuous flow Φ in a locally compact metric space M. Then an index pair is a pair of compact sets (N 1 , N 0 ) in M such that:(1) N 0 ⊂ N 1 and N 1 \ N 0 is a neighborhood of S and S is the maximal invariant set in cl(N 1 \ N 0 );(2) (positive invariance of N 0 ) If x ∈ N 0 and x · [0, t] ⊂ N 1 for some t 0, then x · [0, t] ∈ N 0 ;(3) (exit set property of N 0 ) If x ∈ N 1 and x · [0, ∞) N 1 , then there exists a t 0 with x · [0, t] ⊂ N 1 and x · t ∈ N 0 .

In the present contribution we develop a single invutsingle o-utput fuzzy model-based metti-od for~he de'sign and synthesis of optimal tracking control systems. Optimal tracking control systems re-present a particular class of quadratic control that are based on a pure quadratic trackin..g error, criterion . The resulting.control strategy maintains the output close to a pre-specffied setpoint.-The control strateg~r is continuously adapted by a receding horizon predictive algorithm that computes at each time step an adjustment for the extended control sequence and applies effectively only the first value of this sequence. The ~,ey idea of the algorithm, is the derivation of the ad~bint system associated with a fuzzy dynamic system. This adjoint system can be represented as a fuzzy system so that, only linguistic models are used. The main advantage of this approach is the extension of optimal control results in order to handle fuzzy dynamic systems that can be the only available description of a possibly highly nonlinear plant. An adaptive version of this algorithm can be. easily developed and implemented. Numerical simulation results illustrate the applicability of the approach to a simple indoor climate ~dntrol system.

Abstract. We calculate the Fuglede-Kadison determinant for operators of the form n i=1 M fi L gi where L gi are unitaries or partial isometries coming from Borel (partial) isomorphisms g i on a probability space which generate an ergodic equivalence relation, and M fi are multiplication operators. We obtain formulas for the cases when the relation is treeable or the f i 's and g i 's satisfy some restrictions.

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