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By A.J. Barker

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75) coupled with the inequality fo t u dw>~tPp(w, [0, t]) u e]0, T]. 2 VECTOR RELAY Let us denote by 0 "- (01, 02) the angular coordinates (longitude and colatitude, say) on the unit sphere, $2; by an obvious identification, we can then assume that 0 ranges over S2. For any p "- (Pl, P 2) E R 2 with Pl < P 2 and any 0 e S2, we introduce the vector-relay operator: -' ) . 78), and is multivalued. 77) of scalar hysteresis operators can easily be extended to vectors. O -~dw>~f p(w, [0, t]) u e]0, T].

An algorithm based on an appropriate selection of the input functions allows one to reduce the number of measurements [6-8,51]. 5 MICROMAGNETISM single-domain particles. Although the properties that can be derived from the Preisach model are in good qualitative agreement with the physical evidence, for several ferromagnetic materials there are quantitative discrepancies. This also applies to the vector Preisach model. Physicists and engineers then proposed several variants of the originary Preisach model, in order to provide a more adequate model of ferromagnetic hysteresis.

As u increases (decreases, respectively) certain relays are switched to the state I (-1, respectively), cf. (b) and (c) respectively. The regions characterized by the states -1 and 1 are then separated by a maximal antimonotone graph, Bw(t), provided that the same holds at t - 0; cf. (d). this condition is fulfilled w h e n e v e r # is not singular with respect to the ordinary bidimensional Lebesgue measure. Let us a s s u m e that Bw(O) coincides with the bisectrix of the second quadrant: Bw(O) - B v := {p ~ 7~ : Pl q- P 2 "-" 0}.

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