By Daniel J. Velleman

**Read or Download American Mathematical Monthly, volume 117, June July 2010 PDF**

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As before, let I be the set 502 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 of all intervals I of the form [−h, η] (with 0 < η < ∞) or [−h, η) (with 0 < η ≤ ∞) such that W (I ) = ∅ and recall that T := sup{α > 0 : W [−h, α] = ∅} > 0. Define D := {(t, u) ∈ R+ × C[−h, 0] : (t + s, u(s)) ∈ ∀ s ∈ [−h, 0]}. 7). Moreover, for I ∈ I and u ∈ C(I ), we have gr u ⊂ ⇔ (t, u t ) ∈ D ∀ t ∈ I+ . If is a Cartesian product, that is, = [−h, α) × G, where 0 < α ≤ ∞ and G ⊂ R N is open, then D is also a Cartesian product, namely D = [0, α) × C([−h, 0], G), where C([−h, 0], G) is the subset of all functions in C[−h, 0] with values in G.

The function Iμ gives the deviation of μ from being a measure on the sets xi , x j and hence is an indicator of the interference between xi and x j . Notice that Iμ can have positive or negative values. For instance, in Example 3, Iμ (x2 , x3 ) = −1, while in Example 2, Iμ (x1 , x2 ) = 4. 3, μ is determined by the numbers μ(xi ) and Iμ (xi , x j ), i, j = 1, . . , n. We extend Iμ to a signed measure λμ on P (X × X ) by defining λμ (B) = Iμ (xi , x j ) : (xi , x j ) ∈ B . Since Iμ (xi , x j ) = Iμ (x j , xi ) it follows that λμ is symmetric in the sense that λμ (A × B) = λμ (B × A) for all A, B ∈ P (X ).

V. Fomin, Introductory Real Analysis, Dover, New York, 1975. 12. M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Berlin, 1989. 13. X. Tan, J. S. Baras, and P. S. Krishnaprasad, Control of hysteresis in smart actuators with application to micro-positioning, Systems Control Lett. 54 (2005) 483–492. 09. 013 14. A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. 15. W. Walter, Ordinary Differential Equations, Springer, New York, 1998. D. at the University of Bremen (Germany) under the guidance of Diederich Hinrichsen.