By Daniel J. Velleman
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Extra info for American Mathematical Monthly, volume 117, June July 2010
3. Let α ∈ [0, T ) and w ∈ W [−h, α]. 2) holds. Let b ∈ (α, β], c ∈ (0, γ ], and u, v ∈ W(w; α, b, c). Define u, ˜ v˜ ∈ W(w; α, β, γ ) by u(t), t ∈ [−h, b] u(b), t ∈ (b, β], u(t) ˜ := v(t) ˜ := v(t), t ∈ [−h, b] v(b), t ∈ (b, β]. 2), F(u) − F(v) ≤ F(u) ˜ − F(v) ˜ L 1 [α,b] The claim now follows, since β α L 1 [α,β] ≤λ u˜ (s) − v˜ (s) ds = b α β α u˜ (s) − v˜ (s) ds. u (s) − v (s) ds. 4. Let α ∈ [0, T ) and w ∈ W [−h, α]. 3) holds. 3 applies mutatis mutandis to yield that, for every b ∈ (α, β], b F(u) − F(v) ≤ρ L p [α,b] u (s) − v (s) ds α ∀ u, v ∈ W(w; α, b, γ ).
Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. 10. B. Jayawardhana, H. Logemann, and E. P. Ryan, PID control of second-order systems with hysteresis, Internat. J. Control 81 (2008) 1331–1342. 1080/00207170701772479 11. A. N. Kolmogorov and S. 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.
Examples of quantum measure spaces such as “quantum coins” and particle-antiparticle pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don’t need quantum mechanics or measure theory to understand this article. 1. INTRODUCTION. Measure and integration theory is a well-established field of mathematics that is over a hundred years old. The theory possesses many deep and elegant theorems and has important applications in functional analysis, probability theory, and theoretical physics.