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Read e-book online American Mathematical Monthly, volume 117, June July 2010 PDF

By Daniel J. Velleman

ISBN-10: 0883851857

ISBN-13: 9780883851852

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Additional resources for American Mathematical Monthly, volume 117, June July 2010

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M, gr v ⊂ ∪i=1 Ii × Bi , and g(s, x) ≤ bi (s) for all (s, x) ∈ Ii × Bi , i = 1, . . , m. Defining b˜i : I → R+ by b˜i (s) := bi (s), s ∈ Ii ∩ I 0, s ∈ I \Ii , it follows that g(s, x) ≤ i b˜i (s) for all (s, x) ∈ gr v. Therefore, since i b˜i ∈ L q (I ) and since the function s → g(s, v(s)) is measurable (by statement (1)), the result follows. 1. We make the following connections with the notation of Section 4: for h = 0, we set C[−h, 0] = R N and = D = G. 1) coincide. Furthermore note that, for all I ∈ I, I+ = I .

We now illustrate these ideas in Example 4. All the results in the rest of this section apply to the quantum measure space (X, μ) of Example 4. 4. AμB if and only if xi ∈ A ∩ B implies that yi ∈ B ∩ A and yi ∈ A ∩ B implies that xi ∈ B ∩ A . Proof. The condition is equivalent to the following: if {xi , yi } ⊆ A B, then {xi , yi } ⊆ A ∩ B or {xi , yi } ⊆ B ∩ A . Suppose the condition holds. We may assume without loss of generality that {x1 , y1 , . . , xr , yr } ⊆ A ∩ B , {xr +1 , yr +1 , . . , xs , ys } ⊆ B ∩ A , and there are no other destructive pairs in A μ(A B) = |A B.

Moreover, it follows easily 500 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 from the rate independence of H that H (0) (where 0 denotes the zero function on R+ ) is a constant function, implying that the function F(0) is constant. 16) where L F denotes the Lipschitz constant of F. It follows that (H3) also holds. 7 applies. 15). To complete the proof, we have to show that τ = ∞. Seeking a contradiction, suppose that τ < ∞. 7 that u is unbounded. 16) leads to t u(t) ≤ u(0) + ct + L F u(s) ds ∀ t ∈ [0, τ ).

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American Mathematical Monthly, volume 117, June July 2010 by Daniel J. Velleman

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