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**Extra resources for Algebraic Theory of the Bianchi Groups (Chapman & Hall/CRC Pure and Applied Mathematics)**

**Example text**

8. 19), and let ξ· again represent the corresponding trajectory. 2 Viscosity Solutions t V (s, x) ≤ l(r, ξr ) − s γ2 |ur |2 dr + V (t, ξt ) + ε. 42) By continuity and the supposition that (s, x) is a local maximum of V − g, ˜ such that there exists δˆ ∈ (0, δ] ˆ V (s, x) − g(s, x) ≥ V (t, ξt ) − g(t, ξt ) ∀ t ∈ [s, s + δ]. 43), one has t −ε ≤ l(r, ξr ) − s γ2 |ur |2 dr + g(t, ξt ) − g(s, x). 2 Letting ε < (θt)/2, one has t l(r, ξr ) − s θ(t − s) γ2 |ur |2 dr + g(t, ξt ) − g(s, x) > − . 41) yields t s l(r, ξr ) − γ2 θ(t − s) |ur |2 dr + g(t, ξt ) − g(s, x) ≤ − .

The above discussion was kept highly general to indicate that few assumptions are required for proof of the DPP. As noted earlier, we will have a few problem classes for which we will prove all our results. 1 Dynamic Programming Principle 35 proved for more general problems, but then the assumptions would need to be more abstract; for instance, we assumed the existence of J(x, T ; u) for all x, T, u, and W (x) for all x just above. By focusing on speciﬁc problem classes, we will be able to make the assumptions much more concrete and easily checkable.

27 yields the following representation result. 28. Let X be a complete max-plus space. Any continuous linear functional, f , on X op has the form f (φ) = φ, φ for a unique φ ∈ X . It is natural to refer to this property as reﬂexivity. 20 that CR is a complete max-plus space, we see that it is reﬂexive in −+ this sense. The dual space is again CR but with the opposite operations being given there. ) The second dual is once again −+ CR but with operations (⊕op )op = ⊕ and (⊗op )op = ⊗ being the original max-plus operations again.