≥ 0 which implies J is increasing in q. Therefore, q∗ =min(q(k, t), p) = q(k, t, p). iii) Note that E(V |V > q) = q + E(V −q) + F̄(q) , then the optimal utility is given by J(k, t) = J(k, t, q∗) = E(V |V > q∗)min(tF̄ (q∗), k)+E(V −p) (tF̄(q )−k) F̄(q) = [q∗ + E(V −q ) F̄(q) ]min(tF̄ (q∗), k)+E(V −p) (tF̄(q )−k) F̄ (q) . If q∗ = q(k, t) then F̄ (q(k, t)) = k/t and J(k, t, q(k, t)) = [q(k, t)+ tE(V −q(k,t...

Solution. Suppose p ∈ E. If p ∈ E, then f(p) ∈ f(E) ⊆ f(E). Otherwise, p is a limit point of E, and we claim that if f(p) / ∈ f(E), then f(p) is a limit point of f(E). Let ε > 0 be given, and choose δ > 0 such that for all q ∈ X, if 0 < dX(p, q) < δ, then dY (f(p), f(q)) < ε. Since p is a limit point of E, there exists a q ∈ E such that 0 < dX(p, q) < δ. Then f(q) ∈ f(E) (so in particular f(q) ...

Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p ≤ dim C X we have a sheaf Ω p (E) whose sections are holomorphic (p, 0)-forms with coefficients in E. We set and we define the holomorphic Euler characteristics χ p (X, E) := q≥0 (−1) q h p,q (X, E). It is convenient to introduce the generating function of these numbers χ y (X, E) := p≥0 y p ...

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