By Shan F.

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Xa eiθa(t) αa +u(t), xa+1 eiθa+1(t) αa+1 −u(t), . . , xm−1 eiθm−1 (t) αm−1 −u(t), xm + 21 u(t)−iAt : 2 2 −· · ·−xm−1 +2xm = 0 . t ∈ (− , ), xj ∈ R, x12 +· · ·+xa2 −xa+1 Then N is a special Lagrangian submanifold in Cm . As in Sect. 3, if we assume that θ (t) ∈ (−π/2, π/2) for t ∈ (− , ) then u is an increasing function of t, and we can choose to regard everything as a function of u rather than of t. 4: 794 D. 3. Suppose θ (t) ∈ (−π/2, π/2) for all t ∈ (− , ). 1 is given explicitly by √ √ √ x1 eiθ1 (u) α1 + u, .

6. 9) where µ = (α1 + α3 )1/2 , ν = (α3 − α2 )1/2 (α1 + α3 )−1/2 and S 5 is the unit sphere in C3 . Then Φ is a conformal, harmonic map. We made the assumption above that α2 α3 . If α2 > α3 then we can apply the same method, but swapping over x2 and x3 , and α2 and α3 , so that x1 = (α1 + α3 )−1/2 dn(µs, ν), x2 = (α1 + α2 )−1/2 sn(µs, ν) and x3 = (α1 + α3 )−1/2 cn(µs, ν), where µ = (α1 + α2 )1/2 and ν 2 = α2 − α 3 . α1 + α 2 Note also that all of our expressions for xj (s) depend only on the linear combinations α1 + α2 , α1 + α3 and α2 − α3 of α1 , α2 , α3 .

Xm ) → w1 (t)x1 , . . , wm−1 (t)xm−1 , xm + β(t) , where w1 , . . , wm−1 : (− , ) → C \ {0} and β : (− , ) → C are differentiable functions. Following the method of Sect. e. upon w1 , . . , wm−1 and β. 1. 1. Let (m−1)/2 a m−1. Suppose w1 , . . , wm−1 : (− , ) → C \ {0} and β : (− , ) → C \ {0} are differentiable functions satisfying dwj w1 · · · wj −1 wj +1 · · · wm−1 , = dt − w1 · · · wj −1 wj +1 · · · wm−1 , and 1 j a, a < j < m, dβ = w1 · · · wm−1 . 3) Define a subset N of Cm by N= w1 (t)x1 , .

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A UV -decomposed method for solving an MPEC problem by Shan F.


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