Combinatorial optimization II: proceedings of the CO79 by Rayward-Smith

By Rayward-Smith

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Additional info for Combinatorial optimization II: proceedings of the CO79 conference held at the University of East Anglia, Norwich, England, 9th-12th June 1979

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Rimane da provare l’indipendenza lineare delle soluzioni trovate. Nel caso di n radici distinte μ1 , . . , μn baster` a provare l’indipendenza lineare dei primi n valori delle successioni, cio`e ⎡ ⎤ 1 μ1 μ21 . . μn−1 1 n−1 2 ⎢ 1 μ2 μ2 . . μ2 ⎥ ⎢ ⎥ det ⎢ . . . ⎥=0 ⎣ .. .. . ⎦ 1 μn μ2n . . μn−1 n 38 2 Equazioni alle differenze lineari disuguaglianza vera se μj sono distinti perch´e tale determinante (determinante di Vandermonde2 ) vale (μn − μn−1 ) (μn − μn−2 ) · · · (μn − μ1 ) (μn−1 − μn−2 ) · · · (μn−1 − μ1 ) · · · (μ2 − μ1 ) .

32. 16) si esprimono come Y + X ove X `e una soluzione dell’equazione omogenea corrispondente. 16). Una volta individuata la soluzione generale dell’equazione omogenea corrispondente, `e sufficiente determinare una soluzione particolare dell’equazione completa: la somma di queste due soluzioni fornisce la soluzione generale dell’equazione completa. 16) `e anche una soluzione particolare. 33. Determiniamo la soluzione generale dell’equazione Xk+2 − 4Xk+1 + 3Xk = 2k k ∈ N. L’equazione caratteristica dell’equazione omogenea λ2 − 4λ + 3 = 0 ammette come soluzioni λ1 = 1 e λ2 = 3.

Dunque α = 1 risulta stabile e attrattivo. La soluzione generale `e Xk = c1 2−k + c2 3−k + 1. 3 sono riportati i primi valori della soluzione corrispondente alle condizioni iniziali X0 = 0 e X1 = 2: Xk = 23−k − 32−k + 1. 25. Consideriamo l’equazione Xk+2 − 2Xk+1 + 2Xk = 0. L’equilibrio `e α = 0. Le soluzioni dell’equazione caratteristica λ2 − 2λ + 2λ = 0 sono √ λ1,2 = 1 ± i. Poich´e |λ1,2 | = 2 l’equilibrio `e instabile. La soluzione generale dell’equazione `e: π π k k Xk = c1 (1 + i) + c2 (1 − i) = 2k/2 c1 cos k + c2 sin k k ∈ N.

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