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By Graves L. (ed.)

ISBN-10: 0821813080

ISBN-13: 9780821813089

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Extra resources for Calculus of Variations and Its Applications

Example text

Pr . The following example shows the decomposition of a doubly stochastic matrix as explained in the proof to Birkhoff’s theorem. 20. Consider the doubly stochastic matrix ⎛ ⎞ 1/2 1/3 1/6 X = ⎝ 1/3 1/3 1/3 ⎠ . 1/6 1/3 1/2 We choose as first permutation matrix ⎛ 1 P1 = ⎝ 0 0 0 1 0 ⎞ 0 0 ⎠. 1 The corresponding value λ1 becomes λ1 = min(1/2, 1/3, 1/2) = 1/3. Thus we get ⎛ ⎞ 1/6 1/3 1/6 X − λ1 P1 = ⎝ 1/3 0 1/3 ⎠ . 1/6 1/3 1/6 ⎛ Next we choose 1 P2 = ⎝ 0 0 0 0 1 ⎞ 0 1 ⎠ 0 and get for the corresponding value λ2 = min(1/6, 1/3, 1/3) = 1/6.

Naddef [508] proved that the Hirsch conjecture is true for any 0-1 polytope, while Klee and Walkup [420] showed that it is false for unbounded polyhedra. For general bounded book 2008/12/11 page 34 34 Chapter 2. Theoretical Foundations polyhedra this conjecture is still open. For a survey on the Hirsch conjecture see Klee and Kleinschmidt [419]. A polytope is called Hamiltonian if there exists a path along the edges of the polytope which visits all vertices exactly once and returns to the original starting point.

Instead of trees we consider labeled rooted trees on n vertices: a labeled rooted tree is a tree with one distinguished vertex as the root. The arcs are oriented such that all paths in the tree lead to the root. Every arc has a label from {1, 2, . . , n − 1} and no two arcs have the same label. There are (n − 1)! n possibilities to transform a tree in a labeled rooted tree as there are n choices for the root and (n − 1)! choices to distribute the labels on the arcs. If we denote by Tn the total number of different spanning trees in Kn , we get as the total number of different labeled spanning rooted trees (n − 1)!

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Calculus of Variations and Its Applications by Graves L. (ed.)

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