Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Textbook Question

Chapter 2.10, Problem 29P

Show that the center of mass of three identical particles situated at the points z 1 , z 2 , z 3 i s ( z 1 + z 2 + z 3 ) / 3 .

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Chapter 2.10, Problem 28P

Chapter 2.10, Problem 30P

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Question 4 Consider the graph G given by the following drawing. U6 U5 U4 Ա3 Աշ U1 VA V5 V6 V3 V₁ V2 (a) Determine whether G is bipartite. Show your working. (b) For each of the following sets, state whether it is a matching of G. Justify your answers. (i) {} (ii) {u2v2, U3V3, U4V4, U5V5} (iii) {u1u2, U3u4, V3 V4 V5 V6} Now consider the graph H given by the following drawing. x2 x3 x4 x5 X6 x1 Y1 Y2 Y3 Y4 Y5 Y6 (c) Find a maximum matching of H. Show your working. (d) Use Hall's theorem to show that H does not have a perfect matching.

Question 2 Let (G, w) be the network with V(G) = {a, b, c, d, e, f}, E(G) = {ab, ac, ae, af, bc, bd, bf, cd, ce, de, df, ef}, and w(ab) = 2, w(ac) = 3, w(ae) = 2, w(af) = 2, w(bc) = 3, w(bd) = 2 w(bf) = 3, w(cd) = 2, w(ce) = 1, w(de) = 2, w(df) = 2, w(ef) = 2 (a) Draw the network. (b) For each of the following trees, state whether the tree is a minimum spanning tree of (G, w). Justify your answers. (i) a b C a (iii) a b d b (c) Does there exist a minimum spanning tree of (G, w) that contains the edge af? Justify your answer. Now consider an arbitrary network (G, w), and let e = E(G). (d) Give an efficient algorithm to determine whether (G, w) has a unique minimum spanning tree. Explain briefly why the algorithm is correct and efficient.

Question 3 Consider the directed network (D, c) given by the following drawing, where each arc e = A(D) is labelled by its capacity c(e) and two vertices s and t have been identified. 3 7 3 4 5 t 3 3 8 2 (a) Give an s-t-flow of (D, c) that is not a maximum s-t-flow. Justify your answer. (b) Give an s-t-cut of (D, c) that is not a minimum s-t-cut. Justify your answer. (c) Use the Ford-Fulkerson algorithm to find a maximum s-t-flow of (D,c). Draw the residual network after each iteration of the algorithm and give the size of the maximum flow. (d) Show that the flow you have found is indeed a maximum s-t-flow.

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Mathematical Methods in the Physical Sciences

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