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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. A rat is put into the following maze: 3 moves. 1 2 The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n (a) Explain why {X₂} is a MC and find the transition matrix P. (b) Explain why the chain is irreducible, aperiodic and positive recurrent. (c) What is the limit of Pn? Explain. (d) Find the probability that the rat is in compartment 3 after two moves. (e) In the long run, how many times that the rat enters in compartment 4 in 100 movements?
Consider the network with adjacency matrix A given by 1 1 1 A = 1 1 1 0 0 1 How many weakly connected components does the network contain? Select one: O a. 0 O b. 1 O C. O d. None of the above
Find the weighted adjacency matrix of the following directed graph and represent that as a CSR-matrix. B 2 5 7 D 5 F 3 E 6 G H 1
(b) Find the adjacency matrix for one of the following directed graphs: (i) V1 e3 e2 V3 02 (ii) V4 C V1 es e6 e4 e2 e3 02
Consider the adjacency matrix 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 of a graph with nodes labeled a,b,c,d,e and f and lexiographically ordered rows and columns. 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0. Suppose the above adjacency matrix identifies the network in a small company, where nodes represent workers and edges indicate that those workers interact during the week. The company has a single team leader and two different projects, one larger than the other. Workers in the same project interact often during the week. a) State which node represents the team leader, justifying your answer. b) State whether all workers are involved in at least one project and whether there is any worker, apart from the team leader, who works simultaneously on both projects. c) Explain what happens to the graph if the team leader suddenly resigns. Your response should be written in sentences.
Consider the adjacency matrix 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 of a graph with nodes labeled a,b,c,d,e and f and lexiographically ordered rows and columns. 1 0 1 1 0 0 Lo 1 0 1 0 0 Suppose the above adjacency matrix identifies the network in a small company, where nodes represent workers and edges indicate that those workers interact during the week. The company has a single team leader and two different projects, one larger than the other. Workers in the same project interact often during the week. b) State whether all workers are involved in at least one project and whether there is any worker, apart from the team leader, who works simultaneously on both projects.
G.6
Suppose the coefficient matrix associated with a linear system has a pivot in each row. Can we say, with certainty, that the linear system is consistent or inconsistent? Defend your response.
1.28 Show that the nonzero rows of an echelon form matrix form a linearly indepen- dent set.
Consider the network with adjacency matrix A given by How many weakly connected components does the network contain? Select one: O a. 1 O b. 2 O c. 3 O d. None of the above A = 0 0 0 0 0 0000 0 0 0 1 0 0 0 0 0 00 0 1 0 0

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