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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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12. Let A = 1 1 12 1 3 -[]. and b = (1) Use the Existence and Uniqueness Theorem to show Ax = b is an inconsistent linear system. (2) What is the normal equation? (3) Find a least-squares solution to the inconsistent system Ax = b.
Consider the following project information: Activity A depends on nothing and has an expected time of 8 days, B depends on A with an expected time of 3 days, C depends on nothing with an expected time of 9 days, D depends on A with an expected time of 6 days, finally, activity E depends on B, C, and D and its expected time is 4 days. The critical path for the network activities is: Select one: a. There is no critical path b. C-E c. A-D-E d. A-B-D e. A-B-E
2. Consider the following distance matrix for an undirected graph and solve the 1-center problem (∞ means no edge exists). 2 4 Ce || |
1 4 6 8) 10 a. Construct the adjacency matrix. Why is it a block-diagonal matrix? b. Construct the adjacency matrix of its two projections, on the purple and on the green nodes, respectively. c. Calculate the average degree of the purple nodes and the average degree of the green nodes in the bipartite network. d. Calculate the average degree in each of the two network projections. Is it surprising that the values are different from those obtained in part e? 3.
(4) Consider a mini-web as a dynamical system. Let A be the coefficient matrix of this system and i be a distribution vector. Prove that if Ax = Xã, then X = 1.
1.28 Show that the nonzero rows of an echelon form matrix form a linearly indepen- dent set.
The degree distribution of a network has the following functional form: Pk = Cak, Where a < 1 is a constant, and k ≥ 1 is the degree. a. Find out the value of C. b. Find out its probability generating function go(z) in compact form. C. Find out the corresponding generating function g₁(z) for its excess degree distribution in compact form. d. Find out the probability that a node in this network has zero 2nd neighbors. condition for a such that a giant component is present. e. Find out the f. When 1 - fraction of nodes are randomly removed, find out the critical value of when giant cluster emerges. g. What is the size of the giant cluster if it exists?
In at tropical forest, the populations of falcons and mice are related toeach other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule. * At the beginning of 2023, there were 10 falcons and 40 mice. a) Approximately how many falcons and mice will there be at the beginning of 2068? b) c) Find a diagonalization, A = BDB-1, of the transition matrix A from part (b), where D is a 2x2 diagonal matrix, and B is an invertible2x2 matrix. d) Use your answer to (c) to estimate the (approximate) numbersof falcons and mice there will be at the beginning of 2058. e) what restrictions on the initial population sizes ensures the long time survival of both species in the forest
(4) Consider a mini-web as a dynamical system. Let A be the coefficient matrix of this system and a be a distribution vector. Prove that if A = Aã, then A = 1.
Section 3.4:Number 12

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