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Unformatted text preview: Practice exam I: MAP 4305 September 11, 2006.
Name: Student ID: This is a closed book exam and the use of calculators is not allowed. 1. Let A= (a) Calculate A231 . (b) Discuss stability for the system x(t + 1) = Ax(t). 2. Consider a network with two nodes (call them 1 and 2) and assume that there is a link from 1 to 2 (but not back), and that both 1 and 2 have a self-link. Calculate the pagerank for both nodes. (To determine the stochastic matrix, use the same rule we used on the HW problem). 3. Suppose that the vector functions x1 (t) x2 (t) where t R, are solutions of x= Let X(t) be defined as: x1 (t) y1 (t) x2 (t) y2 (t) and define the quantity m(t) as the determinant of the matrix X(t). Show that m(t) satisfies the following differential equation: m(t) = (a + d)m(t). Show that this implies that m(t) = e(a+d)t m(0). Hint: Write m(t) explicitely in terms of x1 (t), x2 (t), y1 (t) and y2 (t), and calculate the derivative with respect to time t. 4. Solve the following IVP: x= -1 1 x, x(1) = 0 2 1 2 a b x. c d and y1 (t) , y2 (t) 1 2 2 . 1 Instructor: Patrick De Leenheer. 1 ...
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This note was uploaded on 04/07/2008 for the course MAP 4305 taught by Professor Deleenheer during the Spring '06 term at University of Florida.
- Spring '06