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Unformatted text preview: CAAM 335 MATRIX ANALYSIS Examination 1 Posted Wednesday, 6 February 2008. Due no later than 5pm on Monday, 11 February 2008. Instructions: 1. Time limit: 3 uninterrupted hours . 2. There are four questions worth a total of 100 points. Please do not look at the questions until you begin the exam. 3. You may not use any outside resources, such as books, notes, problem sets, friends, calculators, or MATLAB. 4. Please answer the questions thoroughly and justify all your answers. Show all your work to maximize partial credit. 5. Print your name on the line below: 6. Time started: Time completed: 7. Indicate that this is your own individual effort in compliance with the instructions above and the honor system by writing out in full and signing the traditional pledge on the lines below. 8. Staple this page to the front of your exam. CAAM 335 MATRIX ANALYSIS 1. [24 points: 6 points per part] Information transfer between neurons is accelerated by wrapping thin axons with a myelin sheath. In addition to the usual compartment model studied in class, with interior resistance R i and membrane resistance R m , this sheath presents a transverse resistance R s , while the annular interstitial region presents an axial resistance R a . If we divide the length of the myelinated axon into two compartments we arrive at the circuit depicted in the figure below. This problem asks you to work through the derivation of the Strang Quartet for this circuit by answering the four questions given below; please specify all entries in your matrices and vectors. (a) Write the voltage drops, e , in terms of the potentials, x , as e = Ax . (b) Write the currents, y , in terms of the voltage drops, e , as y = Ge . (c) Express Kirchhoffs Current Law via A T y = f . (d) Compute A T GA . To expedite this calculation, you may use that (1) A T GA is a symmetric matrix; and (2) premultiplication by the diagonal matrix G scales the rows: GA = g 1 g 2 . . . g m row 1 row 2 . . . row m = g 1 row 1 g 2 row 2 . . . g m row m . i R m R s R m R s R m R s R i R i R a R a x 1 x 3 x 5 x 2 x 4 x 6 y 1 y 5 y 9 y 2 y 6 y 10 y 3 y 7 y 4 y 8 1 CAAM 335 MATRIX ANALYSIS Solution: (a) Voltage drops: e = Ax , where e 1 = x 1 x 2 e 2 = x 2 e 3 = x 1 x 3 e 4 = x 2 x 4 e 5 = x 3 x 4 e 6 = x 4 e 7 = x 3 x 5 e 8 = x 4 x 6 e 9 = x 5 x 6 e 10 = x 6 . Thus, e = e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 , A =  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , and x = x 1 x 2 x 3 x 4 x 5 x 6...
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 Spring '08
 CAVALLARO

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