2009 School Year

Graduate School

Entrance Examination Problem Booklet

Mathematics

Examination Time: 10:00 to 12:30

Instructions

1. Do not open this problem booklet until the start of the examinationis announced.

2. If you nd missing, misplaced, and/or unclearly printed pages in theproblem booklet, ask the examiner.

3. Answer all of three problems appearing in this booklet, in Japanese orEnglish.

4. You are given three answer sheets. You must use a separate answersheet for each problem. You may continue to write your answer on theback of the answer sheet if you cannot conclude it on the front.

5. Fill the designated blanks at the top of each answer sheet with yourexaminee's number and the problem number you are to answer.

6. The blank pages are provided for rough work. Do not detach themfrom this problem booklet.

7. An answer sheet is regarded as invalid if you write marks and/or sym-bols and/or words unrelated to the answer on it.

8. Do not take the answer sheets and the problem booklet out of theexamination room.

Examinee's number No.

Fill this box with your examinee's number.

(blank page for rough work)

Problem 1

Suppose that three-dimensional column vectors xn satisfy the recurrenceequation:

xn = Axn1 + u (n = 1; 2; : : :);

where

A =

0@ 1 0 00 1 a0 a a2

1A ; u =0@ bc

0

1A and x0 =0@ 00

d

1A :Let a, b, c and d be real constants and assume a 6= 0. Answer the followingquestions.

(1) Express the eigenvalues of A with a.

(2) Express A's eigenvectors p, q and r with a. Let p and r correspond tothe largest and the smallest eigenvalues, respectively, and set kpk =krk = 1p

a2 + 1and kqk = 1.

(3) Express u and x0 as linear combinations of p, q and r.

(4) Suppose xn = np+ nq + nr. Describe n, n and n using n1,n1 and n1.

(5) Obtain xn.

(6) Suppose that n denotes the angle between xn and vector s, where

s =

0@ 100

1A :Describe a necessary and sucient condition on a, b, c and d forlimn!1 n = 0

Problem 2

Consider a system whose temperature is controlled by a heater.Let x(t) denote the temperature of the system at time t. While the

switch of the heater is o, temperature x(t) follows the dierential equation

dxdt

= x: ()

While the switch of the heater is on, temperature x(t) follows the dierentialequation

dxdt

= 1 x: ()First, consider the behavior of the system when the switch is kept intact.

(1) Find the solutions of the dierential equations () and () with theinitial condition x(0) = x0.

Next, given the thresholds 0 and 1 for the temperature that satisfy0 < 0 < 1 < 1, consider automatically manipulating the switch accordingto the following rule to keep the temperature between 0 and 1. Turn onthe switch when the temperature is not higher than 0 and the switch iso. Turn o the switch when the temperature is not lower than 1 and theswitch is on. Otherwise, keep the switch intact.

Assume that the temperature x0 at time 0 is higher than 0, and theswitch is o at time 0.

(2) Find time t1 when the switch is turned on for the rst time after time0.

(3) The temperature evolves periodically after time t1. Find the period, .

(4) Consider the time average of the temperature during one period, x =1

Z t1+t1

x(t)dt, and the mean of the thresholds, =0 + 1

2. The

dierence between these values x = 1

Z t1+t1

(x(t) )dt can beexpressed with a certain function f(x; ) as follows:

x = 1

Z 10

f(x; )dx:

Find the function f(x; ).

(5) Suppose that 0 and 1 are varied so that the mean takes a constant

value. Prove that if