Entrance Examination for Master’s Program
Graduate School of Mathematics
Nagoya University
2020 Admission
Part 2 of 2

July 27, 2019, 13:00 〜16:00
Note:
1. Please do not turn pages until told to do so.
2. The problem sheet consists of the cover page and 4 single-sided pages. Afterthe exam has begun, please first confirm that the number of pages and theirprinting and order are correct. Please report any problem immediately.
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3. There are a total of 4 problems labeled ✂1 ✁, ✂2 ✁, ✂3 ✁, and ✂4 ✁, respectively.
Please answer all 4 problems.
4. The answering sheet consists of 4 pages. Please confirm the number ofpages, and please do not remove the staple.
5. Please
write✄ the answers
to problems ✂1 ✁ , ✂2 ✁ ， ✂3 ✁ ，and ✂4 ✁ on pages✄✄✄✂1 ✁ , ✂2 ✁ , ✂3 ✁ , and ✂4 ✁ of the answering sheet, respectively.
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6. Please write name and application number in the space provided
on each of the 4 pages in the answering sheet．
7. The back side of the 4 pages in the answering sheet may also be used. Ifused, please check the box at the lower right-hand corner on the front side.
8. If the answering sheet staple is torn, or if additional paper is needed forcalculations, please notify the exam proctor.
9. After the exam has ended, please hand in the 4-page answering sheet. The
problem sheet and any additional sheets used for calculations may be taken
home.

Notation:
The symbols Z, Q, R, and C denote the sets of integers, rational numbers, real
numbers, and complex numbers, respectively.

(Master’s Program Entrance Exam for 2020; Part 2)☛1✟

1 ✠Let A, B be n×n complex matrices satisfying A+ B = I, where I is the n×n identity✡

matrix, and suppose that A is diagonalizable.
(1) Show that B is also diagonalizable.
(2) Show that, for an arbitrary eigenvalue λ of A, there exists an eigenvalue µ of B
such that λ + µ = 1.
(3) Show that, if the sum of the ranks of A and B equals n, then each of A and B
has no eigenvalues other than 0, 1.

(July 27, 2019)(over)

(Master’s Program Entrance Exam for 2020; Part 2)☛2✟

2 ✠Answer the following questions.

✡

(1) For an arbitrary x ∈ R, show that the improper integral
Z ∞2e−t cos(xt) dt

0converges absolutely.
For (2)–(4), consider the function f (x) =Z ∞2e−t cos(xt) dt defined on R.

0(2) Show that f (x) is differentiable on R.
(3) Show that y = f (x) satisfies the ordinary differential equation y ′ = −xy.
2(4) Compute the value of f (2).

(July 27, 2019)(over)

(Master’s Program Entrance Exam for 2020; Part 2)☛3✟

3 ✠Let t ∈ R and consider the complex function✡

f (z) =eitz,(z 2 + 1)2

where i is the imaginary unit.
(1) Find all singularities of f (z) in the complex plane C.
(2) For each of the isolated singularities in C with positive imaginary part, find theprincipal part (the part with negative powers) of the Laurent series of f (z) atthat point.
(3) Compute the improper integraleitxdx.
22−∞ (x + 1)

Z ∞(July 27, 2019)(over)

(Master’s Program Entrance Exam for 2020; Part 2)☛4✟

4 ✠Suppose that a point sequence a(k) (k = 1, 2, . . .) in the n-dimensional Euclidean✡

space Rn converges to a ∈ Rn . Here, consider the standard topology on Rn , that is,pPn
2
the topology that arises by defining the distance d(x, y) =i=1 (xi − yi ) between

two points x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn . For the subset A = {a(k) ; k =1, 2, . . . , } of Rn , answer the following questions.
(1) Find the closure A of A in Rn .

(2) For a closed subset B of Rn satisfying A ∩ B = ∅, determine whether
inf{d(x, y) ; x ∈ A, y ∈ B} > 0holds. If it does, then give a proof. Otherwise, give a counterexample and showthat it is indeed a counterexample.

(July 27, 2019)(end)