Entrance Examination for Master’s Program
Graduate School of Mathematics
Nagoya University
2019 Admission
Part 2 of 2

July 28, 2018, 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 for
calculations, 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 2019; Part 2) y1x1n∑
 ..  .. For x =  .  and y =  . , let hx, yi =xi ȳi be the standard Hermitian inner

11

xn

yni=1

product in Cn .
(1) Show that an n × n complex matrix A with the condition (I) is an Hermitianmatrix:(I) For an arbitrary x ∈ Cn , hAx, xi is a real number.
(2) Show that an n × n complex matrix B with the condition (II) is invertible:(II) For an arbitrary x ∈ Cn , hBx, xi is a real number and, in addition,hBx, xi = 0 implies x = 0.
(3) Show that n×n complex matrices C, D with the condition (III) satisfy rank C ≥rank D:(III) For an arbitrary x ∈ Cn , both hCx, xi and hDx, xi are real numbers and,in addition, hCx, xi ≥ hDx, xi ≥ 0.

(July 28, 2018)(over)

(Master’s Program Entrance Exam for 2019; Part 2)2

2 Determine whether or not each of the following statements holds. If it does, thenprove it. Otherwise, show that it is false by finding a counterexample.

(1) If∞∑an is a convergent series, then lim an = 0.
n→∞n=1(2)If {an }∞n=1 is a sequence satisfing

lim an = 0, then the series

n→∞

∞
∑

an converges.

n=1(3) Supposes that {an }∞n=1 is a monotone decreasing sequence with an > 0 (∀n =1, 2, . . . ). If∞∑

an converges, then lim nan = 0.
n→∞

n=1

(4) If a positive term series∞∑

an converges, then lim nan = 0. (A positive term

n=1n→∞

series is a series whose terms are nonnegative.)(July 28, 2018)(over)

(Master’s Program Entrance Exam for 2019; Part 2)3

3 Answer the following questions.
(1) For x ∈ [0, π], y ≥ 0, (x, y) 6= (0, 0), write each of the real part and imaginary
part of log(sin(x + iy)) using only real numbers. (For the complex logarithm,take the principal value.)(2) Let x, y be real numbers. Compute the limit(∫ πlimy→∞){log(sin(x + iy)) − y}dx .

0(Again, take the principal value for the complex logarithm.)(3) Compute the integral∫ πlog(sin x) dx.
0(July 28, 2018)(over)

(Master’s Program Entrance Exam for 2019; Part 2)

4

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4

Let f (x) be a continuous function defined on the half-open interval (0, 1]. Define asubset I ⊂ R as follows: a ∈ I if and only if there exists a sequence {xn }∞n=1 in theinterval (0, 1] such that lim xn = 0 and lim f (xn ) = a.
n→∞

n→∞

(1) Obtain I when f (x) = sin x1 . No explanation is needed.
(2) Show that, in general, if I is nonempty, then it is a connected closed set.

(July 28, 2018)(end)