Entrance Examination for Master’s Program
Graduate School of Mathematics
Nagoya University
2017 Admission
Part 2 of 2

July 30, 2016, 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 2017; Part 2)☛1✟1 ✠Let A and B be n × n diagonalizable matrices with complex entries.

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(1) Show that, if there is a matrix that diagonalizes both A and B, then AB = BA.
For (2) and (3), assume that AB = BA. Also, for an eigenvalue α of A, let W (A, α)
be the corresponding eigenspace.
(2) For an arbitrary vector v ∈ W (A, α), show that Bv ∈ W (A, α).
(3) (i) Show that there is a basis for W (A, α) consisting of eigenvectors of B.
(ii) Using (i), show that there is a matrix that diagonalizes both A and B.

(July 30, 2016)(over)

(Master’s Program Entrance Exam for 2017; Part 2)☛2✟

2 ✠Suppose that f is a nonnegative continuous function defined on the interval [0, ∞)✡

such thatlim f (x) = αx→∞for some α > 0.
(1) Show that f is bounded on [0, ∞).
(2) Show that the improper integral

Z ∞f (x) dx diverges.

0

Next, suppose that g is a nonnegative continuous function defined on the interval[0, ∞) such thatZ ∞0g(x) dx < ∞.

(3) Show that there exists a sequence {an }∞n=1 ⊂ [0, ∞) satisfying
lim an = ∞ andn→∞

lim g(an ) = 0.

n→∞

If necessary, consider the above improper integral in terms of

In =Z n+1g(x) dx (n = 0, 1, 2, . . . ).

n(4) Does lim g(x) = 0 hold? If so, give a proof. Otherwise, give a counterexamplex→∞and show that it is indeed a counterexample.

(July 30, 2016)(over)

(Master’s Program Entrance Exam for 2017; Part 2)☛3✟

3 ✠Let a > 0 be a constant and consider the complex function✡

f (z) =1(z 2 + a2 ) sin(πz).

(1) Find all singularities of f (z) in C.
(2) Let N be a natural number with N > a. Let CN be the circle in the complex
1plane centered at the origin with radius N + . The circle CN is given by the2counter-clockwise orientation. Find the value of the integral
Zf (z) dz.
CN∞X(−1)n(3) Find the value of the series. If necessary, the fact
n2 + a2n=−∞

1| sin(πz)| > √ ,2z ∈ CN , N = 1, 2, . . .

may be used.

(July 30, 2016)(over)

(Master’s Program Entrance Exam for 2017; Part 2)☛4✟

4 ✠ (1) Define that a subset K of a topological space X is compact in terms of open✡

cover.
(2) Suppose that a topological space X is compact and a subset A of X is a closed
set. Is A compact? If so, give a proof. Otherwise, give a counterexample andshow that it is indeed a counterexample.
(3) Define that a topological space Y satisfies the Hausdorff separation axiom interms of open set.
(4) Suppose that a topological space Y satisfies the Hausdorff separation axiom anda subset B of Y is compact. Is B a closed set? If so, give a proof. Otherwise,
give a counterexample and show that it is indeed a counterexample.

(July 30, 2016)(end)