Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 5
20150534 Lee Kyungmin
Real Analysis, MAS540
May 17, 2017
Problem 1. Consider the Mellin transform defined initially for continuous functions f of compact
support in R+ = cfw_t R : t > 0 and x R by
Z
f (t)tix1 dt.
(1)
Mf (x) =
0
Prove that (2
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS101 2015 FRI 11AM Quiz 6
ID:
Name:
1 Answer the following statements with True or False. You dont have to justify your
10 Points answer. For each correct answer 2 points.
(a) The vectors kbka + kakb and kbka kakb are orthogonal.
True
Sol)
(kbka+kakb)(k
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS101 2015 FRI 11AM Quiz 6
ID:
Name:
1 Answer the following statements with True or False. You dont have to justify your
10 Points answer. For each correct answer 2 points.
(a) The vectors kbka + kakb and kbka kakb are orthogonal.
(b) For every vector a
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I 11AM sol Quiz 4
ID:
Name:
Signature:
1 Find the series interval of convergence and, within this interval, the sum of the series
10 Points as a function of x.
Solution. Since
X
(x 1)2n
4n
n=0
xn
n=0 4n
2
P
converges absolutely for x < 4, given
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I Fri 1PM Quiz 5
ID:
1 Show that
Name:
Signature:
X
x2 + x
n2
=
xn
(x 1)3
n=1
10 Points
for x > 1
Solution:
Let y =
1
. Then y < 1
x
X
yn
=
1
1y
ny n
=
y
(1 y)2
n2 y n
=
y(1 + y)
(1 y)3
=
(1 + 1/x)
x(1 1/x)3
=
x2 + x
(x 1)3
n=0
X
n=1
X
n=1
X
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I 1PM Quiz 4
ID:
Name:
Signature:
1 Find the series interval of convergence and, within this interval, the sum of the series
10 Points as a function of x.
X
(x2 + 1)n
3n
n=0
2 Find the Taylor series generated by f at x = /4.
10 Points
f (x) = cos
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I Fri 11AM Quiz 5
1 Find a power series of
ID:
Name:
Signature:
x2
for x < 1
(1 + x)3
10 Points
2 Show that the equation x = r cos , y = r sin transform the polar equation r =
10 Points
May. 22, 2015
k
into the cartesion equation (1 e2 )x2 + y
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I Fri 1PM Quiz 5
1 Show that
10 Points
ID:
Name:
Signature:
X
x2 + x
n2
=
xn
(x 1)3
n=1
for x > 1
b
a
2 Show that the vertical distance between y = x and the upper half of the right hand
10 Points
May. 22, 2015
branch y =
y2
x2
bp 2
x a2 of the
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS101 2015 FRI 1PM Quiz 6
ID:
Name:
1 Suppose that you are given nonzero vectors a, b, and c in R3 . Use dot and cross
10 Points products to give expression for vectors satisfying the following geometric descriptions.
For each correct answer 2 points.
(a
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Calculus I 11AM Quiz 4
ID:
Name:
Signature:
1 Find the series interval of convergence and, within this interval, the sum of the series
10 Points as a function of x.
X
(x 1)2n
4n
n=0
2 Find the Taylor series generated by f at x = 0.
10 Points
f (x) =
May.
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Matrix Completion has no Spurious local minima  generalization
Lee Kyungmin
January 24, 2017
1 Full Observation : Easy case
Let M Rmn be target matrix, and x Rn , y Rm . Then we have following loss function
1
1
f (x, y) = kM x y T k = kuv T x y T k
2
2
(
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 1
20150534 Lee Kyungmin
Real Analysis, MAS540
March 27, 2017
Problem 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In
other words, given two points x, y C, there is a point z
/ C that lies between
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 2
20150534 Lee Kyungmin
Real Analysis, MAS540
April 5, 2017
Problem 6. Integrability of f on R does not necessarily imply the convergence of f (x) to 0 as x .
Prove the following:
(a) There exists a positive continuous function f on R so that f
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
KAIST
Proof of General Case
Kyungmin Lee
January 18, 2017
1 F ULL O BSERVATION : E ASY CASE
Let M R mn be target matrix, and x R n , y R m . Then we have following loss function
1
1
f (x, y) = kM x y T k = kuv T x y T k
2
2
(1.1)
Where u R n and v R m are
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 3
Due date: March 28, 2017 (8:59 am)
Problem 1. (5 points) Let f be analytic in a region containing the annulus cfw_z : r1 z z0  r2
where 0 < r1 < r2 . Show that there are constants an C such that
X
f (z) =
an (z z0 )n ,
n=
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS101 2015 FRI 1PM Quiz 6
Solution
Name:
1 Suppose that you are given nonzero vectors a, b, and c in R3 . Use dot and cross
10 Points products to give expression for vectors satisfying the following geometric descriptions.
For each correct answer 2 point
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 7
Due date: May 9, 2017 (8:59 am)
Problem 1. (5 points) Let be the path z = 3ei along 0 4. Evaluate
Z
z2
dz
3z + 2
only by using the Residue Theorem.
Problem 2. (5 points) Find the number of zeroes of f (z) = 2z 5 + 5z 2 + 2 i
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 4
20150534 Lee Kyungmin
Real Analysis, MAS540
May 8, 2017
Problem 1. Prove that the following are dense subspaces of L2 (Rd ).
(a) The simple functions.
(b) The continuous functions of compact support.
Proof. (a). Let S be set of simple functio
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 10
Due date: June 8, 2017 (6:00 pm)
Problem 1. (4+4+4 points) Find a conformal mapping between A and B.
(a) A = cfw_z = x + iy : 0 < x < 1 and 0 < y < 1, B = cfw_z C : 1 < z < e2 \(1, e2 )
(b) A = D(0; 1)\[0, 1), B = D(0; 1)
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 9
Due date: May 30, 2017 (8:59 am)
Problem 1. (5 points) Evaluate
Z
e2ix
for 0 < a < 1 where coshz =
sinhz =
ez +ez
2
sina
dx
coshx + cosa
for z C. (Hint: you may represent the solution using
ez ez
)
2
Solution. Let
f (z) = e2i
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 6
Due date: May 2, 2017 (8:59 am)
Problem 1. (5 points) Suppose g(z) is analytic in Dr (z0 ) cfw_z0 for some r > 0. Prove that if
g(z) Az z0 1+
for z Dr (z0 ) cfw_z0
with some > 0 and A > 0, then the singularity of g at z
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
EE807: Advanced Stochastic Processes
Prof. Jinwoo Shin
Final Exam
(i)
(ii)
(iii)
(iv)
(v)
Write your name and student id.
Write your solution only on the sheet provided.
Justify your answers.
Time: 3 hours.
The exam is openbook, but you cannot use any el
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 6
20150534 Lee Kyungmin
Real Analysis, MAS540
June 7, 2017
Problem 1. Let (X, M, ) be a measure space. One can define the completion of this space as
follows. Let M be the collection of sets of the form E Z, where E M, and Z F with F M
and (F )
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Problem 1. Prove that
Z
0
sin x
dx =
x
2
Proof. We note by the symmetry of f (x) = x1 sin x, 2
R R ix
Im R eix , and so we only have to calculate
Z
lim
0,R
l
R
0
sin x
x
eix
dx +
x
R +
=
sin x
x .
We also know that
RR
R
eix
dx
x
Z
l+
sin x
x dx
=
(1)
wher
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 8
Due date: May 23, 2017 (8:59 am)
Problem 1. (5 points)
Let f and g be analytic inside and on a smooth regular closed curve . Suppose that f (z) 6= 0
for all z on . Prove that there is > 0 such that Z(f ) = Z(f + g) inside , w
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 10 Solution
Due date: June 8, 2017 (6:00 PM)
Problem 1. (4+4+4 points) Find a conformal mapping between A and B.
(a) A = cfw_z = x + iy : 0 < x < 1 and 0 < y < 1, B = cfw_z C : 1 < z < e2 \(1, e2 )
(b) A = D(0; 1)\[0, 1), B =
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 1 Solution
Due date: March 14, 2017 (8:59 am)
Problem 1. (5 points) Let P (z) be a nonconstant complex polynomial. Show that P (z) as
z .
(Solution) Let P (z) =
n
P
n1
P
k=0
n1
P
ak z k with an 6= 0. Then, P (z) = z n  an
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 8
Due date: May 23, 2017 (8:59 am)
Problem 1. (5 points)
Let f and g be analytic inside and on a smooth regular closed curve . Suppose that f (z) 6= 0
for all z on . Prove that there is > 0 such that Z(f ) = Z(f + g) inside , w
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Assignment 3
20150534 Lee Kyungmin
Real Analysis, MAS540
April 24, 2017
R
Problem 11. Suppose that is an integrable function on Rd with Rd (x)dx = 1. Set K (x) =
d (x/), > 0.
(a) Prove that cfw_K is a family of good kernels.
(b) Assume in addition that
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
3
Dierntiation
Problem 11 (Exercise 3.1). Suppose that is an integrable function on Rd with
Set K (x) = d (x/), > 0.
Rd
(x)dx = 1.
(a) Prove that cfw_K is a family of good kernels.
(b) Assume in addition that is bounded and supported in a bounded set. Ve
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 6
Due date: May 2, 2017 (8:59 am)
Problem 1. (5 points) Suppose g(z) is analytic in Dr (z0 ) cfw_z0 for some r > 0. Prove that if
g(z) Az z0 1+
for z Dr (z0 ) cfw_z0
with some > 0 and A > 0, then the singularity of g at z
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 7 Solution
Due date: May 9, 2017 (8:59 am)
Problem 1. (5 points) Let be the path z = 3ei along 0 4. Evaluate
Z
z2
dz
3z + 2
only by using the Residue Theorem.
1
1
1
1
=
. Since
is analytic at z = 1, Res(f ; 1) =
3z + 2
z2 z1
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
2017 Spring MAS341 Homework 9
Due date: May 30, 2017 (8:59 am)
Problem 1. (5 points) Evaluate
Z
e2ix
for 0 < a < 1 where coshz =
sinhz =
ez +ez
2
sina
dx
coshx + cosa
for z C. (Hint: you may represent the solution using
ez ez
2
Problem 2. (5 points) Evalu