Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Community detection wit
h labelled edges
1
1.Problem Setup and summary of results
Consider extended SBM model with labelled(weighted, colored) edges
Goal : What is the fundamental limit to exactly recover the communitie
s?
Setup : for vertex
p and q a
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Solutions to Homework 3
TA. Kihyun Kim
Ex2.2 The set
[
1
1
,
2k 2k 1
k=1
is a disjoint union of open sets. Therefore, it is open and has measure
X
k=1
1
1
= log 2.
2k 1 2k
Indeed, from the Taylor expansion of log(1 + x), we observe that
log(1 + x) =
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Solutions to Homework 4
TA. Kihyun Kim
Ex2.17 For any subset S of E, we have
0 m (S) m (S) m (E) = 0.
Ex2.24 () Let > 0 be given. Then, we can choose a closed set F and open set G for which
F E G and
m(G) < m (E) +
= m(E) +
2
2
and
m(F ) > m (E)
= m(E) .
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS 441 HOMEWORK 2
1. Let (X, A) be a measurable space. Let A A. Show that the function f (x) := 1,
if x A, and f (x) := 0, if x < A, is measurable.
2. Let (X, A) be a measurable space. Let f , g : X R be measurable functions.
Show that for every A A the
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 1 Solutions
(3,4,5,6,15,18,20,22,30,31,41)
3. (i)
lim sup An Bn An Bn i.o.
An i.o. or Bn i.o.
lim sup An lim sup Bn .
Therefore lim sup An Bn = lim sup An lim sup Bn .
(ii) Note that An A implies lim inf An = lim sup An .
lim inf An Bn lim sup
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Solutions to Homework 2
TA. Kihyun Kim
Prob1 For convenience, we will abbreviate cfw_x X
in R. Then, we observe that
X \ A
[f A] =
A
X
 f (x) A by [f A]. Let A be an open set
if 0, 1
/A
if 0 A, 1
/A
if 1 A, 0
/A
if 0, 1 A
Since , A, X \ A, X are all i
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Solutions to Homework 1
TA. Kihyun Kim
Prob1 In 3adic notation,
Therefore,
1
4
1
= 0.0202 (3) .
4
belongs to the Cantor set.
Prob2 Suppose that uncountably many positive real numbers cfw_ai iI are given. For each j N,
define
1
Ij := cfw_i I  ai .
j
By A
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Math 202B Solutions
Assignment 4
D. Sarason
13. Let (X, A, ) be a finite measure space and let f be a nonnegative measurable function on X. Prove that
f is integrable if and only if
X
(cfw_f > n) < .
n=1
P
Proof: Let En = cfw_f > n, and consider the funct
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Exact Recovery in the SBM(previous results)
To fully recover the communities, consider where and
To establish the infotheoretic threshold for full recovery, we
consider ML estimator, the minbisection problem.
First, if, ML doesnt coincide with true p
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Optimization Theory Solution
TA : Lee, Cheolhyoung
Ex 8.8
Solution. X Rn,m
+ , p, r [1, +], with p r
(X)p,r = max kXv kr : kvkp 1.
v
(1) fX : Rm
t R, fX (u) =
n X
m
1
X
(
Xij ujp )r is concave when p r?
i=1 j=1
Claim g(u) = (
m
X
1
p
ak uk )r is concave.
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
MAS 441 LEBESGUE INTEGRAL THEORY HOMEWORK 1
1. Show that
1
4
belongs to the Cantor ternary set.
2. Show that a sum of uncountably many positive real numbers is always infinite.
3. Show that the open interval (0, 2) is a countable union of closed intervals
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 2 Solutions
(1, 3, 4, 6, 9, 10, 13, 14, 16, 17)
1. Let X be a random variable with distribution function F , and take a monotone
decreasing sequence cfw_an that converges to 0. Since F (x) is rightcontinuous, F (x +
an ) F (x) 0. In addition, F
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 3 Solutions
(1, 2, 5, 6, 7, 8(c), 9, 13, 14, 15, 16)
1. For any > 0, we have P (Xn X ) E[Xn Xk ]/k by Chebyshevs
P
P
k
inequality. Since we have
E[X
X
]
<
,
n
n=1
n=1 P (Xn X ) < .
Thus, by the Theorem 3.4., Xn X a.s.
P
2. Suppose n P [An
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Universitext
Sergei Ovchinnikov
Measure, Integral,
Derivative
A Course on Lebesgue's Theory
Universitext
Universitext
Series Editors:
Sheldon Axler
San Francisco State University
Vincenzo Capasso
Universit`
a degli Studi di Milano
Carles Casacuberta
Unive
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 4 Solutions
(3,5,8,9,10,14,16,23,27,28)
3. P
T
An
=P
n=1
lim
k
T
k n=1
An
= lim P
k
k
T
An
= lim
k
Q
k n=1
n=1
P (An ) =
Q
P (An ).
n=1
5.(a) () Since X is independent of itself, P [X a, X a] = P [X a] P [X a] for
any a R, i.e., P [X a] = (P [X a]
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 5 Solutions
(5,6,14,16,22,24)
5. (a)
Z
Z Z
1cfw_x1 x2 F (dx1 )F (dx2 )
F (x)F (dx) =
R
ZR
ZR
ZR
R
=
1cfw_x1 x2 F (dx2 )F (dx1 ) by Fubini
(1 F (x1 )F (dx1 )
Z
= 1 F (x1 )F (dx1 ).
=
R
R
So,
R
R
F (x)F (dx) = 1/2.
Z
P [X1 X2 ] =
1[X1 X2 ] dF F
Z
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 6
PageRank
References
1. B. White, Math 51 Lecture Notes: How Google Ranks Web Pages.
2. Mor HarcholBalter, Performance Modeling and Design of Computer
Systems: Queueing Theory In Action, Cambridge University Press,
2013.
3. S. Brin and L. Page,
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Mathematics 5051, Fall 2013
Cantor Function
The function F we are about to describe is also often called the CantorLebesgue function or sometimes
called the Devils Staircase; the name Volterra is also occasionally associated with it.
Let C denote the usu
Korea Advanced Institute of Science and Technology
electromagnetic
EE 204

Fall 2016
Chapter 6 Solutions
(1(a),1(b),2,5,6,10,11(a),11(b),11(c), 15, 16)
P
1.(a) Since Xn
X, each subsequence cfw_Xnk contains a further subsequence cfw_Xnk(i)
which converges almost surely to X. Since cfw_Xn is a monotone sequence, cfw_Xn and
a.s.
cfw_Xn