'YI=\ % Obvious
Asswm M 49w. avg/meme Md: 42* out,
1'9 (mi-H, ddc 0 0 o O -0 cfw_Jar Oo -, -01
lo 0 mo _0 11- (o -0\.
1.1- 0| 0 .0 'm l 1
I. 5 00 "a
o | '
a 0 I "olhq
~11
+00 an Ad ~I x
~ 1
-I
42-!
= 44.er ~+~+an + Hm. cs)
T
B) WJMCH'VQ Lyyrszs
= at +m+-
Homework 4
Choi SeongJin
October 20, 2016
Q
3. Suppose cfw_X1 , X2 , . . . is a sequence in the
Qproduct space X .
a. Show that the sequence converges to X X iff the sequence cfw_ (X1 ), (X2 ), . . .
converges to (X) for each index .
b. Is the statement i
Homework 2
Choi SeongJin
October 13, 2016
1. Suppose X and Y are spaces and y0 Y . Let c : X Y be the constant
function defined by c(x) = y0 . Show that c is continuous.
pf . For any open subset V in Y ,
(
X, if V contains y0
1
c (V ) =
, otherwise
, so c
Homework 1
Choi SeongJin
October 12, 2016
3.
Suppose cfw_T is a collection of topologies T on a fixed set X.
a. Show that T is a topology on X.
b. Is T a topoogy? Prove or disprove by a counterexample.
sol.
a. Clearly , X T . We have to show that
A T
is
Homework 3
Choi SeongJin
October 18, 2016
3. Prove the following.
a. A B = A B.
b. A A .
pf .
lemma. If A B, then A B. (It is clear since any closed set(e.g. B)
containing B also contains A)
b. Suppose x A , then x A for some . Since A A ,
A A by the abov
Q. Let X = cfw_(x, y) R2 | x2 + y 2 1 and Y = cfw_(x, y) R2 | x2 + y 2
1, x 0 be subspaces of R2 . Show that X and Y are homeomorphic.
pf . Define functions f : X Y, g : Y X by
p
p
x + 1 y2
f (x, y) := (
, y), g(x, y) := (2x 1 y 2 , y).
2
Then g f = IdX
Name _
FORL 216 English Grammar
Winter 2011
January 12, 2011
I. Choose the best answer to complete each sentence and then fill in the answer on the answer sheet.
1. My friend was curious if I had done my lab report _.
a) already
b) still
c) just
d) not ye
Chapter 2
Probability & Statistics Review
2.1
Random Variables and Probability
Distribution and Density Functions
For a random variable X,
R
FX (x) P (X x), x
( R
x
fX (t) dt, continuous case
P
=
discrete case
kx fX (k),
where fX () is the probability d
Name _
GEDU 133 Grammar Quiz #2
I. All of the sentences below have errors. Find the errors and correct them.
October 6, 2014
4 points each
* The errors are usually a one-word or two-word error (beside each other) that needs to be fixed.
(went or was in)
1
Name _
GEDU 133 Grammar Quiz #1 (Units 1-2)
March 24, 2014
I. All of the sentences below have errors. Underline the errors and correct them.
* If a sentence has an error, it usually has a one-word or two-word error (beside each other) that needs to be fix
Name _
FORL 216 English Grammar
Spring Final 2010
Date _
I. Choose the best answer to complete each sentence and then fill in the answer on the answer sheet.
1. I really hope you _ the job tomorrow because you worked very hard for it.
a) got
b) would get
Name _
GEDU 133 Grammar Quiz #1 (Units 1-2)
March 24, 2014
I. All of the sentences below have errors. Underline the errors and correct them.
* If a sentence has an error, it usually has a one-word or two-word error (beside each other) that needs to be fix
Name _
GEDU 133 Grammar Quiz #1 (Units 1-2)
September 22, 2014
I. All of the sentences have errors. Underline the errors and correct them.
* If a sentence has an error, it usually has a one-word or two-word error (beside each other) that needs to be fixed
Name _
GEDU 133 Grammar Quiz #1 (Units 1-2)
December 31, 2013
I. All of the sentences below have errors. Underline the errors and correct them.
* If a sentence has an error, it usually has a one-word or two-word error (beside each other) that needs to be
Name _
GEDU 133 Grammar Quiz #6
I. Circle the best answer to complete the sentences below.
1. The young man finished the marathon _ his lack of training.
June 9/10, 2014
5 points each
a) but
b) in spite
c) although
d) despite
e) however
2. If I _ a more r
Name _
FORL 216 English Grammar
Spring Final 2010
Date _
I. Choose the best answer to complete each sentence and then fill in the answer on the answer sheet.
1. I really hope you _ the job tomorrow because you worked very hard for it.
a) got
b) would get
Name _
GEDU 133 Grammar Quiz #3
April 14, 2014
I. Circle the best answer to complete each sentence. (5 points each)
1. A: How did you do on the test?
B: Not so well. I _ much better but I misread the directions for writing.
a) will have done
b) should do
One Hundred1 Solved2 Exercises3 for the subject:
Stochastic Processes I4
Takis Konstantopoulos5 1. In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard
Mathematics 414, Spring 2008
Solutions to assignment 5
Problem 10.1.6: Let f : D R be dierentiable at y, and suppose f (y) = 0. Show
that du f (y) as u varies over all unit vectors (|u| = 1) attains its maximum value when
u = f (y) for some > 0 and du f (
Mathematics 414, Spring 2008
Solutions to assignment 4
x
Problem 9.3.17: Let T : C([0, 1]) C([0, 1]) be dened by T f (x) = x + 0 tf (t)dt. Prove
that T satises the hypothesis of the contractive mapping principle. Show that the xed
point is a solution to t