Topic Course on
Probabilistic Methods
(Week 1)
Linearity of Expectation (1)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly bas
Topic Course on
Probabilistic Methods
(Week 11)
Random Graphs (I)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the tex
44 Chapter One. Linear Systems
Canceling squares u, . . . , v5 and dividing by 2 gives a formula for the angle.
11.119] -|- 1.1.2132 + LL3V3
ll WI
1
9 : arccos(
In higher dimensions we cannot draw pic
Section II. Linear Geometry 43
cant prove anything about intuitionin this subsection well observe that a
result familiar from R2 and IE3, when generalized to arbitrary Iii, supports the
idea that a li
Section II. Linear Geometry 49
2.39 The geometric mean. of two positive reals any is Macy. It is analogous to the
arithmetic mean [it + y] / 2. Use the Cauchy-Schwars inequality to show that the
geome
Section III. Reduced Echelon Form 53
Again, the point of view that we are developing, supported now by the lemma,
is that the term reduces to is misleading: where A > B, we shouldnt think
of B as afte
52 Chapter One. Linear Systems
element solution set case the single element is in the column of constants. The
next example shows how to read the parametrization of an innite solution set.
1.4 Example
Section II. Linear Geometry 45
Exercise 18.) Since all the numbers are positive, the inequality holds if and only
if its square holds.
That, in turn, holds if and only if the relationship obtained by
50 Chapter One. Linear Systems
III Reduced Echelon Form
After developing the mechanics of Gausss Method, we observed that it can be
done in more than one way. For example, from this matrix
(1 i)
we co
Section III. Reduced Echelon Form 51
and then to a third stage that uses the leading entries to eliminate all of the
ether entries in each column by combining upwards.
1102 1001
23E: 0101 EP 0101
P3+0
48 Chapter One. Linear Systems
As always, you must back any assertion with either a proof or an example.
2.19 Verify the equality condition in Corollary 2.6, the Cauchy-Schwars Inequal-
ity.
(a) Show
Sectien II. Linear Geernetry 47
the angle is
[11(0) + (US) + [01(2) J: mm[ 3
am 02+32+22 x/Z/
appreximatelv 0.94 radians. Netice that these vecters are net erthegenal. Al
theugh the yz-plane may appea
Topic Course on
Probabilistic Methods
(Week 6)
Correlation Inequalities
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based t
Topic Course on
Probabilistic Methods
(Week 5)
Lovsz Local Lemma
a
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the te
Topic Course on
Probabilistic Methods
(Week 2)
Linearity of Expectation (2)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly bas
Math778P Homework 4 Solution
1. Let G = (V, E ) be a simple graph and suppose each v V is associated
with a set S (v ) of colors of size at least 10d, where d 1. Suppose, in
addition, that for each v
Math778P Homework 3
due Oct. 5, Friday lecture
Choose any 5 problems to solve.
1. Use alteration method to prove the Ramsey number
2
t
ln t
R(4, t) c
for some constant c > 0.
Proof by Edward Boehnlein
Math778P Homework 1 Solution
1. Prove Ramsey number R(4, t) (t3/2 /(ln t)3/2 ).
Proof by Heather Smith:
Claim: For a probability p, if
n.
n
4
4
p(2) +
n
t
t
(1 p)(2) < 1 then R(4, t) >
Randomly two co
Math778P Homework 2 Solution
Choose any 5 problems to solve.
1. Let Sn = n Xi where X1 , . . . , Xn are n independent uniform cfw_1, 1
i=1
random variables. Prove that
E(|Sn |) = n21n
n1
n1
2
.
Proof
Topic Course on
Probabilistic Methods
(Week 3)
Alterations
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the textbook T
Topic Course on
Probabilistic Methods
(Week 4)
Second Moment Method
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the t
Topic Course on
Probabilistic Methods
(Week 9)
Large deviation inequalities (III)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is most
Topic Course on
Probabilistic Methods
(Week 10)
Poisson Paradigm
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly based the text
Topic Course on
Probabilistic Methods
(Week 8)
Large deviation inequalities (II)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostl
Topic Course on
Probabilistic Methods
(Week 7)
Large deviation inequalities (I)
Linyuan Lu
University of South Carolina
Univeristy of South Carolina, Fall, 2012
Introduction
The topic course is mostly
46 Chapter One. Linear Systems
2.6 Corollary (Cauchy-Schwarz Inequality) For any 11,1? E lit,
|*| a, llll
with equality if and only if one vector is a scalar multiple of the other.
Phone The Triangle