Exercise 2
Probability
Math tools
October 22, 2013
1
Paths in a bipartite random graph
Let Gbi be a bipartite graph with n vertices on each side, where each of the possible n2 edges is in the
n,p
graph with probability p, independently of the other edges.
Exercise 12
Linear programming and the simplex algorithm
Math tools
January 7, 2014
1
Dierent paths in the simplex algorithm
Consider the following linear program.
Maximize x + y subject to
x0
y1
yx0
yx1
(a) Convert it to standard form.
(b) Run the simple
Exercise 7
Inequalities
Math tools
December 1, 2013
1
A generalized Hlders inequality
o
Let p, q, r > 1 such that
1
p
+
1
q
+
1
r
= 1. Show that for x, y, z Rn there holds:
n
1
p
n
i=1
2
|zi |r
i=1
i=1
1
r
n
|yi |q
|xi |p
xi yi zi
1
q
n
i=1
Jensens inequ
Exercise 5
More Linear Algebra
Math tools
November 19, 2013
1
The inner product and the induced norm
Let , be an inner product on a space X . Prove that the induced norm dened by x :=
is indeed a norm.
2
x, x
Norms on matrices
Prove that the following are
Exercise 8
The SVD
Math tools
December 3, 2013
1
Calculating the SVD
Compute the singular value decomposition of the matrix
10
A = 1 1 .
0 1
2
Frobenius norm vs operator norm
Show that if A is an n n real matrix whose rank is 1, then its Frobenius norms a
Exercise 8
Eigenvalue theorems and random walks
Math tools
December 10, 2013
1
Bounding the maximal clique
For a graph G = V , E , let K (G) denote the maximal number of vertices in a subgraph of G that is a
clique, and let 1 (G) be the maximal eigenvalue
Exercise 11
Expander graphs, polytopes and linear programming
Math tools
January 9, 2014
1
Vertices of polytopes
Let P Rn be a convex polytope of full dimension dened by P = cfw_x Rn : Ax b.
(a) Show that if x P is a unique maximum (in P ) of the linear f
Exercise 10
Expander graphs, random walks and perfect matchings
Math tools
December 24, 2013
1
The random walk on perfect matchings
Consider the bipartite graph G = V1 V2 , E dened by V1 = cfw_v1 , ., vn , V2 = cfw_u1 , ., un , and
cfw_vi , uj E if |i j
Exercise 4
Linear Algebra
Math tools
November 10, 2013
1
Almost orthonormal vectors
(a) Prove that if v1 , ., vk are orthonormal (that is, vi , vj = 0 for every pair vi , vj and vi
every i) then k n.
2
= 1 for
(b) Let > 0 be a (small) constant. Show that
Exercise 5
More Linear Algebra
Math tools
November 13, 2013
1
Cauchy-Schwarz for k vectors
Prove that for k > 2 and x1 , x2 , ., xk Rn there holds
2
n
x1
i
x2
i
.
xk
i
n
i=1
2
n
(x1 )2
i
i=1
(xk )2
i
.
i=1
An inner product space for polynomials
Let V
STIRLINGS FORMULA
KEITH CONRAD
Our goal is to prove the following asymptotic estimate for n! due to Stirling (1730).
nn
n!
Theorem 1. As n , n! n 2n. That is, lim
= 1.
n (n/e)n 2n
e
We will rst show n! (nn /en ) nC for some unknown constant C . In this f
Course Requirements
Math tools
October 17, 2013
The math tools course covers three major topics that have proven to be of great use
in computer science. These are probability, linear algebra, and linear programming.
There will be an exercise each week. Y
Final Exam
Math tools
February 2, 2014
Regulations
You must solve the exam on your own. You may not consult with any other person.
Do not distribute the exam answers.
Write your id and the questions you decided to solve on your submission.
You have un
Exercise 1
Probability
Math tools
October 10, 2013
1
The Binomial distribution
We ip n independent coins. Each coin comes out heads with probability p <
1 p. Let X be the total number of heads.
1
2
and tails with probability
(a) Compute X s expectation a
Exercise 3
Yet more probability
Math tools
October 29, 2013
1
The exponential random variable
A random variable X is exponential with parameter if it has the density function f (x) = ex for
x 0 and f (x) = 0 for x < 0.
Compute X s expectation and variance
Ex2 Solutions
Probability
Math tools
November 4, 2013
Due to popular demand, I have decided to publish a solution to questions 2 and 3 from exercise 2.
As with most mathematical problems, there are many possible solutions and the one published here is
not
1.Global Metalworks can produce its own hinges for a weekly cost of $96,400 and a cost per hinge of
$0.10. Currently an outside supplier is used for a weekly cost of $82,500 and a cost per hinge of $0.24.
Current weekly demand is 98,000 units. How much la