Homework 7, Math 3012 A, Summer 2011
Due Fri. July 8, in class
1. Use generating functions to solve the recurrence relation rn = 2rn1 + 5n , r0 = 0.
Multiply by xn1 and sum:
rn x
Let F (x) =
n=1
n1
=
n=1 rn1 x
n1
=2
rn1 x
n=1
n
n=0 rn x . Then
n1
(5x)n1 .
Homework 6, Math 3012 A, Summer 2011
Due Fri. July 8, in class
1. Find and solve a recurrence for the number gn of ternary strings of length n that do not
contain 20 as a substring.
By Example 3.3 in the book,
gn = 3gn1 gn2 ,
where g1 = 3, g2 = 8. The cha
Homework 5, Math 3012 A, Summer 2011
Due Fri. July 8, in class
1. How many integers between 1 and 100 are divisible by none of 2, 3 or 5? Use inclusion/exclusion.
X = [100]. Let Pi be the property that an integer is divisible by i. Then
N (cfw_2) = 50,
N
Homework 4, Math 3012 A, Summer 2011
Due Fri. June 24, in class
1. Let G be a connected graph and s = t V (G). An Eulerian walk in G from s to t is
a list of vertices s = x0 , x1 , x2 , . . . , xk = t (with repetition), where (xi , xi+1 ) E (G) for all
i
Homework 3, Math 3012 A, Summer 2011
Due Mon. June 13, by midnight
1. Give a recursion for the number g (n) of ternary strings of length n that do not contain
102 as a substring.
Consider what the nal character is. If the nal character is a 1 or a 0 then
Homework 2, Math 3012 A, Summer 2011
Due Wed. June 1, in class
1. In how many ways can you read o the word MATHEMATICS from the following tables:
MATHEM
ATHEMA
THEMAT
HEMAT I
EMAT IC
MATICS
MATHEM
ATHEMA
TH
MAT
HEMAT I
EMAT IC
MATICS
Each copy of MATHEMAT
Homework 1, Math 3012 A, Summer 2011
Due Wed. May 25, in class
1. Bob is designing a website authentication system. He knows passwords are most secure
if they contain letters, numbers, and symbols. However, he doesnt quite understand that
this additional
Homework 1, Math 3012 A, Summer 2011
Due Wed. May 25, in class
1. Bob is designing a website authentication system. He knows passwords are most secure
if they contain letters, numbers, and symbols. However, he doesnt quite understand that
this additional
MATH 3012G Test III
Spring 2010
Name:
GTid (9xxxxxxxx):
Group:
Instructor: Mitchel T. Keller
There are 6 questions on this exam on 4 pages (not counting this coverpage). Answer each
question in the space provided. If you need additional space, additional
MATH 3012G Test II
Spring 2010
Name:
GTid (9xxxxxxxx):
Group:
Instructor: Mitchel T. Keller
There are 6 questions on this exam on 4 pages (not counting this coverpage). Answer each
question on a separate solution sheet. Be sure to explain your answers, as
MATH 3012G Test I
Spring 2010
Name:
GTid (9xxxxxxxx):
Group:
Instructor: Mitchel T. Keller
There are 6 questions on this exam on 4 pages (not counting this coverpage). Answer each question
on a separate solution sheet. Be sure to explain your answers, as
Group
Name
MATH 3012G TEST III
TAKE-HOME PROBLEM
SPRING 2010
Instructions. You are to work on this problem completely alone. You are permitted to contact the instructor with questions, but otherwise are not to communicate with any other individual about t
Group
Name
MATH 3012G TEST II
TAKE-HOME PROBLEM
SPRING 2010
Instructions. You are to work on this problem completely alone. You are permitted to contact the instructor with
questions, but otherwise are not to communicate with any other individual about th
Group
Name
MATH 3012G TEST I
TAKE-HOME PROBLEM
SPRING 2010
Instructions. You are to work on this problem completely alone. You are permitted to contact the instructor with
questions, but otherwise are not to communicate with any other individual about thi
MATH 3012G Final Exam
Spring 2010
Name:
GTid (9xxxxxxxx):
Group:
Instructor: Mitchel T. Keller
There are 14 questions on this exam on 14 pages (not counting this coverpage). Answer
each question in the space provided. If you need additional space, additio
MATH 3012B Test I
Fall 2008
Name: GTid (9xxxxxxxx): Instructor: Mitchel T. Keller There are 11 questions on this exam on 7 pages (not counting this coverpage). Answer questions in order on the provided solution sheets. Be sure to explain your answers, as
MATH 3012B Quiz #3
Name: 11 November 2008
1. (5 points) A new local bank is being created and will establish a headquarters H, two branches B1 and B2 , and four ATMs A1 , A2 , A3 , and A4 . They need to build a computer network such that the headquarters,
MATH 3012B Quiz #2
Name: 9 October 2008
1. (5 points) If G = (V, E) is a graph, its degree sequence is the list of the degrees of its vertices. That is, if V = cfw_v1 , v2 , . . . , vn , the degree sequence of G is (degG (v1 ), degG (v2 ), . . . , degG (v
MATH 3012B Quiz #1
Name: 4 September 2008
1. (5 points) The setting for this problem is the ctional town of Mascotville, which is laid out as a grid. Mascots are allowed to travel only on the streets, and not as the yellowjacket ies. Buzz wants to go visi
MATH 3012B Final Exam
Fall 2008
Name:
GTid (9xxxxxxxx):
Group:
Instructor: Mitchel T. Keller
There are 15 questions on this exam on 3 pages (not counting this coverpage). Answer questions 1 and 2 on a single solution sheet. Answer each of the remaining qu
Homework 4 Solutions, 3012 Summer 2010
July 29, 2010
1. Find the number of codewords of length k from the alphabet cfw_a, b, c, d, e
such that b occurs an even number of times.
(a.) Solve using generating functions.
k
The solution is the coecient of x ! i
2. Prove the following by mathematical induction: If S is a set of n real
0 < a1 < a2 < < an , where ai+1 2ai , then all the subset sums of the
elements of S are distinct. A subset sum is just a sum of the form
ai1 + ai2 + + aik ,
where i1 < i2 < < ik , a
Problem. 1 Determine the number of integers of the form
2a1 3a2 5a3 7a4 9a5 11a6 13a7 15a8 ,
where each ai is an integer satisfying 02 a1 2 a2 a8 ; and for i =
1, ., 8, ai i.
Each of these integers corresponds to a Catalan path from (0, 0) to (9, 9),
wher
Chapter 4 - Introduction to Complexity Theory
Amanda Pascoe Streib
School of Mathematics
Georgia Institute of Technology
16 May 2011
university-logo
Amanda Pascoe Streib (Georgia Tech)
Chapter 4
16 May 2011
1 / 20
Complexity Theory
Generally in CS we try
Chapter 13 Notes, Math 3012 A, Summer 2011
Amanda Pascoe Streib
July 26, 2011
Lecture Notes from Applied Combinatorics class Summer 2011. These lecture notes are
based on Applied Combinatorics by Keller and Trotter. Other exercises are pulled from
Discret
Chapter 12 Notes, Math 3012 A, Summer 2011
Amanda Pascoe Streib
July 22, 2011
Lecture Notes from Applied Combinatorics class Summer 2011. These lecture notes are
based on Applied Combinatorics by Keller and Trotter. Other exercises are pulled from
Discret
Chapter 9 Notes, Math 3012 A, Summer 2011
Amanda Pascoe Streib
July 14, 2011
Lecture Notes from Applied Combinatorics class Summer 2011. These lecture notes are
based on Applied Combinatorics by Keller and Trotter. Other exercises are pulled from
Discrete