Math 459, Spring 2009
Homework 1
Due: Tuesday, 4/7
1. Prove that Z2 G is not simple where G is any group.
2. Let G be a group with binary operation . Prove that -inverses are unique
(i.e., if a, b G such that a b = e, then b = a1 .
3. Write out the group
Math 459
Senior Seminar
Fall 2012
Instructor: Dr. Dylan Retsek
Oce: 314 Faculty Oces East
email: dretsek@calpoly.edu
Web: www.calpoly.edu/ dretsek
Phone: (805) 756-2072
Course Objective
Our objective this quarter is to parlay the mathematical knowledge an
Math 459
Challenge Problems: -algebras
Fall 2012
Abstract -algebras
In our course notes, we focus on -algebras on R. In general, a -algebra
on a set X is a collection of subsets of X that is closed under complementation
and countable unions. The set X nee
Chapter 3
Constructing a Measure
When the going gets tough, the tough use denitions.
As we begin to develop our measure meeting expectations I-III of chapter one,
we will need to house given sets snugly within more familiar sets. In particular
we will nee
Chapter 1
An Unmeasurable Set
Never ght fair with a stranger, boy. Youll never get out of the jungle that way.
From Death of a Salesman, by Arthur Miller
1.1
Great Expectations
We wish to measure subsets of the real numbers R. Formally, we seek a function
Chapter 2
Measures
Learn the rules so you know how to break them properly.
Dalai Lama
2.1
The Right Setting
Its not the idea of measuring subsets of R thats untenable, its the idea of measuring
all subsets of R. In this section, the right domain for a mea
CHAPTER 3. CONSTRUCTING A MEASURE
3.3
15
Aspiring to Measure
Though A0 is not a -algebra, it is possible that a countably innite disjoint union
of sets in A0 is again in A0 .
1
Exercise 3.32. Let Ej = ( j+1 , 1 ] for each j N. Show that
j
j=1
Ej A0 .
When
Math 459
Presentations
Fall 2012
Presenter Guidelines
The object is to clearly, succinctly and correctly present mathematics to your
classmates. Strive to write legibly in complete sentences and to speak clearly
and condently. A good presentation is a gif
Math 459
Portfolios
Fall 2012
Portfolio Guidelines
The object is to maintain a current accounting of the work we do. Every
exercise, proposition, lemma, theorem and corollary that we encounter is to be
included in your portfolio. Each entry in the portfol
MATH 459 Senior Seminar
1.
Catalog Description
MATH 459
Senior Seminar
(4)
Written and oral analyses and presentations by students on topics from advanced mathematics and
mathematical modeling. 4 seminars. Prerequisite: MATH 306, and completion of at leas
Chapter 8
Complex Numbers
Homework
1) Find k 2 such that M(k) = 0 and prove it.
2) Calculate the Farey sequence F6
3) Find D(5), the sum of the differences between the Farey sequence of 5 and each
1/A(i).
4) A complex number is said to be in trigonometric
Chapter 7 HW Questions & Senior Project Proposals
1. Prove it is not possible to draw a network with 5 vertices so that each
vertex is connected to all the others (hint: see figure 41).
2. Given an odd-by-odd grid with each node occupied by an individual,
Chapter 3
Homework Questions:
1. Find 3 other primes that
2. Show that
generates.
has inverses. (The requirement for an integral domain to be a field.)
3. Using the laws of quaternions, calculate
4. Show that
.
is not a unique factorization domain. (Hint:
Math 459, Spring 2009
Homework 2
Due: Friday, 4/10
1. Let R be a commutative ring and f1 , . . . , fn R. Show that f1 , . . . , fn =
cfw_r1 f1 + + rn fn | r1 , . . . , rn R is an ideal of R.
2. Let R and S be rings. Then a function f : R S is called a rin
Math 459, Spring 2009
Homework 3
Due: Tuesday, 4/14
1. Show that 1 5 10 15 14 18 14 12 12 is a valid Hilbert function
(i.e., show that there is an ideal in R[x1 , . . . , x5 ] such that H (R/I ) = 1 5
10 15 14 18 14 12 12).
2. Compute the Hilbert functio
Math 459, Spring 2009
Homework 4
Due: Friday, 4/17
1. Prove (as we did in class) that x, y, z is a primitive Pythagorean triple if and
only if there are s, t N>0 such that s > t, gcd(s, t) = 1, and one of s and t is
even while the other is odd.
2. Show th
Homework Problems
1. Given the similarity ratio and the number of pieces the figure is cut into, find the
dimension for the following figures
a) r=2, N=6
b) r=4, N=7
c) r=6, N=8
2. Is the Fibonacci Sequence a simple feedback loop? Why or why not?
3. Suppo
Chapter 6 Homework Problems:
1. Prove that the Diophantine equation ax + by = c has integer
solutions x and y if and only if the greatest common divisor of a and b
divides c.
2. X is a member of what set of naturals, with corresponding
Diophantine equatio
Chapter 9 Homework
1. Compute the Alexander Polynomial of the following knot. Show your work.
2. Show a sequence of Reidemeister moves that transform the following knot into a simple
loop.
3.Which of the surfaces have the same topology. Give an explanatio
Homework problems: Chapter 11 (The Efficiency of Algorithms)
Problem 1:
Starting at location A what is the shortest path to visit the rest of the points: B, C, D,
E?
Notice, there is (n-1)! possible permutations for paths.
Problem 2:
Suppose that you want
GO
Projects:
1. Research and extend the topics in Mathematical Go Endgames by Elwyn Berlekamp and
David Wolfe.
2. Research combinatorial game theory and its applications to already developed Go
strategies. (i.e. Middle-Game, Life & Death, and Power vs. Te
Math 459
Senior Seminar
Spring 2009
Senior Project Ideas:
1. Discuss/Analyze the differences and benefits of public vs. private keys.
2. Consider the proof of the Polynomial Runtime Algorithm for determining primality, and
attempt to make it more efficien
CHAPTER 3. CONSTRUCTING A MEASURE
18
Theorem 3.40. If cfw_Ej is a sequence of disjoint sets in A0 whose union is also
j=1
in A0 , then
0
Ej
=
j=1
0 (Ej ) .
j=1
Now that we know the premeasure 0 is behaving properly on A0 , in the next
section we take up