Summary of Linear Theory
Statistical Model:
Assumptions:
Covariance Estimator and Estimate:
Least Squares Estimator and Estimate:
Variance Estimator and Estimate
Statistical Properties:
HOMEWORK #9, DUE THURSDAY APRIL 25TH
1. Herstein, Chapter 4, 4, 2: Let R be the Gaussian integers and
let M be the subset of Gaussian integers a + bi such that a and b are
divisible by 3. Show that M is an ideal and the quotient R/M is a eld
with 9 elemen
1. (15pts) (i) Give the denition of a normal subgroup.
(ii) Give the denition of a group homomorphism.
(iii) Give the denition of An , the alternating group.
1
2. (15pts) (i) Exhibit a proper normal subgroup V of A4 . To which
group is V isomorphic to?
(i
SECOND PRACTICE MIDTERM
You have 80 minutes. This test is closed book, closed notes, no calculators.
There are 6 problems, and the total number of
points is 100. Show all your work. Please make
your work as clear and easy to follow as possible.
Points wil
HOMEWORK #8, DUE THURSDAY APRIL 18TH
1. Herstein, Chapter 4, 3, 1.
2. Herstein, Chapter 4, 3, 2.
3. Let I and J be ideals of a ring R.
(i) Show that I J is an ideal of R.
(ii) Show that
I + J = cfw_ i + j | i I, j J
is an ideal of R.
(iii) Let IJ be the
1. (30pts) Give the denition of a group.
(ii) Give the denition of an automorphism of groups.
(iii) Give the denition of Dn , the dihedral group.
1
(iv) Give the denition of an ideal.
(v) Give the denition of a principal ideal domain.
(vi) Give the deniti
The following material can be found in a number of sources, including Sections
7.3 7.5, 7.7, 7.10 7.11, 7.15 16 of Stanleys Enumerative Combinatorics Volume
2.
1 Elementary and Homogeneous Symmetric Func
tions
A polynomial in n variables, P (x1 , x2 , .
Todays lecture notes cover the Oriented Matrix Theorem, which is discussed in
Sections 9 and 10 of Richard Stanleys Topics in Algebraic Combinatorics lecture
notes. The proof presented in class more closely resembles the bijective proof in
Section 4.4 of
The following is an outline of the material covered April 6th and 8th in class.
This material can be found in Chapter 5 of Stanleys Enumerative Combinatorics
Volume 2. Proofs of most of the results are in class notes.
1
Exponential Generating Functions
De
The following material can be found in several sources including Sections 14.9
14.13 of Algebraic Graph Theory by Chris Godsil and Gordon Royle, as well as
the papers Chip-ring Games on Graphs by Anders Bjrner , Lsl Lovsz, and
o
a o
a
Peter Shor (1991), a
1
Introduction to Partitions
A partition of n is an ordered set of positive integers = (1 , 2 , . . . , n ) such that
i i = n and 1 2 k .
Let P (n) denote the set of all partitions of n with p(n) = |P (n)| and p(0) = 1. For
example, the partitions of 4 ar
The material for this lecture can be found in several sources, for example see
Section 4.1 of William Fultons book Young Tableaux.
1
Proof of Schensteds Theorem
Theorem (Schensted). Let be a permutation of cfw_1, 2, . . . , n written in one-line
notation.
1
Mbius Function on Posets
o
This material closely follows selections from Chapter 3 of Enumerative Combina
torics 1 by Richard Stanley.
1.1
New Posets from Old
If P and Q are posets on disjoint sets, then the disjoint union (or direct sum) of
P and Q is
1
Perfect Matchings and Domino Tilings
Deniton. A Perfect Matching of a graph G = (V, E) is a set M E of
distringuished edges such that each vertex v V is incident to exactly one edge of
M.
Notice that in particular that this implies that |M | = |V |/2 an
1
Recurrence Relations and Generating Functions
Given an innite sequence of numbers, a generating function is a compact way of
expressing this data. We begin with the notion of ordinary generating functions.
To illustrate this denition, we start with the
1
Partially Ordered Sets II: Dilworths Theorem
Denition. An anitchain A of a poset P is a subset of elements of P such that
for all x, y A, x y and y x.
We denote the levels of a graded poset P as Pi where Pi = cfw_x P : rank(x) = i.
Remark. Observe that
Algebraic Combinatorics
Homework # 9
Due Tax Day, Wednesday April 15, 2009
You may discuss the homework with other students in the class, but please write
the names of your collaborators at the top of your assignment. Please be advised
that you should not
Algebraic Combinatorics
Homework # 7
Due Wednesday April 1, 2009
You may discuss the homework with other students in the class, but please write
the names of your collaborators at the top of your assignment. Please be advised
that you should not just obta
3) Let G = Cn be a cycle graph on n vertices and let v0 be one of Gs vertices.
(5 points) How many critical congurations does Cn have, letting v0 be the
sink vertex?
(10 points) Describe the critical congurations of Cn . (Hint: To get started,
try writing
Algebraic Combinatorics
You may discuss the homework with other students in the class, but please write
the names of your collaborators at the top of your assignment. Please be advised
that you should not just obtain the solution from another source. Ple
3) Recall that
of size k.
n
k
, binomial coecient, counts the number of subsets of cfw_1, 2, . . . , n
(10 points) Give an algebraic proof of the identity, i.e. use generating functions:
2
n
n
2n
=
n
k=0 k
.
(1)
(10 points) Give a combinatorial proof
Algebraic Combinatorics
You may discuss the homework with other students in the class, but please write
the names of your collaborators at the top of your assignment. Please be advised
that you should not just obtain the solution from another source. Ple