SOLUTIONS TO PROBLEM SET 8
MATH 122
1 The number of Sylow 5subgroups must divide 4 and must be congruent to 1 modulo
5. So, the number is 1. The Sylow 5subgroup is cyclic of order 5, so it contains 4
elements of order 5.
2 By First Sylow Theorem, G cont
Math 122 / Midterm 1 Solutions.
1. a. Since the group Z/4Z is cyclic with generator 1, any automorphism of Z/4Z is determined by where it sends 1; this can be either
of the elements 1 and 3 Z/4Z of order 4. Thus there are exactly two
automorphisms, so Aut
Math 122 / Practice exam 2
Note: This practice exam is a bit harder and certainly longer than the
actual midterm. We would suggest to allow yourselves 100 minutes for it. It
will be discussed in class on Monday but we strongly urge you to actually try
it
Math 122 Practice Final Exam
Part I (Each problem here is worth 5 points.)
1. Let H be a subgroup of a group G. Is it true that the intersection of
all conjugates of H is a normal subgroup of G?
2. Is it true that the intersection of normal subgroups of a
Note: the actual exam will have only 6 problems.
Problem 1. a) Find all elements of order 2 in the group
G=
a b
0 c
 a, c R, ac = 0 .
b) Show that it is possible to have two elements a and b in a group G which
have nite order, but such that ab has innite
Math 122 Course Outline (Fall 2006)
Meeting 1 September 18 (rst day of class)
Briey review linear algebra. Matrix multiplication, associativity, inverses.
Dene a group using GLn as an example.
Laws of composition. Associativity. Commutativity. Composit
Math 122 Course Outline (Fall 2006)
Meeting 1 September 18 (rst day of class)
Briey review linear algebra. Matrix multiplication, associativity, inverses.
Dene a group using GLn as an example.
Laws of composition. Associativity. Commutativity. Composit
Math 122 / Announcements
Welcome to Math 122! This class will meet on Mondays, Wednesdays, and
Fridays at 10 in Jeerson 250. The course website is:
www.courses.fas.harvard.edu/ math122. It can also be accessed from your
account at my.harvard.edu once you
Math 122 / Problem Set 1
Written problems due Monday, September 25
Monday, September 18
1. A square matrix A is called nilpotent if Ak = 0 for some k > 0. Prove
that if A is nilpotent, then I + A is invertible.
2. Dene the trace of an n n matrix A = (aij
MATHEMATICS 122. ALGEBRA I: THEORY OF GROUPS AND VECTOR SPACES
MATH 122

Winter 2015
Name: Helena Rabelo Freitas
Air Resistance Effects on Projectile Motion
Purpose
Determine the air resistance effects when a steel ball is launched with different angles by a
gun mechanism.
Procedure
The experiment was done by following the instructions on
CHAPTER 7
EQUITY MARKETS AND STOCK VALUATION
CHAPTER 7 QUIZ
CHAPTER ORGANIZATION
7.1
Common Stock Valuation  FinSim
Cash Flows
Stock valuation is more difficult than bond valuation because the cash flows are uncertain, the life is
forever, and the requir
MATHEMATICS 122. ALGEBRA I: THEORY OF GROUPS AND VECTOR SPACES
MATH 122

Spring 2016
Option Pricing under ARMA Processes
Theoretical and Empirical prospective
ChouWen Wang
1
Astract
Motivated by the empirical findings that ass
et returns or volatilities are predictable, this
paper, extending Huang and Wu (2007), stu
dies the pricing of E
Free Energy and Chemical
Potential
We will introduce the Helmholtz energy and the
Gibbs free energy and Maxwells relationships
among natural variables.
We will define the chemical potential that is a
partial molar quantity and fugacity as a necessary
desc
The First Law of Thermodynamics
The first law is about
conservation of energy (in the
form of work and heat).
The laws of thermodynamics are macroscopic
rules.
The first law of thermodynamics is based on the
relationship between three quantities: work,
he
Physical Chemistry
Spring 2016
Instructor:
Prof. Eui Jung Kim
Office: 1307
Email: [email protected]
Office Hours: Mon 1:003:00
(or by appointment)
Gases and the Zeroth
Law of
Thermodynamics
What is Physical Chemistry?
Physical chemistry is the study o
The Second and Third Laws of
Thermodynamics
The second law is about entropy and its role in
determining whether a process will proceed
spontaneously.
The third law is about the impossibility of attaining
the absolute zero of temperature in a
thermodynamic
Equilibria in SingleComponent
Systems
We will apply the concepts of equilibrium to
systems that consist of a single chemical
component.
We will define component and phase, derive new
expressions that we can use to understand the
equilibria of singlecomp
Equilibria in MultipleComponent
Systems
We will extend some of the concepts of the
previous chapter in a limited fashion. We will build
on the previous chapters ideas and develop new
ideas (and equations) that apply to multiplecomponent systems.
We will
1. One mole of a monatomic ideal gas initially at 30.0C and 1 atm pressure is
heated and allowed to expand reversibly at constant pressure until the final
temperature is 350.0C. (1) Calculate the work done by the gas in this
expansion. (2) What is U and H
Introduction to Chemical
Equilibrium
Chemical equilibrium : that point during the
course of a chemical reaction where there is no
further net change in the chemical composition of
the system.
A system in equilibrium is a system we can
understand using the
MATHEMATICS 122. ALGEBRA I: THEORY OF GROUPS AND VECTOR SPACES
MATH 122

Spring 2016
ACRN Journal of Finance and Risk Perspectives
Vol. 3, Issue 2, June 2014, p. 83 132
ISSN 23057394
WHAT DOES THE VIX ACTUALLY MEASURE? AN ANALYSIS
OF THE CAUSATION OF SPX AND VIX
Merav Ozair
Department
of Finance and Risk Engineering, Polytechnic Institu
Math 122 / Problem Set 5
Written problems due Monday, October 23.
Monday, October 16
1. Let V be the vectorspace of polynomials of degree 3 over R. Let
T : V V be the linear operator mapping
T (P (X) =
d2
P (X).
dX 2
Write the matrix for T with respect to
Math 122 / Problem Set 2
Written problems due Wednesday, October 11.
Monday, October 2
1. Find all solutions to the congruence 3x 7 (a) modulo 25 and (b) modulo
15.
2. Prove the wellknown rule for divisibility by 9: that every positive integer
is congrue
Math 122 / Problem Set 2
Written problems due Monday, October 2.
Monday, September 25
1. Let : G G be a group homomorphism, and let x G be an element
of order r. What can you say about the order of (x)?
2. Let : G G be a surjective homomorphism.
(a) Prove
Math 122 / Problem Set 5 Solutions
1. A ring homomorphism : Z Z has to send 1 to 1, and this determines
the image of each integer n, namely, (n) = n, so the identity map is the only
ring homomorphism Z Z.
2. Let f (x) = xp . It is clear that f (ab) = f (a
Math 122 / Problem Set 5 Solutions
Problem 1. The columns of the matrix of T with respect to the given
basis are the images under T of the basis vectors:
T 1 = 0, T x = 0, T x2 = 2, T x3 = 6x,
so the matrix of T with respect to the
00
0 0
0 0
00
given bas
Math 122 / Problem Set 1 Solutions
Problem 1.
Solution 1. If Ak = 0, consider the matrix B = I A + A2 +
(1)k1 Ak1 . Since powers of A commute, we can write
(I + A)B = I + (1)k1 Ak = I.
Similarly, B (I + A) = I . Thus, B is an inverse of A, hence A is inve