1
Cayleys Theorems
As for everything else, so for a mathematical theory: beauty can be perceived
but not explained.
.
Arthur Cayley
An introduction to group theory often begins with a number of examples
of nite groups (symmetric, alternating, dihedral, .)
Burgers Equation
The simplest nonlinear diffusion equation is known as Burgers equation
ut + u ux = uxx ,
(22.35)
and is obtained by appending a linear diffusion term to the nonlinear transport equation
(22.13). In fluids and gases, one can interpret the
Note to the reader
This is not a sophisticated text. In writing it, I have assumed no more mathematical knowledge than might be acquired from an undergraduate degree at an
ordinary British university, and I have not assumed that you are used to learning m
Contents
Note to the reader
Introduction
page vii
1
1
Categories, functors and natural transformations
1.1
Categories
1.2
Functors
1.3
Natural transformations
9
10
17
27
2
Adjoints
2.1
Definition and examples
2.2
Adjunctions via units and counits
2.3
Adju
C A M B R I D G E S T U D I E S I N A D VA N C E D M AT H E M AT I C S 1 4 3
Editorial Board
B . B O L L O B S , W. F U LT O N , A . K AT O K , F. K I R WA N ,
P. S A R N A K , B . S I M O N , B . T O TA R O
BASIC CATEGORY THEORY
At the heart of this shor
A SHORT PROOF OF THE EXISTENCE OF JORDAN
NORMAL FORM
MARK WILDON
Let V be a finite-dimensional complex vector space and let T : V V be
a linear map. A fundamental theorem in linear algebra asserts that there is
a basis of V in which T is represented by a
Problems
47
(a) Show that T is a topology. (It is called the topology generated by A,
and A is called a subbasis for T .)
(b) Show that T is the coarsest topology for which all the sets in A are
open.
(c) Let Y be any topological space. Show that a map f
CHAPTER II
TRANSFORMATIONS
32. Linear transformations
We come now to the objects that really make vector spaces interesting.
DEFINITION.
A linear transformation (or operator) A on a vector space
'0 is a correspondence that assigns to every vector x in '0
An Introduction to Braid Theory
Maurice Chiodo
November 4, 2005
Abstract
In this paper we present an introduction to the theory of braids. We lay down
some clear definitions of a braid, and proceed to establish the braid group Bn . We
show that both the w
ON EMBEDDINGS OF SPHERES
BY
B. C. MAZUR
Princeton, U.S.A. (1)
Introduction
If we embed an
( n - 1 ) - s p h e r e in an n-sphere, the complement consists of two
components. Our problem is to describe the components more exactly.
For n = 2, there is a clas
Ryan Walch
Abstract Project
The Properties of Free Groups
1
Introduction
Combinatorial Group theory is the study of free groups and group presentations. Group
presentations are a representation of a group as a set of relations on a generating set. Combina
GROUP REPRESENTATIONS AND CHARACTER THEORY
DAVID KANG
Abstract. In this paper, we provide an introduction to the representation
theory of finite groups. We begin by defining representations, G-linear maps,
and other essential concepts before moving quickl
Free groups
Contents
1 Free groups
1.1 Definitions and notations . . . . . . . . . .
1.2 Construction of a free group with basis X
1.3 The universal property of free groups. . . .
1.4 Presentations of groups . . . . . . . . . . .
1.5 Rank of free groups .
26
TOM SANDERS
a projection on X, and so W is complemented in X. It follows that there is a subspace
U X such that X W U and hence
X W U `p U p`p `p q U `p p`p U q `p X.
On the other hand `p X V and so
`p `p p`p q `p pX V q
`p pXq `p pV q
pX `p pXqq `p p
HW14 solutions
10.8-1 (a) The easiest way to solve this problem is to transform it to the standard Dirichlet problem
for a rectangle, the solution of which we already know (see Section 10.8, pp. 660661).
Lets write this standard problem in terms of new va
HW12 solutions
10.6-12 (a) Mathematically, the problem is
2 uxx = ut , ux (0, t) = 0 = ux (L, t), u(x, 0) = sin( x).
L
Use the PDE method of separation of variables to write
2 X 00 T = XT 0
T0
X 00
= 2 = ,
X
T
which yields the two ordinary differential
HW13 solutions
10.7-12 (a) The 1D wave equation is given by
a2
2u
2u
=
.
x2
t2
Using the chain rule for partial derivatives, we change the variable x to s = x/L :
u
u ds
1 u
=
=
x
s dx
L s
It follows that
1 2u
2u
=
.
x2
L2 s2
Hence, in terms of the new va
HW11 solutions
10.5-3 Seek a solution of form u(x, t) = X(x)T (t) and the equation becomes
uxx + uxt + ut = X 00 T + X 0 T 0 + XT 0 = 0 X 00 T = T 0 (X 0 + X)
X 00
T0
=
.
X0 + X
T
For the last equality to hold, the two sides must equal to a constant, say
MATH 2930 : HW10 SOLUTIONS
Chapter 10.3 (page 612)
(10.3.1). (a) The function is assumed with period 2L = 2.
Z
Z 0
Z 1
1 L
f (x)dx =
(1)dx +
(1)dx = 0,
a0 =
L L
1
0
and for n > 0,
1
an =
L
bn =
1
L
Z
Z
L
nx
f (x) cos
dx =
L
L
L
f (x) sin
L
nx
dx =
L
Z
Z
MATH 2930 : HW8 SOLUTIONS
Chapter 4.2 (page 234)
(4.2.29). The characteristic equation is r3 + r = 0. We can factor out an r from this equation to
get r(r2 + 1) = 0. In this form, we see that the roots are r = 0, i
So the general solution is of the form y
HW2 solutions
t
2.1-16 y 0 + 2t = cos
is of the formR y 0 + p(t)y = g(t), so if we multiply the equation by the
t2
2
integrating factor (t) = e t dt = e2 ln |t| = t2 , we have t2 y 0 + 2ty = cos t = (t2 y)0 .
Integrate both sides and we obtain t2 y = sin
Diy Qs Section 5 Problems
Peter Tseng
March 1, 2010
9
We notice that the given equation x2 y 5xy + 9y = 0 (x > 0) ts the form
of the homogeneous second-order linear ODE y + p(x)y + q(x)y = 0 by rst
9
5
dividing by x2 , giving us p(x) = x , q(x) = x2 . As
HW 38 Solutions
May 3, 2013
14.(a) Find the solution of Laplaces equation in the rectangle 0 < x < a
and 0 < y < b that satises the boundary conditions
ux (0, y) = 0, ux (a, y) = 0, 0 < y < b
u(x, 0) = 0, u(x, b) = g(x), 0 x a.
Solution: The general form
MATH 2930 - Spring 2010
Worksheet 9
MATH 2930 - Dierential Equations - Spring 2010
Worksheet 9 - Select Solutions
1. (E&P 8.2, #24)
Use the power series method (i.e., assume solution y(x) =
order ODE
(x2 1)y + 2xy + 2xy = 0
cn xn ) to solve the 2nd-
Apart
MATH 2930 - Spring 2010
Worksheet 10
MATH 2930 - Dierential Equations - Spring 2010
Worksheet 10 - Select Solutions
For each of the following periodic functions f (t) whose value in one complete period is
given, nd its Fourier series.
1. f (t) = t for 0 <