Notes on Random Graphs: The purpose of these notes is to explain
what is meant by Paul Erdos result that any two random graphs are isomorphic. These notes are structured in such a way that we avoid talking
about randomness and probability until the last s
Leibnizs Formula: Below Ill derive the series expansion
(1)n
arctan(x) =
n=0
x2n+1
;
2n + 1
0 x 1.
(1)
Plugging the equation = 4 arctan(1) into Equation 1 gives Leibnizs famous
formula for , namely
=
4 4 4 4 4
+ +
1 3 5 7 9
(2)
This series has a special
Multiplication and the Fast Fourier Transform
Rich Schwartz
October 22, 2012
The purpose of these notes is to describe how to do multiplication quickly,
using the fast Fourier transform. As usual, nothing in these notes is original
to me.
1
The Discrete F
Dehns Dissection Theorem
Rich Schwartz
February 1, 2009
1
The Result
A polyhedron is a solid body whose boundary is a nite union of polygons,
called faces. We require that any two faces are either disjoint, or share a
common edge, or share a common vertex
Notes on Rayleighs Principle
Rich Schwartz
March 16, 2012
Here is a proof of the Short-Cut Principle in electric networks. The
principle is also known as Reyleighs Principle. The proof is contained in
Doyle and Snells book, Random Walks and Electric Netwo
Notes on Link Universality. Rich Schwartz: In his 1991 paper, Ramsey Theorems for Knots, Links, and Spatial Graphs, Seiya Negami proved
a beautiful theorem about linearly embedded complete graphs. These notes
give a more straightforward proof. Up to the l
Notes on NP Completeness
Rich Schwartz
November 10, 2013
1
Overview
Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas
Wests graph theory book, and also from wikipedi
Notes on Weierstrass Uniformization
Rich Schwartz
October 6, 2013
1
Introduction
In the Spring of 2011, I taught Math 1540 at Brown. This is the second
semester of our undergraduate algebra sequence. As a portion of the class,
I taught about elliptic curv
Notes on Fourier Series and Modular Forms
Rich Schwartz
August 19, 2011
This is a meandering set of notes about modular forms, theta series,
elliptic curves, and Fourier expansions. I learned most of the statements of
the results from wikipedia, but many
Math 153 Final Assignment: Prof. Schwartz
Instructions: Do as many or as few problems as you like. Either turn it
in or not. This doesnt count towards your grade. These problems are not
indicative of the nature of the nal exam for the class.
1. Let R be t
The purpose of these notes is to prove Lindemanns Theorem. The proof
is adapted from Jacobsons book Algebra I , but I simplied some of the
assumptions in order to make the proof easier. Also, I improved the proof
somewhat.
1
The Main Result
Here is Lindem
Math 153: Sample Proofs
Rich Schwartz
December 23, 2013
Here is a result which is pretty obvious.
Lemma 0.1 If A and B and C are all sets, then
(A B) C = (A C) (B C).
Proof: There are two halves to this result. First we must show that
A B) C (A C) (B C)
a