6-6-2008
Sums, Products, and Binomial Coecients
ai is used to
ai is used to denote a sum of terms; product notation
Summation notation
i
i
denote a product of terms.
If n is a positive integer, n-factorial is n! = 1 2 n. By convention, 0! = 1.
n
k
Bino
6-6-2008
The Ring of Integers
The integers Z satisfy the axioms for an algebraic structure called an integral domain.
The integers have an order relation <; they also satisfy the Well-Ordering Axiom, which is the
basis for induction and the Division Alg
6-16-2008
Congruences and Modular Arithmetic
a is congruent to b mod n means that n | a b. Notation: a = b (mod n).
Congruence mod n is an equivalence relation. Hence, congruences have many of the same properties
as ordinary equations.
Congruences prov
6-6-2008
Mathematical Induction
Induction is a method for proving an innite sequence of statements. In its easiest form, you start by
proving that the rst statement is true. Next, you show that if an arbitrary statement in the list is true,
then the next
6-14-2008
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic says that every integer greater than 1 can be factored
uniquely into a product of primes.
Euclids lemma says that if a prime divides a product of two numbers, it must
6-14-2008
Divisibility Tests and Factoring
There are simple tests for divisibility by small numbers such as 2, 3, 5, 7, and 9. These tests involve
performing operations on the decimal representation of the number to be tested for divisibility.
Fermat fa
6-13-2008
The Extended Euclidean Algorithm
The Extended Euclidean Algorithm nds integers a and b such that (m, n) = am + bn.
The backward recurrence is an implementation of the Extended Euclidean Algorithm. This implementation is well suited for hand co
6-6-2008
Divisibility
a | b means that a divides b that is, b is a multiple of a.
An integer n is prime if n > 1 and the only positive divisors of n are 1 and n. Prime numbers are
important in number theory and its applications.
The Division Algorithm
7-17-2008
Linear Diophantine Equations
A Diophantine equation is an equation which is to be solved over the integers.
A linear Diophantine equation of the form ax + by = c has solutions if and only if (a, b) | c. There is a
similar result for linear Dio
1-24-2013
Calendar Algorithms
Given a date in history, what day of the week was it?
First, some terminology. A sidereal year is the time is takes for the Earth to make 1 revolution around
the sun relative to the xed stars. It is approximately 365.25636 da
Geophysical Turbulence: Problem Set
1. Calculate the skewness v 3 / v 2
Gaussian random variable v.
3/2 ,
where v = v v , and kurtosis v 4 / v 2
2
of a
2. Given functions u = u( x, y), v = v( x, y) and w = w( x, y) that are either periodic or
vanishing at
function [dist,dispvec] = randwalk(N,d,showplot);
%
%
%
%
%
%
[DISTANCE,DISPLACEMENTVEC] = RANDWALK(N,d,showplot) Perform N
random steps of unit length in d-dimensions, where d is a positive
integer, beginning at origin. Returns displacement vectors
(size
Problem Set 2: Geophysical Fluid Dynamics
Due: Thursday, October 19, 2010
1. Vallis 2.2, 4, 7, 12, 14, 16, 17, 18, 19, 23
2. Starting from the zonal momentum equation on the sphere (without the shallow uid approximation), show that zonal angular momentum
Problem Set 1: Geophysical Fluid Dynamics
Due: Monday, September 27, 2010
1. Vallis problems 1.2, 1.6, 1.7, 1.9, 1.10, 1.15, 1.18, 1.20
2. Derive the equation of state for an ideal gas, p = RT , from rst principles.
3. Assuming the atmosphere is a dry ide
The code for the dynamics (MOORDYN.M and TOWDYN.M) solve for the positions
of each mooring and towed body element iteratively until the positions converge,
usually only a few iterations are required for sub-surface, weakly sheared
configurations. In a str
Specifically, the solution is obtained as follows. First the velocity (current) and
density profiles and wire/chain sections are interpolated to approximately one metre
vertical resolution using linear interpolation. Then the drag on each mooring element