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Preliminaries about Maple
Notes for MAT 331
Mathematical Problem Solving with Computers
Summer 2002
Santiago R. Simanca & Scott Sutherland
The University at Stony Brook
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Preliminaries about Maple
Starting a Maple Session
Basic Maple
Maple for numeric calculations
Maple for symbolic manipulations
Graphing in Maple
The Maple worksheet
Introducing the worksheet
Worksheet basics: the front-end and the kernel
Online Maple help
Documenting and structuring your worksheet
Assignments, Functions and Constants
Assignment statements
Variables and subsequent assignments: unassignments
Functions known to Maple. How to define your own functions
Special constants and reserved names
The limit command
The diff (and Diff) command
Higher order derivatives
Implicit differentiation
The int (and Int) command
Symbolic integrals
Numerical integration
Approximations through Riemann Sums
Multiple integrals
The subs command
Why is subs useful?
Plotting with Maple
Plotting several functions (or curves) on the same axes
Fancier plotting
Exercises
If the Curve Fits, Wear It
Interpolation
Polynomial Interpolation
Connect-the-Dots and Splines
When the data is approximate
Fitting a line to data
Fitting a cubic to data
Fitting other types of funtions
Fitting a circle
Robust fitting
A nod toward statistics
The Art of Phugoid
The Phugoid model
What do solutions look like?
Existence of Solutions
Numerical Methods
Euler's method
Numerical Solutions, Numerical Integration, and Runge-Kutta
Seeing the flight path
Fixed Point Analysis
Linear Systems of ODEs
Fixed Points for the Glider
Qualitative Classification of Solutions
Dealing with the Singularity
Bibliography
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Introduction to Cryptography
Simple Ciphers
Simple substitution
The Caesar cipher, and the ASCII encoding
Treating characters as numbers
Render unto Caesar
Defining functions with proc; Local and global variables
Caesar cipher redux
Improved Caesar-like ciphers
The Vignère cipher
One-time pads
Multi-character alphabets
Reading and Writing from a file
Affine enciphering
When do affine encodings fail?
Implementing and using an affine encoding
Breaking an affine cipher
Enciphering matrices
Treating text as vectors
Affine encoding with matrices
A Known-plaintext attack on an affine matrix cipher
Modern cryptography
Secure cryptosystems
Message digests
Public Key cryptography
Some Number Theory
The greatest common divisor and the Euclidean algorithm
The Chinese Remainder Theorem
Powers modulo n
The Euler
-function and Euler's Theorem
The RSA Public key cryptosystem
Details of the RSA algorithm
Implementing RSA in Maple
Implementing the basics of RSA
Making it Useful
RSA encoding a file
Bibliography
A turtle in a fractal garden
Turtle Graphics
A fractal
Recursion and making a Koch Snowflake with Maple
Recursive functions.
A recursive procedure to generate
K
n
The Koch Snowflake
Some variations on the Koch curve
Making a tree
Fractal Dimension
Box counting dimension
Computing the box dimension of some examples
Cantor sets
The Sierpinski gasket
Inside the turtle's shell
Extending the turtle's commands
About this document ...
Translated from LaTeX by Scott Sutherland
2002-08-29
-function and Euler's Theorem