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THE MATHWRIGHT LIBRARY NEWSLETTER, January 2003, VOL
5, #1
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
In this issue:
Linear Algebra Trilogy: Three Views of Linear Equations
Interactive Web Article: Periodic Points of Some Discrete Dynamical Systems
Three
Views of Linear Equations
by Dan Kalman, The American
University
This
new set of three Microworlds is a web presentation of three interpretations
of a linear system: as simultaneous scalar equations, as a single vector equation,
and as a matrix-vector equation. Visit the Microworld on-line in your browser
with MathwrightWeb, or download each one to your own machine so that you can
view it whenever you like off-line.
For
each view there is an interactive visualization exercise, embedded in a webpage
which explains the context for the exercise. The first exercise is illustrated
below.
In
this exercise, two equations in two unknowns are defined by user input (typed
into the colored boxes). A button click with the mouse produces the graphs
of the equations. The object is to verify that a point in the plane satisfies
one of the equations just when that point lies on the corresponding line.
Of course, the solution to the pair of equations is the intersection of the
lines. The user selects test points by clicking the graph with the mouse,
and the results of testing each point in the equations is shown numerically
in the yellow text window.
The second exercise looks like this:
This
time the system is cast into a vector form. Here, the unknowns are viewed
as scalar coefficients for two specific constant vectors, and the goal is
to choose these coefficients so as to produce a vector result equal to a third
given constant vector. The three constant vectors are entered by the user
in the red, blue, and yellow boxes. A button click shows a graphical representation
of the situation, with the three vectors depicted as red, blue, and yellow
line segments. The user can stretch or shrink the red and blue vectors with
the mouse, in effect selecting values for the unknown coefficients. The goal
is to make the pale yellow resultant vector coincide with the bright yellow
constant vector.
This is the third exercise:
Now
the system is expressed as a matrix-vector equation. Here the matrix is made
up of fixed numerical constants (entered by the user). The unknowns make up
one vector, and a second vector is defined by specific numerical constants
(also entered by the user). The object is to find a variable vector which,
when multiplied by the matrix, produces the specified constant vector. On
the graph, the variable vector appears as a red line, its product with the
matrix is shown as a blue line, and the desired outcome is shown as a yellow
line. The interaction involves dragging the red line with the mouse, while
watching the blue line move in response. The goal is move the red line in
such a way that the blue line exactly matches the yellow line.
Interactive
Web Article:
Periodic Points of Some Discrete Dynamical Systems
by Samad Mortabit and
Juan Estrada, Metropolitan State University
Interested
in Dynamical Systems and Chaos? Do you want to see some surprising relations
between Dynamical Systems and Number Theory? Print out the short Article from
your browser, and then open the Microworld to do the experiments indicated
therein. The free article (written using Design Science MathPage™) that
accompanies this Microworld may be read here.
One
of the manifestations of chaotic behavior in discrete dynamical systems is
the existence of a dense set of periodic points. In this short paper and in
the context of some specific classes of one-dimensional discrete dynamical
systems, we address the following question: given a periodic point, what is
its period? To that end, we establish a general result and then proceed with
its refinement in various ways. One of the most attractive aspects of this
work is the connection we made to modern algebra and number theory. In addition,
interactive explorations are included to visualize and illustrate the main
ideas.
Below
is a snapshot from the article. The actual article may be printed directly
from the browser, so that you may read along as you experiment with the principal
ideas.
