Iterated Linear Maps in the Plane

Iterated Linear Maps in the Plane.   That sounds complex, doesn’t it?   In fact, it’s something we are already very familiar with.  At the beginning of the semester we investigated repeated iteration of a single linear equation $f(x)= ax+b$.   This time we are investigating the repeated iteration of a matrix, which is a nice way to write a system of equations.

We can take the matrix $A$= $\left( \begin{array}{cc} a & b \\ c & d \\\end{array} \right)$

and initial values $x_0, y_0$ placed into a vector $x_n$ = $\left( \begin{array}{c} x_0 \\ y_0 \end{array} \right)$

We then investigate repeated iterations of $Ax_n = x_{n+1}$

Analysis of the convergence of these matrices  sees a connection to the direction of the matrix and a special number called an $eigenvalue (\lambda)$ and its corresponding $eigenvector$



Parametric Curves-Luke

The lab I have been looking at is on Parametric Curves. These are where you have two variables and a function that graphs them. But instead of using one function in terms of x and/or y:

$y=f(x)$ or $z=f(x,y)$,

parametric curves give the values for x and y as a function of t. So:

$x=x(t)$ and $y=y(t)$

This way of representing curves can be very useful with circles and polar coordinates. The questions that they ask deal mostly with parameters of $sin(at)$ and $cos(bt)$. The first few questions start exploring functions and seeing what they do with different values, how they intersect, etc. And then forming conjectures based on the data.

Sarah and Sequences and Series

When we were doing the lab on iteration, there was a proof that we encountered that had heavy use of some of the theorems involving infinite series. While there was a chapter on this in my Calculus II class, I think it would be interesting to explore more about series in the context of a more proof-based class. I think this will also be a nice addendum to the first lab on iteration, because we are going to be looking in much more detail into the question of general convergence and divergence. It doesn’t look like we’re going to be doing a lot of strict proving in this lab, but instead we are gong to given the opportunity to explore series in more detail and really be able to explore some of the facets that may have been glossed over in a busy Calculus class. We will be exploring convergence, divergence, the harmonic series, the natural logarithm and Euler’s constant. I’ve never encountered Euler’s constant outside of some proofs that I saw in other labs.

Carrie, Chapter 6: Randomized Response Surveys

A randomized response survey is a method used in research to help researchers find the answers to controversial questions while keeping the participants’ identity anonymous. The theory is that if the participants know that there is no risk involved because of the anonymity, they are more likely to answer honestly, therefore producing a more reliable outcome. Researchers found that when participants are asked a direct question, they are less likely to answer honestly and the results of the studies were skewed because of this. So S. L. Warner first proposed the idea of randomized responses in 1965 and B. G. Greenberg modified the method in 1969.

Human psychology, as I have said in my earlier post on randomized response, is very interesting to me. I appreciate the difficulty in conducting an accurate survey in order to study human behavior. In this lab we are asked to explore different aspects in randomized response such as the margin of error and probabilities. I think that it would be incredibly interesting to conduct our own survey using the randomized response method and use that to study the different aspects of the method that the lab talks about. I would also just like to better understand the process of conducting a randomized response survey in order to be able to utilize it in the future.

Experiments in Periodicity

When we wake up tomorrow, we know that the night has just past and day is creeping upon us. We know this because such event takes place everyday like a broken record. Plays over and over again. Observing such a phenomenon from a mathematical perspective, we say that we are looking at periodic functions.

As we can all remember from Trigonometry, the sine and cosine functions oscillate in a orderly fashion when we look at it graphically. By orderly fashion, I mean to say that we know the trend of the oscillation after observing it for a short period of time. We can also combine them, take the derivative, and integrate such functions to make a whole new function.

For this lab, we observe such periodic functions by calculating the area under the functions. In other words, we evaluate the definite integrals.

Iteration of Quadratic Functions

We have already looked at iterations of linear functions, so a natural extension would be to look at quadratic functions in the form $f(x)=ax(x-1)$.  There are some familiar questions, such as what happens when $a$ is varied with a fixed $x_0$ and vice versa.

There is a little twist, though.  Iteration around a fixed may cause nearby points to converge towards that fixed point.  That point is then called an attractor.  Nearby points could also diverge away from the fixed point.  If so, the point is then called a repeller.  Sometimes there is no pattern at all and can look quite chaotic……..

Chaos will be looked at and that is the major draw for me to consider this chapter.  Let’s have some random fun!!!

Iterated Linear Maps in the Plane

The next lab that has caught my attention and would like to explore further is the second to last chapter in our book: Iterated Linear Maps in the Plane. I like this one because, although very similar to our first lab, is very graphical and includes matrix operations.

Before, we were iterating the function $f(x) = ax+b$. This sort of iteration is called an affine map. In this lab we will be doing linear maps or maps where the constant term $b = 0$. Further, because we are in the plane, our function will have vector valued inputs and will have vector-valued output. So our function will look something like:

$$ f(x, y) = (a_{11}x + a_{12}y, a_{21}x + a_{22}y) $$

Or similarly, we can write our equation:
x_{n+1} \\{}
f(x_n, y_n)
x_n \\{}

where $ A = \left({}
a_{11} & a_{12} \\{}
a_{21} & a_{22}
\right){} $.

Some questions this lab seeks to answer are similar to our first lab: it will ask us to try some different variations of our matrix $A$ and/ or our initial values and see if we can notice a pattern.

Randomized Response: Jordan

This subject is interesting to me for a few reasons. One is the fact that this is a fairly new concept only being about 50 years old and with an estimation method only being about 30 years old. With these newer concepts I believe that there is a lot more to study. Another reason why this subject is appealing to me is that it is a way to gather usable data without invading a persons privacy. For example, if a survey was to go out with questions such as “Have you used illegal drugs?” it is a possibility that some may not answer honestly. This is where the randomized response can help. It can give a more accurate answers for sensitive questions such as those.

Marc on Randomized Response

My name is Marc and I have an interest in randomized response which is a technique developed by Stanley Warner in 1965 that is used to obtain accurate results on surveys that ask delicate questions. For example, if you want to ask people in a classroom if they are sexually active, you can create a decoy question involving the flipping of a coin. The more delicate question can then be asked in tangent with this decoy question and nobody really knows if the participants are answering yes to the decoy question or the real question. This is done in order to make people feel more comfortable about their privacy. A program is used in this lab that simulates a randomized response survey.

One of the questions involves finding a function for a particular survey. By using a proportion concerning the number of yes answers versus the number of total answers, it is asked to find an equation that can help in estimating what the true number of yeses should be to the real question. Further questions build on determining how to uncover accurate results using the probabilities in possible answers from the real question and the decoy question.

I think this is a very useful lab for statistical analysis because you want people to be honest when gathering data but you simply cannot rely on honesty. Finding an accurate way to collect data while helping others to feel comfortable in being honest helps in gathering accurate data.