Author Archives: Ken Coiteux

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!!!

Absolute values on Q (Part II)

Continuing on from absolute values on Q part I/ , I am going to show that the definition of p-adic absolute value satisfies the three properties of absolute values.

Property 1

Let $x=0$ and p be a fixed prime.  By definition then, $|x|_p = 0$.  This satisfies the first property ($|x|_p=0$ if and only if $x=0$).

Suppose now $x \ne 0$.  Then $|x|_p = p^{-ord_p(x)}$.  Using Conjecture 1, $|x|_p=|p^n \cdot a|_p =p^{-n}$.  Since $p^{-n}$ cannot be 0, $p^{-n} > 0$.

Property 2

Let $x=0$ and $y \ne 0$ and p be a fixed prime.  Then $|x \cdot y|_p = |0|_p = 0 = |x|_p \cdot |y|_p$.

Suppose $|x|_p = |p^n \cdot a|$ and $|y|_p = |p^n \cdot b|$.  In other words, x and y have the same exponent on p. Then, by definition,

$|x \cdot y|_p = |p^n \cdot a \cdot p^n \cdot b|_p = |p^{2n} \cdot ab|_p = p^{-2n}$.

By the second property,

$|x \cdot y|_p = |x|_p \cdot |y|_p = |p^n c\dot a|_p \cdot |p^n \cdot b|_p = p^{-n}p{-n}=p{-2n}$

Suppose x and y have different exponents on the p.  $|x|_p=|p^n \cdot a|_p and |y|_p=|p^m cdot b|$.  Multiply x by y….

$|xy|_p=|p^n \cdot a \cdot p^m \cdot b|_p=|p^{n+m} \cdot ab|_p=p^{-n-m}$

By the second property,

$|xy|_p=|x|_p \cdot |y|_p = p^{-n} \cdot p^{-m}=p^{-n-m}$.

Property 3

Let $x=y=0$ and p be a fixed prime.  Then by definition $|x+y|_p = 0$.  This partly satisfies the property for $0 \le 0+0$  Now consider $x \ne 0$.  Then $|x+y|_p = |x|_p = |p^n \cdot a|_p = p^{-n}$.    This contributes to satisfying the property for $p^{-n} \le p^{-n} + 0$.

Now let x and y have the same exponent on p.

$|x+y|_p = |p^n \cdot a + p^n \cdot b|_p = |p^n(a+b)|_p=p^{-n}$

If the definition satisfies the property, then $p^{-n}$ should be less than $|x|_p+|y|_p$.  In fact this is true because $|x|_p+|y|_p = p^{-n}+p^{-n}$ which is larger than $p^{-n}$.

Last case is when x and y have different exponents on p.

$|x+y|_p=|p^n \cdot a + p^m \cdot b|_p=p^{-n}$ where $n<m$

Now consider property 3.

$|x|_p+|y|_p=p^{-n}+p^{-m}$

Now that p-adic absolute values do satisfy the properties of absolute values, there is one more thing to look at:  Archimedean absolute values.  A little warning for you, the following definition is counter-intuitive.

Definition 2:  An absolute value on Q is said to be non-Archimedean if the properties of absolute values are satisfied along with an additional property:

$|x+y| \le max(|x|,|y|)$ for all x,y \in Q.

Absolute values that satisfy the three properties but not the fourth are said to be Archimedean.

Theorem 1:  p-adic absolute values are non-Archimedean.

It has already been shown that p-adic absolute values satisfy the first three conditions.  Now all that needs to be shown is that p-adic absolute values satisfy the fourth property.  Assume p is a fixed prime

Case 1: $x=y=0$

     $|x+y|_p = |0|_p = 0    \le        max(|x|_p,|y|_p) = 0$

 Case 2: $x \ne 0, y=0$

    $|x+y|_p = |x|_p = p^{-n}     \le     max(|x|_p,|y|_p) =  max(p^{-n}, 0) = p^{-n}$

Case 3:  x and y have the same exponent on p

     $|x+y|_p = p^{-n}             \le                max(|x|_p,|y|_p) = max(p^{-n}, p^{-n}) = p^{-n}$

Case 4:  x and y have different exponents on p (n<m)

    $|x+y|_p = p^{-n}              \le               max(|x|_p,|y|_p) = max(p^{-n}, p^{-m}) = p^{-m}$

QED

Absolute Values on Q (Part I)

Real numbers, Q, can be constructed from the rational numbers Q={a/b where a,b ϵ the integers Z}.  Construction of p-adic numbers, Qp, is done exactly the same way.

Let’s now consider absolute values.  An absolute value is a map from Q to [0,∞) that has the following properties for any x,y ϵ Q:

  1. |x| ≥ 0, and |x| = 0 if and only if x = 0,
  2. |x∙y| = |x|∙|y|,
  3. |x+y| ≤ |x|+|y| (The triangle inequality).

Now that we have some properties, we can look at types of absolute values that can be put on the rationals.  Alexander Ostrowski showed in 1935 that there are only three types.

Definition 1:  Let x be a rational number.

The trivial absolute value of x, denoted |x|0, is defined by

  \[ |x|_0 = \left\{
\begin{array}{l l}
1 & \quad \text{if $x\ne0$ }\\
0 & \quad \text{if $x=0$}
\end{array} \right.\]

The usual absolute value of x, denoted |x|, is defined by

\[ |x|_{\infty} = \left\{
\begin{array}{l l}
x & \quad \text{if $x\ge0$ }\\
-x & \quad \text{if $x\le0$}
\end{array} \right.\]

The p-adic absolute value of x, denoted |x|p, is defined for a given prime p by

\[ |x|_p = \left\{
\begin{array}{l l}
1/p^{ord_p(x)} & \quad \text{if $x\ne0$ }\\
0 & \quad \text{if $x=0$}
\end{array} \right.\]

 

The quantity ordp(x) is called the order of x.  It is the highest power of p that divides x.  For example, let p=5 and let x=75.  Then ord5(75)=2.  Since 25=52 and 25 is the highest power of 5 that divides 75, then the order of x is 2.  Confused yet?  Here are some examples of p-adic absolute values:

|75|5 = |52 ∙ 3|5 = 5-2

|10|5 = |51 ∙ 2|5 = 5-1

|13|5 = |50 ∙ 13|5 = 1

$|\frac{2}{75}|5 = |5-2 ∙ \frac{2}{3} |5 = 52$

|-375|5 = |53 ∙ -3|5 = 5-3

In these examples, we were looking at 5-adic absolute values.  We broke the number up into the form 5n ∙ a.  The 5-adic absolute value of 5n ∙ a is 5-n.

Conjecture 1:  Given a rational number x, the p-adic absolute value of x is p-n, where pn divides x.

|x|p = |pn ∙ a|p = p-n

In the introduction post (http://math.boisestate.edu/m287/quick-intro-to-p-adics/), it was mentioned that large p-adic numbers are closer to 0 than smaller p-adic numbers.  If one was to think of absolute value as a measure of distance, notice in the above examples that the larger the exponent on p, the resulting absolute value (or distance from 0) gets smaller.

Quick Intro to p-adics

What are p-adic numbers?  They are a different set of numbers first introduced by Kurt Hensel in 1897.  The motivation at that time was an attempt to bring the ideas and methods of power series methods into number theory.  They have been used in proving Fermat’s Last Theorem and have other applications in number theory.  See http://mathworld.wolfram.com/p-adicNumber.html for more information.

A little terminology needs to be introduced.  The p in p-adic represents any prime number.  For each prime, there is a new and different set of p-adic numbers.  Q2 identifies the 2-adics, Q5 represents the 5-adics, Q17 represents the 17-adics.  To keep the same notation, Q will represent the real numbers.

Another term to consider is “close.”  The basic idea is that given a number n, it is close to 0 if it is highly divisible by a prime p.  Consider the numbers 25 and 625.  Relatively speaking, 625 has more factors of 5 than does 25, or in other words, 625 has a higher divisibility by 5.  Therefore 625 is closer to 0 than 25.  This idea will be made a little bit clearer in future postings.

http://math.boisestate.edu/m287/arithmetic-on-p-adics/

Iterations of Quadratic Functions

Iteration of linear functions was the first lab we covered.  Towards the end of the lab, I felt like this was a precursor to fractal geometry.  After all, the Mandelbrot set is based on an iteration of a quadratic function with the use of complex numbers.  Chapter 14 deals with iteration of quadratic functions.  This is one step closer to fractal geometry…..and a precursor to chaos theory, both of which are very interesting to me.

Early definitions in the chapter are:

  • fixed point:  given a function $f(x)$,  a point $u$ is a fixed point of  $f$ if $f(u)=u$.
  •  attractor:  a fixed point is an attractor when all nearby points move towards it under iteration.
  • repeller:  a fixed point is a repeller when all nearby points move away from it under iteration

Here’s a type of question from the chapter:

Given a function $f(x)=ax(x-1)$, how do various values of $a$ affect fixed points, attractors, repellers, and zeroes.  What about changing initial values?

Tous!!!

Tous.  Ken nee’eesih’inoo.  Neeyeiheihinoo heesou’sii’ii.

Hi!  My name is Ken.  I am studying math.

A couple things that interest me are fractals and cultural math.

http://en.wikipedia.org/wiki/Fractal

The equation Mandelbrot used to create his famous picture is $z_{n+1}=(z_n)^2+c$

http://www.amazon.com/Native-American-Mathematics-MIchael-Closs/dp/0292711859

This book mostly covers various number systems found among Native American tribes……one group had a base-8 number system!!!!