# Football of World Cup

The match between Portugal and Brazil yesterday night inspired me something.

I see the “Four Colors Problem” there!

And, YES. I heard Cristiano Ronaldo asked the people there why at most 4 colors are needed to color the subsurfaces of football such that any two neighboring subsurfaces are colored differently.

“Why not more than 4 colors? What is the proof?”, Ronaldo asked the FIFA officer.

A little history – Augustus De Morgan, a professor in University College, London first discussed (gossip?) the four colors problem with his friend Sir William Rowan Hamilton in 1852. Since then many proofs have been proposed but there isn’t any proof that explains the theorem beautifully (my believe).

By the way, the match between Portugal and Brazil yesterday is 0-0. That’s why he wanted to make the football colorful. (** Cristiano Ronaldo part is just a joke **)

# 1+1 is not 2?

Let $a + b = c$ be the basic equation.

Then, both sides multiply with $(a+b)$. $(a + b)^2 = c (a+b)$. $a^2 + 2ab +b^2 = ac + bc$.

Move $ab$ from left to right; and $ac$ and $b^2$ from right to left. $a^2 + ab - ac = bc - b^2 -ab$.

Factoring … $a (a+b-c) = -b (a + b -c)$

Removing $(a+b-c)$
And we have: $a = -b$.

Oops. It contradicts with our basic equation.

Obviously something is wrong here.

Guess what is the mistake here? It is a primary school’s question in fact 🙂

# Why Mathematicians Keep Proving?

I came across a nice saying from a book named Linear Algebra: Ideas and Applications (3rd Edition) by Richard C. Penney, pg 13.
There is a fundamental difference between mathematics and science.

Science is founded on experimentation.
If certain principles (such as Newton’s laws of motion) seem to be valid every time experiments are done to verify them, they are accepted as ‘law’.
However, they will still remain a law only as long as they agree with experimental evidence.
Thus, Newton’s laws were eventually replaced by the theory of relativity when they were found to conflict with the experiments of Michelson and Morley.

Mathematics, on the other hand, is based on proof.
No matter how many times some mathematical principle is observed to hold, we will not accept it as a theorem until we can produce a logical argument that shows the principle can never be violated.”

That’s why I like mathematician.

# Taylor Series

Came to the Taylor series while I was studying the fundamental element of chaos theory.

So what is Taylor series?

Taylor series is a representation of a function as an infinite sum of terms, calculated from the values of its derivative at a single point ［Wikipedia］.

Let’s look at the definition.

Given a function $f(x)$ and $f(x)$ is differentiable infinitely at a point $a$ of a real or complex domain. Then, we say that, for a value $x$ near $a$, $f(x)$ can be approximated with Taylor series. That is, $\displaystyle \sum^{\infty}_{n=0} \frac{f^{(n)}(a) }{n !} (x-a)^n$.

where $n!$ denotes the factorial of $n$ and $f^{(n)}(a)$ denotes the $n$th derivatives of $f$ at point $a$.

#

OK. Let’s see for an example.

We all know that the derivative of $e^x$ is also $e^x$. In other words, $e^x$ can be differentiated infinitely.

Every algebra has a value of 1 when power with 0, except $0^0$ which is undefined. So, how about $e^{0.001}$ ?

Of cause we can use calculator to find out the value. But, let’s try with Taylor series method, i.e. $f(0.001) = \frac{f^{(0)}(0)}{0!}(0.001)^0 + \frac{f^{(1)}(0)}{1!}(0.001)^1 + \cdots + \frac{f^{(n)}(0)}{n!}(0.001)^n$

#

# Chaos Control over Logistic Map (One-Dimensional Chaos Control)

A work on the chaos control over the low-period orbit of logistic map at [0.3737, 0.8894], with OGY method.

The first two graphs (delta=0.000) are the logistic maps without OGY control; the trajectories of last two graphs (delta=0.001) are tampered with OGY method.

As you may see, I am exploiting the low period orbit at around t=267, which cause them to fluctuate at [0.3737, 0.8894] (cf. the third and fourth graphs).

Again. These graphs are generated with Python and Matplotlib.

# Attractor Reconstruction of Lorenz Attractor

Attractor reconstruction of Lorenz attractor with Python.

# Breaking WIFI WEP

Just found that WEP can be decrypted within 30mins (depends on how fast the sniffer works). It seems like it is time to change to WPA encryption 😦

15-March-2011: If you WIFI card has been patched (by some means), you can break WEP WIFI within 10 seconds.

# Lorenz Attractor

My salutation to Edward Lorenz, the founder of Chaos Theory. These graphs are generated through Python and SciPy. Try dumping the following codes into the IPython and have fun changing the view of the Lorenz attactor.

The following is my Python source code.

# IEEE 802.15.5 – Standard for both High-Rate and Low-Rate Wireless Personal Area Networks (WPAN)

There are two famous IEEE standards for mesh technology – IEEE 802.15.3x for high-rate WPAN and IEEE 802.15.4x for low-rate WPAN. Question is always arisen on the interoperability between these two standards. To answer that question, IEEE has introduced a new standard, IEEE 802.15.5 that is built on  the mesh sublayer on top of both IEEE 802.15.3x and IEEE 802.15.4x (refer to the figure).

More discussion on such standard can be found in IEEE Communication magazine in .

Reference:
  M. Lee, R. Zhang, C. Zhu, T. R. Park, C. S. Shin, Y.A. Jeon, S. H. Lee, S. S. Choi, Y. Liu, S. W. Park, “Meshing Wireless Personal Area Networks: Introducing IEEE 802.15.5”, IEEE Communication Magazine, January 2010, page 54 – 61.

# Time Delay of An Underdamped Second Order System A student asked me how to find the time delay, $t_d$  for a second-order system by applying mathematical equation.

Let $c_{final}$ denote the final value of the waveform. Then, the rise time $t_r$ will be the time duration between $0.1c_{final}$ to $0.9c_{final}$.

Since the time delay $t_d$ equals to the time for the waveform to reach $0.5c_{final}$. My rough idea for the mathematical equation is: $t_d \approx 1.25 * t_r / 2$ .

Just an approximation as the waveform between $0.1c_{final}$ to $0.9c_{final}$ is not a linear line.