Sunday, November 28, 2010

Thanksgiving Dinners

It is three years in a row that we had Thanksgiving dinner at a friend's house. Whether the dinner is in our house or other's house, it is usually a multi-family affair because for first generation immigrants like us, friends are our extended family. One advantage of having Thanksgiving dinner at a friend's house is to meet people out of our circle - friends' friends. This time I met a guy who is a manager of a factory, another guy who loves hunting!

It was a very typical Thanksgiving dinner. On top of traditional Thanksgiving food, turkey, cranberry sauce, mashed potatoes and gravy, sweet potato pie (in place of pumpkin pie), we had many many side dishes - salmon, chickens, pork rib, soup ..... all delicious. After dinner, the group naturally divided into three groups - wives were together in dinning room talking about kids, kids' education, religion ....; The husbands formed another group in kitchen area talking about politics, sports, and hunting - a rare topic to me...; Kids were glued together by the video games.

I have always enjoyed Thanksgiving dinner with family and friends. Over the years, two of the dinners are most memorable to me - my first ever thanksgiving dinner, our first Thanksgiving dinner after moving to Texas.

My first thanksgiving dinner was some 20 years ago, arranged by international student office at my school and hosted by a local family. It was a formal traditional Thanksgiving dinner. The host and hostess, along with their children and grand children, plus two international students sat by a huge dinning table. The host started the dinner with a prayer and giving thanks for a good year they had had. We had turkey, cranberry sauce, mashed potatoes and gravy, pumpkin pie and bread, with plates passing around just like what I saw in movies. It was a cold Thanksgiving evening, I was however warmed by the host family's kindness and generosity. How I wished then that I would one day celebrate Thanksgiving with my own family.

Several years ago, we moved to Texas, starting a new chapter in our life. As our first Thanksgiving at Texas was approaching, our old friends from our graduate school years, invited us to have Thanksgiving with them at their Oklahoma home. On the way to their home, I felt like I was on the way to visit a relative, to visit my brother and sister, just like millions other Americans on the road that day. The Thanksgiving dinner was a formal one. Their family including grandparents and ours, total of 10 people sat by the dinning table celebrating Thanksgiving. Since we had not seen each other after we graduated. it was also a reunion. We took many pictures during that visit. One photo I liked the most from the bunch is one that the two couples, each with their younger kid in laps, sat in a living room sofa. We looked young, healthy and content. Sadly, our hostess passed away earlier this year.

To me, Thanksgiving dinner symbolize "Family togetherness", "Enjoy our life" and "Be thankful for what we have and cherish them".

Be Thanksful and Give Thanks!

Saturday, November 20, 2010

Autumn, Harvest and Thanksgiving

Autumn comes to North Texas later than usual this year. However, when it comes, it is as good as always - crisp air, blue sky, and splendid foliage. Last Sunday morning, I walked around our blocks to capture the magic of the Autumn.

As I stepped out of the house, I was instantly refreshed by the crisp cool early morning air, and the bright sunshine. My feathered neighbors were busy collecting wigs, nuts, ... and they were chirping all around. The splendid foliage looked so much more brilliant under the clear blue sky.

This red foliage is very common in our neighborhood

The blue sky, and the bright sunshine made this bright yellow leafed bush so much more beautiful. They also made mixed colored brown-yellow leaves look great.

The tall grass with white blossom at a neighbor's backyard is a standout among the common foliage of red, yellow, brown and green.


red fruit

pine cone - seed


yellow fruit

white nut - seed

What really drew my attention this morning was the different kind of fruits or seeds on the trees this autumn. They really symbolize what the season is all about - a season of fruition, a season for harvest.

What a beautiful Autmn we have!

I am thankful to Mother Nature for this brilliant Autumn Sunday morning. I am thankful to my neighbors for their plants/bushes/trees that bring the splendor near to my home.

Let us cherish and protect the planet we rely on, the community we live in.

Saturday, November 13, 2010

Fermat's Last Theorem - The Odyessy

It took 360 years for many generations of great mathematicians to finally prove Fermat's last theorem. The theorem itself has no practical significance even in a mathematical sense. During the struggle to prove the theorem, many new techniques were developed, new theorems were proved, in particular the Taniyama-Shimura Conjecture was proved which has tremendous implications in mathematics.

Here is the timeline of major milestones in the establishment and proof of the Fermat's last theorem according to the book "Fermat's Enigma" by Simon Singh.

600BC Pythagorean theorem

1637 Fermat's last theorem proposed

1753 First breakthrough in proving Fermat's last theorem by
Leonhard Euler for n=3

1825 Using a method proposed by Sophie Germain, Gustav Dirichlet
and A. Legndre proved Fermat's theorem for n=5

1839 Gabriel Lame proved the theorem for n=7

1847 Spectacular failures - Proof of the Fermat's last theorem
proposed by G. Lame and A. L. Cauchy was shown to be wrong by Ernst Kummer

1908 Paul Wolfskehl found a mistake in Kummer's paper and corrected it. He created Wolfskehl Prize for the first person who proves the Fermat's last theorem

1955 Taniyama-Shimura Conjecture proposed by Japanese mathematician Yutaka Taniyama and Goro Shimura about the relation between modular form and elliptic equation

1984 Gehard Frey showed that if Taniyama-Shimura Conjecture can be proved, then Fermat's last theorem is automatically proved

1986 Andrew Wiles started working on proving Taniyama-Shimura Conjecture for purpose of proving Fermat's last theorem

1988 The claimed proof by Yoichi Miyaoka was shown to be wrong

1993 Andrew Wiles announced his proof of Fermat's last theorem
1993 An error was found in Wiles' proof by referees

1994 Error fixed by Andrew Wiles and Richard Taylor

1995 The proof published in Annals of Mathematics

1997 Andrew Wiles was awarded Wolfskehl Prize

Saturday, November 6, 2010

Fermat's Last Theorem - Numbers

The history of number theory decribed in the book "Fermat's Enigma" provided additional excitment to the reading. It described how common number concepts we learn from elementary school to high school, such as prime number, negative number, irrational number, imaginary number, and the concept of infinity, were developed hundreds and thousands of years ago, along with the evolving story of proving Fermat's last theorem. (Note: Fermat's last theorem: The equation x^n+y^n=z^n (*) has no integer solution if n>2. n is an integer)
After the proof for n=3 by Euler, there was no progress in the proof for nearly 75 years until a woman mathematician Sohpie Germain proposed an approach for certain type of prime numbers, such that p and 2p+1 are both primes. For example 2,3,5 are such primes, 7 is not.

Prime Numbers
Prime numbers are those which can be divided only by itself and 1, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, ...7919 (the 1000th prime number) ...., as one can see from the list, there are much less prime numbers than whole numbers. The Fermat's last theorem seemed to be not that daunting any more once ancient mathematicians realized that if they could prove the Fermat's theorem for prime numbers then they proved the theorem because all numbers can be factorized into prime numbers.
Unfortunately there are still infinite prime numbers!! The proof was initially given by the Greek mathematician Euclid of Alexandria, i.e. Euclid. The proof is remarkably easy to understand. Here is the excerpt of the proof.
Assume there are only finite number of prime numbers, p1, p2,, then we can construct a new number q
q=p1 x p2 x p3 x ...x pn +1.
If q is not a prime number, then it must be factorized by prime numbers, bu it can not be factorized by existing prime numbers due to its construction. q must be either factorized by new prime numbers not in the known finite list, or q itself is a prime number. So there are infinite prime numbers!
The method of the proof is called proof by contradiction. It is a powerful method.
Perfect Numbers

Almost opposite to prime numbers, the perfect number is a number which equals to the sum of all of its divisors excluding itself, e.g. 6=1+2+3. The first 4 perfect numbers are
6 = 1+2+3
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
8128 = 1+ 2+4 +8+16+32+64+127+254+508+1016+2032+4064
There are remarkable less perfect numbers than prime numbers. It is very hard to find perfect numbers, up to 2009, only 47 perfect numbers were found, the 47th perfect number has 25956377 digits! (more than 25 million) - here is the current list
Pythagoras and followers studied perfect number extensively and found many interesting properties they posses, such as all perfect numbers can be written as the sum of consecutive counting numbers, e.g. 8128=1+2+3+4+...+127. Another properties they posses is that all perfect numbers can also be written as 2^n * (2^(n+1)-1), 8129= 2^6 (2^7 -1).
Perfect numbers are perfect in themselves. One claim went as far as saying: "6 is a number perfect in itself, and not because God created all things in 6 days; rather the inverse is true; God created all things in 6 days because this number is perfect."

Finding perfect number is hard, verification of a large perfect number is not easy either. I used excel to verify 8128, it would take a computer program with a smart algorithm to check on the very large perfect numbers. Verification of reasonably large perfect numbers in Excel would be good mental exercises.
Mathematical Revolution due to Power of Logic

Typical thinking is that if a person is logic in nature, he is less inventive because being logic means you follow existing rules. This is not true in the world of math, the the revolutionary concepts in the history of math came from exact logic and deep thinking.

In mathematics, there is need for completeness, which simply says that for a given rule of calculation, however you apply it, you should get a results. The concept of negative number was developed by Hindus because of the need of completeness. Say 5-3=2 was natural for ancient people, what about 3-5? which can not be expressed as a natural number (1,2, 3, 4, ...) - this need of completeness and logic in math led to the development of negative number concept!

Similar story occurred to the discovery of irrational number in ancient Greece. They believed that all numbers were rational. When people pondered what a number would be square root of 2. It was found that no rational number p/q could represent square root of 2, logic demanded the expansion of math concept to irrational number, a revolutionary concept at the time, which caused the life of its first discoverer. The development of the concepts of zero and imaginary number happened in striking similar fashion. Learning how ancient great mathematicians developed the math concepts we take for granted is really exciting. To replicate what they did is fun for amateurs like me. I learned the concept of irrational number in high school, but I never asked why. Now I can easily prove that square root of 2 is irrational using the method proof by contradiction. - it is exhilarating!