Happy New Year

The first day of 2009 is a raining and freezing cold day, but wish the year will be full of sunshine, happiness, and productive for myself and for you, the reader of this blog if there is any.
Twenty years ago (I cannot believe this long!!), I wrote my wishes for the new year at the end of a notebook and sealed the pages for years. Well, I have all the wishes fulfilled. For the past two years, I really made no much progress in my career. I guess I have not worked hard enough. Nevertheless, I have not lost faith in myself and my future and I believe I reach the dream I kept and the life ahead is going to be meaningful, colorful and worthy to pursue.
Here is a short list for my wishes in 2009.
1) Next stop, Paris, Berlin, London or Iceland.
2) Able to explore chemical space and get in touch with the first livable planet in it.
3) The first fund from CNSF.
4) Some meaningful advancement in understanding nano-device.
5) Well, I have more wishes but should not be too greedy. So leave them to the future.

Merry Christmas

Wish everyone a merry Christmas and a happy new year. No matter how hard has 2008 meant to you,the new year is full of wonders and waiting to be picked.

Background image, Scanning tunneling microscope (STM) image of the complex of hexa-para-tert-butylphenylbenzene-Cu2 on Cu(111) surface due to Dr. Leo Gross, IBM Research, Switzerland.

"Marriage Theorem" Revisted

The marriage theorem discussed in a preivous post has been established in Graph Theory. The following theorem appeared in the book titled Introduction to Graph Theroem by Gary Chartrand and Ping Zhang.

In a collection of r women and s men, where 1<=r<=s, a total of r marriages between acquainted couples is possible if and only if for each integer k with 1<=k<=r, every subset of k women is collectively acquainted with at least k men.

The theorem deals with the matching problem between two sets in Graph Theory. Anoother form of the theorem is as following,

Let G be a bipartite graph with partite sets U and W such that r=|U|<=
|W|. Then G contains a matching of cardinality r if and only if U is neighborly.

Neighborly is defined as following,

Let G be a bipartite graph with partite sets U and W such that |U|<=|W|. For a nonempty set X of U, the neighborhood N(x) of X is the union of the neighborhoods N(x), where x blelongs to X. Equivalently, N(x) consists of all those vertices of W that are neighbors of one or more vertices in X. The partite set U is said to be neighborly if |N(x)|>=|X| for every nonempty subset X of U.

In plain words, if there is a subset of U which colletively know fewer members of W than the number of themselves, it will be impossible to find a 1-to-1 match for U from W.