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is there any formula can calculate it directly?


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We originally planed to drive to Miao-Li(苗栗) appreciating the tung flower (油桐花). But the distance is a little far for us and there are many tung tree in Taipei also. Therefore we change our journey that we first appreciate tung flowers in Pin-Hsi(平溪) and turn to the northeast coast to look for the "wild lily" (野百合).


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Last week, I tried to solve the problem by "edge isoperimetric inequality". But today I found it failed. It really depressed me...

I originally can't understand the proof of lower bound of diagnosability in hypercubes illustrated in the latest paper. So I buried my nose in calculating the relationship of degrees and size of connected component in hypercubes. After a week passing by, when I finally found out an great inequality of this two things, I found the result is exactly the same of the proof in that paper. Oh my godness! what a big round I circled these days. I would like to cry so much...

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It is a wonderful website for our wonderful experience of greek tour last year (2003). If you like the atmosphere of Mediterranean or you are planning to have a trip there, we strongly recommend you to visit this website.


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counting the boundary and internal edges of a connected component of hypercubes

If we obtain two values which represent the number of vertices and edges in hypercubes respectively, do we can determine the size of all possible connected components? For example, there are totlal 16 vertices and 32 edges in a 4-dimensional hypercube. Now we assume 10 vertices and 13 edges are remained in this hypercube. What are all possible cases?

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在作研究的過程中,我們時常要想一些證明來支持我們的理論,但是大部分的些證明不是都那麼的直覺簡單,有些甚至找不到證明的方法,也就成了open problem。我最近正好遭遇到這樣的問題,想了將近兩個星期,一點頭緒都沒有,中研院資訊所的馬自恆老師於是建議我使用probabilistic method來試試。在許多離散數學的應用中probabilistic method 是個很有用的證明工具(雖然我才剛剛接觸),是由大師Paul Erdos將其發揚光大,其中證明Ramsey Theory是一個非常典型的例子(詳細的內容我們先不去管他),但是根據先前學者們的描述,要找出Ramsey number R(m,n) 的lower bound是件"不太容易"的事,不過光是說"不太容易"這個形容詞無法讓一般人真正體會瞭解這些問題的困難,於是大師們便會在這個時候發揮幽默作一個淺顯的比喻:

如果有一天外星人攻擊地球,因為武力科技實力懸殊的關係,地球人先決定議和,不過外星人開了個條件,如果地球人能達到的話外星人就會撤軍。如果這個條件是希望地球人能在一個星期內求出 R(5,5) 的lower bond的話,那麼地球人應該盡快召集全人類的菁英並使用最快的電腦一起來破解這個問題。但若是條件改成 R(6,6) 的話,嘿嘿... 地球人應該考慮的是如何準備跟外星人好好打一場仗囉...

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The "probabilistic method" is one of the most powerful and modern tools in combinatorics. The basic idea of this method is simple and elegant: In order to prove the existence of an object with a specific property, prove that a randomly chosen object satisfies that property with positive probability.


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