Abstract:
In order to promote peaceful development in cross-Strait relations, this article proposes that both sides of the Taiwan Strait sign a “Basic Agreement on Peaceful Cross-Strait Development” – a temporary agreement (modus vivendi) to determine political relations and future development across the Strait. Three major points should be included in this agreement: first, both sides of the Taiwan Strait belong to one “Whole China” and both sides have no intention to separate from this “Whole China”; furthermore, both sides pledge not to split the “Whole China”, but to work in unison to maintain the territorial integrity and sovereignty of the “Whole China”; second, both sides of the Taiwan Strait share constitutionally guaranteed equal relations, and normal relations across the Strait will develop on the basis of this constitutional equality; and third, both sides decide to establish communities in areas of common agreement in order to promote mutually cooperative relations.

Abstract:
The cross-strait relations are an important factor of influencing cross-strait economic development and social progress. To build peaceful and stable cross-strait relations is the expectations of both sides as well as an important work for both governments. The cross-strait peace agreement is very critical in the process of the development of cross-strait relations. The two sides are to start negotiations and signing of peace agreement, which is not only beneficial to the two sides, but also helpful to regional and world peace. This essay is some thinking about the development of cross-strait relations based on cross-strait peace agreement. Key words: Peace agreement; Cross-strait Relations; Chinese Communist Party; Chinese Kuomintang (KMT)

Abstract:
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizations of the above two theorems. We prove the Chung-Feller theorems of Dyck type for these three classes of lattice paths and the Chung-Feller theorems of Motzkin type for two of these three classes. From the obtained results, we find an interesting fact that many lattice paths have the Chung-Feller properties of both Dyck type and Motzkin type.

Abstract:
In this paper, we shall prove the Chung-Feller Theorem in several ways. We provide an inductive proof, bijective proof, and proofs using generating functions, and the Cycle Lemma of Dvoretzky and Motzkin.

Abstract:
Because of the implementation of global strategy by Soviet Union and the expansion of regional hegemonism of Vietnam, some Cambodian issues came into being at the end of 70s in the Twentieth Century. As a regional great nation, China positively supported the war of anti-Vietnam and self-defense of Cambodia and united the other permanent member states of the United Nations Security Council as well as the Association of Southeast Asian Nations to facilitate the peaceful settlement of Cambodian issues, and so maintained the peace and security of Asia.

Abstract:
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths. Mohanty's book [5] devotes an entire section to exploring Chung-Feller theorem. Many Chung-Feller theorems are consequences of the results in [5]. In this paper, we consider the $(n,m)$-lattice paths. We study two parameters for an $(n,m)$-lattice path: the non-positive length and the rightmost minimum length. We obtain the Chung-Feller theorems of the $(n,m)$-lattice path on these two parameters by bijection methods. We are more interested in the pointed $(n,m)$-lattice paths. We investigate two parameters for an pointed $(n,m)$-lattice path: the pointed non-positive length and the pointed rightmost minimum length. We generalize the results in [5]. Using the main results in this paper, we may find the Chung-Feller theorems of many different lattice paths.

Abstract:
This paper gives sufficent and necessary conditions on a kind of limit results to hold on the precise convergent rate of an infinite series of probabilities on the Chung type law of the iterated logarithm.

Abstract:
“Liu Hui’s Commentary of the Chiu Chang Suan Shu” is the best commentary for the most important basic book in Chinese ancient mathematics, the Chiu Chang Suan Shu, and it contains large numbers of logical reasoning and logical ideas. Abundant logical ideas and logical contents in Chinese ancient mathematics were contained in Liu Hui’s Commentary. Based on analyzing the items in Liu Hui’s Commentary and relative ancient literatures, abundant logic contents in the Commentary and the logical contents in ancient mathematics were demonstrated from three angles including concept, logical method and symbolic logic in the article. The relations between calculation and proof, arithmetic and logic in ancient mathematics were also expounded in the article, and the conclusions showed that there was proof logic or arithmetic logic existing in the ancient times of China.

Abstract:
Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=2X_n+b_n (mod p) where X_0=0, p is odd, and b_n for n=0,1,2,... are i.i.d. random variables on {-1,0,1}. If Pr(b_n=-1)= Pr(b_n=1)=\beta and Pr(b_n=0)=1-2\beta, they asked which value of \beta makes X_n get close to uniformly distributed on the integers mod p the slowest. In this paper, we extend the results of Chung, Diaconis, and Graham in the case p=2^t-1 to show that for 0<\beta<=1/2, there is no such value of \beta.