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A geometric problem and the Hopf Lemma. I

Yan Yan Li;  Louis Nirenberg — 2006

Journal of the European Mathematical Society

A classical result of A. D. Alexandrov states that a connected compact smooth n -dimensional manifold without boundary, embedded in ℝ n + 1 , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X n + 1 = const in case M satisfies: for any two points ( X ' , X n + 1 ) , ( X ' , X ^ n + 1 ) on M , with X n + 1 > X ^ n + 1 , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for n = 1 . Some variations...

Adaptive finite-time synchronization of cross-strict feedback hyperchaotic systems with parameter uncertainties

Hai-Yan Li;  Yun-An Hu;  Rui-Qi Wang — 2013

Kybernetika

This paper is concerned with the finite-time synchronization problem for a class of cross-strict feedback underactuated hyperchaotic systems. Using finite-time control and backstepping control approaches, a new robust adaptive synchronization scheme is proposed to make the synchronization errors of the systems with parameter uncertainties zero in a finite time. Appropriate adaptive laws are derived to deal with the unknown parameters of the systems. The proposed method can be applied to a variety...

Bigraphic pairs with a realization containing a split bipartite-graph

Jian Hua Yin;  Jia-Yun Li;  Jin-Zhi Du;  Hai-Yan Li — 2019

Czechoslovak Mathematical Journal

Let K s , t be the complete bipartite graph with partite sets { x 1 , ... , x s } and { y 1 , ... , y t } . A split bipartite-graph on ( s + s ' ) + ( t + t ' ) vertices, denoted by SB s + s ' , t + t ' , is the graph obtained from K s , t by adding s ' + t ' new vertices x s + 1 , ... , x s + s ' , y t + 1 , ... , y t + t ' such that each of x s + 1 , ... , x s + s ' is adjacent to each of y 1 , ... , y t and each of y t + 1 , ... , y t + t ' is adjacent to each of x 1 , ... , x s . Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n , respectively. The pair ( A ; B ) is potentially SB s + s ' , t + t ' -bigraphic if there is a simple bipartite graph containing SB s + s ' , t + t ' (with s + s ' vertices x 1 , ... , x s + s ' in the part of size m and t + t ' vertices...

Linear Congruence Relation and Complete Residue Systems

Xiquan Liang;  Li Yan;  Junjie Zhao — 2007

Formalized Mathematics

In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

Gauss Lemma and Law of Quadratic Reciprocity

Li Yan;  Xiquan Liang;  Junjie Zhao — 2008

Formalized Mathematics

In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems [14], [12]. Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven.MML identifier: INT 5, version: 7.8.05 4.89.993

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