Convergence of iterates of a nonlinear operator arising in statistical mechanics
Convergence of iterates of a nonlinear operator arising in statistical mechanics
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统计力学中非线性算子的迭代收敛
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发表时间：
2002
期刊：
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通讯作者：
Nussbaumt
中科院分区：
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作者：
Nussbaumt
A b d.在对无限原子链的研究中，每个原子通过“弹簧”与其最近的相邻原子连接，整个原子链位于周期势场中，一些作者被引导研究非线性算子（F.x）(s) = minlo(s， I) + x (r)]和有限维版本TA: E？+ W '（这里D（S， I）是一个给定的函数a是一个给定的n x n矩阵）在运筹学的问题中也会出现算子TA。如果T = TA在R ‘ ’中有一个点火点，那么我们就有兴趣讨论迭代T* (T)的收敛性问题。我们将证明T* (T)的收敛性，即T* (T)的收敛性。l G j 4 n)使得C:，，，！~nand使得p=Icm（S!）S中整数的最小公倍数，则Iimk-Tkn(x) = &)存在于每一个x E R”。更进一步，对于每一个x ew '存在k，使得T*(x) = E(x)对于所有k 3k，。如果T(0) = 0，我们将根据a的简单性质给出p的显式公式。我们的结果是最好的，并且证明了R B Griffiths在回答笔者的一个问题时所作猜想的一个改进版本。
A b d. In studies of an infinite chain of atoms, each atom connected by a 'spring' to its nearest neighbours and the whole chain lying in a periodic potential field, several authors have been led to study the nonlinear operator (F.x)(s) = minlo(s, I) + x (r) ] and a finite dimensional version TA : E?" + W" of a given by (Here D(S, I) is a given function and A is a given n x n matrix). The operator TA also arises in questions from operations research. If T = TA has a fired point in R", it is of interest to ask about convergence properties of iterates T* of T. We shall prove that t h e r e i s a s e t S z { j ! l G j 4 n) such that C:,,!~nandsuchthatifp=Icm(S!. theleast common multiple of the integers in S, then Iimk-Tkn(x) = &) exists for everyx E R". Furthermore, for each x E W" there exists k, such that T*(x) = E(x) for all k 3 k,. If T(0) = 0, we shall provide explicit formulas forp in terms of simple properties of A. Our results are best possible and prove a sharpened version of a conjecture which was made by R B Griffiths in response to a question by this author.