控制论的数论部分

IF 1.4 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys Pub Date : 2022-01-01 DOI:10.1070/RM10050

Aleksandr Iosifovich Ovseevich

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引用次数: 0

摘要

Q的重要性主要是由于它为两个稳定矩阵M = - diag(1,2，…)定义了一个公共Lyapunov函数。， N)和A + BC，其中对于i = 1, Aei =−iei+1，…， N, B = e1，且C =−(1/2)B * Q。这里是ei, i = 1，…， N，构成R和eN+1 = 0的标准基。[6]中证明了Q是一个偶数矩阵，即Qij∈2Z，并推测了矩阵的所有元素都可以被N(N + 1)整除。[6]中的证明是基于考虑正交多项式的。这里我们用类似的方法证明了这个猜想。

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A number-theoretic part of control theory

The importance of Q is primarily due to the fact that it defines a common Lyapunov function for the two stable matrices M = −diag(1, 2, . . . , N) and A + BC, where Aei = −iei+1 for i = 1, . . . , N , B = e1, and C = −(1/2)B∗Q. Here the ei, i = 1, . . . , N , form the standard basis of R and eN+1 = 0. In [6] it was shown that Q is an even integer matrix, that is, Qij ∈ 2Z, and it was conjectured that all elements of the matrix are divisible by N(N + 1). The proofs in [6] were based on considering orthogonal polynomials. Here we prove this conjecture using similar methods.

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来源期刊

Russian Mathematical Surveys 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Mathematical Surveys is a high-prestige journal covering a wide area of mathematics. The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. The survey articles on current trends in mathematics are generally written by leading experts in the field at the request of the Editorial Board.