Pick's Theorem—What's the Big Deal?
Pick's Theorem—What's the Big Deal?
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匹克定理——有什么大不了的？
DOI：
10.1080/00029890.2003.11919935
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
John E. McCarthy
中科院分区：
文献类型：
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作者：
John E. McCarthy
当然，找到下确界在计算上比回答下确界是否小于或等于某个固定值的问题更难，但在数学上，这些问题是等价的。通过一个标准的紧性论证（一个复杂的分析家称之为正规族论证），（IP）中的下确界将被获得。通过重新缩放，我们可以假设该下确界为1，那么Pick矩阵（2）就不能严格正定（因为如果是的话，如果所有wi都增加了某个略大于1的因子，那么它仍然是这样，这意味着第一部分
Of course, finding the infimum is computationally harder than answering the question of whether the infimum is less than or equal to some fixed value, but mathematically the problems are equivalent. By a standard compactness argument (what a complex analyst would call a normal family argument), the infimum in (IP) will be attained. By rescaling, we can assume this infimum is 1, and then the Pick matrix (2) cannot be strictly positive definite (for if it were, it would remain so if all the wi s were increased by some factor slightly bigger than one, which would mean by the first part