On the Application of Clifford's Graphs to Ordinary Binary Quantics
On the Application of Clifford's Graphs to Ordinary Binary Quantics
复制标题
论克利福德图在普通二元量子学中的应用
DOI：
10.1112/plms/s1-17.1.107
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发表时间：
1885
影响因子：
1.8
通讯作者：
A. Kempe
中科院分区：
文献类型：
--
作者：
A. Kempe
克利福教授死后，他的图论在一个重要的方面有缺陷；他没能证明它直接适用于普通物理量的情况。海量存储系统(MSS)中。他在这个问题上留下的理论都局限于二元量的情况，因此我将单独讨论它。由于已故的斯波蒂斯伍德先生的慷慨，他在我们的《论文集》第x卷第204页对这一理论作了极好的描述，这些MSS。以传真的形式复制，标题为“数学片段，是已故的克利福德未完成的关于图论的论文的传真”（伦敦：麦克米伦公司，1881），并将副本赠送给许多数学家，希望能填补这一空白。据我所知，迄今为止还没有人试图这样做。当斯波提斯伍德先生把《片段》拿给我看时，他知道数学形式的图形表示这一主题引起了我的注意，他表示希望我在某一时刻能在这方面做点什么。我一直忙于其他的研究，直到最近才有机会研究这个问题，但终于有了必要的机会。西尔维斯特教授在《美国数学杂志》第一卷第66页第14行发表的关于图的论文中的一段话，立刻给这个困难提供了线索。Clifford忽略了一个事实，他指出，任何协变/(xy)都可以被认为是f （xy）和xy - yx两个量的不变量，这样协变就合并为不变量，变量x， y失去了它们独特的特征，变成了单纯的系数。记住这一点，我们立刻就可以看出，在图的方法的符号中，一个普通的二进制量的适当的l-表示，以三次为例，不是在0符号A中找到的，而是在符号o»中找到的，其中线性形式ou在代数上是xi^ -y«2。事实上，回想一下MJMJ= - 1，同样的方程也适用于其他极性量子集
The theory of graphs, as it fell from the hands of Professor Clifford on his death, was deficient in one important particular; he had failed to show that it was directly applicable to the case of ordinary quantics. The MSS. which he left on the subject are confined to the case of binary quantics, with which alone I shall accordingly deal. By the liberality of the late Mr. Spottiswoode, who wrote an excellent account of the theory in our Proceedings, Vol. x., p. 204, these MSS. were reproduced in facsimile, under the title," MathematicalFragments, being Facsimiles of his Unfinished Papers relating to the Theory of Graphs, by the late WK Clifford"(London: Macmillan & Co. 1881), and copies were presented to many mathematicians in the hope that the gap might be filled up. As far as I am aware, no one has hitherto made even an attempt to do so. When presenting me with a copy of the" Fragments," Mr. Spottiswoode, knowing that the subject of the graphical representation of mathematical form was occupying my attention, expressed a wish that I should at some time do what I could in the matter. I have been too much engaged with other reseai* ches to be able to look into the question until quite recently, but have at length had the necessary opportunity. A passage in Professor Sylvester's paper on graphs, in the American Journal of Mathe'inatics, Vol. I., p. 66, line 14, at once afforded a clue to the difficulty. Clifford had overlooked the fact there pointed out, that any covariant/(xy) may be regarded as an invariant of the two quantics f (XY) and Xy—Yx, so that covariants become merged in invariants, and the variables x, y lose their distinctive chai'acter, becoming mere coefficients. Bearing this in mind, it is at once seen that, in the notation of the method of graphs, the proper l-epresentation of an ordinary binary quantic is, taking the cubic as an example, not to be found in the o symbol A, but in the symbol o» where the linear form ou is algebraically xi^—y «2. In fact, recollecting that MJMJ=—1, and that the same equation holds in the case of the other sets of polar quanti-