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Ellipsis

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Last week the following trivial fact became more apparent to me: It's hard to not write something. And that's not related to me being overly wordy here but rather in the sense of it being harder to write something that's not supposed to be there than to write it.

In everyday language it's quite easy. We may use the ellipsis or an apostrophe to indicate some gap has to be filled here – in the spirit of is left as an easy exercise to the reader

In maths things may become a bit more annoying as you don't want to leave things to the reader's imagination and need to be precise about what has to be left out. Suppose you have n objects $\varphi_1\otimes\ldots\otimes\varphi_n$ and want to consider the (tensor) product of all but the k-th of them. The straightforward way to write it is this: $\varphi_1\otimes\ldots\otimes\varphi_{k-1}\otimes\varphi_{k+1}\otimes\ldots\otimes\varphi_n$ Of course that's quite a pain to write, writing your way around what's missing. Also you have to look at the indices carefully when reading to see what's going on. Thus, people devised other methods of denoting what's missing: $\varphi_1\otimes\ldots\otimes\widehat{\varphi_k}\otimes\ldots\otimes\varphi_n$ with the 'hat' denoting the omission. Another notation I saw recently is even shorter $\varphi_1\otimes\ldots\otimes\stackrel{k}{\vee}\otimes\ldots\otimes\varphi_n$

Usually you actually want to sum over these elements with varying k. And maybe even want to do this twice, with two elements missing in the k-th and l-th place. Writing that down carefully and precisely can be quite a pain as you'd have to take into account k could be large or smaller than l and that they can't be equal. Mostly people just end up writing something like $\sum_{k,l} \varphi_1\otimes\ldots\otimes\widehat{\varphi_k}\otimes\ldots\otimes\widehat{\varphi_l}\otimes\ldots\varphi_n$ which ignores all those notational details, leaving it to the reader to know what's written there.

May 4, 2003, 14:51

Tagged as uni.

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