Each mathematician knows that good notation or symbolism – which seems to be irrelevant from a purely logical point of view – makes theorems more plausible and motivates results which would otherwise be overlooked. Examples abound. Let me only mention such diverse things as decimal notation, $B^A$ for the set of mappings from $A$ to $B$, $\frac{{df}}{{dx}}$ for differentiation, commutative diagrams, umbral calculus, etc. I would be interested in a list of examples of the most useful notations or symbolic devices together with hints for the reason of their usefulness.
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8A related MO question http://mathoverflow.net/questions/42929/suggestions-for-good-notation – Andrey Rekalo Sep 13 '11 at 12:43
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2Is the other MO question essentially a duplicate or is there a different focus here? – Qiaochu Yuan Sep 13 '11 at 16:15
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Group conjugation: $a^b$ means $b^{-1}ab$, so $(a^b)^c$ is $a^{bc}$ and $(a^b)^{-1}$ is $(a^{-1})^b$.
maproom
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I really like the multi-index notation for derivatives which is quite often found in the theory of PDEs.
$D^\alpha f = \frac{\partial^{\vert\alpha\vert} f}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}$
Where $\alpha$ is defined as:
$\alpha = (\alpha_1,\cdots,\alpha_n),$ $\vert \alpha\vert = \alpha_1 + \cdots +\alpha_n$
It makes definining Sobolev Spaces (and their corresponding Norms) so much cleaner and straight forward:
$W^{k,p} = \lbrace f\in L^p : D^\alpha f\in L^p \text{ for all } \vert\alpha\vert \le k \rbrace$
Michael Kissner
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I was going to mention the differential operator D used in ODEs. So, for example, $Dy=\frac{d}{dx}y$. I originally thought it was a useless notation, until I realized that it greatly simplified things. For example, $(D^4+4D^3+6D^2+4D+5)y=0$ is difficult to solve until you express it as $(D^2+1)(D^2+4D+5)y=0$. – JRN Sep 14 '11 at 02:39
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For certain problems, suppression of a common entity or entry and making it understood by other means. For a lecture involving manipulation of some algebra (ternary groups, maybe?) a form similar to matrix multiplication was used. Instead of spelling out the matrices with all the entries, those entries that were zero were omitted, giving a more eye-friendly appearence. I (and many others, I'm sure) use it for small incidence matrices which are not very dense; the symmetry and other relationships seem much more obvious without the clutter of zeroes. And this is just one example of many.
Gerhard "Ask Me About System Design" Paseman, 2011.09.13
Gerhard Paseman
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