Usefulness of symbolic devices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:25:20Z http://mathoverflow.net/feeds/question/75306 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75306/usefulness-of-symbolic-devices Usefulness of symbolic devices Johann Cigler 2011-09-13T12:37:39Z 2011-09-13T16:06:32Z <p>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.</p> http://mathoverflow.net/questions/75306/usefulness-of-symbolic-devices/75309#75309 Answer by maproom for Usefulness of symbolic devices maproom 2011-09-13T13:15:24Z 2011-09-13T14:44:59Z <p>Group conjugation: $a^b$ means $b^{-1}ab$, so $(a^b)^c$ is $a^{bc}$ and $(a^b)^{-1}$ is $(a^{-1})^b$.</p> http://mathoverflow.net/questions/75306/usefulness-of-symbolic-devices/75310#75310 Answer by Michael Kissner for Usefulness of symbolic devices Michael Kissner 2011-09-13T13:39:10Z 2011-09-13T13:39:10Z <p>I really like the multi-index notation for derivatives which is quite often found in the theory of PDEs.</p> <p>$D^\alpha f = \frac{\partial^{\vert\alpha\vert} f}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}$</p> <p>Where $\alpha$ is defined as:</p> <p>$\alpha = (\alpha_1,\cdots,\alpha_n),$ $\vert \alpha\vert = \alpha_1 + \cdots +\alpha_n$</p> <p>It makes definining Sobolev Spaces (and their corresponding Norms) so much cleaner and straight forward:</p> <p>$W^{k,p} = \lbrace f\in L^p : D^\alpha f\in L^p \text{ for all } \vert\alpha\vert \le k \rbrace$</p> http://mathoverflow.net/questions/75306/usefulness-of-symbolic-devices/75322#75322 Answer by Gerhard Paseman for Usefulness of symbolic devices Gerhard Paseman 2011-09-13T15:42:02Z 2011-09-13T16:06:32Z <p>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.</p> <p>Gerhard "Ask Me About System Design" Paseman, 2011.09.13</p>