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A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$

My question is: is there an analogue of this notation for ratios?

In other words, what's the best symbol to use for $[?]$ in $[?]x = \dfrac{x_1}{x_0}$?

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    @Henry if you'd like some context, I'm using this to express changes in a product in terms of changes of its individual terms. The reason I'm using ratios is that the product of ratios is the ratio of the products.2011-06-11

2 Answers 2

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Not entirely standard, but in Peter Henrici's discussion of the (justly famous) quotient-difference (QD) algorithm in the books Elements of Numerical Analysis (see p. 163) and Essentials of Numerical Analysis (see p. 155), he defines the quotient operator as

$Q\,x_n=\frac{x_{n+1}}{x_n}$

in complete analogy with the (forward) difference operator $\Delta$.

Henrici's a pretty sharp mathematician, so I wouldn't mind borrowing notation from him if I were in your shoes...


Here's a screenshot of the relevant page of the first book (sorry, I don't have a digital copy of the other book):

Henrici

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The best symbol to use is $\exp\Delta\log$: $\exp\Delta\log x = \exp(\log x_1-\log x_0) = \frac{x_1}{x_0}.$ The point is that this operation isn't "qualitatively different" from $\Delta$, so it may be reduced to $\Delta$. So far, I haven't used any new symbols but if you want some multiplicative new creative symbols, see e-percentages and units of evidence:

http://motls.blogspot.com/2010/01/exponential-percentages-useful-proposed.html
http://motls.blogspot.com/2010/08/units-of-evidence.html

There is no compact symbol for $\exp\Delta\log$. If you want an ally who would endorse the idea to introduce such a symbol, you may count on me. What about $\Delta^\times$?

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    Thank you for the input, I was hoping there was something more compact than $\exp \Delta \log$ and while $\Delta^{\times}$ has some appeal it may confuse some readers because of the association of $\Delta$ with differences. I think I might just end up switching all the expressions to log-space or coming up with some other symbol (perhaps an unambiguous placement of $\div$?)2011-06-12