Wikipedia:Reference desk/Archives/Mathematics/2022 April 3

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April 3

Generic name for a smallish factor?

I want to define something like the following (over real numbers): $x$ is not significantly greater than $y$ (with respect to positive real numbers $\Delta$ and $\mathrm {I}$) if $x\leq (\mathrm {I} y+\Delta )$. I think I'm happy with the concept, but I'm not happy with $\mathrm {I}$. Is there a name commonly used for smallish factors? Also: Is this something that already has been defined as a standard concept and I'm just reinventing the wheel? --Stephan Schulz (talk) 21:54, 3 April 2022 (UTC)

The notation confused me for a second; my first impression was that $\mathrm {I}$ and $\Delta$ stood for matrices, with $\mathrm {I}$ the identity matrix. In mathematical texts the term factor usually refers to a multiplicative factor in a product, but $x$ is not involved in a multiplication. Now to the question itself. I cannot quite grasp when you'd use this concept. When $\mathrm {I} =99,\Delta =1,$ $100$ is not significantly greater than $1$, but $101$ is? The context in which this is used might give some inspiration for a name for the relation. Is it in some way good when $x$ is smallish (it will fit), or rather bad (insufficient)? The (only loosely defined) relation $x\gg y$ is vocalized as "$x$ is much greater than $y$", so its (hardly used) negation $x\not \gg y$ should become "$x$ is not much greater than $y$". In any case, avoid the term significantly, which also has a special meaning and might be confusing.  --Lambiam 09:46, 4 April 2022 (UTC)

Sure. My use case is measuring the performance of a system on a number of test cases, where the system can either succeed, with a concrete CPU time used, or time out completely. I want to be able to determine if one parameterisation subsumes another, i.e. succeed on at least the same problems, and not significantly slower. The distribution of success times is somewhat weird - to a useful approximation, half the successes happen in the first second, the next quarter in the second second, the next eights in the third second, and so on. Also, there is a large number of cases that are simple for many parameterisations, but some are simple for only a few (or none). CPU time measurement is noisy, so I want to cut the potentially subsuming parameter set some slack - I'd consider a problem "solved in the same time or better" even if the time is slightly bigger - 0.01 seconds vs. 0.02 seconds is not significant (so the Delta to capture small absolute differences), and I think 100 seconds vs. 102 seconds is also not significant (so the Iota, to capture small relative differences). --Stephan Schulz (talk) 10:11, 4 April 2022 (UTC)

The distribution sounds like the familiar exponential distribution. So Delta Iota would be slightly larger than 1, no? In mathematical formulations, $\varepsilon$ and $\delta$ are commonly used variable names for small quantities, so an idiomatic mathematical formulation might be $x\leq (1{+}\varepsilon )y+\delta ,$ where $\varepsilon$ and $\delta$ are positive but small. You could then say that "$x$ is not decisively larger than $y$".  --Lambiam 10:39, 4 April 2022 (UTC)

Perfect, thanks a lot. Also formulating the factor as $1{+}\varepsilon$ makes me much happier than my own original idea! --Stephan Schulz (talk) 10:51, 4 April 2022 (UTC)

The usual theoretical language for things like this is Big O notation (which may or may not suit your particular purposes) and its variants. -- JBL (talk) 15:09, 4 April 2022 (UTC)

Yes, I was thinking about that, but it did not quite fit. My constraint are more or less the other way round (I only look at a finite segment of time, and only allow for a maximal deviation). --Stephan Schulz (talk) 15:55, 4 April 2022 (UTC)

Do you want little-o notation? 2602:24A:DE47:B8E0:1B43:29FD:A863:33CA (talk) 10:59, 7 April 2022 (UTC)

Big O and little o are about the behaviour of functions in the limit, usually as the argument of the function goes to infinity. The situation here has nothing to do with limits.  --Lambiam 13:22, 7 April 2022 (UTC)