Terminologie of y1/y0 [closed]

Question

If I set a list of points: {2,3,5}

"Derivative" will be {3-2=1, 5-3=2}

How about {3/2=1.5, 5/3~=1.666}. What is the name of this ?

Answer 1

To nuance and slightly correct @Mark S.'s answer, there is more depth to your question than it seems.

In fact, you can truly consider the sequence of differences of a sequence (the sequence defined by the "difference equation") to be what we should call the "discrete derivative". In this context, recurrent sequences are actually 'discrete differential equations' and series are 'discrete integrals', $2^x$ is the discrete equivalent of $e^x$, etc. All of these operations are linked to the operations of the arithmetic mean.

However, there also exist a bunch of derivatives and integrals linked to the operations of the geometric mean. This is generally referred to as "non-newtonian calculus". See https://en.wikipedia.org/wiki/Product_integral or https://www.calculushowto.com/non-newtonian-calculus/ for example. In this context, the name you are looking for is "discrete multiplicative derivative", corresponding to $u^{[1]}_n = \frac{u_{n+1}}{u_n}$. I have personally encountered uses of it in statistics, in some methods of analysis of time series (though the authors did not seem to know about non-newtonian calculus in general).

You can also take a look at this very interesting table: https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi

And here, for an article which makes use of these techniques: https://d1wqtxts1xzle7.cloudfront.net/46356975/khatami.pdf?1465464788=&response-content-disposition=inline%3B+filename%3DAn_Analytical_Method_for_Some_Nonlinear.pdf&Expires=1612625066&Signature=c~tREtS5~4f3yRRWrVyRQeLpnq46rgFfgCYQk2DFssvCNB13YcOlvgqB-8gxXfkCsWfpOqQJL2mYwZpThx9-qgurW9vlZR39jSBhasmZdJQJcOUmtvBaO4o4i6W0yBpkEpeACrOiNaUPMz3ECJWQ~5Mi2AuthIXHKgQUiv-MSUjwj9GDE6S~aXkHRuXAFRiW4gJvyU1~8~ynH3umlwRpvYUoc67lIamDquyupmsgcHzYihCXOrtwgo8DHMja4kaozRKvHLpIXeDO0QKQAfI3KOGEZE9rAt1WjdJYqERWhC6~uQh0Z66dOizd1my6nfjmr96V8QQoxtQhEJh5exY5UQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA

Answer 2

In math, we usually use curly braces to denote an unordered set, so that $\{2,3,5\}$ is not a list. But $(2,3,5)$ would be fine for a list/"finite sequence".

$(1,2)$ is not the "derivative", but "the (sequence of) (first) differences (between adjacent terms)". (That said, there are connections between finite differences and derivatives.)

I don't recall seeing a calculation like $(\frac32,\frac53)$ before. But I might call it "the list of quotients" if I could easily give an example to explain, or something like "the sequence of quotients of adjacent terms" if I wanted to be very precise in words alone.