51–60 of 125 posts
It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators. Interestingly, the abs (absolute value) function is non-elementary. I wonde…
It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators. Interestingly, the abs (absolute value) function is non-elementary. I wonde…
abs(0)
= f(0) ; by defn
= exp(1/2 log 0) ; by defn
= exp(-∞/2) ; log 0 rule
= exp(-∞) ; extended real arith
= 0 ; exp(-∞) rule
https://en.wikipedia.org/wiki/Template:Mathematical_expressi...
The original article explicitly acknowledged this limitation, that while in "the classical differential-algebraic setting, one often works with a broader notion of elementary function, defined relative to a chosen field of constants and allowing algebraic adjunctions, i.e., adjoining roots of polynomial equations," the author works with the less general definition. Neither the present article, nor the original one ha…
Many things that in retrospect seem immediately obvious weren't obvious before, let alone immediately obvious.
Earlier quoted context omitted.
Introduction to ...
That's code for 101.
See e.g. Petzold, et al.
Earlier quoted context omitted.
The author discusses this in his third paragraph, and states explicitly in his fourth that he considers the result faulty for its unrealistically narrow definition of elementarity. (I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)
Well the author saysin that paragraph: > In layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates. But is there actually a combination of NANDs that find the roots of an arbitrary quintic? I always thought the answer was no but admittedly this is above my math level.
Earlier quoted context omitted.
That's code for 101.
No. It's code for the thickest, densest book on the subject that you're ever gonna not read, as it actually assumes you're experienced in the subject and goes into everything except intro level topics. See e.g. Petzold, et al.
> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold. > Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them. If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I've actually just lea…
I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?
Definitions are either a bit fuzzy, or not universally agreed on.
Though interestingly https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.
The original article explicitly acknowledged this limitation, that while in "the classical differential-algebraic setting, one often works with a broader notion of elementary function, defined relative to a chosen field of constants and allowing algebraic adjunctions, i.e., adjoining roots of polynomial equations," the author works with the less general definition. Neither the present article, nor the original one ha…
Maybe. But I found it a nice piece of recreational mathematics nevertheless.