Timeline for Non-real constants
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Oct 16, 2023 at 10:57 | comment | added | Anixx | @JesseElliott another possible constant, an uncountable one (but we can express it as integrals $\alpha=\int_0^1\omega dx=\int_0^\infty\ln \omega dx$, the later equality in itself is fascinating): math.stackexchange.com/questions/4787181/… | |
| Oct 7, 2023 at 3:06 | comment | added | Anixx | @JesseElliott in surreal numbers, $\ln \omega=\omega^{1/\omega}$. In divergent integrals, $\ln \omega=-\lambda-\gamma$, a different thing. The germ at infinity of the function $x^{1/x}$ is infinitely close to 1. | |
| Oct 7, 2023 at 3:02 | comment | added | Jesse Elliott | Here is a great reference. The potential connection between Hardy fields and surreal numbers is via transseries. texmacs.org/joris/ln/ln.html | |
| Oct 7, 2023 at 2:47 | comment | added | Anixx | @JesseElliott one more link: math.stackexchange.com/a/4556566/2513 I also was thinking about uncountable numerocities math.stackexchange.com/a/4770070/2513 but this last post I think, has errors in proposed integration rules in the end. Nevertheless, it introduces one more constant, $\alpha$ which cannot be expressed via $\omega$ in closed form. | |
| Oct 7, 2023 at 2:39 | comment | added | Jesse Elliott | I think I saw somewhere that the connection (which may still be conjectural) is deeper than one might naively think. | |
| Oct 7, 2023 at 2:32 | comment | added | Anixx | @JesseElliott well, I think, surreal $\omega$ is the same as $\omega$ in this post except the operations on surreal numbers are defined differently. So, the connection is there but not too much. | |
| Oct 7, 2023 at 2:26 | comment | added | Anixx | @JesseElliott some links: mathenchant.wordpress.com/2018/09/16/a-new-game-with-infinity math.stackexchange.com/questions/4552603/… math.stackexchange.com/questions/4627326/… | |
| Oct 7, 2023 at 2:23 | comment | added | Anixx | @JesseElliott well, 1) it allows to represent divergent integrals and series (and not only divergent at infinity but also, via Laplace transform, divergent at a point) as such numbers. 2) Extending reals with logarithm of zero, etc 3) All monotonous sequences have limit in this extended field (a kind of compactification) 4) representing numerocities of sets (counting measures of sets), even of infinite ones. | |
| Oct 7, 2023 at 2:20 | comment | added | Jesse Elliott | I guess there are connections to surreal numbers. | |
| Oct 7, 2023 at 2:19 | comment | added | Jesse Elliott | I mean, what advantages does one gain by thinking of germs of Hardian functions as "numbers" of a sort? | |
| Oct 7, 2023 at 2:17 | comment | added | Anixx | @JesseElliott I think, one can look up formal power series (evaluated at infinity). It is great if you are writing a book on this! I would be glad to contribut and otherwise help! | |
| Oct 7, 2023 at 2:15 | comment | added | Jesse Elliott | I'm going to have to give this some further thought. I'm finishing up writing a book, a chapter of which is on Hardy fields, but I hadn't thought of this particular perspective. Is it written up anywhere? | |
| Oct 7, 2023 at 1:48 | history | edited | Anixx | CC BY-SA 4.0 |
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| Oct 7, 2023 at 1:37 | comment | added | Anixx | @JesseElliott basically if $f(x)$ has germ at infinity (satisfies Hardy fields requirements), then the germ of this function at infinity is $f(\omega)$, in other words (up to infinitesimals), $f(\omega)=f(a)+\int_a^\infty f'(x)dx$ | |
| Oct 7, 2023 at 1:34 | history | edited | Anixx | CC BY-SA 4.0 |
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| Oct 7, 2023 at 1:26 | comment | added | Jesse Elliott | This seems neat but is mostly all new to me. Can one somehow bootstrap these "infinite" constants into (divergent) asymptotic expansions? | |
| Oct 6, 2023 at 22:55 | history | edited | Anixx | CC BY-SA 4.0 |
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| S Oct 6, 2023 at 22:35 | history | answered | Anixx | CC BY-SA 4.0 | |
| S Oct 6, 2023 at 22:35 | history | made wiki | Post Made Community Wiki by Anixx |