Timeline for Sigmoid functions in a particular function set
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2017 at 22:38 | review | Close votes | |||
| Feb 10, 2017 at 3:22 | |||||
| Feb 9, 2017 at 22:29 | comment | added | Arthur B | The one at the very beginning: corrected a mistake in the definition of the set of function I'm interested in. The one about entire functions: it seemed like an interesting lead for a little bit, but actually there are plenty of entire functions which are sigmoids. | |
| Feb 9, 2017 at 22:16 | comment | added | Christian Remling | This version of the question doesn't seem equivalent to the one you asked originally (at least not obviously). Is there a reason for the change? | |
| Feb 9, 2017 at 22:02 | comment | added | Arthur B | Right, right and so was $exp(-exp(-x)) - exp(-exp(x))$ | |
| Feb 9, 2017 at 22:01 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 9, 2017 at 20:18 | comment | added | Arthur B | Yes, yes it is, d'oh! | |
| Feb 9, 2017 at 19:30 | comment | added | Gro-Tsen | Isn't the error function a pretty sigmoid function that extends to an entire function? | |
| Feb 9, 2017 at 19:30 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 9, 2017 at 19:24 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 9, 2017 at 19:08 | comment | added | Arthur B | I edited the question to reflect that lead. | |
| Feb 9, 2017 at 19:08 | history | edited | Arthur B | CC BY-SA 3.0 |
made the question more concise and more explicitely about entire functions
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| Feb 8, 2017 at 23:48 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 8, 2017 at 6:17 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 7, 2017 at 16:58 | comment | added | Arthur B | You're right, sign error for $f''$. This is an interesting angle to tackle it. To be sure, there are some sigmoid like functions in this set: $exp(-exp(-x+log(log(2))))$ satisfies all properties except for $f(x)+f(-x)=1$. $exp(-exp(sinh(-x)+log(log(2))))$ doesn't satisfy the convexity property, nor $f(x)+f(-x)=1$ but not by much. | |
| Feb 7, 2017 at 16:49 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 7, 2017 at 11:31 | comment | added | Kevin Buzzard | Disregarding the assumption about the second derivative, let me make another comment. All the functions in $S$ will extend to holomorphic functions from the complexes to themselves. However none of the examples of sigmoids on Wikipedia seem to have this property. Could one prove that a sigmoid cannot extend to a holomorphic function on the complexes, or is this asking too much? I have no feeling as to whether one should expect this sort of thing to be true, but if it is true it would serve as a way to attack the question. | |
| Feb 7, 2017 at 9:48 | comment | added | Kevin Buzzard | Is there still a mistake in this question? Surely for $x>0$ one wants $f(x)>f(0)$ and $f''(x)<0$, contradicting the last hypothesis in the question -- or have I misunderstood what a sigmoid is? | |
| Feb 7, 2017 at 4:56 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 7, 2017 at 4:56 | comment | added | Arthur B | Sorry I meant to add that x is in S | |
| Feb 7, 2017 at 3:56 | comment | added | Alex Meiburg | What do you mean, $S$ is "the smallest set of functions"? There are no "seed" elements. For instance, $S$ being the empty set, or the set of constant functions, both satisfy your constraint. Does e.g. $S$ necessarily contain $f(x) = x$? | |
| Feb 7, 2017 at 2:38 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 7, 2017 at 2:25 | history | edited | Arthur B | CC BY-SA 3.0 |
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| Feb 7, 2017 at 2:18 | history | asked | Arthur B | CC BY-SA 3.0 |