Timeline for Smallest positive zero of Weierstrass nowhere differentiable function
Current License: CC BY-SA 3.0
14 events
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| Jul 8, 2021 at 16:47 | comment | added | Dave L Renfro | @Hans: I don't think the graph is self-similar, but don't know for sure --- the notion of self-similarity is not very relevant to the issues here. Also, note that for the zeros you're looking at just one level set of the function, and it can change simply by adding a constant to the function, and as such is not even a geometrically useful notion (because vertical translations can change it). More relevant are the properties of level sets in general, and this is dealt with in many papers for various pathological types of functions, of which [14] is a useful starting point. | |
| Jul 8, 2021 at 1:24 | comment | added | Hans | Got you. Is the set of zeros self-similar, given that the function itself is a fractal and self-similar? | |
| Jul 7, 2021 at 18:52 | comment | added | Dave L Renfro | @Hans: No phone. Never seemed to need one, so I've never gotten one. Maybe by 2030 I'll have a (cell or smart) phone . . . There seems to be a pattern: Never used email before Dec. 1992, years after most everyone I knew (in math) had been using email. Never had internet at home until July 1999, years after most everyone I knew had internet at home. | |
| Jul 7, 2021 at 18:40 | comment | added | Hans | I see. That is so unfortunate. Do you use iPhone? If so, you can download some PDF scanner such as Adobe Scan apps.apple.com/us/app/adobe-scan-mobile-pdf-scanner/… . It works well. Have a try if you care to. Thank you very much. | |
| Jul 7, 2021 at 17:00 | comment | added | Dave L Renfro | @Hans: I have a photocopy of Prasad's paper (no translation of Zahorski that I know of), but unfortunately can't make scans now -- a printer I purchased a while back (HP MFP M428fdw) isn't scanning to flash drives (says memory device not working, but that's incorrect) and I haven't yet tried to figure out what to do; the google "answers" I get are gobbledygook to me, and being entirely by myself with no support staff to ask, I've kept putting off trying to contact the company (which experience tells me will probably be fruitless, assuming I can even get a person). | |
| Jul 7, 2021 at 16:36 | comment | added | Hans | Thank you, Dave. Do you have a copy of [1] Ganesh Prasad's paper On the zeroes of Weierstrass's non-differentiable function? Is there an English translation of the [11] (Zahorski's paper)? | |
| Jul 7, 2021 at 15:38 | comment | added | Dave L Renfro | @Hans: Yes. See [11] (Zahorski's paper) in the continuation answer. I'm fairly certain the result was known well before this (Zahorski's paper deals with more general intersections of the Weierstrass function's graph than its intersection with the $x$-axis), but I don't know off-hand (and don't have time now to delve into this) when it might have be first observed, or at least a much earlier observance of this than Zahorski's paper. Incidentally, once you get "uncountably many", then "continuum many" is automatic since the zeros form a closed set. | |
| Jul 7, 2021 at 15:12 | comment | added | Hans | +1 for a great and detailed answer. By "continuum many" do you mean as many as the real number? Where is the statement that the zeros are as many as the real numbers proved? | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Apr 7, 2014 at 13:45 | comment | added | Dave L Renfro | @Todd Trimble: I was not able to find out anything about the least positive zero of the Weierstrass function. I thought I might find something about this buried in the papers I discussed (and many other less relevant papers that I also looked at), but I didn't. However, I believed having a detailed record of exactly what these papers contain would be useful, since to my knowledge most appear to have never been discussed anywhere (internet, published literature, etc.) except in some of these papers. | |
| Apr 6, 2014 at 15:38 | comment | added | Todd Trimble | Quite a wealth of scholarly information here; thanks for your efforts, Dr. Renfro. Unfortunately, several users flagged this answer as either being excessively long or as not actually answering the precise question. If you could point out the parts that would either explicitly or in principle provide an answer to the precise question, that would surely address those concerns. | |
| Apr 6, 2014 at 15:07 | review | Low quality posts | |||
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| Apr 6, 2014 at 14:49 | history | answered | Dave L Renfro | CC BY-SA 3.0 |