bio | website | |
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location | Iowa City, IA, USA | |
age | 55 | |
visits | member for | 2 years, 10 months |
seen | 2 days ago | |
stats | profile views | 305 |
Primary Mathematical Interests:
(1) Study and use of negligible sets, especially involving porosity notions and fractal dimension notions.
(2) Application and refinements of the Baire category method for proving existence.
(3) Classical point set theory and real function theory.
(4) History of nowhere differentiable continuous functions and history of the topics above.
Apr 8 |
comment |
Level sets of a Weierstrass nowhere-differentiable function
I was not able to find anything in Young's paper On infinite derivates about the nature of the two countable level sets of the Weierstrass function. |
Apr 7 |
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Smallest positive zero of Weierstrass nowhere differentiable function
@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 7 |
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Level sets of a Weierstrass nowhere-differentiable function
@Todd Trimble: Some information about the Hausdorff and packing dimensions of the level sets of the Weierstrass function is in Fraydoun Rezakhanlou, The packing measure of the graphs and level sets of certain continuous functions, Mathematical Proceedings of the Cambridge Philosophical Society 104 (1988), 347-360. |
Apr 7 |
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Level sets of a Weierstrass nowhere-differentiable function
@Todd Trimble: In reference [11] (Zahorski), the corollary implies that all but possibly two of the level sets are either empty or have cardinality continuum. I don't know for sure about the two exceptional level sets, but I strongly suspect they are countable. There seems to be nothing about this in the papers I have with me. There may be something relevant in G. C. Young's On infinite derivates [Quarterly Journal of Pure and Applied Mathematics 47 (1916), 127-175], but I don't have access to this paper now. I'll look at my copy of it (at home) tonight. |
Apr 6 |
answered | Level sets of a Weierstrass nowhere-differentiable function |
Apr 6 |
answered | Smallest positive zero of Weierstrass nowhere differentiable function |
Mar 25 |
comment |
Smallest positive zero of Weierstrass nowhere differentiable function
I suggest that you wait until I provide descriptions and/or exerpts from them, since several are a bit weak mathematically. At least wait before making any interlibrary loan requests. Of course, if your library actually carries the journals, then by all means take a look at them. |
Mar 25 |
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Smallest positive zero of Weierstrass nowhere differentiable function
Regarding the papers I mentioned in my previous comment, I went through several of them early this morning summarizing relevant aspects of them. However, because I can only do this in the early morning hours (roughly before 7:30 a.m., as I have a full-time non-academic job), it will take me a few days to get everything typed and proof-read for posting an answer. |
Mar 24 |
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Smallest positive zero of Weierstrass nowhere differentiable function
At the Jahrbuch Database, if you enter "non-differentiable function" into the Title window, then select "Expression" from the drop-down menu, then click the tab labeled "Search", you'll find 3 papers with the title On the zeros of Weierstrass's non-differentiable function. I have copies of these (and others) at home and will look at them, and report back tomorrow. I reviewed these three and many other similar papers in my Ph.D. Dissertation's bibliography, so I'll also look there to see what I said. |
Mar 24 |
revised |
Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension?
The answer did not explicitly address the Hausdorff dimension 1 requirement. I have fixed this. |
Mar 24 |
suggested | suggested edit on Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension? |
Feb 20 |
revised |
Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Correction of Denjoy citation [6] and further explanation pertaining to it. |
Feb 20 |
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Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
See also this paper: Isidor Pavlovich Natanson and Garal'd Isidorovich Natanson, On the mutual relation between the integral of Denjoy in the narrow and in the wide sense (Russian), Uspehi Matematicheskih Nauk (N.S.) 12 #6 (1957), 161-168. See MR 20 #949 (by Casper Goffman, in English) and Zbl 78.04503 (by Solomon Marcus, in French). |
Feb 20 |
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Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Based on your interests (given on your Berkeley web page), you might be interested in the following paper: John Charles Burkill and Frederick William Gehring, A scale of integrals from Lebesgue's to Denjoy's, Quarterly Journal of Mathematics (Oxford) (2) 4 #1 (1953), 210-220. |
Feb 19 |
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Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Thanks for the link to the 1937 edition of Saks book. I knew the 1933 1st edition (in French) was freely available on-line, but I didn't know that the 1937 edition was freely available. I've included your link into my answer. |
Feb 19 |
revised |
Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
I've included a link to a freely available version of Saks' 1937 book |
Feb 19 |
awarded | Yearling |
Feb 19 |
revised |
Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
URL for Luzin paper was not correctly linked-up to the journal title |
Feb 19 |
answered | Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$? |
Jan 23 |
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Characterization of Angles Trisectable with Straightedge and Compass
In addition to the 3 references I gave in my answer, some of the references in "7. References: General and Historical" (pp. 36-41) of my manuscript A Detailed and Elementary Solution to $x^{17} = 1$ might also be useful. |