bio | website | |
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location | Iowa City, IA, USA | |
age | 55 | |
visits | member for | 3 years, 2 months |
seen | 6 hours ago | |
stats | profile views | 341 |
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.
Aug 19 |
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Is there an algebraic number that cannot be expressed using only elementary functions?
@Misha Verbitsky: Tito Piezas III's answer to the math StackExchange question Solving 5th degree or higher equations appears to give some information relating to your question. |
Aug 19 |
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Condensation points of orbits of roots of unity
Are you sure "condensation point" is the concept you want? If the set $S$ is countable, there are no condensation points, unless you're using "condensation point" in a way that differs from its standard meaning. |
Aug 18 |
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First Description of how to Remove Radicals from Equations
I don't really have an answer (hence the comment), but for some publications that are possibly worth looking at, see some of the older references I cite at the very end of this 16 November 2010 ap-calculus at Math Forum and see the references I give in my answer to the math StackExchange question History of the theory of equations: John Colson. |
Aug 6 |
answered | Name of a generalized version of semi-continuity |
Jul 21 |
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Tools for Removing Radicals from Equations
The answers at the math StackExchange question Rationalizing radicals describe algebraic procedures you can use. It can be a bit tedious, but you can use these methods along with various on-line devices that multiply-out and combine like terms of factored algebraic expressions. In the case of the expression you gave, you can rewrite with a $0$ on one side and then multiply both sides by an appropriate rationalizing factor (see André Nicolas's answer and my answer in that StackExchange question). |
Jul 8 |
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Are proper linear subspaces of Banach spaces always meager?
For a survey of related issues (Borel types of linear subspaces in infinite dimensional Banach spaces), see my answer to the math StackExchange question Does there exist a linearly independent and dense subset?. |
Jun 19 |
awarded | Necromancer |
May 27 |
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Are norms intrinsically $\mathbb{R}$-valued?
On a lark I googled "Banach valued norm" and got some hits that deal with norm functions that have outputs in a Banach space (that does not have to be the reals). My initial impression is that this generalization is internally motivated (to write papers) rather than externally motivated in order to unify existing special cases. However, I didn't put any effort towards looking into this. |
May 16 |
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Everywhere differentiable function that is nowhere monotonic
For more extreme examples, see my answer at How discontinuous can a derivative be?. Note that regarding the "levels of pathology" I list, each of #2, 4, 5, 6, 7 implies the differentiable function is nowhere monotone. (Recall that monotone in an interval implies at most countably many discontinuities in that interval.) |
Apr 30 |
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Is this property equivalent to Lusin's property (N) for continuous functions?
I haven't had a chance to think about this, and now I see that Christian Remling has posted an answer. Since I've already tracked down a couple of places I was going to suggest looking, I may as well stick them here in case anyone is interested. For Luzin's (N) property, see my 24 November 2006 sci.math post and two others in the same thread. Also, my Bibliography for Singular Functions might have some use. |
Apr 28 |
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The Hidden Aspect of Set Theory
I read the Ulam paper this weekend, and probably the most relevant parts for the original poster is the bottom of p. 343 to the top of p. 344, and the top fourth of p. 348. Several of Ulam's speculations (e.g. the universe possibly being like a Cantor set at very great distances and at very small distances) are in other writings by Ulam, such as his autobiography. Reading Ulam's speculations reminded me of another paper that might be of interest: van Vleck's 1915 Bull. AMS paper The role of the point-set theory in geometry and dynamics. |
Apr 25 |
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The Hidden Aspect of Set Theory
@bof: My first few attempts showed that I don't have access (I'm not associated with a college or university), but then I found an apparently bootleg copy. I haven't yet done more than glance at it, but my initial feeling is that Ulam's knowledge of mathematics probably influences his "gut level view" much more than was the case for Bridgman. |
Apr 25 |
answered | The Hidden Aspect of Set Theory |
Apr 8 |
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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 |
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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. |