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location Iowa City, IA, USA
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Primary Mathematical Interests:

Study and use of negligible sets, especially involving porosity notions and fractal dimension notions.

Application and refinements of the Baire category method for proving existence.

Classical point set theory and real function theory.

History of nowhere differentiable continuous functions and history of the topics above.

My email address has the form "first-name period last-name at YAHOO period COM".


Aug
27
revised Determine if these correspondences on ${\mathbb Q}$ define functions
Made title less tautologous sounding and more specific to question being asked
Aug
26
comment Book on calculus of several variables.
@Bungo: I too like Lang's Calculus of Several Variables (see here), and there has been at least one math StackExchange question about Lang's calculus books -- Would it be fine to use Serge Lang's two Calculus books as textbooks for freshman as Maths major?.
Aug
25
comment What should I learn next?
You might want to look over some of the calculus references I gave in answer to the Mathematics Educators StackExchange question Extremely “hard” books (or handouts) for undergrad studies.
Aug
25
comment Countable and uncountable sets in Riemann integration
I think this is one of those things that you need to correctly and explicitly rephrase. For example, "a set" is singular tense, and since you didn't include any quantifiers in your (final) question, I assumed $i$ was fixed. Also (as another example of apples and oranges), you wrote near the beginning "the number of terms in the sum above ...", but when I look at "the sum above" I see a sum with $n$ terms (the index goes from $i=0$ to $i=n).$ In this regard, see Jonas Meyer's first comment. FYI, the equivalent method of using upper and lower Darboux sums might make more intuitive sense.
Aug
25
comment Characterization of sets of differentiability
I am deleting my first two comments here, and my first comment in my answer, because I think they could be too much of a confusing distraction for others coming to this page in the future, in light of the fact that I used (by accident) the symbol $D(f)$ to denote the complement what you used $D(f)$ to denote.
Aug
25
comment Countable and uncountable sets in Riemann integration
You asked: What is the number of elements in a set $[x_i, \, x_{i+1}]$ as $\delta \rightarrow 0.$ By the Cantor Intersection Theorem, the limit of a strictly decreasing sequence of closed and bounded intervals is a singleton set. That said, I think you're trying to mix apples and oranges here.
Aug
22
revised Characterization of sets of differentiability
added 8239 characters in body
Aug
22
comment Characterization of sets of differentiability
Oops, sorry about the notation mix-up. I did in fact get things backwards from what you originally posted! I've fixed this in the update I'm now posting.
Aug
21
answered Characterization of sets of differentiability
Aug
21
comment Countable and uncountable sets in Riemann integration
The limit of the number of elements is $c$ (cardinality of the continuum), while the number of elements in the "limiting set" is $1.$
Aug
19
revised Is there a theory of integration in elementary terms for definite integrals?
A relevant additional reference is added
Aug
19
comment Assumptions in Word Problems (Calculus)
How about this? "The volume of a spherical balloon that is being inflated with gas is increasing at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters, assuming the stated rate of increase of the balloon's volume holds for all radii belonging to some open interval containing 30 and 60?" Note this takes care of what is probably a more significant issue, namely the fact that as the pressure in the balloon increases, the density of the air in the balloon increases.
Aug
18
answered Is there a theory of integration in elementary terms for definite integrals?
Aug
18
answered Readings on more general/abstract notions of induction related to logic
Aug
18
comment I need help finding a rigorous Pre-calculus textbook
Fairly rigorous is the very widely used (in the U.S., from the late 1960s through the 1970s) Dolciani text Modern Introductory Analysis. Also worth looking at is Olmsted's Prelude to Calculus and Linear Algebra (1968).
Aug
18
comment I need help finding a rigorous Pre-calculus textbook
FYI, I taught 6 precalculus courses in the 1990s using an earlier version of this book: Brown/Robbins, Advanced Mathematics: A Precalculus Course, revised edition, 1987.
Aug
11
comment where can I get math book reviews?
Notices of the American Mathematical Society (freely available) has book reviews. Also, Bulletin of the American Mathematical Society has a lot of book reviews (freely available back to the 1890s).
Aug
7
comment Jordan Measure and Lebesgue Measure
Regarding "no characterisation of Riemann integrable functions", Jordan measure can be used to give a characterization and in fact the pre-Lebesgue characterizations used this: A bounded function $f:[a,b] \rightarrow {\mathbb R}$ is Riemann integrable if and only if for each $\epsilon > 0$ the set of points at which the oscillation of $f$ is greater than or equal to $\epsilon$ has Jordan measure zero. Indeed, it is this characterization that H. J. S. Smith used in showing that certain functions were not Riemann integrable in his 1875 paper where the first Cantor set (essentially) appeared.
Aug
6
comment Product of Borel $\sigma$-algebras vs Borel $\sigma$-algebra of product
See my 5 May 2002 sci.math.research post. I'm not sure if your specific question is answered there (I don't have time right now to check), but even if it doesn't, it should give you enough leads to find an answer.
Aug
6
comment Elementary ways to calculate the arc length of the Cantor function (and singular function in general)
See Richard Brian Darst, Some Cantor sets and Cantor functions, Mathematics Magazine 45 #1 (January 1972), 2-7. For a summary of this paper, see item #11 in my Bibliography for Singular Functions.