9,377 reputation
1030
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location Iowa City, IA, USA
age 55
visits member for 2 years, 9 months
seen 6 hours ago

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".


7h
comment Definition of General Associativity for binary operations
I don't have time right now to give this question much attention (and it looks like several others do), but maybe my 12 September 2006 sci.math post combinatorics of associativity will be of interest. If nothing else, you'll find a lot of literature references there.
8h
comment Set of rational functions is which category
Can you determine whether each of your functions has at least one point at which the function is differentiable? (I'm guessing, however, that if this is homework, then you're supposed to do this from scratch, not deduce the result from a big-gun result.)
13h
comment What is the “Principle of permanence”?
@Mauro ALLEGRANZA: My guess is that the complex function version arose like a lot of such things do: someone uses the term in a class or in a paper and for various reasons it spreads. Examples of this I know of are "Ruler function" (early 1960s classes, then an Amer. Math. Monthly paper), "Thomae's function" (more recent, when it became more widely known the function appeared in an 1870s booklet by Thomae), "typical" for Baire category (Bruckner in an early 1970s Amer. Math. Monthly paper), "Smith–Volterra–Cantor set" (I believe this is due to David Bressoud, who wanted to acknowledge others).
13h
comment What is the “Principle of permanence”?
@Anupam: Wikipedia is written by individuals who vary in their depth and breadth of expertise. Indeed, I sometimes see math articles I think put incorrect emphasis on certain things or have a fairly narrow point of view. An example of the 1st is Vaidyanathaswamy ("He contributed extensively to point-set topology": he contributed far more to several other fields than to point-set topology), and an example of the 2nd is porous set which entirely ignores the more used upper porous notions.
1d
comment Are “most” continuous functions also differentiable?
For those interested in pursuing this a lot further, see my answer at Generic Elements of a Set..
1d
comment Real Analysis and dynamics
It sounds like you want an updated version of Whittaker/Watson's A Course of Modern Analysis.
1d
comment terms/names for these *things*
alex, $4$-dimensional space is often called hyperspace, especially in older literature. For example, go to the JFM reviews lookup site and enter "hyperspace" for TITLE and then select SEARCH. I don't know of any common names for spaces with dimension greater than $4.$
1d
comment What is the “Principle of permanence”?
I'm not sure what you are asking, but it's clear Fine is talking about Peacock's usage of the term. Peacock's usage of the term appears in a lot of 1800s literature, some of which shows up in a google-books search for "permanence" and "algebra". It may be of interest to point out that extending factorials of positive integers to the gamma function and extending $n$'th order differentiation to fractional differentiation are examples of this meta-mathematical principal.
1d
comment What is the “Principle of permanence”?
@Anupam: What that Wikipedia article deals with is something I thought was (usually) called the Identity Theorem (for complex analytic functions). Regarding Fine and Peacock, I briefly discussed them and the principal of permanence in these math-teach posts at Math Forum: 14 October 2008, 12 May 2009, and 12 May 2009.
2d
answered How do you find the derivative of the $\int_{1-x}^{1+2x} e^{t^2} dt$?
2d
comment Show that if the projection of a set is negligible, then the set is negligible as well
You wrote: "By negligible I mean that the jordan measure is zero. meaning for any $\epsilon >0$ I can cover the set with $\aleph_0$ many open intervals and the sum of the lengths of the intervals is less than $\epsilon$" This sounds like Lebesgue measure zero to me, not Jordan measure zero.
2d
comment Real world tangent functions
I used to emphasize (both high school and college teaching) that the tangent function allows you to translate between slope (rise over run) and angular measurements involving lines via $\tan{\theta} = m.$ Using this (and a calculator), you can find the measures of the $4$ angles made by a pair of intersecting lines if you know their slopes. I even had a worksheet of problems on this . . .
Apr
14
answered Why limits work
Apr
14
revised Meagre and dense sets
corrected parenthetical explanation in item 2
Apr
14
answered Meagre and dense sets
Apr
10
answered Finding the limit of this sequence using the fastest growing term
Apr
10
comment Relationship between the Weierstrass function and other fractals
@Foo Barrigno: Perhaps the following property will be of interest. Given ANY continuous function $f:[0,1] \rightarrow {\mathbb R}$ and ANY $\epsilon > 0,$ there exists a continuous nowhere differentiable function $g(x)$ (in fact, continuum many of them) such that for all $x \in [0,1]$ we have $|f(x) - g(x)| < \epsilon.$ That is, any given any continuous function and given any $\epsilon$-tube placed around that continuous function, there is a continuous nowhere differentiable function that lies within this tube. (This is because the complement of a co-meager set is dense.)
Apr
10
comment Relationship between the Weierstrass function and other fractals
@Foo Barrigno: The problem to me seems to be that you're insisting on the example being definable without saying what that means. Do you mean constructive in the Erret Bishop sense, constructive in the Russian Markov's sense, constructive in some type of recursive theory sense, definable by infinite series of elementary functions with rational coefficients defined by a finite term recursion form, or what? The possibilities are endless . . .
Apr
9
comment Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts?
A useful thing to observe and remember is that the squares of each of the $6$ trig. functions is easy to integrate. As for the trig. functions themselves (first powers), sine and cosine are super easy, tangent and cotangent are easy $(u$-substitution with $u$ equal to the denominator when you express the function as a quotient involving a sine and cosine), and the secant and cosecant are a bit involved.
Apr
9
comment Is $\frac{2^x}{4x+5}$ rational?
Each rational function $R(x)$ has the property that for some positive integer $n$ we have $\lim_{x \rightarrow \infty}\frac{R(x)}{x^n} = 0.$ (This is not hard to show once limits have been covered.) However, no such positive integer $n$ works for the function you gave (for each positive integer $n,$ using $n-1$ applications of L'Hopital's rule shows the limit to be $+\infty),$ so the function you gave is not rational.