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location Amherst, MA
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visits member for 2 years, 6 months
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I'm an about-to-retire mathematics professor who's used numerous programming languages over the years, including FORTRAN, Pascal, and APL. I've dabbled a bit with Java. My current programming interests are in Mathematica and J (the jsoftware.com free product).


Dec
12
comment Genius mathematicians who never published anything
@Michael Hardy: could you indicate the gist of Dame Cartwright's proof? (Although back in graduate school I heard her lecture about her work on dynamical systems growing out of the British radio problem, I never before heard about the proof you mention.)
Dec
7
comment What is a simple example of an unprovable statement?
There's a second alternative: neither $X$ nor $Y$ is bigger than the other!
Dec
7
comment Why are integers subset of reals?
2 and 2.0 are the same mathematically because the latter means $2 + 0/10$.
Dec
1
comment An open set in $\mathbb{R}$ is a union of balls of rational radius and rational center.
I think you're confusing that with the fact that $\mathbb{R}$ and, more generally, any separable metrizable space, is second-countable. Look at the proof of that.
Dec
1
comment “Honest” introductory real analysis book
Exactly what real analysis topics do you expect to cover? Presumably either construction of the reals from integers or rationals, or at least an axiomatic treatment (which ought to include uniqueness up to order isomorphism); convergence of sequences and numerical series along with Taylor series and, more generally, convergence & uniform convergence of sequences and series of functions. The critical issue is which integral: just the Riemann integral? Riemann-Stieltjes? or Lebesgue? The 1st edition just did Riemann-Stieltjes integration, as I recall; the current, 3rd edition includes Lebesgue.
Oct
26
comment Solving/simplifying a trig function
What does a, whose tangent and quadrant is given, have to do with B or $\theta$?
Sep
24
awarded  Autobiographer
Aug
12
comment ZFC and apples described using only fundamental axioms (complete expanded reasoning)
We do not really "know" that the number of elements in the disjoint union of two finite sets is the sum of the number of elements in each until we prove it! And that really involves a use of mathematica induction. (Actually, it requires, too, proof that there is such a thing as "the number of elements of a finite set.)
Aug
9
comment If a function $f$ is decreasing on its domain then would its inverse be increasing or decreasing?
Simple "test case": $f(x) = 1/x$ on $(0,\infty)$. What's $f^{-1}(y)$?
Aug
9
comment (a+b)^1/2 another question is Square root (-4)^2=?
Since $(-4)^2 = 16$, then the ordinary, real-number square root $((-4)^2)^{1/2} = 16^{1/2} = \sqrt{16} = 4$.
Aug
9
comment What is the orthogonality condition for associate legendre polynomials with different indexes?
The question may refers to the "associated" Legendre polynomial $P_n^m(x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_n(x)$, where $P_n(x)$ is the ordinary Legendre polynomial.
Aug
5
comment How to evaluate the following indefinite integral? $\int e^{e^x}\mathrm dx$
Since every polynomial is an elementary function, there are infinitely many elementary functions.
Aug
4
comment Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis
The usual definition in topology of "locally P" is: each point has a local base of neighborhoods having property P. Nonetheless, in a Hausdorff space, the existence of some compact neighborhood implies locally compact in the preceding sense.
Aug
4
comment Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis
To the contrary, you do "know" that -- it's a standard result of topology -- provided the space is a Hausdorff (which is usually part of the definition of "locally compact"): Given a point $p \in U$, then $U$ is a neighborhood of $p$. By definition of local compactness, there is a compact neighborhood $W$ of $p$ with $W \subset U$. Since $W$ is a neighborhood of $p$, there exists an open set $V_p$ with $p \in V_p \subset W$. Then $\bar{V_p} \subset \bar{W}$. Now $W$, being compact, is closed, so that $\bar{W} = W$. Thus $\bar{V_p} \subset W \subset U$, and so $\bar{V_p} \subset U$.
Aug
3
answered Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis
Aug
3
comment Evaluate $\int x\sqrt{(a^2 - x^2)}dx$
That's the sort of problem that sadistic calculus instructors throw in with a bunch of others just after teaching trig substitution!
Aug
3
answered Given $\sum_{n=0}^\infty \frac{1}{2^n}$ and $\sum_{n=0}^\infty \frac{1}{4^n}$, what is the Cauchy product?
Aug
3
comment In calculus, which questions can the naive ask that the learned cannot answer?
The original question was easily stated and understood calculus questions whose answers are unknown. Which elementary functions have elementary antiderivatives is known (Risch's decision procedure) -- although the answer is quite complicated.
Aug
2
awarded  Critic
Aug
2
comment Mathematical description of bagel slicing into interlinked tori?
I didn't find an explicit mathematical description of the torus-splitting there.