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As of August 2011 I am an assistant professor of mathematics at Loras College in Dubuque, Iowa, USA.


5h
awarded  Constituent
10h
comment To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
@ChristianBlatter: I corrected it to "$\leq$" and "nonnegative".
10h
revised To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
added 7 characters in body
10h
comment $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ dense in $\mathbb{R}$?
Related: Proving that $m+n\sqrt{2}$ is dense in R, math.stackexchange.com/q/889952, math.stackexchange.com/questions/136665/…, math.stackexchange.com/questions/852210/…
10h
comment To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
OK, so this example isn't $C^2$, and Andrew has shown the conjecture is true. Still an interesting example/technique. And I'm curious, do you know if $f$ is twice differentiable? (If so we know $f''(0)=0$ but $\lim\limits_{x\to 0+}f''(x)$ doesn't exist.)
20h
comment To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
Another detail I don't know: How do we show that twice differentiability extends to $0$? We need $\lim\limits_{x\to 0+}f(x)=\lim\limits_{x\to0+}f'(x)=\lim\limits_{x\to0+}f''(x)=0$, requiring upper bounds of $Q_n$, $Q_n'$, and $Q_n''$ on $[1/(n+1),1/n]$ going to $0$ as $n\to\infty$.
21h
comment To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
According to Mathematica, $T_n(x) = 2 \left(3 n^4-14 n^2-15 n-5\right)-3 n^2 (n+1)^2 \left(2 n^3-3 n^2-5 n-2\right) x^3+n (n+1) \left(18 n^4-12 n^3-68 n^2-56 n-15\right) x^2-\left(18 n^5+3 n^4-81 n^3-110 n^2-56 n-10\right) x$. I don't know why it's positive on $[1/(n+1),1/n]$, but otherwise this looks great.
1d
comment Is every continuous function that preserves (ir)rationality a rational function?
$x^2$ is a rational function with rational coefficients. It sends some irrational numbers to rational numbers.
1d
comment Is every continuous function that preserves (ir)rationality a rational function?
It isn't true that rational functions always work, e.g., $x^2$.
1d
awarded  Good Answer
1d
answered How prove there exsit $\xi\in (0,1)$ such $|f(\xi)|\le|f'(\xi)|$
1d
revised If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.
added 86 characters in body; edited title
1d
answered If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.
1d
revised To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.
edited title
2d
revised $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
added 222 characters in body
2d
revised $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
include problem not only in title
2d
comment $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
This reminded me of a thread where more general techniques for similar problems are described: math.stackexchange.com/questions/3215/…
2d
answered $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
Dec
16
comment To evaluate limit of sequence
Related: math.stackexchange.com/q/997763 (This is the geometric mean of the numbers in the interval $[1,2]$.)
Dec
16
revised Does $\exp(2ir\pi)$ equal $1$? What's wrong?
added 11 characters in body