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visits member for 4 years, 5 months
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As of August 2011 I am an assistant professor of mathematics at Loras College in Dubuque, Iowa, USA.


24m
comment Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.
You could try $C_n(x-x^2)^{1/n}\left(1-\frac\pi8+\sqrt{\frac14-(x-\frac12)^2}\right)$ with $n$ large ($C_n$ is a number a little larger than $1$ to make the area come out to $1$).
1h
comment Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.
Using continuous piecewise linear functions, you can get arbitrarily close to $3$ by going up quick, then horizontal, then down quick.
2h
comment A finite subset of an ordered set contains an $\inf$ and $\sup$
See also math.stackexchange.com/q/548806
2h
comment A finite subset of an ordered set contains an $\inf$ and $\sup$
Even though this is phrased in more generality, the answer here applies (as would any answer to that question): math.stackexchange.com/q/259893 That answer shows specifically how to use induction to show that $\sup A$ exists and is in $A$, and it can be adapted to prove the same about the $\inf$.
2h
revised A finite subset of an ordered set contains an $\inf$ and $\sup$
changed irrelevant title and tag (don't know if the tag choice is good though)
3h
comment Applying the Stone-Weierstrass Theorem to approximate even functions
Very closely related: math.stackexchange.com/q/284996
4h
comment Applying the Stone-Weierstrass Theorem to approximate even functions
@trb456: Starting with a polynomial function $p$ on $[-1,1]$, one can define another polynomial $q$ on $[-1,1]$ by the formula $q(x) = \frac12(p(x)+p(-x))$. This $q$ is called the even part of $p$, and you can see that what it does is remove all the odd degree terms from $p$. It is easy to check that $q$ is even, whether or not $p$ is even. Typically, $q$ and $p$ are different polynomials. In case $f$ is even, it turns out that if $p$ approximates $f$ on $[-1,1]$, then so does $q$ (in a sense made precise in orangekid's answer). But this does not mean that $p$ is even.
4h
comment Applying the Stone-Weierstrass Theorem to approximate even functions
@trb456: It is incorrect that $p$ is forced to be even. And you did write "show that $p$ must also be even" so I cannot understand your first sentence. The fact is, an approximating polynomial obtained from Weierstrass's theorem for an even function need not be even. But, one can take the even part of the polynomial, which is even, and still approximates. That is what orangekid's answer does, except without using those words. It would be redundant for me to post an answer.
4h
comment Applying the Stone-Weierstrass Theorem to approximate even functions
@trb456: It isn't true that $p$ must be even, but its even part will still approximate $f$.
4h
comment Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$
Using that equality means that this answer bridges the gap left by Paul's answer, so it is a nice supplement.
5h
comment Lebesgue Measure of a set satisfying infinitely many solutions of this inequality
If what you described were accurate, it would be clear that $A$ has positive measure. However, I would interpret your definition of $A$ to contradict your later statements. Does that mean $x$ is in $A$ if and only if there exist infinitely many pairs $(p,q)$ such that $x\in H_{p,q}$? That sounds like a limsup of sets, not a union. So $A=(0,1)\cap \left(\cap_{n=1}^\infty \cup_{|p|+q=n}^\infty H_{p,q}\right)$?
9h
comment Linear dependence of $\left\{x^{n}\,\colon\, n\in\mathbb{N}\right\}$
Related: math.stackexchange.com/q/152133
9h
comment Positive Real Numbers forming a subring
Yes, you need additive inverses and don't have them, and that is it.
11h
comment Show a sequence is bounded (therefore has convergent subsequence by Bolzano-Weierstrass)
It was unclear, bad notation, $.5$ with the $.$ easy to miss. I changed it to $1/2$.
11h
revised Show a sequence is bounded (therefore has convergent subsequence by Bolzano-Weierstrass)
added 3 characters in body
17h
reviewed Leave Open Prove that the determinant of $ A^{-1} = \frac{1}{det(A)} $- Linear Algebra
17h
reviewed Close Determinant of the inverse matrix
17h
reviewed Leave Open Is there an obvious reason why $4^n+n^4$ cannot be prime for $n\ge 2$?
1d
reviewed Close reducible measure functions
1d
reviewed Close Determine average rate of change of function