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the mean square The Mean Square
(with one standard deviation and several unusual ones)

aka Rob Johnson

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1h
revised Given a curve, such as $1/x$, how to find which tangent is closest to its OWN interception with the y-axis
add x-intercept hint
2h
answered Given a curve, such as $1/x$, how to find which tangent is closest to its OWN interception with the y-axis
2h
comment Evaluation of the integral $\int 3x \cos x^2 \, dx$
This looks like a perfectly good hint. Why the downvote?
3h
comment Evaluation of the integral $\int 3x \cos x^2 \, dx$
To get the answer you are heading for, it would seem that the integral should be written as $$\int3x\cos\left(x^2\right)\,\mathrm{d}x$$
3h
revised Showing unstablity of differential equation.
explicitly state that $(7)$ answers both parts of the question
3h
comment Showing unstablity of differential equation.
@Artem: Thanks for your comments. I will look into Floque theory. I had been considering trying to adapt the method of integrating factors to this problem. The form of the answer you give looks reminiscent.
3h
comment Showing unstablity of differential equation.
@Artem: If you are so inclined, would you post an answer and show how I could have fixed my answer using the trace directly? Thanks.
4h
comment Showing unstablity of differential equation.
@Artem: I was agreeing with you ("but I may change that part as well") and changing my answer ("I will edit soon"). Please read my edited answer.
4h
revised Showing unstablity of differential equation.
add another approach
4h
comment Showing unstablity of differential equation.
@Artem: In that paper, only some of the eigenvalues have positive real part. In the answer above, I only show that the sum of the eigenvalues have positive real part, although it is the case that both of the eigenvalues have positive real part. I am working on a reply to the second part of the question in which I show that both eigenvalues have positive real part, but I may change that part as well. I will edit soon.
5h
comment Showing unstablity of differential equation.
@Artem: is there an example of a time dependent matrix with constant positive eigenvalues that are stable?
5h
comment Showing unstablity of differential equation.
@Artem: locally (that is, for $t$ close to some $t_0$), the propagation of $(x,y)$ will be very close to that given by the matrix given above evaluated at $t_0$. Since the action of that constant matrix is unstable at every point, we get the same instability.
8h
answered Showing unstablity of differential equation.
22h
revised Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
fix typo
23h
comment Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk
You might want to mention why it is important for this question that $\sqrt2$ is irrational; why the result is not true if $\sqrt2$ is replaced by a rational number.
23h
revised Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
add an alternate result
1d
answered Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
1d
revised Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$
add some local color
1d
answered Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$
1d
comment Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
@Ant: I've added a small circle about $1$. Since $\log(x)$ is locally integrable, integrating up to the singularity can usually be ignored, but there's nothing wrong with being a bit more careful.