# Tag Info

## Hot answers tagged real-analysis

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### Transforming two smooth densities to the same density

This is impossible if $f$ is injective, without further assumptions such as bijective, differentiable, etc. Let $Q_1,Q_2$ be probability measures on a measurable space $(\Omega, \mathcal{F})$, and ...
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### If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

Here are some details on Sam Forster’s construction . To make the computation simpler I’d take powers of $4$ instead, i.e. define $g:= \sum_{k=1}^\infty \frac{f_k}{4^k},$ where $f_n$ and $C_n$ have ...
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### Approximation of Incomplete elliptic integral of first kind

The series expansion in powers of $k$ of the incomplete elliptic integral of the first kind $$F(\varphi,k)= \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$ can be simply obtained by ...
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### An inequality about the second-order difference

$\newcommand\de\delta\newcommand\lhs{\text{lhs}}\newcommand\rhs{\text{rhs}}$No. E.g., let $$f(x):=x^3\sin\frac1x$$ for $x\in(0,1/2]$, with $f(0):=0$. Then $f$ can be obviously extended to a ...
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This answer to Geometrically showing $\frac{\alpha}{\beta} > \frac{\sin\alpha}{\sin\beta}$, for $0 < \beta < \alpha < 90^\circ$ shows using only elementary geometry that $\sin \alpha / \... 1 vote ### Ekeland's standardness-property inheritable? The principle for getting examples of nonapplicability of Ekeland's theorem is described by the following simple Example. With$\mathbb I=[0,1]$consider the Frechet space$E=F=C^\infty(\mathbb I)\$ ...
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