## Welcome!

PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community. On February 13th 2013, PlanetMath.org was updated to use the new software system Planetary. Some release notes are here. Please report bugs in the Planetary Bugs Forum or on Github.

## Error message

Notice: Undefined index: id in planetmath_legacy_urls_init() (line 5 of /home/joe/staging/beta/sites/all/modules/planetmath_legacy_urls/planetmath_legacy_urls.module).

## Latest Messages

Mar 7
Hi, I've written an article and I'd like to attach a file (a program) to the article. I couldn't figure out how to do that -- any ideas? best, Robert Dodier>

Mar 5
They always exist provided you consider the extended real line, i.e, adding the two infites to the line. They exist because, for instance, lim inf is an increasing sequence, that is bounded from above (by infinity). So it converges (eventually it may converge to infinity). The same argument is valid for lim sup, which is decreasing

[p] Solution by Filipe Mar 5
The expectation E(X)-E(min(X,M)) can be written $$\sum_x (x-min(x,M))p(x)$$ Instead of summing over all the values of the random variable X, you can sum just over the values of the random variable for which x-min(x,M) is strictly positive. Then you must have that the probability of the random variable X taking such a x is zero. So the probability $P(X > M)$ must be zero.

Mar 5
Thanks for your suggestion. I have studied the page you recommended, but it does not explicitly address what I want to establish.

Mar 5
Would you have some hint in http://planetmath.org/conjugatediametersofellipse ? Unfortunately, the picture is corrupt.

Mar 4
I am searching for a source demonstrating that, for any set of parallel chords spanning an ellipse, the longest chord passes through the center of the ellipse. I am not referring to the major and minor axes, which I know are the longest and shortest diameters. Rather, I am referring to any set of parallel chords and want to show that the longest chord is a diameter that passes through the center. Any sources would be much appreciated!>

Mar 4
Why not add some observations to the definition as is done, for instance, in the page about Anosov diffeomorphisms? For instance, adding that all Anosov diffeomorphisms are Axiom A, (as well as Morse-Smale for instance, or Smale horseshoe’s); or that the conditions in the definition are not redundant, in the sense that the first condition doesn’t imply the second (it does in dimension 2);

Feb 25
That's because the set containing only a single point has measure zero. As with all resoults involving integrals, here "same function" means "same up to a set of measure zero".

Feb 22
Hi, There seems to be a simple counter-example. Let $f$ be 0 on $\mathbb{R}$, except at 0, where it shall be 1. The link to "integrable" in the text does not help, so I cannot verify whether my idea of it matches the one in the text. However, $f$ is Riemann- and Lebesgue-integrable. It's Fourier transform exists and is 0. The sums exist and are absolutely convergent. So the prerequisites of this theorem seem to be fulfilled. But the sums differ: 1 on the left side, 0 on the right. Regards Wolf

Feb 22
However, Rautenberg on p 265 mentions variant Delta-0 's and one variant is (the above) prim. rec.. He also says that higher index things are ''fairly stable'' under minor changes in Delta-0.

[P] Delta 0 by visu Feb 19
Rautenberg p239 says Delta-0 is pr but that pr need not be Delta-0--he gives hyperexponentiation as example

Feb 19
Cantor's Delusions: http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-556.html#post27247

Feb 18
[url=http://greenvillestoragellc.com/][/url]

Feb 18
http://greenvillestoragellc.com/