How is a "riffle shuffle" mathematically defined? According to http://mathworld.wolfram.com/RiffleShuffle.html : (1) 8 out-shuffles return an ordinary deck of 52 cards to its original order. (2) 7 riffle shuffles are needed to get to close to random. [Hence 8 riffle shuffles would be even better, I think.] So clearly the "riffle shuffles" of (2) are not meant to be identical to "out-shuffles". But then how is a "riffle shuffle" of (2) mathematically defined?

A riffle shuffle is defined to take the deck, cut it into an initial segment $A$ and a final segment $B$, and then mix $A$ and $B$ together, preserving the ordering within $A$ and $B$. 
There is a standard model of a random riffle shuffle due Gilbert: Let there be $n$ cards in the deck. The probability that the deck is cut at position $k$ is $\frac{1}{2^n} \binom{n}{k}$; then all $\binom{n}{k}$ shuffles of $A$ and $B$ are equally likely. The first condition says that, with high probability, the cut point is near $n/2$, with an error of $\approx \sqrt{n}$. The second condition says that, when I have $a$ cards in my left hand and $b$ in my right, the odds that the next card will drop from the left hand is $a/(a+b)$, so the thicker pack of cards drops faster.
This condition is a decent model for real shuffling, according to experiments by Diaconis. IIRC, these experiments only used two shufflers, Diaconis and a friend, and Diaconis is a practiced magician, so one might wonder if this is a fair sample. We had an undergrad, Alex Cope, who was paying random people here at Michigan to shuffle cards and checking them against the model; he doesn't seem to have published his work yet. ADDED JULY 2014 Cope's data still doesn't seem to be public, but you can read an analysis of it here.
It is also a very convenient theoretical model because there is a very simple description for the inverse process: To produce the inverse of a random shuffle, go through the deck and random place each card in your left hand or your right, according to an independent coin flip; then stack up the decks in your two hands. 
For lots more on this model, including the 7 shuffles to randomize result, see Trailing the Dovetail Shuffle to its Lair by Bayer and Diaconis.
