Prékopa–Leindler inequality: Difference between revisions

Latest revision as of 17:45, 20 April 2013

In abstract algebra, the triple product property is an identity satisfied in some groups.

Let $G$ be a non-trivial group. Three nonempty subsets $S, T, U \subset G$ are said to have the triple product property in $G$ if for all elements $s, s^{'} \in S$ , $t, t^{'} \in T$ , $u, u^{'} \in U$ it is the case that

s^{'} s^{- 1} t^{'} t^{- 1} u^{'} u^{- 1} = 1 \Rightarrow s^{'} = s, t^{'} = t, u^{'} = u

where $1$ is the identity of $G$ .

Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. Template:Arxiv. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.

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+In [[abstract algebra]], the '''triple product property''' is an identity satisfied in some [[group (mathematics)|groups]].
+Let <math>G</math> be a non-trivial group.  Three nonempty subsets <math>S, T, U \subset G</math> are said to have the ''triple product property'' in <math>G</math> if for all elements <math>s, s' \in S</math>, <math>t, t' \in T</math>, <math>u, u' \in U</math> it is the case that
+: <math>
+s's^{-1}t't^{-1}u'u^{-1} = 1 \Rightarrow s' = s, t' = t, u' = u
+</math>
+where <math>1</math> is the identity of <math>G</math>.
+It plays a role in research of [[fast matrix multiplication algorithms]].
+==References==
+* Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. {{arxiv|math.GR/0307321}}. ''Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science'', 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp.&nbsp;438&ndash;449.
+==See also==
+[[Category:Finite groups|*]]
+[[Category:Properties of groups]]