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Hypercubes As Direct Products

SIAM Journal on Discrete Mathematics, 2005
Summary: Let \(G\) be a connected bipartite graph. An involution \(\alpha\) of \(G\) that preserves the bipartition of \(G\) is called bipartite. Let \(G^\alpha\) be the graph obtained from \(G\) by adding to \(G\) the natural perfect matching induced by \(\alpha\). We show that the \(k\)-cube \(Q_{k}\) is isomorphic to the direct product \(G \times H\)
Brešar, Boštjan   +3 more
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Direct Product Testing

2014 IEEE 29th Conference on Computational Complexity (CCC), 2014
A direct product function is a function of the form g(x_1, ldots, x_k)=(g_1(x_1), ldots, g_k(x_k)). We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability.
Irit Dinur, David Steurer
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Direct summands of products

Archiv der Mathematik, 2002
Let \(R\) be a ring. All modules considered are right modules. A module \(M\) is said to be (finitely) product-rigid if any (finitely presented) direct summand of a product of copies of \(M\) having a local endomorphism ring is isomorphic to some indecomposable direct summand of \(M\) itself.
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