We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.

It is proved that if E, F are infinite dimensional strictly convex Banach spaces totally incomparable in a restricted sense, then the Cartesian product E × F with the sum or sup norm does not admit a forward shift. As a corollary it is deduced that there are no backward or forward shifts on the Cartesian product `p1 × `p2 , 1 < p1 6= p2 < ∞, with the supremum norm thus settling a problem left o...

The game LIGHTS OUT! is played on a 5× 5 square grid of buttons; each button may be on or off. Pressing a button changes the on/off state of the light of the button pressed and of all its vertical and horizontal neighbors. Given an initial configuration of buttons that are on, the object of the game is to turn all the lights out. The game can be generalized to arbitrary graphs. In this paper, C...

This paper is concerned with the fast computation of a relation d on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of d is the relation δ, whose convex closure yields the product relation σ th...

We use the link between the existence of tilings in Manhattan metric with {1}-bowls and minimum total dominating sets of Cartesian products of paths and cycles. From the existence of such a tiling, we deduce the asymptotical values of the total domination numbers of these graphs and we deduce the total domination numbers of some Cartesian products of cycles. Finally, we investigate the problem ...

Under suitable conditions of connectivity or non-bipartiteness, each of the three standard graph products (the Cartesian product, the direct product and the strong product) satisfies the unique prime factorization property, and there are polynomial algorithms to determine the prime factors. This is most easily proved for the Cartesian product. For the other products, current proofs involve a no...