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On graphs G for which both g and G̅ are claw-free

Shinya Fujita (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are K 1 , t -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.

Multiplicative free square of the free Poisson measure and examples of free symmetrization

Melanie Hinz, Wojciech Młotkowski (2010)

Colloquium Mathematicae

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We compute the moments and free cumulants of the measure ρ t : = π t ⊠ π t , where π t denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of ρ t . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.

ℵ k -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of ℵ k -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is ℵ k -free if every subset of size < ℵ k is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...

On the Lukacs property for free random variables

Kamil Szpojankowski (2015)

Studia Mathematica

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The Lukacs property of the free Poisson distribution is studied. We prove that if free and are free Poisson distributed with suitable parameters, then + and ( + ) - 1 / 2 ( + ) - 1 / 2 are free. As an auxiliary result we compute the joint cumulants of and - 1 for free Poisson distributed . We also study the Lukacs property of the free Gamma distribution.