Free powers of the free Poisson measure

Melanie Hinz; Wojciech Młotkowski

Displaying similar documents to “Free powers of the free Poisson measure”

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.

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.

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.

The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution

Łukasz Jan Wojakowski (2007)

Banach Center Publications

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We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by $μ ⊞_{T} ν = T^{- 1} (T μ ⊞ T ν)$ . We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution...

On some generalization of the t-transformation

Anna Dorota Krystek (2010)

Banach Center Publications

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Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{}$ of probability measures on the real line ℝ, which is another possible generalization of the t-transformation. Using that deformation we define a new convolution by deformation of the free convolution. The central limit measure with respect to the -deformed free convolutions is still a Kesten measure, but the Poisson limit depends on the...

New limit theorems related to free multiplicative convolution

Noriyoshi Sakuma, Hiroaki Yoshida (2013)

Studia Mathematica

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Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of ${(μ^{⊠ N})}^{⊞ N}$ as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by...

Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...

The square-free kernel of $x^{2^{n}} - a^{2^{n}}$

Paulo Ribenboim (2002)

Acta Arithmetica

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$ℵ_{k}$ -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...

Probability measures corresponding to Aval numbers

Wojciech Młotkowski (2012)

Colloquium Mathematicae

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We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers $(m - k) / (m + k) (\binom{m + k}{m})$ .

S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case

Josefina Alvarez, Martha Guzmán-Partida, Urszula Skórnik (2003)

Studia Mathematica

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We obtain real-variable and complex-variable formulas for the integral of an integrable distribution in the n-dimensional case. These formulas involve specific versions of the Cauchy kernel and the Poisson kernel, namely, the Euclidean version and the product domain version. We interpret the real-variable formulas as integrals of S’-convolutions. We characterize those tempered distribution that are S’-convolvable with the Poisson kernel in the Euclidean case and the product domain case....

On the global regularity of subcritical Euler–Poisson equations with pressure

Eitan Tadmor, Dongming Wei (2008)

Journal of the European Mathematical Society

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We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $γ$ -law pressure, $γ \geq 1$ , admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2 \times 2$ $p$ -system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

A free stochastic partial differential equation

Yoann Dabrowski (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra $R^{ω}$ embeddable. This includes an $N$ -tuple of $q$ -Gaussian random variables e.g. for $| q | N \leq 0.13$ .

Maximal free sequences in a Boolean algebra

J. D. Monk (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $〈 a_{ξ} : ξ < α 〉$ of elements of $A$ , with $α$ an ordinal, such that for all $F, G \in {[α]}^{< ω}$ with $F < G$ we have $\prod_{ξ \in F} a_{ξ} \cdot \prod_{ξ \in G} - a_{ξ} \neq 0$ . A free sequence of length $α$ exists iff the Stone space $Ult (A)$ has a free sequence of length $α$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $𝔣_{sp} (A) = {| α | : A has an infinite maximal free sequence of length α}$ and the associated min-max function $𝔣 (A) = min (𝔣_{sp} (A)) .$ Among...

The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras

K. R. Goodearl, S. Launois (2011)

Bulletin de la Société Mathématique de France

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The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar, Mouna Chaabouni, Mabrouka Hfaiedh (2007)

Bollettino dell'Unione Matematica Italiana

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We consider Kontsevich star product on the dual $\mathfrak{g}^{*}$ of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.

$\exists$ -свободные группы.

В.Н. Ремесленников (1989)

Sibirskij matematiceskij zurnal

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On fixed point free involutions of $R^{1} \times S^{2}$

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action

Denise de Mattos, Thaís F. M. Monis, Edivaldo L. dos Santos (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let (X,A) be a pair of topological spaces, T : X → X a free involution and A a T-invariant subset of X. In this context, a question that naturally arises is whether or not all continuous maps $f : X \to ℝ^{k}$ have a T-coincidence point, that is, a point x ∈ X with f(x) = f(T(x)). In this paper, we obtain results of this nature under cohomological conditions on the spaces A and X.

Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of $S_{\infty}$ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty}$ can be extended to a large free subgroup of $S_{\infty}$ , and, under Martin’s Axiom, any free subgroup of $S_{\infty}$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty}$ . Finally, if Gₙ are countable...

On some free semigroups, generated by matrices

Piotr Słanina (2015)

Czechoslovak Mathematical Journal

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Let $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}], B_{λ} = [\begin{matrix} 1 & 0 \\ λ & 1 \end{matrix}] .$ We call a complex number $λ$ “semigroup free“ if the semigroup generated by $A$ and $B_{λ}$ is free and “free” if the group generated by $A$ and $B_{λ}$ is free. First families of semigroup free $λ$ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $λ$ ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common...