Free probability aspect of irreducible meander systems, and some related observations about meanders

We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of NC(n). As a consequence, we show that the even generating function for irreducible meandric systems is the R-transform of XY, where X and Y are classically (commuting) independent random variables, and each of X,Y has centred semicircular distribution of variance 1. Following this point of view, we make some observations about the symmetric linear functional on polynomials which has R-transform given by the even generating function for meanders.Comment: Two new remarks in Sections 3 and 5, and an additional referenc

Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of "subordination distribution of \mu \boxplus \nu with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of free additive convolution on D_k. The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space Multi-variable subordination turns out to have nice relations to a process of evolution towards \boxplus-infinite divisibility on D_k that was recently found by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.Comment: Minor modifications and reference added to 1st version. 34 pages, no figure

Posets of annular non-crossing partitions of types B and D

We study the set \sncb (p,q) of annular non-crossing permutations of type B, and we introduce a corresponding set \ncb (p,q) of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between \sncb (p,q) and \ncb (p,q) is a poset isomorphism, where the partial order on \sncb (p,q) is induced from the hyperoctahedral group $B_{p+q}$, while \ncb (p,q) is partially ordered by reverse refinement. In the case when $q=1$, we prove that \ncb (p,1) is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between \sncd (p,q) and \ncd (p,q). For $q=1$, the poset \ncd (p,1) coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.Comment: Revised version (shortened Introduction, corrected typos), 31 pages, 4 figures, to appear in Discrete Mathematic

Double-ended queues and joint moments of left-right canonical operators on full Fock space

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (A, phi), a family of "(l,r)-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple (A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the left and B_1, ... , B_d canonical operators on the right. This extends a known one-sided formula for free cumulants of A_1, ..., A_d, which establishes a basic operator model for the R-transform of free probability.Comment: In this (final) version, the introduction was re-written to better show the motivation for the question considered in the pape

Infinitesimal non-crossing cumulants and free probability of type B

Free probabilistic considerations of type B first appeared in a paper by Biane, Goodman and Nica in 2003. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A_1, ..., A_k in an infinitesimal noncommutative probability space (A, phi, phi'), and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices of non-crossing partitions of type B appear in the combinatorial study of (A, phi, phi'), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion functional" phi'.Comment: 38 pages, 1 figur

PUBLIC POLICIES IN THE BUCHAREST METROPOLITAN AREA – INERTIAS AND CHALLENGES FOR LOCAL ADMINISTATION

Bucharest, the most developed city from Romania, covers perhaps most superlatives in any field for describing its importance in the national territory. This city is the only European capital without having an integrated and coherent urban policy adopted by the state. There are the short term local interventions of adjusting trend, specific to Romania, that don’t rely on general directions of territorial development, because they are neither intended nor assumed by the competent decisional authorities. In general, the problem is lack of partnership between the project elaborating actors and the implementing actors of those projects. Bucharest municipality has important potential reserves, but the general status is far from corresponding to the demands recommended by the European development policy. The loss of potential, as a result of the decision of modernization postponements, produces both immediate negative effects, and also induces vast losses on a medium and long term. It requires greater costs, with resonance in the entire economic system and in the living conditions of the area. In present study, we try to evaluate the public policies regarding the Metropolitan Area of Bucharest which should be centered on 2 main issues – territorial cohesion and polycentric policy. Because of the difficulties of their understanding by the politicians and thus, the slowness of their use, the evaluation made by us is using a qualitative approach. Our conclusions drift from the result of the efforts made by the CPUMB for convincing the politicians of using and implementing the results drawn by researches and studies made.Bucharest Metropolitan Area (BMA), public policies, territorial cohesion, polycentricity