Free Jacobi process associated with one projection: local inverse of the flow

We pursue the study started in \cite{Dem-Hmi} of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around $z=0$ of the flow determined in \cite{Dem-Hmi}. When the rank of the projection equals $1/2$, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in $(0,1)$, we derive a contour integral representation for the first derivative of the Taylor series which is a major step toward the analytic extension of the flow in the open unit disc.Comment: some misprints are corrected as well as the reasoning at the end of the pape

First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes

We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the $W$-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types $A$, $B$, $D$. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

HAMPARAN JACQUES DERRIDA: TEORI POSTMODERNISME DALAM WACANA DALAM WACANA SENI PERTUNJUKAN

Kanizar1 Abstract: Deconstruction is a strike of postmodernism to the modern thought that is always subjectively functional, structural, and paradigmatic. Postmodernism deconstructs functional thought that maintain stability, deconstructs structuralism over the conventional meaning, and try to find a new meaning so that the postmodernism is also poststructuralist. Postmodernism deconstructs the existing paradigm questioning to the ability of conventional paradigm in solving a new problem so that the postmodernism is also called post-paradigm. Deconstruction in performing arts that becomes the label “difference” is Tabuik ceremony in Pariaman, West Sumatra. Phenomena from the perspective of postmodernism are to refuse the central authority in the meaning of cultural “label”. Cultural meaning-whatsoever form of performing arst-should not be single, but open to other meanings, so that the researcher-other people-may liberally assume the readable discourse. Meaning may exist in anything; in small thing that is less noticeable, less mentioned that might have signi ficant meaning. Thus, Postmodernism approach refuses all assumption that prevents meanings. It does not mean that postmodernism wants to be selfish, disappointed with the previous research paradigm, and or only trapped on the euphoria, but has a congent reason in producing meanings

Lagrange inversion formula, Laguerre polynomials and the free unitary Brownian motion

This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single projection of rank $1/2$ and those of the $S$-transform as well. This generating series lead also to the Taylor expansions of the Schur function of the spectral distribution of the free unitary Brownian motion and of its first iterate.Comment: last version: other typos are correcte

First hitting time of the boundary of a wedge of angle π/4\pi/4 by a radial Dunkl process

In this paper, we derive an integral representation for the density of the reciprocal of the first hitting time of the boundary of a wedge of angle $\pi/4$ by a radial Dunkl process with equal multiplicity values. Not only this representation readily yields the non negativity of the density, but also provides an analogue of Dufresne's result on the distribution of the first hitting time of zero by a Bessel process and a generalization of the Vakeroudis-Yor's identity satisfied by the first exit time from a wedge by a planar Brownian motion. We also use a result due to Spitzer on the angular part of the planar Brownian motion to prove a representation of the tail distribution of its first exit time from a dihedral wedge through the square wave function.Comment: Title is changed, many corrections, new result

Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices

We begin with the study of some properties of the radial Dunkl process associated to a reduced root system $R$. It is shown that this diffusion is the unique strong solution for all $t \geq 0$ of a SDE with singular drift. Then, we study $T_0$, the first hitting time of the positive Weyl chamber : we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of $T_0$. The second and new part deals with the law of $T_0$ for which we compute the tail distribution, as well as some insight via stochastic calculus on how root systems are connected with eigenvalues of standard matrix-valued processes. This gives rise to the so-called $\beta$-processes. The ultraspherical $\beta$-Jacobi case still involves a reduced root system while the general case is closely connected to a non reduced one. This process lives in a convex bounded domain known as principal Weyl alcove and the strong uniqueness result remains valid. The last part deals with the first hitting time of the alcove's boundary and the semi group density which enables us to answer some open questions.Comment: 33 page