Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$)

Functional equation of zeta distributions related to noneuclidean Jordan algebras

The paper is devoted to study certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities are obtained

Huygens' principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators

Let \a be an Euclidean vector space of dimension $N,$ and let $k=(k_\alpha)_{\alpha\in \cal R}$ be a multiplicity function related to a root system $\cal R.$ Let $\Delta(k)$ be the trigonometric Dunkl-Cherednik differential-difference Laplacian. For (a,t)\in \exp(\a)\times \R, denote by $u_k(a,t)$ the solution to the wave equation $\Delta(k) u_k(a,t)=\partial_{tt}u_k(a,t),$ where the initial data are supported inside a ball of radius $R$ about the origin. We prove that $u_k$ has support in the shell \{(a,t)\in \exp(\a)\times \R\;|\; \vert t\vert-R\leq \Vert \log a\Vert\leq \vert t\vert+R\} if and only if the root system $\cal R$ is reduced, $k_\alpha\in \N$ for all $\alpha\in \cal R,$ and $N$ is odd starting from $3.

On the integrability of a representation of sl(2, R)

Soumis à Journal of functional analysisThe Dunkl operators involve a multiplicity function $k$ as parameter. For positive real values of this function, we consider on the Schwartz space $\mathcal S(\R^N)$ a representation $\omega_k$ of \s\l(2,\R) defined in terms of the Dunkl-Laplacian operator. By means of a beautiful theorem due to E. Nelson, we prove that $\omega_k$ exponentiates to a unique unitary representation $\Omega_k$ of the universal covering group \GG of ${SL(2,\R)}.$ Next we show that the Dunkl transform is given by $\Omega_k(g_\circ),$ for an element g_\circ \in \GG. Finally, the representation theory is used to derive a Bochner-type identity for the Dunkl transform

A Paley-Wiener theorem for the Bessel-Laplace transform (I): The case SU(n,n)/GL(n,C)+SU(n,n)/GL(n,C)_+

Soumis à Journal of Lie theoryLet \q be the tangent space to the noncompact causal symmetric space SU(n,n)/SL(n,\C)\times \R^*_+ at the origin. In this paper we give an explicit formula for the Bessel functions on \q, and we then use it to prove a Paley-Wiener theorem for the Bessel-Laplace transform on \q. Further, an Abel transform for \q is defined and inverted

Bessel-type functions of matrix variables

we compute explicitly a ceratin type of hypergeometric function of matrix variables given as an integral of a Gaussian-type kernel. In the case of one variable, this function is related to the modified Bessel function of the third kind

Variation-based Cause Effect Identification

Mining genuine mechanisms underlying the complex data generation process in real-world systems is a fundamental step in promoting interpretability of, and thus trust in, data-driven models. Therefore, we propose a variation-based cause effect identification (VCEI) framework for causal discovery in bivariate systems from a single observational setting. Our framework relies on the principle of independence of cause and mechanism (ICM) under the assumption of an existing acyclic causal link, and offers a practical realization of this principle. Principally, we artificially construct two settings in which the marginal distributions of one covariate, claimed to be the cause, are guaranteed to have non-negligible variations. This is achieved by re-weighting samples of the marginal so that the resultant distribution is notably distinct from this marginal according to some discrepancy measure. In the causal direction, such variations are expected to have no impact on the effect generation mechanism. Therefore, quantifying the impact of these variations on the conditionals reveals the genuine causal direction. Moreover, we formulate our approach in the kernel-based maximum mean discrepancy, lifting all constraints on the data types of cause-and-effect covariates, and rendering such artificial interventions a convex optimization problem. We provide a series of experiments on real and synthetic data showing that VCEI is, in principle, competitive to other cause effect identification frameworks

The wave equations for Dunkl operators

Let $k=(k_\alpha)_{\alpha\in \cal R}$ be a positive-real valued multiplicity function related to a root system $\cal R,$ and $\Delta_k$ be the Dunkl-Laplacian operator. For $(x,t)\in \R^N\times \R,$ denote by $u_k(x,t)$ the solution to the deformed wave equation $\Delta_k u_k(x,t)=\partial_{tt}u_k(x,t),$ where the initial data belong to the Schwartz space on $\R^N.$ We prove that for $k\geq 0$ and $N\geq 1,$ the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if $(N-3)/2+\sum_{\alpha\in \cal R^+}k_\alpha\in \N.$ Here $\cal R^+\subset \cal R$ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius $R>0$ about the origin, the strict Huygens principle implies that the support of $u_k(x,t)$ is contained in the conical shell $\{(x,t)\in \R^N\times \R\;|\; \vert t\vert -R\leq \Vert x\Vert\leq \vert t\vert+R\}.$ Our approach uses the representation theory of the group $SL(2,\R),$ and Paley-Wiener theory for the Dunkl transform. Also, we show that the ($t$-independent) energy functional of $u_k$ is, for large $\vert t\vert$, partitioned into equal potential and kinetic parts

Bessel functions for root systems via the trigonometric setting

By taking an appropriate limit, we obtain the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions . A more general class of Bessel functions is also investigated, which we shall call the \Theta-Bessel functions. Explicit formuals for the \Theta-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown