113 research outputs found
FONCTIONS HARMONIQUES ET SOUSHARMONIQUES ASSOCIÉES A DES SYSTÈMES DE RACINES
No full textIn this thesis, we show that, for any root system in the Euclidean space and for any nonnegative multiplicity function on , we can develop in this geometric framework a Newtonian type or more generally a Riesz type potential theory which coincide with the classical theories when is the zero function.Dans cette th\`{e}se, nous montrons que pour tout syst\`{e}me de racines de l'espace euclidien et pour toute fonction de multiplicit\'{e} positive sur , on peut d\'{e}velopper dans ce cadre g\'{e}om\'{e}trique une th\'{e}orie du potentiel de type newtonien et plus g\'{e}n\'{e}ralement de type Riesz qui co\"{\i}ncident avec les th\'{e}ories classiques lorsque la fonction est identiquement nulle
Green function and Poisson kernel associated to root systems for annular regions
No full textLet ∆ k be the Dunkl Laplacian relative to a fixed root system R in R d , d ≥ 2, and to a nonnegative multiplicity function k on R. Our first purpose in this paper is to solve the ∆ k-Dirichlet problem for annular regions. Secondly, we introduce and study the ∆ k-Green function of the annulus and we prove that it can be expressed by means of ∆ k-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for ∆ k-subharmonic functions and we study positive continuous solutions for a ∆ k-semilinear problem
Solutions aux équations WDVV associées aux variétés de Dubrovin-Frobenius sur les espaces de Hurwitz
No full textDans cette thèse, nous nous intéressons aux solutions des équations d’associativité de Witten-Dijkgraaf-
Verlinde-Verlinde (WDVV) et aux structures de variétés de Frobenius sur les espaces de Hurwitz. La
théorie des variétés de Frobenius, développée par Dubrovin, offre une reformulation géométrique des
équations WDVV. En particulier, chaque variété de Frobenius M donne naissance à une solution de ces
équations, connue sous le nom de prépotentiel de M.
Nous commençons par introduire, dans les trois premiers chapitres, certaines notions préalables essentielles
à la compréhension de la thèse. Ensuite, nous démontrons une nouvelle formule qui fournit les
prépotentiels pour les structures de variétés de Frobenius sur les espaces de Hurwitz. Cette approche est
appliquée pour construire divers nouveaux exemples explicites de solutions aux équations WDVV. En
particulier, pour tout espace de Hurwitz de dimension N en genre un, avec N arbitraire, nous exprimons
explicitement le prépotentiel associé à la différentielle holomorphe normalisée en termes de polynômes de
Bell, de séries d’Eisenstein et de fonctions de Weierstrass. Comme applications, nous discutons des liens
entre les prépotentiels obtenus en petites dimensions et certaines équations différentielles particulières,
y compris l’équation de Chazy et les équations différentielles de Ramanujan
Volume mean operator and differentiation results associated to root systems
No full textLet R be a root system in R d with Coxeter-Weyl group W and let k be a non-negative multiplicity function on R. The generalized volume mean of a function f ∈ L 1 loc (R d , m k), with m k the measure given by dm k (x) := ω k (x)dx := ∏ α∈R | ⟨α, x⟩ | k(α) dx, is defined by: ∀ x ∈ R d , ∀ r > 0, M r B (f)(x) := 1 m k [B(0,r)] ∫ R d f (y)h k (r, x, y)ω k (y)dy, where h k (r, x, .) is a compactly supported nonnegative explicit measurable function depending on R and k. In this paper, we prove that for almost every x ∈ R d , lim r→0 M r B (f)(x) = f (x). MSC (2010) primary: 42B25, 42B37, 43A32; secondary: 31B05, 33C52
New formula for the prepotentials associated with Hurwitz-Frobenius manifolds and generalized WDVV equations
No full textWe consider the Hurwitz spaces of ramified coverings of with
prescribed ramification profile over the point at infinity. By means of a
particular symmetric bidifferential on a compact Riemann surface, we introduce
quasi-homogeneous differentials. By following Dubrovin, we construct on Hurwitz
spaces a family of Frobenius manifold structures associated with the
quasi-homogeneous differentials. We explicitly derive new generating formulas
for the corresponding prepotentials. This produces quasi-homogeneous solutions
to the following generalized WDVV associativity equations:
, where the invertible constant matrix
is a linear combination of the matrices . In particular, our approach
provides another look at Dubrovin's construction of semi-simple
Hurwitz-Frobenius manifolds and establishes an alternative practical method to
calculate their primary free energy functions. As applications, we use our
formalism to obtain various explicit quasi-homogeneous solutions to the WDVV
equations in genus zero and one and give a new proof of Ramanujan's
differential equations for Eisenstein series
WDVV solutions associated with the genus one holomorphic differential
No full textConsider the genus one Hurwitz space of
ramified covering of fixed degree with prescribed poles of order
, respectively. Based on a recent formula proved in
\cite{Rejeb23}, we derive an explicit solution to the WDVV equations associated
with the genus one Dubrovin-Hurwitz-Frobenius manifold structure induced by the
normalized holomorphic differential. The obtained solution is written in terms
of Bell polynomials, Eisenstein series as well as the Weierstrass functions
Propriétés de moyenne pour les fonctions harmoniques et polyharmoniques au sens de Dunkl
No full textSupport properties of the intertwining and the mean value operators in Dunkl's analysis
No full textIn this paper we show that the Dunkl intertwining operator has a compact support which is invariant by the associated Coxeter-Weyl group. This property enables us to determine explicitely the support of the volume mean value operator, a fundamental tool for the study of harmonic functions relative to the Dunkl-Laplacian operator
Radial mollifiers, mean value operators and harmonic functions in Dunkl theory
No full textIn this paper we show how to use mollifiers to regularise functions relative to a set of Dunkl operators in R d with Coxeter-Weyl group W , multiplicity function k and weight function ω k. In particular for Ω a W-invariant open subset of R d , for ϕ ∈ D(R d) a radial function and u ∈ L 1 loc (Ω, ω k (x)dx), we study the Dunkl-convolution product u * k ϕ and the action of the Dunkl-Laplacian and the volume mean operators on these functions. The results are then applied to obtain an analog of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators
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