It is well-known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder's celebrated result on the addition formula.In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra sl(1|2), combined with the equivalence between the L 2 inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra osp(1|2).as well as the Fischer inner productAs in the case of scalar valued polynomials, P (∂) is obtained by replacing the complex variable z j = x j + ix n+j with the derivative 2∂ zj = ∂ xj + i∂ xn+j in P (z). Of special interest is the duality of the vector variablesz andz † with the Dirac operators ∂z and ∂ † z respectively.Lemma 4.2. If P (z) and Q(z) are homogeneous C 2n -valued polynomials then it holds that 2 ∂zP, Q ∂ = P,zQ ∂ ,

<p>An update on our project aiming to provide space weather predictions that will be initiated from observations on the Sun and to predict radiation in space and its effects on satellite infrastructure. Real-time predictions and a historical record of the dynamics of the cold plasma density and ring current allow for evaluation of surface charging, and predictions of the relativistic electron fluxes will allow for the evaluation of deep dielectric charging. The project aims to provide a 1-2 day probabilistic forecast of ring current and radiation belt environments, which will allow satellite operators to respond to predictions that present a significant threat. As a backbone of the project, we use the most advanced codes that currently exist and adapt existing codes to perform ensemble simulations and uncertainty quantifications. This project includes a number of innovative tools including data assimilation and uncertainty quantification, new models of near-Earth electromagnetic wave environment, ensemble predictions of solar wind parameters at L1, and data-driven forecast of the geomagnetic Kp index and plasma density. The developed codes may be used in the future for realistic modelling of extreme space weather events. The PAGER consortium is made up of leading academic and industry experts in space weather research, space physics, empirical data modelling, and space environment effects on spacecraft from Europe and the US.</p>

By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on a constant function. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in C. At the end we compare formulas, obtaining interesting equalities involving the real part of positive and negative powers of the standard Hermitian product.In the two appendices we show how our computations are optimal compared to direct ones.

<p>The particle flux in the near-Earth environment can increase by orders of magnitude during geomagnetically active periods. This leads to intensification of particle precipitation into Earth’s atmosphere. The process potentially further affects atmospheric chemistry and temperature.</p><p>In this research, we concentrate on ring current electrons and investigate precipitation mechanisms on a short time scale using a numerical model based on the Fokker-Planck equation. We focus on understanding which kind of geomagnetic storm leads to stronger electron precipitation. For that, we considered two storms, corotating interaction region (CIR) and coronal mass ejection (CME) driven, and quantified impact on ring current. We validated results using observations made by POES satellite mission, low Earth orbiting meteorological satellites, and Van Allen Probes, and produced a dataset of precipitated fluxes that covers energy range from 1 keV to 1 MeV.</p>

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