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In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical $ \bar {\partial } $ ∂ ¯ -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $ \eta =(\eta _n)_{n\geq ~0} $ η = (η n) n ≥ 0 leading to a special entire function $ \mathsf {E}(z) $ E (z) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes. [ABSTRACT FROM AUTHOR]

In this paper we show that continuous endomorphisms on the space of entire functions with a given order can be expressed as differential operators of infinite order satisfying suitable growth conditions. [ABSTRACT FROM AUTHOR]

By integrating the simplicial model of social aggregation with existing research on opinion leadership and diffusion networks, this article introduces the constructs of simplicial diffusers (mathematically defined as nodes embedded in simplexes; a simplex is a socially bonded cluster) and simplicial diffusing sets (mathematically defined as minimal covers of a simplicial complex; a simplicial complex is a social aggregation in which socially bonded clusters are embedded) to propose a strategic approach for information diffusion of cancer screenings as a health intervention on Facebook for community cancer prevention and control. This approach is novel in its incorporation of interpersonally bonded clusters, culturally distinct subgroups, and different united social entities that coexist within a larger community into a computational simulation to select sets of simplicial diffusers with the highest degree of information diffusion for health intervention dissemination. The unique contributions of the article also include seven propositions and five algorithmic steps for computationally modeling the simplicial model with Facebook data. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded open set in ℍ2, and let K be a compact subset of Ω. In this article we prove that approximate solutions to the Cauchy–Fueter system in two variables on Ω ∖ K can be extended to approximate solutions to the same system in all of Ω. [ABSTRACT FROM AUTHOR]

In [Gentili, G. and Stoppato, C., 2007, Zeros of regular functions and polynomials of a quaternionic variable, n. 1 (Dipartimento di Matematica "U. Dini", Università di Firenze), Preprint; Gentili, G. and Struppa, D.C., 2006, A new approach to Cullen-regular functions of a quaternionic variable. C. R. Acad. Sci. Paris, Ser. I, 342, 741-744; Gentili, G. and Struppa, D.C., 2007, A new theory of regular functions of a quaternionic variable. Adv. Math., 216, 279-301.] the authors offered an alternative definition and theory of regularity for functions of a quaternionic or octonionic variable, inspired by an idea of Cullen [Cullen, C.G., 1965, An integral theorem for analytic intrinsic functions on quaternions. Duke Mathematics Journal, 32, 139-148]. This alternative theory is intriguing because it allows the study of natural power series (and polynomials) with quaternionic or octonionic coefficients, which is excluded when the Fueter approach is followed. In this article the authors follow the same ideas to offer a new definition of regularity on the Clifford algebra Cl(0, 3). It turns out that this setting presents a family of new phenomena, which did not appear in the cases of quaternions and octonions. It is nevertheless possible to find a power series expansion for functions regular on an appropriate subset of Cl(0, 3). [ABSTRACT FROM AUTHOR]

In this article we show that it is possible to construct a Koszul-type complex for maps given by suitable pairwise commuting matrices of polynomials. This result has applications to surjectivity theorems for constant coefficients differential operators of finite and infinite order. In particular, we construct a large class of constant coefficients differential operators which are surjective on the space of regular (or monogenic) functions on open convex sets. [ABSTRACT FROM AUTHOR]

In this article some generalizations to eight dimensional real algebras of the notion of Cauchy-Fueter regularity are introduced, by defining suitable operators. Some properties of the nullsolutions of those operators such as removability of compact singularities are studied and the analogue of the Cauchy formula is provided. Special attention is devoted to the case of regularity in the Clifford algebra on three units. [ABSTRACT FROM AUTHOR]