New Synthesis of Selenophenes and Condensed Ring Systems from Ketene Dithioacetals

Abstract

New selenophenes were prepared from ketene dithioacetals. Isoselenocyanates were used to acceded to aminoselenophenes. Starting from substituted methylsulfanylselenophenes thieno- and selenolo-selenophenes were synthetized.     

Duality for Hermitean Systems in {\mathbb R^{2n}}

In this paper, using the ring structure of the space of circulant (2 × 2)-matrix, we characterize the dual of the (Fréchet) space of germs of left Hermitean monogenic matrix functions in a compact set _boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose R^2n{{\bf E}\subset\mathbb R^{2n}}. As an application we describe the dual space of the so-called h-monogenic functions satisfying simultaneously two Dirac type equations.     

Annihilators for Harmonic Differential Forms Via Clifford Analysis

In this paper we describe completely the annihilators of harmonic differential forms into the Clifford analysis approach.     

Operator Identities in q-Deformed Clifford Analysis

In this paper, we define a q-deformation of the Dirac operator as a generalization of the one dimensional q-derivative. This is done in the abstract setting of radial algebra. This leads to a q-Dirac operator in Clifford analysis. The q-integration on \mathbbR^m{\mathbb{R}^m}, for which the q-Dirac operator satisfies Stokes’ formula, is defined. The orthogonal q-Clifford- Hermite polynomials for this integration are briefly studied.     

A Discrete Bochner�CMartinelli Formula

In the even dimensional case the discrete Dirac equation may be reduced to the so-called discrete isotonic Dirac system in which suitable Dirac operators appear from both sides in half the dimension. This is an appropriated framework for the development of a discrete Martinelli–Bochner formula for discrete holomorphic functions on the simplest of all graphs, the rectangular \mathbbZ^m{\mathbb{Z}^m} one. Two lower-dimensional cases are considered explicitly to illustrate the closed analogy with the theory of continuous variables and the developed discrete scheme.     

Selenium‐Containing Heterocycles from Isoselenocyanates: Cycloaddition of Carbodiimides to Selenazetidines

The reactions of isoselenocyanates with carbodiimides in refluxing hexane afforded 1,3‐selenazetidine‐2,4‐diimines 3a – 3h by a [2+2] cycloaddition in moderate‐to‐good yields. The molecular structure of 3a has been established by X‐ray crystallography.     

Partial Primitives for Clifford Algebra-Valued Functions

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function on , i.e. a solution for the generalized Cauchy-Riemann operator D on , construct a monogenic function such that . In view of the fact that, for monogenic functions g, this can be written as g = f, a straightforward generalization consists in replacing the scalar generator of translations in the x ₀-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.     

Duality for Harmonic Differential Forms Via Clifford Analysis

The space HF _k (Ω) of harmonic multi-vector fields in a domain as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HF _k (E) being a compact set. It is proved that HF _k (E)^* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity. Research of the third author was supported by the FWO Research Network WO. 003. 01N, research of the fourth author was supported by the FWO “Krediet aan Navorsers: 1.5.106.02”