Bochner-Martinelli formula in superspace

In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner-Martinelli formula to superspace by means of the corresponding projections on the space of spinor-valued superfunctions.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1804.0096

An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian

We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related to the Dunkl operators (see [R\"osler M., Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006]) as well as with the basis of the weighted $L^{2}$ space introduced by Dunkl.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

New techniques for the two-sided quaternionic fourier transform

In this paper, it is shown that there exists a Hermite basis for the two-sided quaternionic Fourier transform. This basis is subsequently used to give an alternative proof for the inversion theorem and to give insight in translation and convolution for the quaternionic Fourier transform

Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc

Explicit probabilistic models for databases and networks

Recent work in data mining and related areas has highlighted the importance of the statistical assessment of data mining results. Crucial to this endeavour is the choice of a non-trivial null model for the data, to which the found patterns can be contrasted. The most influential null models proposed so far are defined in terms of invariants of the null distribution. Such null models can be used by computation intensive randomization approaches in estimating the statistical significance of data mining results. Here, we introduce a methodology to construct non-trivial probabilistic models based on the maximum entropy (MaxEnt) principle. We show how MaxEnt models allow for the natural incorporation of prior information. Furthermore, they satisfy a number of desirable properties of previously introduced randomization approaches. Lastly, they also have the benefit that they can be represented explicitly. We argue that our approach can be used for a variety of data types. However, for concreteness, we have chosen to demonstrate it in particular for databases and networks.Comment: Submitte

Handover Mechanisms in ATM-based Mobile Systems

This paper presents two handover mechanisms that can be used in the access part of an ATM-based mobile system. The first handover mechanism, which is called ¿handover synchronised switching¿ is relatively simple and does not use any ATM multicasting or resynchronisation in the network. It assumes that there is sufficient time available such that all data and history information of the old path can be transferred to the mobile terminal (MT) before the actual handover to the new path takes place. It is possible that the time between a handover decision and the actual handover is too short to end the transmission on the old path gracefully (e.g., ending the interleaving matrix, ending transcoder functions, emptying intermediate buffers). A possible solution to this problem is given by the second handover mechanism, where multicast connections to all possible target radio systems (RAS) are used in the core network. This mechanism is called ¿handover with multicast support

Hermite and Gegenbauer polynomials in superspace using Clifford analysis

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae are proven. Finally, an interesting physical application of the super Clifford-Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.Comment: 18 pages, accepted for publication in J. Phys.

Sibling Dependence, Uncertainty and Education: Findings from Tanzania

Primary school enrolment rates are continuously low in many developing countries. The main explanation in the economic literature on schooling is focused on credit constraints and child labour, implying that the indirect cost of schooling in terms of foregone earnings is too high. This paper investigates the effects of future income uncertainty on sibling dependence in the schooling decisions of rural households in developing countries. Schooling tends to direct skills towards future urban employment, whereas traditional rural education or on-farm learning-by-doing tends to direct skills towards future agricultural employment. Given this dichtomy, the question is then: Does future income uncertainty influence the joint educational choice made by parents on behalf of their children and is it possible to test this on simple cross-sectional data? I extend a simple human capital portfolio model to a three period setting. This allows me to explore the natural sequentiality in the schooling decision of older and younger siblings. The model can generate testable empirical implications, which can be taken to any standard cross-sectional data set. I find empirical evidence of negative sibling dependence in the educational decision, which is consistent with a human capital portfolio theory of risk diversification and which cannot be explained by sibling rivalry over scarce resources for credit constrained households. The paper thus provides a complementary explanation to why enrolment rates in developing countries are often continuously low.schooling; human capital investment; specific human capital; sibling dependency; old-age security; uncertainty; risk and income source diversification; liquidity constraints; Tanzania; Africa

Conformal symmetries of the super Dirac operator

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) \subset osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries