5,260 research outputs found
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
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
Formalising the subjective interestingness of a linear projection of a data set : two examples
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.
Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator
We introduce the so-called Clifford-Gegenbauer polynomials in the framework
of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space
. In both cases we obtain several properties of these polynomials, such as
a Rodrigues formula, a differential equation and an explicit relation
connecting them with the Jacobi polynomials on the real line. As in the
classical Clifford case, the orthogonality of the polynomials on must be
treated in a completely different way than the orthogonality of their
counterparts on B(1). In case of , it must be expressed in terms of a
bilinear form instead of an integral. Furthermore, in this paper the theory of
Dunkl monogenics is further developed.Comment: 19 pages, accepted for publication in Bulletin of the BM
Fractional fourier transforms of hypercomplex signals
An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given
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