# Clifford Research Group

• 2 April 2012
Jean-Philippe Michel (University of Luxembourg)
Higher symmetries of the Laplacian via quantization
• 20 February 2012
Frank Sommen
A Cauchy-type theorem in Clifford analysis

#### Analysis of Priority Queues: Asymptotics of Low-Priority Buffer Occupancy through Singularity Analysis of Generating Functions Joris Walraevens (Telin, UGent)

Abstract:
In telecommunication networks, different concurrent applications (sending Emails, making telephone calls, gaming, ...) have different delay requirements. Using a priority scheduling when transmitting information exploits these different requirements and increases network performance. Furthermore, information that cannot be send to the next network node immediately has to be buffered. Hence, a probabilistic priority queueing model is analysed to study the impact of priority scheduling on the low-priority buffer occupancy. In earlier work, high-priority queue capacity was assumed to be infinite for mathematical tractability, enabling an analysis based on generating functions (z-transforms). Singularity analysis of these generating functions yields asymptotics for the corresponding probability distribution. An important property of the generating function of the low-priority buffer occupancy is the presence of branch cuts in the complex plane. Recently, we started studying priority queues with finite high-priority queue capacity. Our preliminary (numerical) results show that the resulting generating function only possesses poles, and demonstrates how branch cuts are formed in the limit for the high-priority capacity going to infinity. We conjecture that the latter can be proved mathematically by linking a recurrence relation for the denominator of the generating function to orthogonal polynomials.

#### sl(2)-actions in Clifford analysis Peter Van Lancker (Hogent)

Abstract:
We will introduce a one-parameter family of sl(2)-actions which arise naturally in the setting of Clifford analysis. The modules for these actions can be expressed in terms of Gegenbauer polynomials.
Amongst this family there is a certain action which also preserves monogenicity and which is closely related tot the notion of axially monogenic functions.
In this talk we will explain how the (generalized) Fueter theorem can be understood in terms of such sl(2)-actions.
The original Fueter theorem is a classical result in quaternionic analysis, which explains how to obtain regular functions (solutions for a generalized Cauchy-Riemann operator), starting from holomorphic functions f(z) in the complex plane C. This result has later been generalized, on several occasions, within the setting of Clifford analysis (see e.g the work of F. Sommen and D. Pena Pena).
This is joint work with D. Eelbode and V. Soucek.

#### Matrix valued analogues of Chebyshev polynomials and SU(2) x SU(2) Erik Koelink (RU Nijmegen)

Abstract:
Recently, matrix-valued orthogonal polynomials are studied intensively from an analytic point of view, but they go back to the work of Krein in the 1950s on differential equations. Matrix-valued orthogonal polynomials satisfy many properties analogous to the classical properties of the scalar-valued orthogonal polynomials, such as three-term recurrence relations, differential operators, Rodrigues formula, etc.
The purpose of the talk is to discuss a particular set of matrix-valued orthogonal polynomials with the matrix-size of arbitrary dimension, which are analogues of the Chebyshev polynomials of the second kind. These polynomials arise from the study of matrix-valued spherical functions on $SU(2)\times SU(2)$ with respect to the diagonal subgroup, and we show how group theory in combination with analytic methods yields the information for these matrix-valued orthogonal polynomials. We end with an outlook and a discussion of other cases.
This is joint work Maarten van Pruijssen (Radboud Universiteit Nijmegen) and Pablo Roman (Universidad Nacional de Cordoba, Argentina).

#### A General Geometric Fourier Transform Roxana Bujack (Leipzig)

Abstract:
The increasing demand for Fourier transforms on geometric algebras has
resulted in a large variety. We introduce one single straight forward
definition of a general geometric Fourier transform covering most
versions in the literature and show which constraints are necessary to
obtain certain features like linearity, a shift theorem, and a convolution theorem.

#### Finite oscillator models and discrete orthogonal polynomials Joris Van der Jeugt (UGent)

Abstract:
The requirement to use a finite version of a quantum oscillator comes from quantum optics.
We shall introduce a well known model for a quantum oscillator in finite dimensions, which is based on the Lie algebra su(2). In that case, the discrete wave functions are in terms of Krawtchouk polynomials, so this model is often referred to as the Krawtchouk oscillator. It can be seen, in plots and by computing limits, that these discrete wave functions tend to the standard oscillator wave functions in terms of Hermite polynomials when the dimension (of the representation) tends to infinity.
Recently, we have investigated one-parameter deformations of su(2) and its representations. New finite oscillator models can be constructed based on these deformed algebras. The discrete spectrum of the position and momentum operators can be determined explicitly, and it is shown that the corresponding discrete wave functions are in terms of Hahn polynomials. We shall present some basic properties of these wave functions. Of interest is also the discrete Fourier transform that maps position wave functions into momentum wave functions.

#### Fueter's theorem: an overview Dixan Pena Pena (UGent)

Abstract:
Fueter’s theorem is a fundamental result in Clifford analysis. It disclosesa remarkable connection existing between the classical holomorphic functions and it higher dimensional counterpart (i.e. the monogenic functions). The aim of our talk is to provide an overview of the most important issues related to this topic, including some generalizations and examples of special monogenic functions generated by this technique.

#### From approach theory to index calculus Bob Löwen (UA)

Abstract:
In this talk we give an overview of approach theory, index calculus and applications in topology, functional analysis and probability theory.

#### Sparse representations in image restoration and reconstruction Aleksandra Pizurica (UGent)

Abstract:
After domination of the wavelet representation in image coding, restoration and analysis, in the last two decades we are witnessing expansion of more general sparse representations. The term sparsity refers to representing a general signal with few elementary functions or “atoms” from a given dictionary. One of the important research directions in this emerging field is design of wavelet-like representations, which are better adapted to discontinuities in two or more dimensions (e.g. better orientation selectivity). These properties naturally lead to better “compression” of multidimensional signals, i.e., ability to reconstruct the signal or image content from fewer elements of the dictionary. Motivated by this, many x-lets arose recently, like curvelets, contourlets, bandlets, grouplets, shearlets, directionlets, etc.

A second important research direction is dictionary learning: learning of elementary functions from the image or a set of images, usually using techniques of L1 optimization. Motivation and roots for this research, which is currently expanding in signal and image processing, are largely in neuroscience and psycho-visual studies of the human visual system.

Finally, talking about sparse representations, the topic of compressed sensing (compressive sampling) should be mentioned. The central idea there is to decrease the sampling frequency much below the Nyquist rate, using the properties of the signal (image, video) itself and smart nonlinear reconstruction algorithms for data recovery from few measurements.

In this talk, I will address the mentioned topics from the point of view of personal research, commenting on applications in image restoration and tomographic reconstructions. A special attention will be devoted to one of the most recent trends in this field: structured sparsity, i.e. how to encode and use the structure of sparse coefficients. Among the concrete application examples, I will discuss noise reduction and tomographic reconstructions using magnetic resonance imaging (MRI) and quantitative microwave tomography.

#### The discrete Gaussian distribution and discrete Clifford-Hermite polynomials Hilde De Ridder (UGent)

Abstract:
Discrete Clifford analysis is a discrete version of Clifford analysis, i.e. a higher dimensional discrete function theory in a Clifford algebra context, on the simplest of all graphs, the rectangular Z^m grid.

In this talk, we introduce discrete distribution theory with as an important example the discrete Gaussian distribution. We will show links and differences with the continuous setting regarding its definition, properties,... By means of this discrete Gaussian distribution, we furthermore introduce discrete Clifford-Hermite polynomials; in particular for the so-called "radial Hermite polynomials", we discuss analoguous properties/formulas to the recurrence relation, Rodrigues' Formula, orthogonality relation and the differential equation which they satisfy.

#### Non-intersecting Brownian motions and complex analysis Steven Delvaux (KULeuven)

Abstract:
In this expository talk, we discuss non-intersecting 1-dimensional Brownian motion paths with prescribed starting and ending positions. We obtain some interesting phase transitions. We also discuss the limiting distribution of the paths. These problems lead to techniques from potential theory, orthogonal polynomials and complex analysis, particularly jump problems for piecewise analytic functions.

#### Wigner quantization of one-dimensional Hamiltonians Gilles Regnier (University of Ghent)

Abstract:
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization.

In this talk, we consider two Hamiltonians describing one-dimensional quantum systems: the Hamiltonian H=xp, which is notorious for its possible connection with the Riemann hypothesis, and the Hamiltonian of the free particle. Solutions for the compatibility conditions attached to these systems are found in terms of generators of the Lie superalgebra osp(1|2). The spectrum of the relevant operators H, x and p can then be obtained with the help of various orthogonal polynomials.

#### Harmonic and monogenic potentials in Euclidean halfspace Fred Brackx (University of Ghent)

Abstract:
In the framework of Clifford analysis a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space R^{m+1}. Their distributional limits at the boundary are computed, obtaining in this way well-known distributions in Rm such as the Dirac distribution, the Hilbert kernel, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.