Hermitian clifford analysis

This paper gives an overview of some basic results on Hermitian Clifford analysis, a refinement of classical Clifford analysis dealing with functions in the kernel of two mutually adjoint Dirac operators invariant under the action of the unitary group. The set of these functions, called Hermitian monogenic, contains the set of holomorphic functions in several complex variables. The paper discusses, among other results, the Fischer decomposition, the Cauchy–Kovalevskaya extension problem, the axiomatic radial algebra, and also some algebraic analysis of the system associated with Hermitian monogenic functions. While the Cauchy–Kovalevskaya extension problem can be carried out for the Hermitian monogenic system, this system imposes severe constraints on the initial Cauchy data. There exists a subsystem of the Hermitian monogenic system in which these constraints can be avoided. This subsystem, called submonogenic system, will also be discussed in the paper

Spherical harmonics and integration in superspace II

The study of spherical harmonics in superspace, introduced in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212], is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces, which also determines the irreducible pieces under the action of SO(m) x Sp(2n). In the second part of the paper, this decomposition is used to describe all possible integrations over the supersphere. It is then shown that only one possibility yields the orthogonality of spherical harmonics of different degree. This is the so-called Pizzetti-integral of which it was shown in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212] that it leads to the Berezin integral.Comment: 18 pages, accepted for publication in J. Phys.

A matrix recurrence for systems of Clifford algebra-valued orthogonal polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory ("Matrix representations of a basic polynomial sequence in arbitrary dimension". Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Guerlebeck, Bock, Lavicka, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.Fundação para a Ciência e a Tecnologia (FCT

Hypercomplex polynomials, vietoris’ rational numbers and a related integer numbers sequence

This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For S a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

Establishing key research questions for the implementation of artificial intelligence in colonoscopy - a modified Delphi method

Background and Aims Artificial intelligence (AI) research in colonoscopy is progressing rapidly but widespread clinical implementation is not yet a reality. We aimed to identify the top implementation research priorities. Methods An established modified Delphi approach for research priority setting was used. Fifteen international experts, including endoscopists and translational computer scientists/engineers from 9 countries participated in an online survey over 9 months. Questions related to AI implementation in colonoscopy were generated as a long-list in the first round, and then scored in two subsequent rounds to identify the top 10 research questions. Results The top 10 ranked questions were categorised into 5 themes. Theme 1: Clinical trial design/end points (4 questions), related to optimum trial designs for polyp detection and characterisation, determining the optimal end-points for evaluation of AI and demonstrating impact on interval cancer rates. Theme 2: Technological Developments (3 questions), including improving detection of more challenging and advanced lesions, reduction of false positive rates and minimising latency. Theme 3: Clinical adoption/Integration (1 question) concerning effective combination of detection and characterisation into one workflow. Theme 4: Data access/annotation (1 question) concerning more efficient or automated data annotation methods to reduce the burden on human experts. Theme 5: Regulatory Approval (1 question) related to making regulatory approval processes more efficient. Conclusions This is the first reported international research priority setting exercise for AI in colonoscopy. The study findings should be used as a framework to guide future research with key stakeholders to accelerate the clinical implementation of AI in endoscopy