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

A critical appraisal of appendage disparity and homology in fishes

Fishes are both extremely diverse and morphologically disparate. Part of this disparity can be observed in the numerous possible fin configurations that may differ in terms of the number of fins as well as fin shapes, sizes and relative positions on the body. Here, we thoroughly review the major patterns of disparity in fin configurations for each major group of fishes and discuss how median and paired fin homologies have been interpreted over time. When taking into account the entire span of fish diversity, including both extant and fossil taxa, the disparity in fin morphologies greatly complicates inferring homologies for individual fins. Given the phylogenetic scope of this review, structural and topological criteria appear to be the most useful indicators of fin identity. We further suggest that it may be advantageous to consider some of these fin homologies as nested within the larger framework of homologous fin‐forming morphogenetic fields. We also discuss scenarios of appendage evolution and suggest that modularity may have played a key role in appendage disparification. Fin modules re‐expressed within the boundaries of fin‐forming fields could explain how some fins may have evolved numerous times independently in separate lineages (e.g., adipose fin), or how new fins may have evolved over time (e.g., anterior and posterior dorsal fins, pectoral and pelvic fins). We favour an evolutionary scenario whereby median appendages appeared from a unique field of competence first positioned throughout the dorsal and ventral midlines, which was then redeployed laterally leading to paired appendages.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151971/1/faf12402_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151971/2/faf12402.pd

Global Diversity of Ascidiacea

The class Ascidiacea presents fundamental opportunities for research in the fields of development, evolution, ecology, natural products and more. This review provides a comprehensive overview of the current knowledge regarding the global biodiversity of the class Ascidiacea, focusing in their taxonomy, main regions of biodiversity, and distribution patterns. Based on analysis of the literature and the species registered in the online World Register of Marine Species, we assembled a list of 2815 described species. The highest number of species and families is found in the order Aplousobranchia. Didemnidae and Styelidae families have the highest number of species with more than 500 within each group. Sixty percent of described species are colonial. Species richness is highest in tropical regions, where colonial species predominate. In higher latitudes solitary species gradually contribute more to the total species richness. We emphasize the strong association between species richness and sampling efforts, and discuss the risks of invasive species. Our inventory is certainly incomplete as the ascidian fauna in many areas around the world is relatively poorly known, and many new species continue to be discovered and described each year

Research Note: On the error behaviour of force and moment sources in simplicial spectral finite elements

The representation of a force or moment point source in a spectral finite-element code for modelling elastic wave propagation becomes fundamentally different in degenerate cases where the source is located on the boundary of an element. This difference is related to the fact that the finite-element basis functions are continuous across element boundaries, but their derivatives are not. A method is presented that effectively deals with this problem. Tests on one-dimensional elements show that the numerical errors for a force source follow the expected convergence rate in terms of the element size, apart from isolated cases where superconvergence occurs. For a moment source, the method also converges but one order of accuracy is lost, probably because of the reduced regularity of the problem. Numerical tests in three dimensions on continuous mass-lumped tetrahedral elements show a similar error behaviour as in the one-dimensional case, although in three dimensions the loss of accuracy for the moment source is not a severe as a full order.Accepted Author ManuscriptApplied Geophysics and Petrophysic