The genesis and early developments of Aitken\u2019s process, Shanks\u2019 transformation, the \u3b5\u2013algorithm, and related fixed point methods

In this paper, we trace back the genesis of Aitken\u2019s \u3942 process and Shanks\u2019 sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Pad\ue9 approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks\u2019 transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis

Modulation of immune responses by targeting CD169/Siglec-1 with the glycan ligand

A fundamental role in the plant-bacterium interaction for Gram-negative phytopathogenic bacteria is played by membrane constituents, such as proteins, lipopoly- or lipooligosaccharides (LPS, LOS) and Capsule Polysaccharides (CPS). In the frame of the understanding the molecular basis of plant bacterium interaction, the Gram-negative bacterium Agrobacterium vitis was selected in this study. It is a phytopathogenic member of the Rhizobiaceae family and it induces the crown gall disease selectively on grapevines (Vitis vinifera). A. vitis wild type strain F2/5, and its mutant in the quorum sensing gene ΔaviR, were studied. The wild type produces biosurfactants; it is considered a model to study surface motility, and it causes necrosis on grapevine roots and HR (Hypersensitive Response) on tobacco. Conversely, the mutant does not show any surface motility and does not produce any surfactant material; additionally, it induces neither necrosis on grape, nor HR on tobacco. Therefore, the two strains were analyzed to shed some light on the QS regulation of LOS structure and the consequent variation, if any, on HR response. LOS from both strains were isolated and characterized: the two LOS structures maintained several common features and differed for few others. With regards to the common patterns, firstly: the Lipid A region was not phosphorylated at C4 of the non reducing glucosamine but glycosylated by an uronic acid (GalA) unit, secondly: a third Kdo and the rare Dha (3-deoxy-lyxo-2-heptulosaric acid) moiety was present. Importantly, the third Kdo and the Dha residues were substituted by rhamnose in a not stoichiometric fashion, giving four different oligosaccharide species. The proportions among these four species, is the key difference between the LOSs from both the two bacteria. LOS from both strains and Lipid A from wild type A. vitis are now examined for their HR potential in tobacco leaves and grapevine roots