By A. Treves, L. Maraschi, M. Abramowicz
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The useful equation of associativity is the subject of Abel's first contribution to Crelle's magazine. Seventy years later, it used to be featured because the moment a part of Hilbert's 5th challenge, and it used to be solved less than successively weaker hypotheses via Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar confirmed that the "triangular norms" brought through Menger in his definition of a probabilistic metric area might be associative; and of their booklet Probabilistic Metric areas, they offered the fundamental houses of such triangular norms and the heavily similar copulas.
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Extra resources for Accretion: A collection of influential papers
Yet many researchers suspect that the inexorable direction of the chemical path is similar on any habitable planet or moon [1, 7, 15]. Such a stepwise scenario informs attempts to define life, because the exact point at which such a system of gradually increasing complexity becomes “alive” is intrinsically arbitrary. The evolutionary path to cellular life must have featured a rich variety of intermediate, complex, self-replicating emergent chemical systems. Each of those steps represents a distinctive, fundamentally important stage in life’s molecular synthesis and organization.
Living cells undergo chemical reactions, not unlike burning in which two chemicals (oxygen and fuel) react and release energy. However, the trick in metabolism, unlike an open fire, is to capture part of that released energy to make new useful molecules that reinforce the cycle. So metabolism requires a sequence of chemical reactions that work in concert . 3 The iron-sulfur world The theoretical and experimental pursuit of the metabolism-first viewpoint is exemplified by Günter Wächtershäuser’s iron-sulfur world hypothesis [44, 45, 46, 47, 48].
One of the clever proposals in Günter Wächtershäuser’s model is that sulfide minerals promote primordial metabolic reactions. In fact, many modern metabolic enzymes have at their core a small cluster of iron or nickel and sulfur atoms – clusters that look like tiny bits of sulfide minerals. Perhaps ancient minerals played the role of enzymes. Wächtershäuser’s iron-sulfur world was manifest as flat life, which predated the emergence of cells. The first self-replicating entity was a thin layer of chemical reactants on a sulfide mineral surface.