AbstractIn two recent papers (Maccone 2013, 2014) as well as in the book (Maccone 2012), this author described the Evolution of life on Earth over the last 3.5 billion years as a lognormal stochastic process in the increasing number of living Species. In (Maccone 2012, 2013), the process used was ‘Geometric Brownian Motion’ (GBM), largely used in Financial Mathematics (Black-Sholes models). The GBM mean value, also called ‘the trend’, always is an exponential in time and this fact corresponds to the so-called ‘Malthusian growth’ typical of population genetics. In (Maccone 2014), the author made an important generalization of his theory by extending it to lognormal stochastic processes having an arbitrary trend mL(t), rather than just a simple exponential trend as the GBM have.The author named ‘Evo-SETI’ (Evolution and SETI) his theory inasmuch as it may be used not only to describe the full evolution of life on Earth from RNA to modern human societies, but also the possible evolution of life on exoplanets, thus leading to SETI, the current Search for ExtraTerrestrial Intelligence. In the Evo-SETI Theory, the life of a living being (let it be a cell or an animal or a human or a Civilization of humans or even an ET Civilization) is represented by a b-lognormal, i.e. a lognormal probability density function starting at a precise instant b (‘birth’) then increasing up to a peak-time p, then decreasing to a senility-time s (the descending inflexion point) and then continuing as a straight line down to the death-time d (‘finite b-lognormal’).(1)Having so said, the present paper describes the further mathematical advances made by this author in 2014–2015, and is divided in two halves: Part One, devoted to new mathematical results about the History of Civilizations as b-lognormals, and(2)Part Two, about the applications of the Evo-SETI Theory to the Molecular Clock, well known to evolutionary geneticists since 50 years: the idea is that our EvoEntropy grows linearly in time just as the molecular clock.
(a)Summarizing the new results contained in this paper: In Part One, we start from the History Formulae already given in (Maccone 2012, 2013) and improve them by showing that it is possible to determine the b-lognormal not only by assigning its birth, senility and death, but rather by assigning birth, peak and death (BPD Theorem: no assigned senility). This is precisely what usually happens in History, when the life of a VIP is summarized by giving birth time, death time, and the date of the peak of activity in between them, from which the senility may then be calculated (approximately only, not exactly). One might even conceive a b-scalene (triangle) probability density just centred on these three points (b, p, d) and we derive the relevant equations. As for the uniform distribution between birth and death only, that is clearly the minimal description of someone's life, we compare it with both the b-lognormal and the b-scalene by comparing the Shannon Entropy of each, which is the measure of how much information each of them conveys. Finally we prove that the Central Limit Theorem (CLT) of Statistics becomes a new ‘E-Pluribus-Unum’ Theorem of the Evo-SETI Theory, giving formulae by which it is possible to find the b-lognormal of the History of a Civilization C if the lives of its Citizens Ci are known, even if only in the form of birth and death for the vast majority of the Citizens.(b)In Part Two, we firstly prove the crucial Peak-Locus Theorem for any given trend mL(t) and not just for the GBM exponential. Then we show that the resulting Evo-Entropy grows exactly linearly in time if the trend is the exponential GMB trend.(c)In addition, three Appendixes (online) with all the relevant mathematical proofs are attached to this paper. They are written in the Maxima language, and Maxima is a symbolic manipulator that may be downloaded for free from the web.In conclusion, this paper further increases the huge mathematical spectrum of applications of the Evo-SETI Theory to prepare Humans for the first Contact with an Extra-Terrestrial Civilization.
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