Claudio Maccone

The statistical Drake equation

SETI is a comparatively new branch of scientific research that began only in 1959. The goal of SETI is to ascertain whether alien civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be.SETI is a comparatively new branch of scientific research that began only in 1959. The goal of SETI is to ascertain whether alien civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be.

SETI, Evolution and Human History Merged into a Mathematical Model

In this paper we propose a new mathematical model capable of merging Darwinian Evolution, Human History and SETI into a single mathematical scheme: (1) Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution. (2) In 2008 this author providedthe statistical generalization of the Drakeequation yielding the numberN of communicating ET civilizations in the Galaxy. N was shown to follow the lognormal probability distribution. (3) We call b-lognormals those lognormals starting at any positive time b (birth) larger than zero. Then the exponential growth curve becomes the geometric locus of the peaks of a one-parameter family of b-lognormals: this is our way to re-define Cladistics. (4) b-lognormals may be also be interpreted as the lifespan of any living being (a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization). Applying this new mathematical apparatus to Human History, leads to the discovery of the exponential progress between Ancient Greece and the current USA as the envelope of all b-lognormals of Western Civilizations over a period of 2500 years. (5) We then invoke Shannons Information Theory. The b-lognormals entropy turns out to be the index of development level reached by each historic civilization. We thus get a numerical estimate of the entropy difference between any two civilizations, like the Aztec-Spaniard difference in 1519. (6) In conclusion, we have derived a mathematical scheme capable of estimating how much more advanced than Humans an Alien Civilization will be when the SETI scientists will detect the first hints about ETs. Received 21 November 2012, accepted 25 February 2013, first published online 23 April 2013

Evolution and mass extinctions as lognormal stochastic processes

In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI . The relevant mathematical tools are:n (1) Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black–Sholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time. (2) The probability distributions known as b -lognormals, i.e. lognormals starting at a certain positive instant b >0 rather than at the origin. These b -lognormals were then forced by us to have their peak value located on the exponential mean-value curve of the GBM (Peak-Locus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics . (3) The (Shannon) entropy of such b -lognormals is then seen to represent the ‘degree of progress’ reached by each living organism or by each big set of living organisms, like historic human civilizations. Having understood this fact, human history may then be cast into the language of b -lognormals that are more and more organized in time (i.e. having smaller and smaller entropy, or smaller and smaller ‘chaos’), and have their peaks on the increasing GBM exponential. This exponential is thus the ‘trend of progress’ in human history. (4) All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics). (5) But the most striking new result is that the well-known ‘Molecular Clock of Evolution’, namely the ‘constant rate of Evolution at the molecular level’ as shown by Kimuras Neutral Theory of Molecular Evolution, identifies with growth rate of the entropy of our Evo-SETI model, because they both grew linearly in time since the origin of life. (6) Furthermore, we apply our Evo-SETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past: (a) one based on GBMs and (b) the other based on a parabolic mean value capable of covering both the extinction and the subsequent recovery of life forms. (7) Finally, we show that the Markov & Korotayev (2007, 2008) model for Darwinian Evolution identifies with an Evo-SETI model for which the mean value of the underlying lognormal stochastic process is a cubic function of the time. In conclusion: we have provided a new mathematical model capable of embracing molecular evolution, SETI and entropy into a simple set of statistical equations based upon b -lognormals and lognormal stochastic processes with arbitrary mean, of which the GBMs are the particular case of exponential growth .

INNOVATIVE SETI BY THE KLT

SETI searches are, by definition, the extraction of very weak radio signals out of the cosmic background noise. When SETI was born in 1959, it was natural to attempt this extraction by virtue of the only detection algorithm well known at the time: the Fourier Transform (FT). In fact: 1) SETI radio astronomers had adopted the viewpoint that a candidate ET signal would necessarily be a sinusoidal carrier, i.e. a very narrow-band signal. Over such a narrow band, the background noise is necessarily white. And so, the basic mathematical assumption behind the FT that the background noise must be white was perfectly matched to SETI for the next fifty years! 2) In addition, the Americans J. W. Cooley and J. W. Tukey discovered in April, 1965 that all FT computations could be speeded up by a factor of N/ln(N) (N is the number of numbers to be processed) by replacing the old FT with their own Fast Fourier Transform (FFT) algorithm. Then, SETI radio astronomers all over the world gladly, and unquestioningly, adopted the new FFT.

New Evo-SETI results about civilizations and molecular clock

In 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 GBMmean 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 themby 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 (BPDTheorem: no assigned senility). This is precisely what usually happens inHistory, 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 CivilizationC if the lives of its CitizensCi 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 trendmL(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 arewritten in theMaxima language, andMaxima 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. Received 11 August 2015, accepted 27 October 2015, first published online 28 March 2016

Optimal Trajectories from the Earth-Moon L1 and L3 Points to Deflect Hazardous Asteroids and Comets

Abstract: Software code named asteroff was recently created by the author to simulate the deflection of hazardous asteroids off of their collision course with the Earth. This code was both copyrighted and patented to avoid unauthorized use of ideas that could possibly be vital to construct a planetary defense system in the vicinity of the Earth. Having so said, the basic ideas and equations underlying the asteroff simulation code are openly described in this paper. A system of two space bases housing missiles is proposed to achieve the planetary defense of the Earth against dangerous asteroids and comets, collectively called impactors herein. We show that the layout of the Earth‐Moon system with the five relevant Lagrangian (or libration) points in space leads naturally to only one, unmistakable location of these two space bases within the sphere of influence of the Earth. These locations are at the two Lagrangian points L1 (between the Earth and the Moon) and L3 (in the direction opposite to the Moon from the Earth). We show that placing missile bases at L1 and L3 would enable those missiles to deflect the trajectory of impactors by hitting them orthogonally to their impact trajectory toward the Earth, so as to maximize their deflection. We show that confocal conics are the best class of trajectories fulfilling this orthogonal deflection requirement. One additional remark is that the theory developed in this paper is just a beginning for a wider set of future research. In fact, we only develop the Keplerian analytical theory for the optimal planetary defense achievable from the Earth‐Moon Lagrangian points L1 and L3. Much more sophisticated analytical refinements would be needed to: (1) take into account many perturbation forces of all kinds acting on both the impactors and missiles shot from L1 and L3; (2) add more (non‐optimal) trajectories of missiles shot from either the Lagrangian points L4 and L5 of the Earth‐Moon System or from the surface of the Moon itself; and (3) encompass the full range of missiles currently available to the US (and possibly other countries) so as to really see which impactors could be diverted by which missiles, even in the very simplified scheme outlined here. Published for the first time in February 2002, our Keplerian planetary defense theory has proved, in just one year, to be simple enough to catch the attention of scholars, in addition to popular writers, and even of someone from the US Military. These recent developments might possibly mark the beginning of an all embracing vision in planetary defense beyond all learned congressional activities, dramatic movies, and unknown military plans covered by secrecy.

So much gain at 550 AU

The gravitational focusing effect of the Sun is one of the most amazing discoveries produced by the general theory of relativity. The first paper in this field was published by Albert Einstein in 1936 [1], but his work was virtually forgotten until 1964, when Sydney Liebes of Stanford University [2] gave the mathematical theory of gravitational focusing by a galaxy located between the Earth and a very distant cosmological object, such as a quasar.

Focusing the Galactic Internet

The gravitational lens of the Sun is an astrophysical phenomenon predicted by Einstein’s general theory of relativity. It implies that, if we can send a probe along any radial direction away from the Sun up to the minimal distance of 550 AU and beyond, the Sun’s mass will act as a huge magnifying lens, letting us “see” detailed radio maps of whatever may lie on its other side even at very large distances. This author’s recent book (Maccone, 2009) studies such future FOCAL space missions to 550 AU and beyond.

A real-time KLT implementation for radio-SETI applications

SETI, the Search for ExtraTerrestrial Intelligence, is the search for radio signals emitted by alien civilizations living in the Galaxy. Narrow-band FFT-based approaches have been preferred in SETI, since their computation time only grows like N*lnN, where N is the number of time samples. On the contrary, a wide-band approach based on the Kahrunen-Lo`eve Transform (KLT) algorithm would be preferable, but it would scale like N*N. In this paper, we describe a hardware-software infrastructure based on FPGA boards and GPU-based PCs that circumvents this computation-time problem allowing for a real-time KLT.

Evolution and mass extinctions as lognormal stochastic processes – CORRIGENDUM

In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI. The relevant mathematical tools are: (1) Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in themathematics of finances (Black–Sholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time. (2) The probability distributions known as b-lognormals, i.e. lognormals starting at a certain positive instant b>0 rather than at the origin. These b-lognormals were then forced by us to have their peak value located on the exponential mean-value curve of the GBM (Peak-Locus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics. (3) The (Shannon) entropy of such b-lognormals is then seen to represent the ‘degree of progress’ reached by each living organism or by each big set of living organisms, like historic human civilizations. Having understood this fact, human history may then be cast into the language of b-lognormals that are more and more organized in time (i.e. having smaller and smaller entropy, or smaller and smaller ‘chaos’), and have their peaks on the increasing GBM exponential. This exponential is thus the ‘trend of progress’ in human history. (4) All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics). (5) But the most striking new result is that the well-known ‘Molecular Clock of Evolution’, namely the ‘constant rate of Evolution at themolecular level’ as shown byKimura’sNeutral Theory ofMolecular Evolution, identifies with growth rate of the entropy of our Evo-SETI model, because they both grew linearly in time since the origin of life. (6) Furthermore, we apply our Evo-SETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past: (a) one based on GBMs and (b) the other based on a parabolicmean value capable of covering both the extinction and the subsequent recovery of life forms. (7) Finally, we show that the Markov & Korotayev (2007, 2008) model for Darwinian Evolution identifies with an Evo-SETI model for which the mean value of the underlying lognormal stochastic process is a cubic function of the time. In conclusion: we have provided a new mathematical model capable of embracing molecular evolution, SETI and entropy into a simple set of statistical equations based upon b-lognormals and lognormal stochastic processes with arbitrary mean, of which the GBMs are the particular case of exponential growth. Received 20 January 2014, accepted 6 March 2014, first published online 21 July 2014