Time filter

Source Type

Maccone C.,International Academy of Astronautics IAA
Proceedings of the International Astronautical Congress, IAC | Year: 2016

Computers for SETI are exploiting the most advanced technologies available nowadays: massive data handling, parallel computing, and advanced signal processing. In addition, understanding an alien message like the messages based on Lincos (Lingua Cosmica) might possibly require the use of Artificial Intelligence (A.I.) to an unprecedented scale. Thus, the role of computers in SETI is ever increasing, and one might wonder whether the time will come when "computers will take over". Ray Kurzweil's famous 2006 book "The Singularity is Near" predicted that the Singularity (i.e. computers taking over humans) would occur around the year 2045. In this paper we show that Kurzweil's prediction is in perfect numerical agreement with the "Evo-SETI" (Evolution and SETI) mathematical model that this author has developed over the last five years in a series of highly mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology. The key ideas of Evo-SETI Theory are: 1) Evolution of Life on Earth over the last 3.5 billion years is a Geometric Brownian Motion (GBM) in the number of the Living Darwinian Species. It thus increases exponentially in time, and is compatible with the Statistical Drake Equation of SETI. 2) The level of advancement of each Living Species is the (Shannon) ENTROPY of the b-lognormal probability density (i.e. a lognormal starting at the positive time b (birth)) corresponding to that Species (Peak-Locus Theorem of Evo-SETI Theory). 3) Humanity is now very close to the point of minimum radius of curvature of the GBM exponential, called "GBM's knee". This knee is precisely Kurzweil's SINGULARITY in that before it the exponential growth was very slow (i.e. animal and human Species made of flesh reproducing sexually over millions of years), whereas after the SINGULARITY the exponential growth is extremely rapid (computers reproducing technologically faster and faster). 4) We then prove mathematically the following Evo-SETI Singularity Theorem: if the Singularity occurs in 2045, then life on Earth MUST have started between 3.5 and 3.8 billion years ago, that is precisely what actually occurred in reality. Therefore, our Evo-SETI Theory and Kurzweil's Singularity prediction match perfectly. Above all, we now have a key to evaluate mathematically HOW MUCH an Alien Post-Biological Civilization will be MORE ADVANCED THAN HUMANS when SETI astronomers will for the first time detect one on nearby Exoplanets in the Milky Way.


MacCone C.,International Academy of Astronautics IAA | MacCone C.,SETI Permanent Study Group of the IAA
Acta Astronautica | Year: 2011

The statistics of habitable planets may be based on a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book "Habitable planets for man" (1964). In this paper, we first provide the statistical generalization of the original and by now too simplistic Dole equation. In other words, a product of ten positive numbers is now turned into the product of ten positive random variables. This we call the SEH, an acronym standing for "Statistical Equation for Habitables". The mathematical structure of the SEH is then derived. The proof is based on the central limit theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that The new random variable NHab, yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the lognormal distribution. By construction, the mean value of this lognormal distribution is the total number of habitable planets as given by the statistical Dole equation. But now we also derive the standard deviation, the mode, the median and all the moments of this new lognormal NHab random variable.The ten (or more) astrobiological factors are now positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our SEH by allowing an arbitrary probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations.An application of our SEH then follows. The (average) distance between any two nearby habitable planets in the Galaxy may be shown to be inversely proportional to the cubic root of NHab. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed "Maccone distribution" by Paul Davies in 2008.Data Enrichment Principle. It should be noticed that ANY positive number of random variables in the SEH is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the SEH we call the "Data Enrichment Principle", and we regard it as the key to more profound future results in the fields of Astrobiology and SETI.A practical example is then given of how our SEH works numerically. We work out in detail the case where each of the ten random variables is uniformly distributed around its own mean value as given by Dole back in 1964 and has an assumed standard deviation of 10%. The conclusion is that the average number of habitable planets in the Galaxy should be around 100 million±200 million, and the average distance in between any couple of nearby habitable planets should be about 88 light years±40 light years.Finally, we match our SEH results against the results of the Statistical Drake Equation that we introduced in our 2008 IAC presentation. As expected, the number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). And the average distance between any two nearby habitable planets turns out to be much smaller than the average distance between any two neighboring ET civilizations: 88 light years vs. 2000 light years, respectively. This means an ET average distance about 20 times higher than the average distance between any couple of adjacent habitable planets. © 2010 Elsevier Ltd. All rights reserved.


MacCone C.,International Academy of Astronautics IAA | MacCone C.,SETI Permanent Study Group of the IAA
Acta Astronautica | Year: 2011

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 the other side of the Sun even at very large distances. The recent book by this author (Claudio Maccone, Deep Space Flight and Communications, 414 pages, Praxis-Springer, 2009) describes such future FOCAL space missions to 550 AU and beyond. In this paper, however, we want to study another possibility yet: how to enable the future interstellar radio links between the solar system and any future interstellar probe by utilizing the gravitational lens of the Sun as a huge antenna. In particular, we compare the bit error rate (BER) across interstellar distances with and without using the gravitational lens effect of the Sun. The conclusion is that only when we will exploit the Sun as a gravitational lens we will be able to communicate with our own probes (or with nearby Aliens) across the distances of even the nearest stars to us in the Galaxy and that at a reasonable bit error rate. Furthermore, we study the radio bridge between the Sun and any other Star that is made up by the two gravitational lenses of both the Sun and that Star. The alignment for this radio bridge to work is very strict, but the power saving is enormous, due to the huge contributions of the two stars' lenses to the overall antenna gain of the system. For instance, we study in detail: The SunAlpha Cen A radio bridge.The SunBarnard's star radio bridge.The SunSirius A radio bridge.The radio bridge between the Sun and any Sun-like star located in the galactic bulge.The radio bridge between the Sun and any Sun-like star located inside the Andromeda galaxy (M 31). The conclusion is that a radio interstellar communications network can indeed be built if the gravitational lenses of all stars involved are exploited. Then, the new question arises: has any advanced civilization already built such a radio telecommunication network? If so, our current and future SETI searches should be tuned-up to match with this newly realized possibility. © 2010 Elsevier Ltd. All rights reserved.


MacCone C.,International Academy of Astronautics IAA
Acta Astronautica | Year: 2010

We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that:The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found.The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course.An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed "Maccone distribution" by Paul Davies.DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random variables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation, we call the "Data Enrichment Principle," and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case, where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) 1 billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization. © 2010 Elsevier Ltd. All rights reserved.


Maccone C.,International Academy of Astronautics IAA
Proceedings of the International Astronautical Congress, IAC | Year: 2015

Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA ?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics ("Black-Sholes models"), this "exponential" stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see ref. [3], Chapter 30, and ref. [4]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: This result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see ref. [1]). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see ref. [4]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b ("birth"), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the 6-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae ("History Formulae") allowing one to find this "finite i-lognormal" when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of i-lognormals. Since A-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the 6-lognormals become more and more "peaky", and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the "level of civilization". This "level of civilization" is what physicists call ENTROPY. Also, in refs. [3] and [4], this author proved that, for all GBMs, the (Shannon) Entropy of the i-lognormals in his Peak-Locus Theorem grows LINEARLY in time. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base- substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed "Darwinian selection laws" (macroscopic laws). This is Kimura's neutral theory of molecular evolution. The conclusion is that the Molecular Clock and the b- lognormal Entropy are the same thing. At last, we reach the new, original result justifying the publication of this paper. On exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially. Thus, Evo-Entropy, i.e. the (Shannon) Entropy of the generic b-lognormal of the Peak-Locus Theorem, provides the Evo-SETI SCALE to measure the evolution of life on Exoplanets. Copyright © (2015) by International Astronautical Federation All rights reserved.


Maccone C.,International Academy of Astronautics IAA
Acta Astronautica | Year: 2014

In a series of recent papers (Refs. [1-5,7,8]) and in a book (Ref. [6]), this author suggested a new mathematical theory capable of merging Darwinian Evolution and SETI into a unified statistical framework. In this new vision, Darwinian Evolution, as it unfolded on Earth over the last 3.5 billion years, is defined as just one particular realization of a certain lognormal stochastic process in the number of living species on Earth, whose mean value increased in time exponentially. SETI also may be brought into this vision since the number of communicating civilizations in the Galaxy is given by a lognormal distribution (Statistical Drake Equation). Now, in this paper we further elaborate on all that particularly with regard to two important topics: The introduction of the general lognormal stochastic process L(t) whose mean value may be an arbitrary continuous function of the time, m(t), rather than just the exponential mGBM(t)=N0 eμt typical of the Geometric Brownian Motion (GBM). This is a considerable generalization of the GBM-based theory used in Refs. [1-8].The particular application of the general stochastic process L(t) to the understanding of Mass Extinctions like the K-Pg one that marked the dinosaursend 65 million years ago. We first model this Mass Extinction as a decreasing Geometric Brownian Motion (GBM) extending from the asteroids impact time all through the ensuing "nuclear winter". However, this model has a flaw: the "final value" of the GBM cannot have a horizontal tangent, as requested to enable the recovery of life again after this "final extinction value".That flaw, however, is removed if the rapidly decreasing mean value function of L(t) is the left branch of a parabola extending from the asteroids impact time all through the ensuing "nuclear winter" and up to the time when the number of living species on Earth started growing up again, as we show mathematically in Section 3. In conclusion, we have uncovered an important generalization of the GBM into the general lognormal stochastic process L(t), paving the way to a better, future understanding the evolution of life on Exoplanets on the basis of what Evolution unfolded on Earth in the last 3.5 billion years. That will be the goal of further research papers in the future. © 2014 IAA. All rights reserved.


Maccone C.,International Academy of Astronautics IAA
Proceedings of the International Astronautical Congress, IAC | Year: 2014

The KLT (acronym for Karhunen-Loève Transform) is a mathematical algorithm superior to the classical FFT in many regards: 1) The KLT can filter signals out of the background noise over both wide and narrow bands. That is in sharp contrast to the FFT that rigorously applies to narrow-band signals only. 2) The KLT can be applied to random functions that are non-stationary in time, i.e. whose autocorrelation is a function of the two independent variables t1 and t2 separately. Again, this is a sheer advantage of the KLT over the FFT, inasmuch as the FFT rigorously applies to stationary' processes only, i.e. processes whose autocorrelation is a function of the absolute value of the difference of t1 and t2 only. 3) The KLT can detect signals embedded in noise to unbelievably small values of the Signal-to-Noise Ratio (SNR), like 10-3 or so. This particular feature of the KLT is studied in detail in this paper. An excellent filtering algorithm like the KLT, however, comes with a cost that one must be ready to pay for especially in SETI: its computational burden is much higher than for the FFT. In fact, it can be shown that no fast KLT transform can possibly exist and, for an autocorrelation matrix of size N, the calculations must be of the order of N2, rather than N∗log(N). Nevertheless, for moderate values of N (in the hundreds) the KLT dominates over the FFT, as shown by the numerical simulations. Finally, an important and recent (2007-2008) development in the KLT theory, called the "Bordered Autocorrelation Method" (BAM) is presented. This BAM-KLT method gets around the difficulty of the N2 brunt calculations and ends up in the following unexpected theorem: the KLT of a feeble sinusoidal carrier embedded into a lot of white stationary noise is given by the Fourier transform of the derivative of the largest KLT eigenvalue with respect to the bordering index. This basic result is fully proved analytically in the final sections of this paper.


Maccone C.,International Academy of Astronautics IAA
Acta Astronautica | Year: 2015

Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics ("Black-Sholes models"), this "exponential" stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see Ref. [3], Chapter 30, and Ref. [4]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see Ref. [1]). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see Ref. [4]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b ("birth"), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae ("History Formulae") allowing one to find this "finite b-lognormal" when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more "peaky", and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the "level of civilization". This "level of civilization" is what physicists call ENTROPY. In Refs. [3] and [4], this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. At last, we reach the new, original result justifying the publication of this paper. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed "Darwinian selection laws" (macroscopic laws). This is Kimura's neutral theory of molecular evolution. The conclusion of this paper is that the Molecular Clock and the b-lognormal Entropy are the same thing. And, on exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially. © 2015 IAA. Published by Elsevier Ltd. All rights reserved.


MacCone C.,International Academy of Astronautics IAA
International Journal of Astrobiology | Year: 2013

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 provided the statistical generalization of the Drake equation yielding the number N 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 Shannon's 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. Copyright © Cambridge University Press 2013 The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence . The written permission of Cambridge University Press must be obtained for commercial re-use. © 2013 Cambridge University Press.


Maccone C.,International Academy of Astronautics IAA
International Journal of Astrobiology | Year: 2014

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 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 Kimura's 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. © Cambridge University Press 2014.

Loading International Academy of Astronautics IAA collaborators
Loading International Academy of Astronautics IAA collaborators