56th International Astronautical Congress of the International Astronautical Federation, the International Academy of Astronautics, and the International Institute of Space Law, 2005
Assume that the history of Astronomy and SETI is the leading proof of the evolution of human inte... more Assume that the history of Astronomy and SETI is the leading proof of the evolution of human intelligence on Earth over the last 3000 years. Then, human intelligence has increased a lot, although not at a uniform pace. A mathematical description of how much has human intelligence increased, however, is difficult to achieve. In this paper, we cast a mathematical model the evolution of human intelligence over the last three thousand years that seems to reflect reasonably well both what is known from past and might be extrapolated into the future. Our model is based on two seminal books: 1) “Cosmos” by Carl Sagan (1980), widely known to millions of people all over the western world because of the corresponding TV series, and 2) “Interstellar Migration and the Human Experience” (by Ben Finney and Eric Jones, Editors, University of California Press, 1983). Our model is based on the use of two cubic curves, representing the evolution of Astronomy and of SETI, respectively. We conclude by extrapolating these curves into the future and reach the conclusion that the ”Star Trek” age of humankind might possibly begin by the end of this century.
Abstract Ray Kurzweil's famous 2006 book “The Singularity Is Near” predicted that the Singula... more Abstract 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 prove that Kurzweil's prediction is in agreement with the “Evo-SETI” (Evolution and SETI)” mathematical model that this author has developed over the last five years in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology. The key ideas of Evo-SETI are: 1) Evolution of life on Earth over the last 3.5 billion years is a stochastic process in the number of living Species called Geometric Brownian Motion (GBM). It increases exponentially in time and is in agreement with the Statistical Drake Equation of SETI (see ref. [1] ). 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 knee”. We claim that this knee is precisely Kurzweil's Singularity, in that before the Singularity the exponential growth was very slow (these are animal and human Species made of meat and reproducing sexually over millions of years), whereas, after the Singularity, the exponential growth will be extremely rapid (computers reproducing technologically faster and faster in time). But how is this paper structured in detail? Well, first of all (Part 1) we describe what the GBM is, and why it reflects the stochastic exponential increase that occurred in Darwinian Evolution for over 3.5 billion years. Please notice that the denomination “Geometric Brownian Motion” (taken from Financial Mathematics) is incorrect since the GBM is NOT a Brownian motion as understood by physicists (i.e. a stochastic process whose probability density function (pdf) is a Gaussian). On the contrary, the GBM is a lognormal process, i.e. a process whose pdf is a lognormal pdf. Next (Part 2) we compute the time when the GBM knee occurs (i.e. the time of minimum radius of curvature) and find what we call the knee equation, i.e. the relationship between t_knee, ts (the time of the origin of life on Earth) and B (the rate of growth of the GBM exponential. This equation holds good for any time assumed to be the Singularity time, either in the past, or now, or in the future. Then (Part 3) Ray Kurzweil's claim that the Singularity is near becomes part of our Evo-SETI Theory in that t_knee is set to zero (i.e. approximately nowadays, when compared to the 3.5 billions of years of past Darwinian Evolution of life on Earth). This leads to a very easy form of the GBM exponential as well as to the discovery of a pair of important new equations: 1) The inverse proportionality between the average number of Species living NOW on Earth and B, the pace of evolution. In other words, it would be possible to find B were the biologists able to tell us “fairly precisely” how many Species live on Earth nowadays. Unfortunately, this is not the case since, when it comes to insects and so on, the number of Species is so huge that it is not even known if it ranges in the millions or even in the billions. 2) More promising appears to be another new equation, that we discovered, relating the time of the origin of life on Earth, ts (that is known fairly precisely to range between 3.5 and 3.8 billion years ago) and the average number of living Species NOW. Finally, the mathematical machinery typical of the Evo-SETI Theory is called into action: 1) The Peak-Locus Theorem stating that the GBM exponential is where ALL PEAKS of the b-lognormals running left-to-right are located, so that the b-lognormals become higher and higher and narrower and narrower (with area =1 as the normalization). 2) The Shannon ENTROPY as EVOLUTION MEASURE of the b-lognormals, more correctly with the sign reversed and starting at the time of the origin of life on Earth, that is rather called EvoEntropy. 3) After this point, one more paper should be written to describe… how the b-lognormal's “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)… 4) …but this is “too much to be done now”, and so we leave it to a new, forthcoming paper.
The international scientific community, and especially the IAA (International Academy of As-trona... more The international scientific community, and especially the IAA (International Academy of As-tronautics) have long been discussing the need to keep the Farside of the Moon free of man-made RFI (radio frequency interference). In fact, the center of the Farside, specifically crater Daedalus, is ideal to set up a future radiotelescope (or phased array) to detect radio waves of all kinds that it is impossible to detect on Earth because of both the ever-growing RFI and of the intervening ionosphere. Nobody, however, seems to have established a precise border for the circular region around the antipode of the Earth (i.e. zero latitude and 180o longitude both East and West) that should be Protected from wild human exploitation when several nations will have reached the capability of easy travel to the Moon. In this paper we propose the cre-ation of PAC, the Protected Antipode Circle, centered around the antipode on the Farside and spanning an angle of 30o in longitude, in latitude and in all radial directions from the antipode. There are sound scientific reasons for this definition: 1) PAC is the most shielded area of the Farside, with an expected attenuation of man-made RFI of 100 dB or higher; 2) PAC is the only area of the Farside that will never be reached by the radiation emitted by future human space bases located at the L4 and L5 Lagrangian points of the Earth-Moon system; 3) PAC does not overlap with other areas of interest to human activity except for possibly a minor common area with the Aitken Basin, the southern depression supposed to have been created 3.8 billion years ago during the "big wham" between the Earth and the Moon. In view of these unique features, we propose PAC to be officially recognized by the United Nations as an Internationally Protected Area where no radio contamination by humans will possibly take place now and in the future for the benefit of all Humankind.
... And FOCAL (an acronym for Fast Outgoing Craft for Astrophysical Lensing), will not just b... more ... And FOCAL (an acronym for Fast Outgoing Craft for Astrophysical Lensing), will not just be looking at a point, but rather at a small line segment in the sky, enormously enlarged. SunLensing the Cosmic Microwave Background from 763 AU Claudio Maccone a "G. Colombo ...
The nonlinear time dependent Brownian motion, B(g(t)), is the object of the present paper. The ti... more The nonlinear time dependent Brownian motion, B(g(t)), is the object of the present paper. The time-rescaling function, g(t), may be any function continuous over a finite positive time interval ranging from O to a fixed T, and such that g(0)=0. We firstly prove B(g(t)) to be equivalent to a white—noise integral having zero mean and a time—dependent variance. Secondly we find the eigenfunction (Karhunen — Loeve) expansion of B(g(t)). The eigenfunctions are Bessel functions of the first kind, with a suitable time function for argument, and they are multiplied by another time function. The eigenvalues virtually are determined by the zeros of the Bessel functions.
Just as Chapter 23 showed the application of the general results obtained in Chapter 22 to the pa... more Just as Chapter 23 showed the application of the general results obtained in Chapter 22 to the particular time-rescaled stochastic process \( {B}(t^{2H})\), the present appendix investigates the application of the general results obtained in Chapter 24 about the square time-rescaled process B2(...) to the particular process \( {B^2}(t^{2H})\).
In two recent papers [1, 2] the author proved that radio communications between any pair of stars... more In two recent papers [1, 2] the author proved that radio communications between any pair of stars within our Galaxy are feasible with modest transmitted powers if the gravitational lenses of both stars are exploited. In this chapter we extend those innovative results to the case of radio communications between nearby galaxies. We show that radio communications between galaxies may become feasible if supermassive black holes, usually located at the center of galaxies, are exploited as gravitational lenses. In other words, a massive black hole may be regarded as a huge focusing device for radio waves being transmitted out of that galaxy and/or being received from another galaxy.
Since the 1990s the international scientific community—especially the IAA (International Academy ... more Since the 1990s the international scientific community—especially the IAA (International Academy of Astronautics)—have been discussing the need to keep the farside of the Moon free from man-made radio frequency interference (RFI). In fact, the center of the farside, specifically the Daedalus crater, is an ideal site to set up a future radio telescope (or, more realistically, a phased array) to detect radiowaves of all kinds, which are impossible to detect on Earth because of the ever-growing RFI.
Let us now change topics completely! Rather than looking for ETs in the Galaxy, we now seek habit... more Let us now change topics completely! Rather than looking for ETs in the Galaxy, we now seek habitable planets for man in the Galaxy. How many are there and how far from us is the nearest such habitable planet?
It is well known that the probability density function(pdf) \( f_{aX+b} (x) \)of the random varia... more It is well known that the probability density function(pdf) \( f_{aX+b} (x) \)of the random variable \( {aX+b}\)where a and b are arbitrary real constants with respect to the independent variable x, is related to the pdf \( f_{\rm{X}}(x)\)of the random variable X by the linear transformation formula for random variables that reads: $$ f_{aX+b}(x)=\frac{1}{\mid a \mid} f_X \left(\frac{x-b}{\mid a \mid}\right).$$ (5.1)
56th International Astronautical Congress of the International Astronautical Federation, the International Academy of Astronautics, and the International Institute of Space Law, 2005
Assume that the history of Astronomy and SETI is the leading proof of the evolution of human inte... more Assume that the history of Astronomy and SETI is the leading proof of the evolution of human intelligence on Earth over the last 3000 years. Then, human intelligence has increased a lot, although not at a uniform pace. A mathematical description of how much has human intelligence increased, however, is difficult to achieve. In this paper, we cast a mathematical model the evolution of human intelligence over the last three thousand years that seems to reflect reasonably well both what is known from past and might be extrapolated into the future. Our model is based on two seminal books: 1) “Cosmos” by Carl Sagan (1980), widely known to millions of people all over the western world because of the corresponding TV series, and 2) “Interstellar Migration and the Human Experience” (by Ben Finney and Eric Jones, Editors, University of California Press, 1983). Our model is based on the use of two cubic curves, representing the evolution of Astronomy and of SETI, respectively. We conclude by extrapolating these curves into the future and reach the conclusion that the ”Star Trek” age of humankind might possibly begin by the end of this century.
Abstract Ray Kurzweil's famous 2006 book “The Singularity Is Near” predicted that the Singula... more Abstract 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 prove that Kurzweil's prediction is in agreement with the “Evo-SETI” (Evolution and SETI)” mathematical model that this author has developed over the last five years in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology. The key ideas of Evo-SETI are: 1) Evolution of life on Earth over the last 3.5 billion years is a stochastic process in the number of living Species called Geometric Brownian Motion (GBM). It increases exponentially in time and is in agreement with the Statistical Drake Equation of SETI (see ref. [1] ). 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 knee”. We claim that this knee is precisely Kurzweil's Singularity, in that before the Singularity the exponential growth was very slow (these are animal and human Species made of meat and reproducing sexually over millions of years), whereas, after the Singularity, the exponential growth will be extremely rapid (computers reproducing technologically faster and faster in time). But how is this paper structured in detail? Well, first of all (Part 1) we describe what the GBM is, and why it reflects the stochastic exponential increase that occurred in Darwinian Evolution for over 3.5 billion years. Please notice that the denomination “Geometric Brownian Motion” (taken from Financial Mathematics) is incorrect since the GBM is NOT a Brownian motion as understood by physicists (i.e. a stochastic process whose probability density function (pdf) is a Gaussian). On the contrary, the GBM is a lognormal process, i.e. a process whose pdf is a lognormal pdf. Next (Part 2) we compute the time when the GBM knee occurs (i.e. the time of minimum radius of curvature) and find what we call the knee equation, i.e. the relationship between t_knee, ts (the time of the origin of life on Earth) and B (the rate of growth of the GBM exponential. This equation holds good for any time assumed to be the Singularity time, either in the past, or now, or in the future. Then (Part 3) Ray Kurzweil's claim that the Singularity is near becomes part of our Evo-SETI Theory in that t_knee is set to zero (i.e. approximately nowadays, when compared to the 3.5 billions of years of past Darwinian Evolution of life on Earth). This leads to a very easy form of the GBM exponential as well as to the discovery of a pair of important new equations: 1) The inverse proportionality between the average number of Species living NOW on Earth and B, the pace of evolution. In other words, it would be possible to find B were the biologists able to tell us “fairly precisely” how many Species live on Earth nowadays. Unfortunately, this is not the case since, when it comes to insects and so on, the number of Species is so huge that it is not even known if it ranges in the millions or even in the billions. 2) More promising appears to be another new equation, that we discovered, relating the time of the origin of life on Earth, ts (that is known fairly precisely to range between 3.5 and 3.8 billion years ago) and the average number of living Species NOW. Finally, the mathematical machinery typical of the Evo-SETI Theory is called into action: 1) The Peak-Locus Theorem stating that the GBM exponential is where ALL PEAKS of the b-lognormals running left-to-right are located, so that the b-lognormals become higher and higher and narrower and narrower (with area =1 as the normalization). 2) The Shannon ENTROPY as EVOLUTION MEASURE of the b-lognormals, more correctly with the sign reversed and starting at the time of the origin of life on Earth, that is rather called EvoEntropy. 3) After this point, one more paper should be written to describe… how the b-lognormal's “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)… 4) …but this is “too much to be done now”, and so we leave it to a new, forthcoming paper.
The international scientific community, and especially the IAA (International Academy of As-trona... more The international scientific community, and especially the IAA (International Academy of As-tronautics) have long been discussing the need to keep the Farside of the Moon free of man-made RFI (radio frequency interference). In fact, the center of the Farside, specifically crater Daedalus, is ideal to set up a future radiotelescope (or phased array) to detect radio waves of all kinds that it is impossible to detect on Earth because of both the ever-growing RFI and of the intervening ionosphere. Nobody, however, seems to have established a precise border for the circular region around the antipode of the Earth (i.e. zero latitude and 180o longitude both East and West) that should be Protected from wild human exploitation when several nations will have reached the capability of easy travel to the Moon. In this paper we propose the cre-ation of PAC, the Protected Antipode Circle, centered around the antipode on the Farside and spanning an angle of 30o in longitude, in latitude and in all radial directions from the antipode. There are sound scientific reasons for this definition: 1) PAC is the most shielded area of the Farside, with an expected attenuation of man-made RFI of 100 dB or higher; 2) PAC is the only area of the Farside that will never be reached by the radiation emitted by future human space bases located at the L4 and L5 Lagrangian points of the Earth-Moon system; 3) PAC does not overlap with other areas of interest to human activity except for possibly a minor common area with the Aitken Basin, the southern depression supposed to have been created 3.8 billion years ago during the "big wham" between the Earth and the Moon. In view of these unique features, we propose PAC to be officially recognized by the United Nations as an Internationally Protected Area where no radio contamination by humans will possibly take place now and in the future for the benefit of all Humankind.
... And FOCAL (an acronym for Fast Outgoing Craft for Astrophysical Lensing), will not just b... more ... And FOCAL (an acronym for Fast Outgoing Craft for Astrophysical Lensing), will not just be looking at a point, but rather at a small line segment in the sky, enormously enlarged. SunLensing the Cosmic Microwave Background from 763 AU Claudio Maccone a "G. Colombo ...
The nonlinear time dependent Brownian motion, B(g(t)), is the object of the present paper. The ti... more The nonlinear time dependent Brownian motion, B(g(t)), is the object of the present paper. The time-rescaling function, g(t), may be any function continuous over a finite positive time interval ranging from O to a fixed T, and such that g(0)=0. We firstly prove B(g(t)) to be equivalent to a white—noise integral having zero mean and a time—dependent variance. Secondly we find the eigenfunction (Karhunen — Loeve) expansion of B(g(t)). The eigenfunctions are Bessel functions of the first kind, with a suitable time function for argument, and they are multiplied by another time function. The eigenvalues virtually are determined by the zeros of the Bessel functions.
Just as Chapter 23 showed the application of the general results obtained in Chapter 22 to the pa... more Just as Chapter 23 showed the application of the general results obtained in Chapter 22 to the particular time-rescaled stochastic process \( {B}(t^{2H})\), the present appendix investigates the application of the general results obtained in Chapter 24 about the square time-rescaled process B2(...) to the particular process \( {B^2}(t^{2H})\).
In two recent papers [1, 2] the author proved that radio communications between any pair of stars... more In two recent papers [1, 2] the author proved that radio communications between any pair of stars within our Galaxy are feasible with modest transmitted powers if the gravitational lenses of both stars are exploited. In this chapter we extend those innovative results to the case of radio communications between nearby galaxies. We show that radio communications between galaxies may become feasible if supermassive black holes, usually located at the center of galaxies, are exploited as gravitational lenses. In other words, a massive black hole may be regarded as a huge focusing device for radio waves being transmitted out of that galaxy and/or being received from another galaxy.
Since the 1990s the international scientific community—especially the IAA (International Academy ... more Since the 1990s the international scientific community—especially the IAA (International Academy of Astronautics)—have been discussing the need to keep the farside of the Moon free from man-made radio frequency interference (RFI). In fact, the center of the farside, specifically the Daedalus crater, is an ideal site to set up a future radio telescope (or, more realistically, a phased array) to detect radiowaves of all kinds, which are impossible to detect on Earth because of the ever-growing RFI.
Let us now change topics completely! Rather than looking for ETs in the Galaxy, we now seek habit... more Let us now change topics completely! Rather than looking for ETs in the Galaxy, we now seek habitable planets for man in the Galaxy. How many are there and how far from us is the nearest such habitable planet?
It is well known that the probability density function(pdf) \( f_{aX+b} (x) \)of the random varia... more It is well known that the probability density function(pdf) \( f_{aX+b} (x) \)of the random variable \( {aX+b}\)where a and b are arbitrary real constants with respect to the independent variable x, is related to the pdf \( f_{\rm{X}}(x)\)of the random variable X by the linear transformation formula for random variables that reads: $$ f_{aX+b}(x)=\frac{1}{\mid a \mid} f_X \left(\frac{x-b}{\mid a \mid}\right).$$ (5.1)
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