Tag Archives: Probability

SETI, evolution and human history merged into a mathematical model [CL]

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http://arxiv.org/abs/2203.10116


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.

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C. Maccone
Tue, 22 Mar 22
7/82

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI [CL]

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http://arxiv.org/abs/2203.10189


The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our “Peak-Locus Theorem” translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The “molecular clock” is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this paper for the first time the EvoEntropy corresponding to the Markov-Korotayev (2007) “cubic” evolution: a curve of logarithmic increase.

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C. Maccone
Tue, 22 Mar 22
59/82

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Tensor- and spinor-valued random fields with applications to continuum physics and cosmology [CL]

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http://arxiv.org/abs/2112.04826


In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic.
Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic.
For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

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A. Malyarenko and M. Ostoja-Starzewski
Fri, 10 Dec 21
23/94

Comments: 2 figures

Explicit form of the random field spectral representation and some applications [GA]

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http://arxiv.org/abs/2107.05070


We present here an explicit form of the random spectral measure element, what allows us to express a stationary random field as a stochastic integral explicitly depending on its power spectrum and a spectral tensor if the field is a vector one. It has been shown here that convergence mechanism of such integral is significantly different from the one of the Fourier transform and that the traditional formalism is a partial limiting case of the one presented here. The fact that there is an explicit expression of a random field makes calculation of higher order statistics of it much more straightforward (see for example Chepurnov et al. 2020). For a vector field such expression contains a projection of an isotropically distributed random vector by a spectral tensor, what makes geometrical interpretation of harmonics behavior possible, simplifying its analysis (see Sect. 2). This spectral representation also makes straightforward numerical generation of a random field, what is extensively used by Chepurnov et al. 2020. We also present here some practical applications of this formalism.

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A. Chepurnov
Tue, 13 Jul 21
49/79

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Multilevel Bayesian Parameter Estimation in the Presence of Model Inadequacy and Data Uncertainty [CL]

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http://arxiv.org/abs/1711.10599


Model inadequacy and measurement uncertainty are two of the most confounding aspects of inference and prediction in quantitative sciences. The process of scientific inference (the inverse problem) and prediction (the forward problem) involve multiple steps of data analysis, hypothesis formation, model construction, parameter estimation, model validation, and finally, the prediction of the quantity of interest. This article seeks to clarify the concepts of model inadequacy and bias, measurement uncertainty, and the two traditional classes of uncertainty: aleatoric versus epistemic, as well as their relationships with each other in the process of scientific inference. Starting from basic principles of probability, we build and explain a hierarchical Bayesian framework to quantitatively deal with model inadequacy and noise in data. The methodology can be readily applied to many common inference and prediction problems in science, engineering, and statistics.

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A. Shahmoradi
Thu, 30 Nov 17
11/77

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Fast generation of isotropic Gaussian random fields on the sphere [CL]

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http://arxiv.org/abs/1709.10314


The efficient simulation of isotropic Gaussian random fields on the unit sphere is a task encountered frequently in numerical applications. A fast algorithm based on Markov properties and Fast Fourier Transforms in 1d is presented that generates samples on an n x n grid in O(n^2 log n). Furthermore, an efficient method to set up the necessary conditional covariance matrices is derived and simulations demonstrate the performance of the algorithm.

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P. Creasey and A. Lang
Mon, 2 Oct 17
28/47

Comments: 13 pages, 3 figures

Stochastic Eternal Inflation in a Bianchi Type I Universe [CL]

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http://arxiv.org/abs/1508.02670


In this paper, we analyze a Bianchi Type I model with a scalar field in a chaotic inflation potential, $V(\phi) = \frac{1}{2}\phi^2$ in the context of stochastic eternal inflation. We use the typical slow-roll approximation in combination with expansion-normalized variables in an orthonormal frame approach to obtain a dynamical system which describes the dynamics of the shear anisotropy and the inflaton field. We first show that the dynamics of the inflaton field can be decoupled from the dynamics of the shear anisotropy. We then use a fixed-points analysis in combination with global techniques from topological dynamical systems theory to prove that the cosmological model under consideration isotropizes irrespective of an inflationary epoch, which has also described by other authors who have investigated a Bianchi Type I model under similar configurations. We then show that for inflation to occur, the amount of anisotropy must be very small.
We also give a description of the stochastic dynamics of the inflaton field by using techniques from stochastic calculus. We show that the Klein-Gordon equation becomes a stochastic differential equation with a highly nonlinear drift term. In this case, the deceleration parameter itself becomes a random variable, and we give details regarding when such a model can undergo inflation. We finally derive the form of the long-term, stationary probability distribution of the inflaton field, and show that it has the form of a double-well potential. We then calculate the probability of inflation occurring based on this approach. We conclude the paper by performing some numerical simulations of the stochastic differential equation describing the dynamics of the inflaton field. We conjecture that even in the case of stochastic eternal inflation, one requires precise initial conditions for inflation to occur.

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I. Kohli and M. Haslam
Wed, 12 Aug 15
50/50

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