The Drake Equation has proven fertile ground for speculation about the abundance, or lack thereof, of communicating extraterrestrial intelligences (CETIs) for decades. It has been augmented by subsequent authors to include random variables in order t
o understand its probabilistic behavior. In this paper, the first model for the number of CETIs with stochastic processes governing both their emergence and quiescence is developed using the Skellam Distribution. Results from this include the possibility that there can still be substantial times multiple CETIs exist even if the Drake Equation terms are approximately zero. In addition, it can give us a basic estimate of the average CETI age gap based on their broadcast time. Finally, we will introduce a definition of how the interaction between CETIs, where possible, can be measured by statistical dependence between the terms N and L in the Drake Equation by indicating how the number of co-existing CETIs affect their relative individual lifetimes.
Two estimates for the inverse binary entropy function are derived using the property of information entropy to estimate combinatorics of sequences as well as related formulas from population genetics for the effective number of alleles. The second es
timate shows close correspondence to the actual value of the inverse binary entropy function and can be seen as a close approximation away from low values of binary entropy where $p$ or $1-p$ are small.
Attempts to accurately measure the monetary velocity or related properties of bitcoin used in transactions have often attempted to either directly apply definitions from traditional macroeconomic theory or to use specialized metrics relative to the p
roperties of the Blockchain like bitcoin days destroyed. In this paper, it is demonstrated that beyond being a useful metric, bitcoin days destroyed has mathematical properties that allow you to calculate the average dormancy (time since last use in a transaction) of the bitcoins used in transactions over a given time period. In addition, bitcoin days destroyed is shown to have another unexpected significance as the average size of the pool of traded bitcoins by virtue of the expression Littles Law, though only under limited conditions.
A new method to measure nonlinear dependence between two variables is described using mutual information to analyze the separate linear and nonlinear components of dependence. This technique, which gives an exact value for the proportion of linear de
pendence, is then compared with another common test for linearity, the Brock, Dechert and Scheinkman (BDS) test.
There is no closed form analytical equation or quick method to calculate probabilities based only on the entropy of a signal or process. Except in the cases where there are constraints on the state probabilities, one must typically derive the underly
ing probabilities through search algorithms. These become more computationally expensive as entropies of higher orders are investigated. In this paper, a method to calculate a joint probability matrix based on the entropy for any order is elaborated. With this method, only first order entropies need to be successfully calculated while the others are derived via multiplicative cascades.
The development and evolution of malware including computer viruses, worms, and trojan horses, is shown to be closely analogous to the process of community succession long recognized in ecology. In particular, both changes in the overall environment
by external disturbances, as well as, feedback effects from malware competition and antivirus coevolution have driven community succession and the development of different types of malware with varying modes of transmission and adaptability.
In this paper, new techniques that allow conditional entropy to estimate the combinatorics of symbols are applied to animal communication studies to estimate the communications repertoire size. By using the conditional entropy estimates at multiple o
rders, the paper estimates the total repertoire sizes for animal communication across bottlenose dolphins, humpback whales, and several species of birds for N-grams length one to three. In addition to discussing the impact of this method on studies of animal communication complexity, the reliability of these estimates is compared to other methods through simulation. While entropy does undercount the total repertoire size due to rare N-grams, it gives a more accurate picture of the most frequently used repertoire than just repertoire size alone.
The relationship between period doubling bifurcations and Feigenbaums constants has been studied for nearly 40 years and this relationship has helped uncover many fundamental aspects of universal scaling across multiple nonlinear dynamical systems. T
his paper will combine information entropy with symbolic dynamics to demonstrate how period doubling can be defined using these tools alone. In addition, the technique allows us to uncover some unexpected, simple estimates for Feigenbaums constants which relate them to log 2 and the golden ratio, phi, as well as to each other.
The distribution of frequency counts of distinct words by length in a languages vocabulary will be analyzed using two methods. The first, will look at the empirical distributions of several languages and derive a distribution that reasonably explains
the number of distinct words as a function of length. We will be able to derive the frequency count, mean word length, and variance of word length based on the marginal probability of letters and spaces. The second, based on information theory, will demonstrate that the conditional entropies can also be used to estimate the frequency of distinct words of a given length in a language. In addition, it will be shown how these techniques can also be applied to estimate higher order entropies using vocabulary word length.
In this paper we analyze Greshams Law, in particular, how the rate of inflow or outflow of currencies is affected by the demand elasticity of arbitrage and the difference in face value ratios inside and outside of a country under a bimetallic system.
We find that these equations are very similar to those used to describe drift in systems of free charged particles. In addition, we look at how Greshams Law would play out with multiple currencies and multiple countries under a variety of connecting topologies.
