Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).

Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction

We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics.

Growth of matter perturbations in the extended viscous dark energy models

In this work, we study the extended viscous dark energy models in the context of matter perturbations. To do this, we assume an alternative interpretation of the flat Friedmann–Lemaître–Robertson–Walker Universe, through the nonadditive entropy and the viscous dark energy. We implement the relativistic equations to obtain the growth of matter fluctuations for a smooth version of dark energy. As result, we show that the matter density contrast evolves similarly to the $$\Lambda $$ Λ CDM model in high redshift; however, in late time, it is slightly different from the standard model. Using the latest geometrical and growth rate observational data, we carry out a Bayesian analysis to constrain parameters and compare models. We see that our viscous models are compatible with cosmological probes, and the $$\Lambda $$ Λ CDM recovered with a $$1\sigma $$ 1 σ confidence level. The viscous dark energy models relieve the tension of $$H_0$$ H 0 in $$2 \sim 3 \sigma $$ 2 ∼ 3 σ . Yet, by involving the $$\sigma _8$$ σ 8 tension, some models can alleviate it. In the model selection framework, the data discards the extended viscous dark energy models.

Exploring the Neighborhood of q-Exponentials

The q-exponential form eqx≡[1+(1−q)x]1/(1−q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1−∑ipiqq−1 (with S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing Sq as particular case.

A generalized q growth model based on nonadditive entropy

We present a general growth model based on nonextensive statistical physics. We show that the most common unidimensional growth laws such as power law, exponential, logistic, Richards, Von Bertalanffy, Gompertz can be obtained. This model belongs to a particular case reported in (Physica A 369, 645 (2006)). The new evolution equation resembles the “universality” revealed by West for ontogenetic growth (Nature 413, 628 (2001)). We show that for early times the model follows a power law growth as [Formula: see text], where the exponent [Formula: see text] classifies different types of growth. Several examples are given and discussed.

Estudo de algoritmos de otimização bio-inspirados aplica à segmentação de imagens

Image segmentation is one of the first steps within the framework for processing scenes. Among the main existing techniques, we highlight the histogram-based binarization, which due to the simplicity of understanding and low computational complexity is one of the most used methods. However, for a multi-threshold process, this method becomes computationally costly. To minimize this problem, optimization algorithms are used to find the best thresholds. Recently, several algorithms inspired by nature have been proposed in a generic way in the area of combinatorial optimization and obtained excellent results, among which we highlight the more traditional ones such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution Algorithm (DE), Artificial Bee Colony (ABC), Firefly Algorithm (FA) and Krill Herd (KH). This work shows a comparison between some of these algorithms and more recent algorithms, from 2014, as Grey Wolf Optimizer (GWO), Elephant Herding Optimization (EHO), Whale Optimization Algorithm (WOA), Grasshopper Optimization Algorithm (GOA) and Harris Hawks Optmization (HHO) . This work compared the thresholds obtained by 7 bio-inspired algorithms in a base composed of 100 images with 1 single object provided by the Weizmann Institute of Science (WIS). The comparison was made using consolidated metrics like Dice/Jaccard and PSNR, as well the recent Hxyz. In the experiments were used the extensive system as an objective function (Kapurs´ Method). Still in the proposal of this experiment, the extensive system was compared with a Tsallis nonadditive entropy, with the Super-extensive system being configured with q ? [0.1, 0.2, . . . 0.9] and the Sub-extensive system with q ? [1.1, 1.2, . . . 1.9]. The image Database contains 100 images with only 1 object on scene. The results show that the Krill Herd (KH) algorithm was the winning algorithm in 35% of executions according to the PSNR metric, 28% in the Dice/Jaccard metric and 35% on the Hxyz metric. The extensive system had the best overall performance and was responsible for the best threshold of 54 images according to the metric PSNR, 30 according to the metric Dice/Jaccard and 39 according to the Hxyz metric

A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics

We consider the thermodynamic and stability problem of Kerr black holes arising from the nonextensive/nonadditive nature of the Bekenstein-Hawking entropy formula. Nonadditive thermodynamics is often criticized by asserting that the zeroth law cannot be compatible with nonadditive composition rules, so in this work we follow the so-called formal logarithm method to derive an additive entropy function for Kerr black holes satisfying also the zeroth law's requirement. Starting from the most general, equilibrium compatible, nonadditive entropy composition rule of Abe, we consider the simplest, non-parametric approach that is generated by the explicit nonadditive form of the Bekenstein-Hawking formula. This analysis extends our previous results on the Schwarzschild case and shows that the zeroth law compatible temperature function in the model is independent of the mass-energy parameter of the black hole. By applying the Poincar&eacute; turning point method we also study the thermodynamic stability problem in the system.

Black hole entropy and the zeroth law of thermodynamics

By mapping the nonadditive entropy composition law of the Bekenstein–Hawking formula to an additive one via the so-called "formal logarithm" operation, a new approach to the black hole entropy problem is considered. The new temperature function satisfies the zeroth law of thermodynamics, and turns out to be independent of the mass-energy parameter of the black hole in the case of the Schwarzschild solution. It is shown that pure isolated black holes are thermodynamically stable against spherically symmetric perturbations within this approach.