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, ζq∑(s)≡∑n=1∞1〈n〉qs=1+1〈2〉qs+⋯ and ζq∏(s)≡∏pprime11−〈p〉q−s=11−〈2〉q−s11−〈3〉q−s11−〈5〉q−s⋯, which, for q<1, generically satisfy ζq∑(s)<ζq∏(s), in variance with the q=1 case, where of course ζ1∑(s)=ζ1∏(s).     

222Rn concentrations of water in the Balaton Highland and in the southern part of Hungary,and the assessment of the resulting dose

In this study the ²²²Rn concentration of mains water in 120 settlements in Hungary (Southern Hungary, the Balaton Highland region) was measured. The average ²²²Rn concentration was 5.56 $(0–24.3)\mathrm{Bq}{\mathrm{l}}^{-1}$. On the basis of the ²²²Rn concentration of mains water inspected in the Southern Great Plain region, it can be stated that the ²²²Rn concentration of mains water here is, as an average, half of the ²²²Rn concentration of fountains in the same region. This decrease in radon probably happens during the water management and storage of mains drinking water. The ²²²Rn concentration of spring-water examined in the region of the Balaton Highland exceeds the average ²²²Rn concentration of drinking water (average $27.1\mathrm{Bq}{\mathrm{l}}^{-1}$).The radiation dose originating from the consumption of mains drinking water in case of adults does not reach the value of $0.1\mathrm{mSv}{\mathrm{year}}^{-1}$, even as a conservative assessment ($1\mathrm{l}{\mathrm{day}}^{-1}$ water consumption and ${10}^{-8}\mathrm{Sv}{\mathrm{Bq}}^{-1}$ dose conversion factor).     

YbNi2: A heavy fermion ferromagnet

Measurements of the magnetization and specific heat of YbNi₂ binary alloy are reported. The DC magnetic susceptibility displays a ferromagnetic behavior with a Curie temperature T_C=10.5 K, one of the highest found in Yb compounds. Moreover, the temperature dependence of the specific heat exhibits a lambda anomaly with a peak of 5.12 J/mol K at 9.4 K. The analysis also shows an additional magnetic contribution around 32 K stemming from the crystalline electric field of a quartet at ${\Delta }_1=72\mathrm{K}$ and a doublet at ${\Delta }_2=126\mathrm{K}$, according to the splitting of the Yb³⁺ ion in cubic symmetry. From the magnetic contribution to the specific heat, a relatively high Kondo temperature $T_K=27\mathrm{K}$ is estimated. Below the magnetic transition, the specific heat shows a huge value of the electronic coefficient ${\gamma }_{\mathit{LT}}=573\mathrm{mJ}/\mathrm{mol}\mathrm{K}$, which is a signature of a heavy fermion behavior. Therefore, this alloy is a fine example of enhanced ferromagnetism and heavy fermion behavior among Yb compounds.     

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.     

Principal and secondary luminescence lifetime components in annealed natural quartz

Time-resolved luminescence spectra from quartz can be separated into components with distinct principal and secondary lifetimes depending on certain combinations of annealing and measurement temperature. The influence of annealing on properties of the lifetimes related to irradiation dose and temperature of measurement has been investigated in sedimentary quartz annealed at various temperatures up to $900{}^{\circ }\mathrm{C}$. Time-resolved luminescence for use in the analysis was pulse stimulated from samples at 470 nm between 20 and $200{}^{\circ }\mathrm{C}$. Luminescence lifetimes decrease with measurement temperature due to increasing thermal effect on the associated luminescence with an activation energy of thermal quenching equal to $0.68\pm 0.01\mathrm{eV}$ for the secondary lifetime but only qualitatively so for the principal lifetime component. Concerning the influence of annealing temperature, luminescence lifetimes measured at $20{}^{\circ }\mathrm{C}$ are constant at about $33\mathrm{\mu }\mathrm{s}$ for annealing temperatures up to $600{}^{\circ }\mathrm{C}$ but decrease to about $29\mathrm{\mu }\mathrm{s}$ when the annealing temperature is increased to $900{}^{\circ }\mathrm{C}$. In addition, it was found that lifetime components in samples annealed at $800{}^{\circ }\mathrm{C}$ are independent of radiation dose in the range 85–1340 Gy investigated. The dependence of lifetimes on both the annealing temperature and magnitude of radiation dose is described as being due to the increasing importance of a particular recombination centre in the luminescence emission process as a result of dynamic hole transfer between non-radiative and radiative luminescence centres.     

Quantum Probes for the Characterization of Nonlinear Media

Active optical media leading to interaction Hamiltonians of the form H=λ˜(a+a†)ζ represent a crucial resource for quantum optical technology. In this paper, we address the characterization of those nonlinear media using quantum probes, as opposed to semiclassical ones. In particular, we investigate how squeezed probes may improve individual and joint estimation of the nonlinear coupling λ˜ and of the nonlinearity order ζ. Upon using tools from quantum estimation, we show that: (i) the two parameters are compatible, i.e., the may be jointly estimated without additional quantum noise; (ii) the use of squeezed probes improves precision at fixed overall energy of the probe; (iii) for low energy probes, squeezed vacuum represent the most convenient choice, whereas for increasing energy an optimal squeezing fraction may be determined; (iv) using optimized quantum probes, the scaling of the corresponding precision with energy improves, both for individual and joint estimation of the two parameters, compared to semiclassical coherent probes. We conclude that quantum probes represent a resource to enhance precision in the characterization of nonlinear media, and foresee potential applications with current technology.     

Magnetized strange quark matter solutions in f(R,T) gravity with cosmological constant

In this research, we have studied magnetized strange quark matter (SQM) solutions for Friedmann-Robertson-Walker (FRW) universe in f(R, T) gravity. To obtain exact solutions of modified field equations we have used $f(R,T)=R+2f\left(T\right)$ and $f(R,T)=f_1\left(R\right)+f_2\left(T\right)$ models given by Harko et al. (Harko et al. in Phys. Rev. D 84:024020, 2011) and $f(R,T)=R+f_3\left(T\right)$ model (here f₃ is an arbitrary function) with cosmological constant Λ. For t → ∞ we obtain $p=?\rho$ dark energy situation with small constant values of cosmological constant in three different f(R, T) gravitation models. In our solutions magnetic field does not observe also we have transformed our solutions from FRW universe to Static Einstein Universe (SEU) and we get f(R, T) gravity results for SEU universe. Finally we discussed our physical solutions.     

Non-extensive statistical entropy,quantum groups and quantum entanglement

In this work, we explore a new connection between quantum groups and Tsallis entropy through the energy spectrum of a Hamiltonian with S_Uq(2)$S{U}_q\left(2\right)$ symmetry. Identifying the deformation parameter of the entropy with the parameter of deformation of the associated quantum group, we deduce Tsallis entropy for states related to such a system with S_Uq(2)$S{U}_q\left(2\right)$ symmetry and conducted an investigation of quantum entanglement.     

Aspartame tablets—gamma dose response and usability for routine radiation processing dosimetry using spectrophotometry

Aspartame tablets were studied for gamma dose response, using spectrophotometric read-out method. The optimum concentration for ferrous ions was $2\times {10}^{-4}\mathrm{mol}{\mathrm{dm}}^{-3}$ and xylenol orange with $2.5\times {10}^{-1}\mathrm{mol}{\mathrm{dm}}^{-3}$ of sulphuric acid for the optimum acidity in FX solution. Wavelength of maximum absorbance is 548 nm. Post-irradiation stability is appreciable i.e. for not less than one month. Dose response is non-linear with third order polynomial fit, in the dose range of 1000–10000 Gy. This system of aspartame was further used for carrying out relative percentage dose profile measurement in Gamma Cell-220. Results obtained were inter-compared with that of a glutamine dosimeter, which showed that maximum difference between the values of aspartame and glutamine systems is within $\pm 10\%$.