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).     

High-Affinity Ratiometric Fluorescence Probe Based on 6-Amino-2,2′-Bipyridine Scaffold for Endogenous Zn2+ and Its Application to Living Cells

Zinc is an essential trace element involved in many biological activities; however, its functions are not fully understood. To elucidate the role of endogenous labile Zn²⁺, we developed a novel ratiometric fluorescence probe, 5-(4-methoxyphenyl)-4-(methylsulfanyl)-[2,2′-bipyridin]-6-amine (6 (rBpyZ)) based on the 6-amino-2,2′-bipyridine scaffold, which acts as both the chelating agent for Zn²⁺ and the fluorescent moiety. The methoxy group acted as an electron donor, enabling the intramolecular charge transfer state of 6 (rBpyZ), and a ratiometric fluorescence response consisting of a decrease at the emission wavelength of 438 nm and a corresponding increase at the emission wavelength of 465 nm was observed. The ratiometric probe 6 (rBpyZ) exhibited a nanomolar-level dissociation constant (K_d = 0.77 nM), a large Stokes shift (139 nm), and an excellent detection limit (0.10 nM) under physiological conditions. Moreover, fluorescence imaging using A549 human lung adenocarcinoma cells revealed that 6 (rBpyZ) had good cell membrane permeability and could clearly visualize endogenous labile Zn²⁺. These results suggest that the ratiometric fluorescence probe 6 (rBpyZ) has considerable potential as a valuable tool for understanding the role of Zn²⁺ in living systems.     

The Odyssey of Entropy: Cryptography

After being introduced by Shannon as a measure of disorder and unavailable information, the notion of entropy has found its applications in a broad range of scientific disciplines. In this paper, we present a systematic review on the applications of entropy and related information-theoretical concepts in the design, implementation and evaluation of cryptographic schemes, algorithms, devices and systems. Moreover, we study existing trends, and establish a roadmap for future research in these areas.     

N‐shell ionization cross sections by proton impact in the BEA

The N‐subshell ionizations cross sections of heavy elements by proton impact have been calculated in the binary‐encounter approximation. The momentum distribution of target electrons is taken into account by the use of the nonrelativistic and relativistic hydrogenic models and the Hartree–Fock–Roothaan and the relativistic Hartree–Fock–Roothaan methods. The obtained subshell ionization cross sections are compared with the experimental data and other theoretical calculations. The electronic relativistic effect and the wave‐function effect on N‐shell ionization cross sections are discussed.     

Atomic and molecular spectroscopy with a continuous-wave, doubly resonant, monolithic optical parametric oscillator

Doppler-broadened atomic and molecular spectra were observed with a one octave tunable, continuous-wave, doubly resonant, monolithic optical parametric oscillator (OPO) using 5% MgO-doped LiNbO₃ as a non-linear crystal with a birefringent phase-matching configuration. By tuning the frequency of a pump laser, longitudinal mode selection over 20 successive modes, corresponding to a 60 GHz span, was possible, owing to the simple structure of the monolithic OPO. Continuous frequency tuning was achieved using an external waveguide-type electrooptic phase modulator (EOM). By changing the modulation frequency of the EOM, frequency tuning of the optical sidebands over 12 GHz was possible, which is larger than the one free spectral range of the monolithic cavity of 3 GHz. We could observe the Cs-D₁ (894 nm), Cs-D₂ (852 nm), Rb-D₁ (795 nm), acetylene R9 (1520 nm) and P9 (1530 nm) transitions with the single monolithic OPO.     

Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization

The vectorial Debye integral shows that tightly focused Laguerre-Gaussian (LG) beams have a residual intensity at the focal point for linear polarization, for a topological charge of m=1 and 2. We measured the shapes of linearly and circularly polarized LG beams and found that a central intensity appeared at m=1 and 2 for linear and right-handed circular polarization, however, it is completely canceled for left-handed circular polarization. In general, when the orbital angular momentum of the LG beam is parallel to the spin angular momentum of the photons, zero intensity is always achieved at the focus.     

Unequally spaced multiple mid-infrared wavelength generation using an engineered quasi-phase-matching device

We propose a novel quasi-phase-matched (QPM) device that can generate unequally spaced multiple wavelengths. Unequally spaced multiple QPM peaks can be obtained by employing the optimized phase modulation of a periodic domain structure. We fabricated a LiNbO3 waveguide device for 3.2-3.4 microm band difference frequency generation based on the design. Using the multiple mid-infrared outputs, we demonstrate the detection of multiple hydrocarbon gases, namely, methane, ethylene, and ethane.     

Apparent diffusion coefficient of line scan diffusion image in normal prostate and prostate cancer—comparison with single-shot echo planner image

Purpose

This retrospective study was designed to evaluate the apparent diffusion coefficient (ADC) of line scan diffusion images (LSDI) in normal prostate and prostate cancer. Single-shot echo planner images (SS-EPI) were used for comparison.

Materials and Methods

Twenty prostate tumors were examined by conventional MRI in 14 patients prior to radical prostatectomy. All patients were examined with a 1.5-T MR imager (Signa CV/i ver. 9.1 GE Medical System Milwaukee, WI, USA). Diffusion-weighted MR imaging (DWI) using LSDI was performed with a pelvic phased-array coil, with b values of 5 and 800 s/mm². DWI using SS-EPI was performed with a body coil, with b values of 0 and 800 s/mm². The ADCs of each sequence for 14 normal prostate and 20 prostate cancers were histopathologically assessed. Signal-to-noise ratio (SNR) on DWI was estimated and compared for each sequence.

Results

The mean ADCs (±S.D.) of normal peripheral zones (PZ), transition zones (TZ) and cancer (in 10⁻³ mm²/s) that used LSDI were 1.42±0.12, 1.23±0.10 and 0.79±0.19, respectively. Those that used SS-EPI were 1.76±0.26, 1.38±0.20 and 1.05±0.27, respectively. Using unpaired t test (P<.05), we found a significant difference in each sequence between normal tissue (both PZ and TZ) and the cancer. Paired t test (P<.05) also registered a significant difference between LSDI and SS-EPI. Mean SNR for DWI using LSDI was 16.49±5.03, while the DWI using SS-EPI was 18.85±9.26. The difference between the SNR of each sequence was not statistically significant by paired t test.

Conclusion

We found that ADCs using LSDI and SS-EPI showed similar tendencies in the same patients. However, in all regions, LSDI ADCs had smaller standard deviations than SS-EPI ADCs.     

Kaluza-Klein theory from the viewpoint of gauge theory of gravity

We consider Kaluza-Klein theory based on the fiber bundle. We obtain the modified Kaluza-Klein metric as an invariant line element of a bundle. Its reduced action includes a higher derivative action in gravitation as well as a term linear inR.     

WIDE RANGE DETECTOR USING PARABOLIC CYLINDRICAL MIRROR FOR THz APPLICATIONS

A new, wide-band, high-speed and high-sensitivity THz detector has been developed. The prototype detector consists of a parabolic cylindrical mirror, a long wire antenna and a Schottky barrier diode. Direct detection measurements have shown a stable sensitivity of 150 ± 50 V/W for 1–2 THz without any adjustments. The long wire antenna was fixed at the focus of parabolic cylindrical mirror then it has been realized less operation steps, easy coupling to the external THz signals and a dramatic enhancement in the practicality of this system. The optically polished mirror and frosted surface one showed comparable sensitivities, thus easy polishing and less cost mirror fabrication can be applied for this system. The radiation pattern showed a maximum radiation angle of approximately 23° with its dominant main lobe, which was attributed to the wire antenna character and confirmed good agreements with classical antenna theory.