Diophantine equations with products of consecutive members of binary recurrences

We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Stochastic Equations and Inclusions with Mean Derivatives and Some Applications

The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing some cost criteria, is proved.     

Carbon-nanotube-based saturable absorbers for near infrared solid state lasers

We demonstrate the passive mode-locking of a diode-pumped Nd⁺³:YAG (central wavelength: 1.32 μm; pulse duration: 50 ps; output energy: up to 70 μJ) laser using a polymer film containing single-wall carbon nanotubes. The mode-locking regime is stable at a pump repetition rate of up to 1 kHz. We also investigate the temporal evolution of the light-induced absorption change of the polymer film containing carbon nanotubes in the spectral range of 1.3–1.5 μm by femtosecond time-resolved pump-probe measurements. The measurements reveal that light-induced transmission exhibits fast and slow components that last 280 fs and more than 10 ps, respectively. The third-order susceptibility of the polymer film containing single wall carbon nanotubes is as high as 10⁻¹¹ esu.     

The system Na2CO3–CaCO3–MgCO3 at 6 GPa and 900–1250°C and its relation to the partial melting of carbonated mantle

In order to constrain the Na₂CO₃–CaCO₃–MgCO₃ T–X diagram at 6?GPa in addition to the binary and pseudo-binary systems we conducted experiments along the Na₂CO₃–Ca_0.5Mg_0.5CO₃ join. At 900–1000°C, melting does not occur and isothermal sections are presented by one-, two- and three-phase regions containing Ca-bearing magnesite, aragonite, Na₂CO₃ (Na₂) and Na₂(Ca_1–0.9Mg_0-0.1)_3-4(CO₃)_4-5 (Na₂Ca_3-4), Na₄(Ca_1–0.6Mg_0–0.4)(CO₃)₃ (Na₄Ca), Na₂(Ca_0-0.08Mg_1–0.92)(CO₃)₂ (Na₂Mg) phases with intermediate compositions. The minimum melting point locates between 1000°C and 1100°C. This point would resemble that of three eutectics: Mgs–Na₂Ca₃–Na₂Mg, Na₂Mg–Na₂Ca₃–Na₄Ca or Na₂Mg–Na₄Ca–Na₂, in the compositional interval of [45Na₂CO₃·55(Ca_0.6Mg_0.4)CO₃]–[60Na₂CO₃·40Ca_0.6Mg_0.4CO₃]. The liquidus projection has seven primary solidification phase regions for Mgs, Dol, Arg, Na₂Ca₃, Na₄Ca, Na₂ and Na₂Mg. The results suggest that extraction of Na and Ca from silicate to carbonate components has to decrease minimum melting temperature of carbonated mantle rocks to 1000–1100°C at 6?GPa and yields Na-rich dolomitic melt with a Na# (Na₂O/(Na₂O?+?CaO?+?MgO))?≥?28?mol%.     

Phase transitions of nonlinear waves in quadratic waveguide arrays

We study two-color parametric nonlinear modes in waveguide arrays with a quadratic nonlinear response. We predict theoretically and observe experimentally a new type of phase transition manifested in an abrupt power-controlled change of the mode structure from unstaggered to staggered, due to the interplay of localization and synchronization in parametrically driven discrete systems.     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).     

Spectral Photoluminescent Parameters of Barley Seeds (Hordéum vulgáre) Infected with Fusarium ssp

We needed effective and sustainable technologies for better microbiological control of crops, including Fusarium. However, photoluminescent UV–Vis methods are potential for diagnosing plant diseases with Fusarium. It has not been sufficiently studied despite the application of these methods for other biological researches. The excitation spectrum of the seeds during infection shifts to the shorter wavelength and a new maximum appears in the region λ ≈ 232 nm. The photoluminescence of infected seeds increases with excitation by radiation of wavelengths λ_e,1 = 232 nm, λ_e,2 = 362 nm and λ_e,3 = 424 nm by 1.33–3.14 times, and λ_e,3 = 424 nm—decreases by 1.1 times. Statistical moments μ₃ and μ₄, asymmetry and kurtosis change only with short-wave excitation. When analyzing the decomposition of the frequency spectrum into Gaussian curves, the most informative ratio is the ratio of right-handed and left-handed Gaussians under excitation λ_e,2 = 362 nm and λ_e,3 = 424 nm. The ratios of their maxima change during infection by 1.36–3.2 times, and for excitation by radiation λ_e,2, the frequency boundaries of Gaussians change. The results of measurements and calculations provide a basis for the development of a method and device for photoluminescence diagnostics of fusarium seeds in UV–Vis ranges.     

Pseudoisocyanine J-aggregate to optical waveguiding crystallite transition: microscopic and microspectroscopic exploration

Using fluorescent microscopy and microspectroscopy, optical properties and morphology transformations in individual pseudoisocyanine (PIC) J-aggregates in aqueous electrolyte solutions have been explored. A stringlike structure of J-aggregates with a string diameter much less than 1 microm has been observed. Photodestruction of the strings under short-wavelength excitation has been revealed. Rodlike PIC crystallites, about 1 microm in diameter, have been observed with time. The fluorescence spectrum of rodlike crystallites has been found to differ from that of stringlike J-aggregate and from PIC crystal powder spectra. The crystallites are very stable, and their photodestruction has not been observed under any excitation conditions. It has been found that rodlike crystallites in contrast to stringlike J-aggregates possess optical waveguide properties. The luminescence of crystallites can be observed only at the excitation spot and at butt ends located up to hundreds of micrometers from the excitation spot.