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New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. [ABSTRACT FROM AUTHOR]

A series of rhenium compounds with the octahedral cluster core {Re6S8-xBrx} (x = 0–4): with molecular and polymeric structure were obtained. In these compounds the cluster core composition varies monotonically, the geometry of the cluster and the rhenium coordination polyhedron are retained unchanged, while the symmetry of the cluster changes. The vibrational spectra (Raman and IR) were recorded and analyzed for compounds with all possible S/Br ratios in the cluster core. The group vibrations of clusters were attributed with the use of DFT calculations of vibrational spectra. It is shown that the set of main characteristic bands is retained in both ionic and polymeric compounds regardless of the composition and the symmetry of the cluster core while the observed vibration frequencies of these bands depend on the S/Br ratio in the cluster core. In particular, the group Re–S stretching vibrations (A1g(S8) and T2g(S8) modes) shifted to higher frequencies with the increase in the number of Br atoms in the cluster. The difference in the connectivity in polymeric compounds leads to an increase in the number of bands in the spectra and to the disappearance of the A1g(Br) modes. [ABSTRACT FROM AUTHOR]

The concept of a β -integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a β -integrated resolving function. The necessary and sufficient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a β -integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufficiently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation. [ABSTRACT FROM AUTHOR]

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media. [ABSTRACT FROM AUTHOR]

In this study, we present the growth of monolayer MoS2 (molybdenum disulfide) film. Mo (molybdenum) film was formed on a sapphire substrate through e-beam evaporation, and triangular MoS2 film was grown by direct sulfurization. First, the growth of MoS2 was observed under an optical microscope. The number of MoS2 layers was analyzed by Raman spectrum, atomic force microscope (AFM), and photoluminescence spectroscopy (PL) measurement. Different sapphire substrate regions have different growth conditions of MoS2. The growth of MoS2 is optimized by controlling the amount and location of precursors, adjusting the appropriate growing temperature and time, and establishing proper ventilation. Experimental results show the successful growth of a large-area single-layer MoS2 on a sapphire substrate through direct sulfurization under a suitable environment. The thickness of the MoS2 film determined by AFM measurement is about 0.73 nm. The peak difference between the Raman measurement shift of 386 and 405 cm−1 is 19.1 cm−1, and the peak of PL measurement is about 677 nm, which is converted into energy of 1.83 eV, which is the size of the direct energy gap of the MoS2 thin film. The results verify the distribution of the number of grown layers. Based on the observation of the optical microscope (OM) images, MoS2 continuously grows from a single layer of discretely distributed triangular single-crystal grains into a single-layer large-area MoS2 film. This work provides a reference for growing MoS2 in a large area. We expect to apply this structure to various heterojunctions, sensors, solar cells, and thin-film transistors. [ABSTRACT FROM AUTHOR]