In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned δ-coupling at the vertices.

We consider a two-particle Schrodinger operator H on the d−dimensional diamond lattice. We find a sufficiency condition of finiteness for discrete spectrum eigenvalues of H.

In this paper, the properties of solutions for the nonlinear system equations not in divergence form:

||x| ^{n ∂u} _∂t = u ^γ1∇  |∇u| ^p−2∇u + |x| ⁿu ^q1 v ^q2 ,

|x| ^{n ∂v} _∂t = v ^γ2∇  |∇v| ^p−2∇v + |x| ⁿv ^q4 u ^q3,

are studied. In this work, we used method of nonlinear splitting, known previously for nonlinear parabolic equations, and systems of equations in divergence form, asymptotic theory and asymptotic methods based on different transformations. Asymptotic representation of self-similar solutions for the nonlinear parabolic system of equations not in divergence form is constructed. The property of finite speed propagation of distributions (FSPD) and the asymptotic behavior of the weak solutions were studied for the slow diffusive case.

This work aims to study the existence and uniqueness of a solution for a problem of the loaded degenerating mixed type equation. We consider the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals. The uniqueness of solution is proved by using the method of integral energy applying an extremum principle. Based on the statement of equivalence between “The existence and uniqueness of solution” and “Solvability of the respectively Fredholm type integral equations”, the existence of a solution was proved.

Once, a referee asked how one can write an oscillator model for a ferroelectric and the total Maxwell equation for a rapidly varying electric field. From what is known about polarization, it is a slow function of time and coordinates, but the optical wave is a fast function. However, there are examples for the interaction of high frequency and low frequency waves in nonlinear wave theory. This means that similar equations can be written for ferroelectric polarization and electromagnetic waves.

The chiral properties of a metasurface made of 2-D array of twisted gammadion crosses was studied at frequencies ranging from 0.1–0.12 THz. The influences of unit cell design on the optical activity and the ellipticity of the metasurface were revealed. The maximal values of ellipticity and polarization azimuth rotation angle were obtained for the gammadion crosses with petals in the form of the truncated circles. The changing of the semi axes’ relation of gammadion cross ellipses allows tuning of the operational frequency and creation of a multiband polarizer.

We propose a theoretical model for the optically controlled terahertz filter based on hybrid graphene/metasurface structure. Such a device has a high accuracy for the frequency adjustment by the different intensities of the optical pumping in the infrared spectral range, fast response time and polarization independence. The tuning by optical pumping of the spectral characteristics of the filter in term of resonant frequency and Q-factor was shown.

We studied the theoretical possibility of generating single-photon states in fibers with active profile dispersion formed by spontaneous four-wave mixing (SFWM) using femtosecond laser pulses. The possibility of eliminating the spectral correlation for single-photon states generated in commercial fibers was assessed. To model the structure of photonic crystal fibers and the modes of SFWM generation, an optimization was performed on the parameters of the pump and the fiber structure.

Transparent Ce:YAG ceramics were fabricated by the solid-state reaction method with an additional round of pre-calcining using nanopowders of Ce_2xY_2−2xO₃ (x = 0.001, 0.01, 0.03 and 0.05) and Al₂O₃ synthesized by laser ablation. Additional pre-calcining of the nanopowder mixture in air was used to partially transform the oxides into the YAG phase before sintering. The transmittance of the obtained 2-mm-thick Ce:YAG ceramics was over 81 % in the wavelength range of 500–900 nm. The average volumes of the scattering centers in the obtained ceramics were evaluated by direct count method to be 34, 74, 53, 50 ppm for 0.1, 1, 3 and 5 % Ce doping concentration, respectively. The absolute light yields, energy resolutions under 662 keV gamma ray, and decay curves of scintillations of the fabricated Ce:YAG samples were measured and compared to those of Ce:YAG and CsI:Tl single crystals.

The distribution of heavily post-critical nucleation centers in the nucleation kinetics subject to the macroscopic flow effect has been successfully obtained. Examples of rotary and convective motions during the centrifugal casting are considered. The obtained distribution has been confirmed by an experimental investigation of the composite material AK12 (boron carbide (B₄C)) crystallization.

We consider the evolution of a light bullet in a fermi-liquid while considering the presence of a magnetic field. The energy spectrum for the Fermi liquid was calculated in the framework of AdS/CFT correspondence. The influence of a magnetic field’s amplitude and its period on light bullet evolution was obtained. The stable character of the light bullet propagation in the Fermi liquid in the presence of a magnetic field was confirmed.

Nanopowders of nickel ferrite doped with cobalt were synthesized by co-precipitation using a 3 % KOH solution as a precipitant. The effects of different annealing regimes on the composition and particle size of Ni_1−xCoxFe₂O₄ were studied. It was established that with annealing at t = 1000 ◦C for 2 h single-phase products were formed with a crystallite size of 30–50 nm. The saturation magnetization and the coercive force increased as the content of the dopant increased from 50.3 emu/g and 51.94 Oe for NiFe₂O₄ to 80.45 emu/g and 848.32 Oe for CoFe₂O₄.

This paper presents the results from a study of the microstructure of the surface of lead sulfide films deposited via the pyrolysis of aqueous [Pb(N₂H₄CS)₂(CH₃COO)₂] aerosols. The influences of temperature and the concentration of thiourea on the synthesized films’ surface topography were investigated. A change in the type of conductivity with increasing temperature was observed.

An important problem in the practical implementation of fiber optical quantum communication systems is to synchronize the sender and receiver modules using a separate optical channel. The signal visibility in the quantum channel, which contributes to quantum bit error rate, is influenced by the synchronization signal delay. In this work, we investigate the dependence of the synchronization signal parameters on the dispersive effects in the fiber for a subcarrier wave quantum communication system (SCWQC), which is promising for quantum networking applications. The ITU-T G.652D standard single mode optical fiber was used for modeling. The maximum calculated phase mismatch of the synchronization signal for the system operating at 100 km fiber length corresponds to 1.7 ps signal time delay. The results show that dispersion causes significant signal distortion, therefore additional phase adjustment at least every 2.3 hours is required for stable system operation.

In this work, a mathematical model to study the indirect interaction in topological insulators was constructed. Analysis of the model was carried out numerically. We have calculated the indirect exchange interaction in the film of a topological insulator, for example Bi₂Te₃, within the s–d model. The calculations showed that the magnetic ordering of the impurity spins varies periodically with increasing distance between atoms, asymptotically decreasing to zero. λ is a parameter associated with hexagonal distortion and is a component of the dispersion relation. The dependence of the constants of the effective exchange interaction upon the λ parameter is shown; this parameter characterizes the crystal lattice geometry for a topological insulator.

This work demonstrates the positive effects of zirconium dioxide nanoparticles on cells in vitro. This is supported by the absence of toxicity, stimulation of metabolic and proliferative activity. The nanoparticles of solid solution of europium oxide in europium dioxide do not exhibit an explicit biological effect. The potentially successful application of zirconium dioxide-based nanoparticles in pharmacology has been demonstrated.

The temperature of water-ice crystallization initiation decreases (∆T) were determined in the binary water solutions of water soluble derivative of light fullerene C70 with amino-acid lysine at 272.99 – 273.15 K. Partial molar excess functions for H₂O were calculated. For the thermodynamic description of our systems, we have elaborated an original semi-empirical model VD-AS (Virial Decomposition Asymmetric Model), based on the virial decomposition of the molar Gibbs energy of the component molar fractions in the solution. With the help of the VD-AS model, partial molar functions of nano-clusters were calculated. Excess and full average Gibbs energies for the solutions and miscibility gaps concentration regions (with the help of diffusional instability equations) were calculated. Thus, the VD-AS model excellently describes pre-delamination or micro-heterogeneous-structure formation in the considered solutions. These calculations were accordingly confirmed by dynamic light scattering data.