Aim: to systematize experimental and theoretical data on the Jahn–Teller effect in metallophthalocyanines (MPc) and identify its role as a universal controlling parameter of structural, electronic, and magnetic properties.

Methodology. Results obtained by X–ray diffraction, electron diffraction, scanning tunneling microscopy, Raman spectroscopy, electron paramagnetic resonance, and density functional theory calculations were analyzed. Experimental and theoretical data from international sources are summarized.

Results. It is shown that the formation of [MPc]⁻ anion radicals causes Jahn–Teller deformation of the macrocycle with an amplitude of R_dist ≈ 0.03–0.07 Å. Direct correlations are established between the deformation amplitude and: (1) the activation energy of electrotransport, (2) the mobility of charge carriers, and (3) the selectivity in CO₂/CO electrocatalysis. Potential–induced Jahn–Teller deformation of the CoN₄ center changes the CO adsorption mechanism (linear 0.52 eV → bridge 1.80 eV), providing control of catalytic selectivity without chemical modification of the molecule.

Research implications. The concept of the Jahn–Teller effect as a fundamental mechanism of MPc functionality is formulated. Methods for targeted control of the structural and electronic properties through electrochemical potential, doping, or fluorination are proposed. Several gaps are identified, and methodological approaches to address them are recommended.

Aim. Solution of the Clausing equation describing rarefied gas flow in a circular pipe.

Methodology. An approximate analytical solution of this equation is proposed, based on the method of self-similar interpolation.

Results. The passage probabilities obtained in the study are compared with the results of known numerical solutions.

Research implications. Calculating gas flow through channels of various geometries in vacuum systems is one of the most important problems in rarefied gas dynamics.

Aim. To study the non-inertial correction to the spectrum of the hydrogen atom. We derive the spectrum of bound states of the hydrogen atom from the stability requirement formulated in non- inertial reference frames and show how the resulting Hamiltonian reduces to the standard Coulomb problem, taking into account small corrections due to the reference frame.

Methodology. In this formulation, we introduce a stability condition for physical trajectories based on the positivity of the second variation of the action function. As an application, this approach is used to analyze bound motion in a Coulomb field. The balance between stochastic input and radiative losses selects a discrete set of stable periodic trajectories. Their characteristic frequencies exhibit scaling behavior similar to the hydrogen spectrum. In the limit of vanishing non-inertial fluctuations, the formalism reduces to the Schrödinger description.

Results of the study consist of a monotonic decrease in the absolute deviation |ΔEn| with increasing principal quantum number n, which we check at low-lying levels and summarize in a compact table, including relative errors. This structure also leads to frequency shifts for several transitions (e. g., 1S–2S and 2S–nD), expressed directly in hertz.

Research implications of the results lies in certain spectral corrections, which can be considered as consequences of dynamic stability in a fluctuating non-inertial background.

Aim: to modify the well-known analytical bimodal model of the Tamm – Mott-Smith shock wave for the case of shock compression of a binary gas mixture.

Methods. Asymptotic and approximation methods of mathematical physics were used.

Results. An analytical nonlinear single-parameter model has been developed for the structure of a shock-compressed binary gas mixture, where, in particular, the classical Tamm-Mott-Smith distribution parameter is used as a parameter.

Research implications: The proposed generalization of the classical bimodal shock wave model to the case of a binary gas mixture retains its simplicity and convenience for use in solving theoretical problems and in gas-dynamic experiments.

Aim. Theoretical modeling of optical parameters of a “dielectric – metal – dielectric” nanostructure taking into account the ellipsoidal metal band structure.

Methodology. The quantum theory of transport phenomena is applied, which involves the determination of the charge carrier density matrix diagonal components by solving the Liouville equation. Charge carrier surface scattering is taken into account by using Soffer boundary conditions.

Results. Analytical expressions for the optical coefficients and polarization parameters of reflected and transmitted waves were derived. It was shown that the isoenergetic surface anisotropy significantly influences the reflected wave polarization shape and the ability of the “dielectric – metal – dielectric” nanostructure to absorb radiation.

Research implications. The results of the work can be used to design polarizers based on layered “dielectric – metal – dielectric” nanostructures.

Aim. To establish the relationship between the intensity of light passing through a loaded sample and the state of polarization and the magnitude of the acting force, taking into account induced polarization dichroism.

Methodology. An experimental study of the azimuthal dependence of the intensity of monochromatic light passing through a transparent sample subjected to uniaxial compression was conducted. Theoretical models were constructed and analyzed, accounting for possible deviation of the sample's optical axis from the vertical and induced polarization dichroism.

Results. Based on the study of the azimuthal dependences of light intensity under various loads, the value of the photoelasticity coefficient of the studied sample and the absorption coefficient due to induced anisotropy were determined.

Research implications. The analysis carried out showed the need to take into account, in addition to the birefringence caused by the load applied to the sample, both induced polarization dichroism and the possible deviation of the optical axis, for a complete analysis of the azimuthal dependences of the light intensity.

Aim. To conduct a comprehensive study of the electrotransport properties and relaxation processes in CdF₂-PbF₂ solid solution crystals, and to establish the relationship between their structure, thermodynamic behavior, and ion transport mechanisms.

Methodology. X-ray phase analysis, differential scanning calorimetry, and dielectric spectroscopy in a broad frequency-temperature range (20 Hz – 10 MHz, -60…+100°C) were used. The composition was determined using Vegard's law; relaxation processes were analyzed using the Havriliak-Negami model.

Results. The synthesized crystals were found to have the composition Cd_0.326Pb0_.674F₂ and a fluorite-type structure. A diffuse superionic transition at ~320°C with pronounced hysteresis was detected by thermal analysis. Dielectric spectroscopy revealed a bulk relaxation process associated with F^– ion hopping. The activation energies for relaxation time and conductivity were ~0.20 eV and ~0.26 eV, respectively, indicating a significant contribution of correlated ion motion.

Research implications. The relationship between structural parameters, the thermodynamics of the phase transition, and the kinetics of ion transport in the CdF₂–PbF₂ system has been established. The results are important for understanding the nature of superionic conductivity in fluoride systems and can be used in the development of new solid-state electrolytes.

Aim. Definition of a special algebraic system based on the field of complex numbers with its addition and multiplication operations and their properties, concepts of norm and argument of new numbers, forms of their representation, special space, as an interpretation of this system.

Methodology. Analysis of algebraic number theory, complex and hypercomplex numbers, theory of vector spaces, theory of functions of a complex variable.

Results. On the basis of complex numbers, an algebraic system of a new type is defined, the operations of which are also invertible, as operations on the field of complex numbers, the properties of operations are defined and proved. This system differs from known systems of hypercomplex numbers and is a special extension of the field of complex numbers. A special space is defined that interprets this system, the concepts of the module and argument of new numbers, the form of their presentation. The possibility of solving in a new system of a certain type of equations insoluble on the set of complex numbers has been proven.

Research implications. The new algebraic system is an element of further development of number theory and the theory of vector spaces, expands the direction of research in other areas (solvability of algebraic equations, function theory, etc.). The results are also applied. In particular, the solvability in the new algebra of previously insoluble complex equations can be used in practical problems, the ability to graphically represent functions of a complex variable can be used in mathematical modeling of spatial objects.