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Abstract The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

Abstract For the equation $${{u}_{{tt}}} - \Delta u - f(x,u) = 0, (x,t) \in {{\mathbb{R}}^{4}},$$ where $$f(x,u)$$ is a smooth function of its variables and is compact in x, the inverse problem of recovering this function from given information on solutions of Cauchy problems for the differential equation is studied. Plane waves with a strong front that propagate in a homogeneous medium in the direction of the unit vector ν and then impinge on an inhomogeneity localized inside some ball B(R) are considered. It is supposed that the solutions of the Cauchy problems can be measured on the boundary of this ball for all ν at times close to the arriving time of the front. The forward Cauchy problem is studied, and the existence of a unique bounded solution in a neighborhood of a characteristic wedge is stated. An amplitude formula for the derivative of the solution with respect to t on the front of the wave is derived. It is demonstrated that the solution of the inverse problem reduces to a series of X-ray tomography problems.

For a hyperbolic equation of the second order, we consider the inverse problem of recovering the coefficient $ q(x,y) $ in this equation. We discuss the scheme of solution of the problem which was proposed by Kabanikhin about 30 years ago. This scheme generalizes the Gelfand–Levitan–Krein method for the solution of the inverse spectral problem to the multidimensional case and reduces the solution of the inverse problem to some infinite system of linear integral equations. No mathematical justification for this scheme has been obtained yet. But numerical experiments based on the $ N $ -approximation produced good results. In this article, we justify some elements of the scheme related to the construction of the infinite system of integral equations in the case when the coefficient $ q(x,y) $ is analytic in $ x $ . In particular, we prove the convergence of the series in these equations and find the conditions for the $ N $ -approximation of the system. We also establish that the infinite system of integral equations is not Fredholm. The question of the solvability of the systems remains open.

Given a hyperbolic equation with variable coefficients, we construct a regularizing algorithm to solve the problem of continuation of the wave field from the boundary of the half-plane inside it. We introduce some $$N$$ -approximate solutions and establish their convergence to the exact solution. Under consideration is the case when the problem data have an error of $$\delta $$ . We find an estimate of the accuracy of the approximate solutions and prove the convergence of the approximate solutions to the unique solution as $$\delta \to 0$$ .

An experimental study of the optical properties of two types of metal–dielectric composites based on opal matrices has been carried out: (1) layered structures, obtained by successive deposition of metal and dielectric films on a monolayer of opal globules, where anomalous transmission and absorption of light was found due to the excitation of surface plasmon polaritons of various types; (2) “massive” opal samples, into which the metal was introduced by the method of electrothermal diffusion, where an asymmetric shape of Bragg resonance curves was observed, due to the Fano resonance.

An Erratum to this paper has been published: https://doi.org/10.1134/S106456242207002X

In yttrium aluminum garnet (YAG) crystals containing terbium and cerium impurities, along with the main EPR signals of Tb3+ and Се3+ ions located in the dodecahedral sites of the YAG lattice in a regular environment, EPR lines with a lower (several percent) intensity were observed. They also belong to the paramagnetic centers of terbium and cerium, but are characterized by slightly altered parameters—the initial level splitting for non-Kramers Tb3+ ions and g-factors for Се3+ ions. It is shown that the nature of such centers and their number can be explained by the presence of YAl antisite defects, i.e. yttrium ions in the octahedral aluminum positions, in the environment of Tb3+ and Се3+.

The methods of high-frequency electron paramagnetic resonance (EPR), electron spin echo (ESE) and optically detected magnetic resonance (ODMR) are used to study the unique properties of nitrogen-vacancy (NV) defects in diamond in strong magnetic fields. It is shown that in strong magnetic fields (∼3–5 T) there occurs an effective optically induced alignment of populations of spin levels resulting in filling the level MS = 0 and emptying of the levels MS = ±1, that makes it possible to record ODMR using the change in the intensity of photoluminescence which reaches 10% at resonance. It is demonstrated that the efficiency of the alignment has the same order as in zero and low magnetic fields. The samples were preliminary studied by the ODMR method in zero magnetic fields that allowed accurate determination of the main parameters of the fine structure and hyperfine interactions with nitrogen nuclei, as well as dipole-dipole interactions between the NV center and deep nitrogen donors in the form of a nitrogen atom replacing carbon, N0. Hyperfine interactions with the nearest carbon atoms (isotope 13C) were observed in the high-frequency ODMR spectra, that opens up opportunities for measuring the processes of dynamic polarization of carbon nuclei in strong magnetic fields using optical methods. It is assumed that narrow ODMR lines in strong magnetic fields can be used to measure these fields with submicron spatial resolution. A new method for recording ODMR of NV centers with microwave frequency modulation has been developed, which simplifies the technique of measuring high magnetic fields. A significant increase in the intensity of the ODMR signal was demonstrated when a strong magnetic field was oriented along the symmetry axis of the NV center.

For electrodynamic equations with permittivity specified by a symmetric matrix $$\varepsilon (x) = ({{\varepsilon }_{{ij}}}(x),i,j = 1,2,3)$$ , the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant $${{\varepsilon }_{0}} > 0$$ outside a bounded domain $$\Omega \subset {{\mathbb{R}}^{3}}$$ , while, inside $$\Omega $$ , it is an anisotropic quantity such that the differences $${{\varepsilon }_{{ij}}}(x) - {{\varepsilon }_{0}}{{\delta }_{{ij}}} = :{{\tilde {\varepsilon }}_{{ij}}}(x),$$ $$i,j = 1,2,3,$$ are small. Here, $${{\delta }_{{ij}}}$$ is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix $$\tilde {\varepsilon }(x) = {{\tilde {\varepsilon }}_{{ij}}}(x),\;i,j = 1,2,3$$ , can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components $${{\tilde {\varepsilon }}_{{ij}}}(x),\;i = 1,2,3,$$ leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.

A survey of recent research concerning phaseless inverse problems for several differential equations is given. Mainly, the surveyed studies were performed over the last five years, although their importance of this subject for quantum scattering theory was noted more than 40 years ago. Problem formulations and results are presented, and the basic ideas underlying the research are described.

A high-frequency electron paramagnetic resonance spectrometer with frequency modulation has been developed. Advantages of the applied method for recording spectra of paramagnetic centers with giant fine-structure splittings are demonstrated by the example of non-Kramers ions in a garnet crystal.

Under study is the integral inequality that has as kernel a nonnegative polynomial in the powers of the difference of arguments and a large parameter N. We establish some inequality whose form agrees with the celebrated Gronwall-Bellman inequality in which the argument of the exponent depends linearly on N.

The advantage of the high-frequency spectroscopy of electron paramagnetic resonance (EPR) for the identification of nitrogen donors and a deep compensating vanadium impurity in various crystallographic positions of the silicon-carbide crystal is shown. Measurements are performed using a new generation EPR spectrometer operating in the continuous wave and pulsed modes at frequencies of 94 and 130 GHz in a wide range of magnetic fields (–7–7 T) and temperatures (1.5–300 K). A magneto-optical closed-cycle cryogenic system (Spectormag PT), highly stable generators (94 and 130 GHz), and a cavity-free system for supplying microwave power to the sample are used.

An Erratum to this paper has been published: https://doi.org/10.1134/S106456242234004X

For a system of electrodynamic equations, the inverse problem of determining an anisotropic conductivity is considered. It is supposed that the conductivity is described by a diagonal matrix σ(x) = $${\text{diag}}({{\sigma }_{1}}(x),{{\sigma }_{2}}(x)$$ , σ3(x)) with $$\sigma (x) = 0$$ outside of the domain Ω = $$\{ x \in {{\mathbb{R}}^{3}}|\left| x \right| < R\} $$ , $$R > 0$$ , and the permittivity e and the permeability μ of the medium are positive constants everywhere in $${{\mathbb{R}}^{3}}$$ . Plane waves coming from infinity and impinging on an inhomogeneity localized in Ω are considered. For the determination of the unknown functions $${{\sigma }_{1}}(x)$$ , $${{\sigma }_{2}}(x)$$ , and $${{\sigma }_{3}}(x)$$ , information related to the vector of electric intensity is given on the boundary S of the domain Ω. It is shown that this information reduces the inverse problem to three identical problems of X-ray tomography.

Electron paramagnetic resonance (EPR) spectra of Ce3+, Yb3+, Cr3+, and Gd3+ impurity ions in yttrium aluminum garnet Y3Al5O12 (YAG) ceramics were detected and identified at frequency of 94 GHz. The advantage of measuring the EPR spectra in the high-frequency range compared to the standard EPR technique is shown, which makes it possible to separate the EPR spectra characterized by different anisotropic g-factors and also to isolate the EPR signals due to the splitting of the fine structure for centers with high-spin states. In ceramics with a high content of magnetic Gd3+ ions, EPR and electron spin echo (ESE) spectra of multi-ionic gadolinium complexes were observed, and EPR spectra of complexes with the maximum number of exchange-coupled gadolinium ions are seen at low temperatures. The temperature dependences of the EPR spectra indicates ferromagnetic ordering of exchange-coupled complexes of gadolinium.

We studied magnetic properties of the In–opal nanocomposite, which is a close-packed system of SiO2 spheres of the same diameter D with metal-filled cavities (series of opal structures with D = 190, 230, and 290 nm were studied). The dependencies of the samples’ magnetization m on temperature T, as well as on external magnetic field H, were measured at various temperatures below the critical temperature Tc of the superconducting (SC) transition. The critical temperature Tc and the critical magnetic field Hc were measured. Magnetization jumps were observed in m(H) dependences at temperatures below 2.6 K. It was found that the jumps are quasiperiodic as functions of H, and the number of jumps increases both with increase in the sweeping rate dH/dt, as well as with decrease in the characteristic size of the In inclusions in the nanocomposite. A model describing the breakup of the SC state with an increase in the external magnetic field is proposed. According to this model, when the screening current reaches a critical value in the near-surface layer of the nanocomposite, an avalanche-like penetration of the field into the volume of the nanocomposite occurs. After this, the SC state is restored, and the magnetic field “freezes” in the structure of the nanocomposite.

A procedure based on precipitation methods for separation and purification of the desired radionuclide was developed for isolating 109Cd from irradiated 107Ag.

For an unmagnetized nonconducting medium, a system of electrodynamic equations corresponding to time-periodic oscillations is considered. For this system, we study the problem of determining the permittivity $$\varepsilon $$ from a given electric field magnitude in the electromagnetic field resulting from the interference of two fields produced by external point current sources. It is assumed that the permittivity differs from a given positive constant $${{\varepsilon }_{0}}$$ only inside a compact domain $${{\Omega }_{0}} \subset {{\mathbb{R}}^{3}}$$ and the electric field magnitude is given for all frequencies, starting at a fixed one $${{\omega }_{0}}$$ , on the boundary $$S$$ of a domain $$\Omega $$ surrounding $${{\Omega }_{0}}$$ . By analyzing an asymptotic expansion of the solution to a direct problem at high frequencies, it is shown that, with the use of given information, the original problem is reduced to the well-known inverse kinematic problem of determining the refraction coefficient inside $$\Omega $$ from the travel times of an electromagnetic wave propagating between boundary points of this domain. This reduction opens up the possibility of obtaining a constructive solution of the original problem.

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.

Samples of a superconducting indium nanocomposite based on a thin-film porous dielectric matrix prepared by the Langmuir–Blodgett method are obtained for the first time, and their low-temperature electrophysical and magnetic properties are studied. Films with thickness b ≤ 5 μm were made from silicon dioxide spheres with diameter D = 200 and 250 nm; indium was introduced into the pores of the films from the melt at a pressure of P ≤ 5 kbar. Thus, a three-dimensional weakly ordered structure of indium nanogranules was created in the pores, forming a continuous current-conducting grid. Measurements of the temperature and magnetic field dependences of the resistance and magnetic moment of the samples showed an increase in the critical parameters of the superconductivity state of nanostructured indium (critical temperature Tc ≤ 3.62 K and critical magnetic field Hc at T = 0 K Hc(0) ≤ 1700 Oe) with respect to the massive material (Tc = 3.41 K, Hc(0) = 280 Oe). In the dependence of the resistance on temperature and the magnetic field, a step transition to the superconductivity state associated with the nanocomposite structure was observed. A pronounced hysteresis M(H) is observed in the dependence of the magnetic moment M of the nanocomposite on the magnetic field at T < Tc, caused by the multiply connected structure of the current-conducting indium grid. The results obtained are interpreted taking into account the dimensional dependence of the superconducting characteristics of the nanocomposite.

Under consideration is the problem of continuation of the wave field from the boundary of a half-plane inside it. We obtain stability estimate for the solution of the corresponding Cauchy problem.

A regularizing algorithm is constructed for solving the problem of extension of a wave field from the boundary to the interior of a half-plane. The cases are considered of the wave equation and hyperbolic equations with variable lower order coefficients.

We consider the inverse problems for differential equations with complex-valued solutions in which the modulus of a solution to the direct problem on some special sets is a given information in order to determine coefficients of this equation; the phase of this solution is assumed unknown. Earlier, in similar problems the modulus of the part of a solution that corresponds to the field scattered on inhomogeneities in a wide range of frequencies was assumed given. The study of high-frequency asymptotics of this field allows us to extract from this information some geometric characteristics of an unknown coefficient (integrals over straight lines in the problems of recovering the potential and Riemannian distances between the boundary points in the problem of the refraction index recovering). But this is physically much more difficult to measure the modulus of a scattered field than that of the full field. In this connection the question arises how to state inverse problems with the full-field measurements as a useful information. The present article is devoted to the study of this question. We propose to take two plane waves moving in opposite directions as an initiating field and to measure the modulus of a full-field solution relating to interference of the incident waves. We consider also the problems of recovering the potential for the Schrodinger equation and the permittivity coefficient of the Maxwell system of equations corresponding to time-periodic electromagnetic oscillations. For these problems we establish uniqueness theorems for solutions. The problems are reduced to solving some well-known problems.

Optical properties of novel metal–dielectric nanocomposite materials based on opal matrices have been investigated. The position of optical resonances of nanocomposites, obtained by embedding of silver into the opal matrix by the electrothermodiffusion method, is explained by the Bragg diffraction, and an asymmetric form of resonance curves is attributed to the Fano resonance. An anomalous transmission and absorption of light by hybrid plasmon-photonic layered heterostructures, which is apparently associated with excitation of surface plasmon-polaritons, propagating along “metal–dielectric” interfaces, was revealed.

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient e from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant e0 only inside some compact domain Ω ⊂ R3 with smooth boundary S. To recover e inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω 0 on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.

The structure and optical diffraction properties of monolayers of monodisperse spheres crystallized on transparent dielectric substrates are studied. Two types of diffraction phenomena are considered: surface light diffraction on the lattice of spheres and waveguide resonances in the monolayer plane. For experimental study of these phenomena, optical retroreflection and transmission spectra are measured as functions of the light incidence angle and azimuthal orientation of the incidence plane. The monolayer structures determined by scanning electron microscopy and light diffraction methods are in quantitative agreement. It is concluded that one-dimensional Fraunhofer diffraction is applicable to describe surface diffraction in the hexagonal lattice of spheres. In the case of oblique light incidence, anisotropy of diffraction and transmission spectra depending on the light incidence plane orientation with respect to the sphere lattice and linear polarization of incident light is detected. Waveguide resonances of the planar two-dimensional photonic crystal are approximated within the light diffraction model in the “empty” hexagonal lattice. The best approximation of the waveguide resonance dispersion is achieved using the effective refractive index, depending on the wavelength. Surface diffraction suppression by waveguide resonances of the photonic crystal is demonstrated. Surface diffraction orders are identified as diffraction at singular points of the Brillouin zone of the planar twodimensional photonic crystal.

The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity e from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant e0 only inside a compact domain Ω ⊂ R 3 with a smooth boundary S. To find e inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω0 and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.

The Upper Volga region was an area of contacts of Finno-Ugric, Slavic, and Scandinavian speaking populations in the 8th–10th centuries AD. However, their role in the formation of the contemporary gene pool of the Russian population of the region is largely unknown. To answer this question, we studied four populations of Yaroslavl oblast (N = 132) by a wide panel of STR and SNP markers of the Y-chromosome. Two of the studied populations appear to be genetically similar: the indigenous Russian population of Yaroslavl oblast and population of Katskari are characterized by the same major haplogroup, R-M198 (xM458). Haplogroup R-M458 composes more than half of Sitskari’s gene pool. The major haplogroup in the gene pool of the population of the ancient town of Mologa is N-M178. Subtyping N-M178 by newest “genomeera” Y-SNP markers showed different pathways of entering this haplogroup into the gene pools of Yaroslavl Volga region populations. The majority of Russian populations have subvariant N3a3-CTS10760; the regular sample of Yaroslavl oblast is equally represented by subvariants N3a3-CTS10760 and N3a4-Z1936, while subvariant N3a4-Z1936 predominates in the gene pool of population of Mologa. This N3a4-Z1936 haplogroup is common among the population of the north of Eastern Europe and the Volga-Ural region. The obtained results indicate preservation of the Finno-Ugric component in the gene pool of population of Mologa and a contribution of Slavic colonization in the formation of the gene pool of the Yaroslavl Volga region populations and make it possible to hypothesize the genetic contribution of the “downstream” (Rostov- Suzdal) rather than “upstream” (Novgorod) Slavic migration wave.

Population biobanks are collections of thoroughly annotated biological material stored for many years. Population biobanks are a valuable resource for both basic science and applied research and are essential for extensive analysis of gene pools. Population biobanks make it possible to carry out fundamental studies of the genetic structure of populations, explore their genetic processes, and reconstruct their genetic history. The importance of biobanks for applied research is no less significant: they are essential for development of personalized medicine and genetic ecological monitoring of populations and are in high demand in forensic science. Establishment of an efficient and representative biobank requires strict observance of the principles of sample selection in populations, protocols of DNA extraction, quality control, and storage and documentation of biological materials. We reviewed regional biobanks and presented the organizational model of population biobank establishment based on the Biobank of Indigenous Population of Northern Eurasia created under supervision of E.V. Balanovska and O.P. Balanovsky. The results obtained using the biobanks in transdisciplinary research and prospective applications for the purposes of genogeography, genomic medicine, and forensic science are presented.

An optical quantum thermometer with submicrometer resolution is proposed. Its operation is based on the physical phenomenon of the optical response in a system of spin centers in silicon carbide under conditions of anticrossing of spin sublevels of the excited quadruplet state (S = 3/2) with temperature-dependent fine-structure splitting.

We describe a series of high-frequency EPR/optically detected magnetic-resonance (ODMR) spectrometers operating in both continuous-wave and pulsed regimes in 2-, 3-, 4-, and 8-mm. The resonance signal can be monitored in both microwave and optical detection channels. The entire series of EPR/ODMR spectrometers has common design features.

The Cauchy problem for a linear second-order uniformly parabolic equation with variable coefficients depending only on the spatial variable is considered. For two- and three-dimensional spaces, asymptotic expansions of the solution as t → +0 are written. It is shown how these expansions can be used in the study of inverse problems of determining the coefficients of a parabolic equation.

Exchange-coupled complexes consisting of Mn ions and holes have been revealed by optically detected magnetic resonance in the narrowest quantum wells of asymmetrical multiple-quantum-well structures (CdMn)Te/(CdMg)Te. Calculations have been performed to estimate the parameters of the complexes (exchange interactions and hole g-factors) and simulate the spectra. The formation of such complexes implies the directional electron tunneling from narrow to larger wells, i.e., an electron cascade, which results in the creation of the excess hole concentration in the narrowest and intermediate-width quantum wells.

An integrated analytical study of 30 Devonian (DIII, DI, D3br, Dl, D3lv) and Carboniferous (A4, B2, V1, V3) oil samples from oilfields in the northern, central, and southern parts of Samara oblast has been performed in order to determine the basic characteristics of the composition and the properties of the oils. The results of the comprehensive study of the composition and properties of the crude oils lead to the conclusion that oils from the oilfields in the northern part of Samara oblast significantly differ from those of the southern and central zones. It has been found that the main factor determining the formation of the physicochemical properties of oils in the northern zone is the concentration of higher molecular weight paraffins and asphaltenes, whereas such a factor for the oils of the southern zone is the concentration of higher molecular weight paraffins only. By cluster analysis, the crudes from oilfields in the south of the Samara oblast have been divided into three clusters.

Long-lasting afterglow of ZnO single crystals grown by hydrothermal method has been studied using high-frequency (94 GHz) optically detected magnetic resonance (ODMR) at helium temperatures. Giant (up to 30%) afterglow intensity-detected electron paramagnetic resonance signals of shallow donors and deep acceptors, which are lithium atoms substituting for zinc in the ZnO lattice, unambiguously indicate a spin-dependent character of the donor-acceptor recombination. The observed variations in the shape of the afterglow-detected ODMR spectra of lithium acceptors with time passed after irradiation are associated with the difference in the recombination rates of shallow donors with axial and non-axial LiZn centers and with a change in the recombination character with time.

Chrysotile samples with macroscopically ordered channels filled by gold and silver have been studied using optical transmission spectroscopy. The channels had inner diameters below 5 nm and lengths up to about 1 cm. The transmission spectra of samples strongly depend on the polarization of probing light. The observed spectral properties are probably related to plasmon resonances of metal particles and to surface plasmon polaritons.

We consider the problem of finding the boundaries of discontinuities and jumps at these discontinuities of a piecewise smooth function for X-ray tomography. We give some algorithm for the reconstruction of the unknown values as well as formulas for their calculation. The original problem of X-ray tomography is reduced to a problem for a continuous piecewise differentiable function.

For the integrodifferential viscoelasticity equations, we study the problem of determining the coefficients of the equations and the kernels occurring in the integral terms of the system of equations. The density of the medium is assumed to be given. We suppose that the inhomogeneity support of the sought functions is included in some compact domain B 0. We consider a series of inverse problems in which an impulse source is concentrated at the points y of the boundary of B 0. The point y is the parameter of the problem. The given information about the solution is the trace of the solution to the Cauchy problem with zero initial data. This trace is given on the boundary of B 0 for all y ∈ ∂B 0 and for a finite time interval. The main result of the article consists in obtaining uniqueness theorems for a solution to the initial inverse problem.

Results of theoretical and experimental studies of regularities in the transformation of the topological and polarization structures of optical vortices by polarization dynamic holograms formed by pulse Gaussian and singular light beams in dye solutions are presented.

Based on colloidal crystals of various dimensionality, hybrid metal-dielectric plasmonic-photonic heterocrystals have been prepared. It has been shown that the spectra of optical transmission of heterocrystals are mostly controlled by the sum of contributions of composing plasmonic and photonic crystals. At the same time, there are a number of phenomena caused by the mutual effect of heterostructure components, which lead to a deviation of observed optical properties from the linear superposition of responses of these crystals. In particular, it has been found that the anomalous transmission controlled by the plasmonic crystal decreases with increasing the dimensionality of the photonic crystal attached to it. At the same time, light reflection on a metallized surface changes light diffraction in photonic crystals and leads to Fabry-Perot oscillation amplification. It has been assumed that an intermediate layer is formed, in which Bloch modes of the photonic crystal and surface plasmon-polaritons of the plasmonic crystal are hybridized.

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ℝ2 | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.

To perform optical detection of high-frequency magnetic resonance (ODMR) and cyclotron resonance (ODCR) with a spatial resolution, we have proposed a noncavity scheme of an ODMR-ODCR spectrometer using a quasi-optical microwave channel. The efficiency of this scheme for obtaining information on the spin properties of ZnO quantum dots and CdMnTe quantum wells and on the effective masses of carriers in crystalline silicon films with a spatial resolution within a focused laser beam is illustrated.