Singularly perturbed spectral problems with Neumann boundary conditions

The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue problem for a singularly perturbed Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and formulate the selection criterium that allows us to choose a solution of the limit Hamilton-Jacobi equation which gives the logarithmic asymptotics of the principal eigenfunction

Polarization Effects in Superfluid 4^4He

A theory of thermoelectric phenomena in superfluid $^4He$ is developed. It is found an estimation of the dipole moment of helium atom arising due to electron shell deformation caused by pushing forces from the side of its surrounding atoms. The corresponding electric signal generated in a liquid consisting of electrically neutral atoms by the ordinary sound waves is found extremely small. The second sound waves in superfluid $^4He$ generate the polarization of liquid induced by the relative accelerated motion of the superfluid and the normal component. The derived ratio of the amplitudes of temperature and electric polarization potential was proved to be practically temperature independent. Its magnitude is in reasonable correspondence with the experimental observations. The polarity of electric signal is determined by the sign of temperature gradient in accordance with the measurements. The problem of the roton excitations dipole moment is also discussed.Comment: 8 pages, no figure

Roton dipole moment

The roton excitation in the superfluid He-4 does not possess a stationary dipole moment. However, a roton has an instantaneous dipole moment, such that at any given moment one can find it in the state either with positive or with negative dipole moment projection on its momentum direction. The instantaneous value of electric dipole moment of roton excitation is evaluated. The result is in reasonable agreement with recent experimental observation of the splitting of microwave resonance absorption line at roton frequency under external electric field.Comment: 5 page

Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary

It was shown in [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497{1531] that in doubly connected domains there exist local minimizers of the simplified Ginzburg-Landau functional with modulus one and prescribed degrees on the boundary, unlike global minimizers that typically do not exist. We generalize the results and techniques of the aforementioned paper to the case of the magnetic Ginzburg-Landau functional.В работе [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497-1531] было показано, что в двусвязных областях существуют локальные минимизанты упрощенного функционала Гинзбурга-Ландау, имеющие модуль один и заданные степени отображения на границе, в отличие от глобальных минимизантов, которые обычно не существуют. Результаты и методы упомянутой выше статьи обобщаются на случай "магнитного" функционала Гинзбурга-Ландау

Great Perm Governance in the Late 16th – Early 17th Century: From Administrators to Voivodes

Introduction. Great Perm (Perm Velikaya) in the late 16th – early 17th centuries included Cherdyn, Solikamsk and Kaigorod districts. It was an important strategic region. Taxes from this region entered the royal treasury almost regularly, while the central part of the state was devastated by military operations during the Time of Troubles. The region provided communication with Siberia, its annexation and development. Historiography is dominated by the opinion that the institution of voivodeship was introduced in Great Perm in the late 16th century. Methods and materials. The research is based on the documents of the archive of the Solikamsk district court, which currently constitute the collection of the fund No. 122 “Acts of Solikamsk” of the Archive of the St. Petersburg Institute of History of the Russian Academy of Sciences for the period from 1605 to 1613. The documentary complex was previously reconstructed using the principle of mutual correspondence of documents. The prosopography method was applied to study the biographies of clerks. Analysis. The article analyzes the argumentation of researchers on the problem when the institute of voivodeship was introduced in Great Perm. The fundamental documents of the late 16th century have been revealed. It is proved that they do not contain information about the voivodeship form of government. The list of administrators and clerks of the late 16th century – early 17th century has been restored. The beginning and end of their service are indicated. Results. There was no voivodeship position in Great Perm until 1609. An administrator and a clerk were appointed to the clerk hut from Moscow. The change of administration took place on average every 2 years. The institute of voivodeship was introduced in Great Perm in 1610. The first voivode of Great Perm was Ivan Ivanovich Chemodanov. The administrators who replaced him also served as voivodes. The control system in Great Perm was strengthened by 1613

Organizing Solikamsk Coach Station and Coachman Service in Great Perm in 1607

Introduction. The city of Solikamsk on the territory of Great Perm was located on the first land road from Moscow to Siberia. This road was built in the late 16th century. The research is devoted to the problem of establishing a permanent coachman service in Solikamsk in 1607. Methods and materials. This issue has not been studied yet. The article is based on documents from Fund no. 21 “Solikamsk Acts” (Archive of St. Petersburg Institute of History of the Russian Academy of Sciences): monarch’s decrees, letters from Permian clerks, petitions. We have reconstructed the Solikamsk archive using the method of mutual compliance of documents and their source study analysis. Analysis. In the course of the study we were able to restore the chronology of events, find out the coachman service form of organization, material support, service operation conditions, staff of the Vyatka coach station by name. The paper reveals the mechanisms of managing Great Perm territory, the ways of solving problems of management and transportation. Results. The article reveals the duality of the situation in Great Perm in the early 17th century. On the one hand, we can see functioning of a strictly centralized management system, on the other hand, we observe a strong local government in provinces. Perm’s clerk Prince S.Yu. Vyazemsky had to clearly execute the orders of Moscow and was prevented from making his own decisions. At the same time, key financial issues influencing stable work of coachman service were not originally planned in Moscow. Decisions arrived late. The central government was more concerned with the timely dispatch of tax collections to Moscow. After Solikamsk coach station coachmen dissolution, the problem of transportation on the Siberian road remained

Theory of a Narrow roton Absorption Line in the Spectrum of a Disk-Shaped SHF Resonator

We calculate the probability of the birth of a circular phonon (c-phonon) in He II by a c-photon of the resonator. It is shown that this probability has sharp maxima at frequencies, where the effective group velocity of the c-phonon is equal to zero; the density of states of c-phonons strongly grows at such frequencies. For He II, these frequencies correspond to a roton and a maxon. From the probability of the c-roton birth, we calculate the roto line width which is found to approximately agree with the experimental one. We conclude that the roton line observed in the super-high-frequency (SHF) absorption spectrum of helium is related to the birth of c-rotons. A possible interpretation of the Stark effect observed for the roton line is also proposed.Comment: 13 pages, 1 figure, v2: journal variant, several minor correction