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In this paper we deal with numerical triangles defined by generating functions in the power of k. We present new approach to study such triangles that allow us to get methods for obtaining generating functions of the diagonals of the triangles. Methods for obtaining generating functions of the central coefficients and the diagonal T2n,nof the triangle Tn,k are discussed. Some further ideas for application are given. ; The author thanks the organizing committee and the program committee of section "Mathematics and Computer Science" of All-Russian Forum of Young Scientists, Ekaterinburg, 2017, for fruitful recommendations and comments. This work was partially supported by the Ministry of Education and Science of Russia, Government Order no. 2.8172.2017/BP (TUSUR).

In the early 21 century the Arctic has become a very important region in international relations. Its vast natural richness, potential oil and gas resources, and transit capabilities make the Arctic attractive for many states of the modern world. Arctic region has gained great importance in matters of ensuring the U.S. national security in the new geostrategic environment. The author examines main documents related to the U.S. military-strategic interests in the Arctic. The main challenges and achievements of the U.S. military activities in the Arctic region are disclosed as well.

Authors consider the need to identify the criteria of true democracy and ambiguity of interpretation of this concept interpreted by various philosophers and ordinary people in very different ways.

Aus osteuropäischer Sicht + Aus sowjetischer Sicht

S.P.K. acknowledges support by the National Academy of Sciences of Ukraine (Project No.0116U002067). Calculations were performed using Latvian Super Cluster (LASC), located in the Center of Excellence at Institute of Solid State Physics, the University of Latvia, which is supported by European Union Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-Teaming. Phase two under Grant Agreement No. 739508, project CAMART2. ; We propose a novel approach for characterising the electron spectrum of disordered crystals constructed from a Hamiltonian of electrons as well as phonons and a diagram approach for Green's function. The system's electronic states were modelled by means of the multi‐band, tight-binding approach. The system's Hamiltonian is described based on the electron wave functions at the field of the atom nucleus. Our novel approach incorporates the long‐range Coulomb interplay of electrons located in different lattice positions. Explicit interpretations of Green's functions are derived using a diagram method.Equations are obtained for the vertex components for the mass operators of the electron–electron as well aselectron–phonon interplays. A system of equations for the spectrum of elementary excitations in the crystal is obtained, in which the vertex components for the mass operators of electron–electron as well as electron–phonon interplays are renormalised. Thismakes it possible to perform numerical computationsfor the system's energy spectrum with a predetermined accuracy. In contrast to other approaches in which electron correlations are only taken into account in the limiting cases of an infinitely large and infinitesimal electron density, in this method, electron correlations are described in the general case of an arbitrary density. We obtained the cluster expansion of the density of states (DOS) of the disordered systems. We demon-strate that the addition of the electron‐scattering mechanismsto the clusters is decreasing. This hap-pens due to a growing number of positions in the cluster, which hang ontothe small parameter. The computing exactness is fixed by a small parameter for cluster expansion of Green's functions of electrons as well as phonons. © 2022 by the authors. Submitted for possible open access.--//-- This is an open access article Kruchinin, S.P.; Eglitis, R.I.; Babak, V.P.; Vyshyvana, I.G.; Repetsky, S.P. Effects of Electron Correlation inside Disordered Crystals. Crystals 2022, 12, 237. https://doi.org/10.3390/cryst12020237; published under the CC BY 4.0 licence. ; European Union Horizon 2020 Framework Programme H2020‐WIDESPREAD‐01‐2016‐2017‐Teaming; National Academy of Sciences of Ukraine 0116U002067; Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union's Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508, project CAMART2.

This report presents the conceptual design of a new European research infrastructure EuPRAXIA. The concept has been established over the last four years in a unique collaboration of 41 laboratories within a Horizon 2020 design study funded by the European Union. EuPRAXIA is the first European project that develops a dedicated particle accelerator research infrastructure based on novel plasma acceleration concepts and laser technology. It focuses on the development of electron accelerators and underlying technologies, their user communities, and the exploitation of existing accelerator infrastructures in Europe. EuPRAXIA has involved, amongst others, the international laser community and industry to build links and bridges with accelerator science — through realising synergies, identifying disruptive ideas, innovating, and fostering knowledge exchange. The Eu-PRAXIA project aims at the construction of an innovative electron accelerator using laser- and electron-beam-driven plasma wakefield acceleration that offers a significant reduction in size and possible savings in cost over current state-of-the-art radiofrequency-based accelerators. The foreseen electron energy range of one to five gigaelectronvolts (GeV) and its performance goals will enable versatile applications in various domains, e.g. as a compact free-electron laser (FEL), compact sources for medical imaging and positron generation, table-top test beams for particle detectors, as well as deeply penetrating X-ray and gamma-ray sources for material testing. EuPRAXIA is designed to be the required stepping stone to possible future plasma-based facilities, such as linear colliders at the high-energy physics (HEP) energy frontier. Consistent with a high-confidence approach, the project includes measures to retire risk by establishing scaled technology demonstrators. This report includes preliminary models for project implementation, cost and schedule that would allow operation of the full Eu-PRAXIA facility within 8—10 years.

This report presents the conceptual design of a new European research infrastructure EuPRAXIA. The concept has been established over the last four years in a unique collaboration of 41 laboratories within a Horizon 2020 design study funded by the European Union. EuPRAXIA is the first European project that develops a dedicated particle accelerator research infrastructure based on novel plasma acceleration concepts and laser technology. It focuses on the development of electron accelerators and underlying technologies, their user communities, and the exploitation of existing accelerator infrastructures in Europe. EuPRAXIA has involved, amongst others, the international laser community and industry to build links and bridges with accelerator science — through realising synergies, identifying disruptive ideas, innovating, and fostering knowledge exchange. The Eu-PRAXIA project aims at the construction of an innovative electron accelerator using laser- and electron-beam-driven plasma wakefield acceleration that offers a significant reduction in size and possible savings in cost over current state-of-the-art radiofrequency-based accelerators. The foreseen electron energy range of one to five gigaelectronvolts (GeV) and its performance goals will enable versatile applications in various domains, e.g. as a compact free-electron laser (FEL), compact sources for medical imaging and positron generation, table-top test beams for particle detectors, as well as deeply penetrating X-ray and gamma-ray sources for material testing. EuPRAXIA is designed to be the required stepping stone to possible future plasma-based facilities, such as linear colliders at the high-energy physics (HEP) energy frontier. Consistent with a high-confidence approach, the project includes measures to retire risk by establishing scaled technology demonstrators. This report includes preliminary models for project implementation, cost and schedule that would allow operation of the full Eu-PRAXIA facility within 8—10 years.

This report presents the conceptual design of a new European research infrastructure EuPRAXIA. The concept has been established over the last four years in a unique collaboration of 41 laboratories within a Horizon 2020 design study funded by the European Union. EuPRAXIA is the first European project that develops a dedicated particle accelerator research infrastructure based on novel plasma acceleration concepts and laser technology. It focuses on the development of electron accelerators and underlying technologies, their user communities, and the exploitation of existing accelerator infrastructures in Europe. EuPRAXIA has involved, amongst others, the international laser community and industry to build links and bridges with accelerator science — through realising synergies, identifying disruptive ideas, innovating, and fostering knowledge exchange. The Eu-PRAXIA project aims at the construction of an innovative electron accelerator using laser- and electron-beam-driven plasma wakefield acceleration that offers a significant reduction in size and possible savings in cost over current state-of-the-art radiofrequency-based accelerators. The foreseen electron energy range of one to five gigaelectronvolts (GeV) and its performance goals will enable versatile applications in various domains, e.g. as a compact free-electron laser (FEL), compact sources for medical imaging and positron generation, table-top test beams for particle detectors, as well as deeply penetrating X-ray and gamma-ray sources for material testing. EuPRAXIA is designed to be the required stepping stone to possible future plasma-based facilities, such as linear colliders at the high-energy physics (HEP) energy frontier. Consistent with a high-confidence approach, the project includes measures to retire risk by establishing scaled technology demonstrators. This report includes preliminary models for project implementation, cost and schedule that would allow operation of the full Eu-PRAXIA facility within 8—10 years.