We present an original algorithm in the MAPLE system for solving the scattering problem in single-channel approximation of the coupled-channel method of the optical model (OM) described by a second-order ordinary differential equation (ODE) with a complex-valued potential and regular boundary conditions. The complex-valued potential consists of the known real part, which is a sum of the nuclear potential, the Coulomb potential, and the centrifugal potential, and the imaginary part, which is a product of the unknown coupling constant g ( E ), depending on the collision energy E of a pair of ions, and the derivative of the real part of the known nuclear potential with respect to the ODE independent variable. The presented algorithm implements the solution of the inverse problem, i.e., calculates the unknown coupling constant g ( E ) and scattering matrix S ( g ( E ), E ) from condition | S ( g ( E ) , E ) | 2 = 1 - | T ( E ) | 2 by means of the secant method. The required amplitudes of transmission T ( E ) and reflection R ( E ) subject also to the condition | R ( E ) | 2 = 1 - | T ( E ) | 2 of the model with incoming wave boundary conditions (IWBCs) are previously calculated by the standard MAPLE implemented KANTBP 4M program. The algorithm provides a one-to-one correspondence between the OM with a complex-valued potential and the model of IWBCs with a real-valued potential. The efficiency of the proposed approach is shown by solving numerically the scattering problem and calculating the reference fusion cross section for a pair of heavy ions 16 O+ 144 Sm in the single-channel approximation of the close-coupling method.

The piecewise polynomial functions constructed from the multivariate Hermitian interpolation polynomials that are continuous together with derivatives on the boundaries of finite elements are used in implementations of the high-accuracy finite element method (FEM). The efficiency of our finite element schemes, algorithms and program GCMFEM implemented in Maple and Mathematica are demonstrated by reference calculations of the boundary value problems (BVPs) for the Geometric Collective Model (GSM) of atomic nuclei. The BVP for GSM is reduced also to the BVP for a system of ordinary differential equations, which is solved by program KANTBP 5M implemented in Maple and compared with solution of algebraic eigenvalue problem in representation of the basis functions associated within irreducible representations of the U(5) ⊃ O(5) ⊃ O(3) chain of groups.

A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrodinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at left- and right-boundary points for the discrete spectrum and scattering problems. The resulting system of such equations, containing potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. The scattering problem is solved with non-diagonal potential matrix elements in the left and/or right asymptotic regions and different left and right threshold values. Benchmark calculations for the fusion cross sections of S-36+Ca-48, Ni-64+Mo-100 reactions are presented. As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions of the exact solvable wave-guide model, and also the fusion cross sections and mean angular momenta of the O-16+Sm-144 reaction. Program summary Program Title: KANTBP CPC Library link to program files: https://doi.org/10.17632/4vm9fhyvh3.1 Licensing provisions: CC BY NC 3.0 Programming language: FORTRAN Nature of problem: In the adiabatic approach [1], a multidimensional Schrodinger equation for quantum reflection [2], the photoionization and recombination of a hydrogen atom in a homogeneous magnetic field [3-6], the three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [7], wave-guide models [8], the fusion model of the collision of heavy ions [9-11], and low-energy fusion reactions of light- and medium mass nuclei [12] is reduced by separating the longitudinal coordinate, labeled as z, from transversal variables to a system of second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present a program based on the use of high-order accuracy approximations of the finite element method (FEM) for calculating energy levels, reflection and transmission matrices and wave functions for such systems of coupled-channel second order differential equations (CCSODEs) on finite intervals of the variable z is an element of [Z(min), Z(max)] with homogeneous boundary conditions of the third-type at the left- and right-boundary points, which follow from the discrete spectrum and scattering problems. Solution method: The boundary-value problems for the system of CCSODEs are solved by the FEM using high-order accuracy approximations [13,14]. The generalized algebraic eigenvalue problem AF = E B F with respect to pair unknowns (E, F), arising after the replacement of the differential eigenvalue problem by the finite-element approximation, is solved by the subspace iteration method [14]. The generalized algebraic eigenvalue problem of a special form (A - E B) F = D F with respect to pair unknowns (D, F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LD L-T factorization of the symmetric matrix and back-substitution methods [14]. Additional comments including restrictions and unusual features: The user must supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSL and ASYMSR (when solving the scattering problem) which evaluate asymptotics of the wave functions at boundary points in the case of a boundary conditions of the third-type for the above problems. (C) 2022 Elsevier B.V. All rights reserved.

We consider fully symmetric quadrature rules with positive weights, and with nodes lying inside the 3,…,6 dimensional simplex (so-called PI-type). PI-type fully symmetric quadrature rules up to 20-th order on the tetrahedron, 16-th order on 4-simplex, 10-th order on 5- and 6-simplexes are presented. The number of nodes of the presented quadrature rules for the corresponding orders does not exceed the known ones, and most of them are new. In the calculation we applied the modified Levenberg-Marquardt methods for solving nonlinear equations with convex constraints. The corresponding programs are implemented in MAPLE-FORTRAN environment, and the weights and nodes are first calculated using a FORTRAN program with an accuracy of 10−25 and refined up to accuracy of 10−50 using a MAPLE program.

We briefly review the analysis of the energy spectrum, the envelope eigenfunctions of electron, hole and exciton states, and the direct interband light absorption in cone-shaped and spheroidal impenetrable quantum dots. We apply high-order finite element method and calculation schemes of Kantorovich method in comparison with the adiabatic approximation (in the strong size quantization limit) for solving boundary-value problems that describe axially symmetric quantum dots. We demonstrate the efficiency of the algorithms and software by benchmark calculations of spectral and optical characteristics of the cone-shaped and spheroidal quantum dots and crossing points in their spectra.

Trong bài báo này chúng tôi trình bày sơ đồ thuật toán và kết quả tính toán các trạng thái siêu bền trong bài toán tán xạ và bài toán trị riêng chứa rào thế năng ở dạng phức. Đối với bài toán tán xạ, các hàm sóng với ma trận tán xạ S được tính toán với năng lượng có giá trị thực xác định của sóng tới, còn đối với bài toán trị riêng, các hàm sóng cùng với các trị riêng tương ứng cũng được tính toán. Sau đó, chúng tôi khảo sát và tính toán hàm sóng của các trạng thái siêu bền trong lân cận của các giá trị năng lượng cộng hưởng cho hai bài toán này. Nghiệm của bài toán biên được tính toán bằng chương trình phần mềm được biên soạn bởi tác giả bài báo cùng các cộng sự khoa học ở Viện Liên hiệp Nghiên cứu Hạt nhân Dubna, Thành phố Dubna, Liên bang Nga. Các thuật toán của chương trình tính toán này dựa trên phương pháp phần tử hữu hạn với độ chính xác cao. Kết quả tính toánđược biểu diễn dưới dạng bảng và đồ thị.

Trong bài báo này chúng tôi trình bày sơ đồ thuật toán và kết quả tính toán hàm sóng tán xạ đối với các trạng thái siêu bền của phân tử lưỡng nguyên tử Beryli trong quang phổ laser. Nghiệm của bài toán biên được tính toán bằng chương trình phần mềm được biên soạn bởi tác giả bài báo cùng các cộng sự khoa học ở Viện Liên hiệp Nghiên cứu Hạt nhân Dubna, Thành phố Dubna, Liên bang Nga. Các thuật toán của chương trình tính toán này dựa trên phương pháp phần tử hữu hạn với độ chính xác cao. Hàm thế năng được cho ở dạng bảng giá trị được nối với hàm thế năng tiệm cận Waals bằng cách sử dụng đa thức nội suy Hermite và đảm bảo tính liên tục của nghiệm hàm cùng đạo hàm của nó. Sự hiệu quả của chương trình tính toán này được thể hiện bằng việc tính toán các giá trị năng lượng cộng hưởng ở dạng phức của các trạng thái siêu bền trong phổ xung động quay của phân tử lưỡng nguyên tử Beryli. Với các trạng thái siêu bền này, các hàm sóng tán xạ tương ứng với năng lượng cộng hưởng mang giá trị thực được tính toán và biểu diễn dưới dạng đồ thị.

We propose a mathematical model of COVID-19 pandemic preserving an optimal balance between the adequate description of a pandemic by SIR model and simplicity of practical estimates. As base model equations, we derive two-parameter nonlinear first-order ordinary differential equations with retarded time argument, applicable to any community (country, city, etc.).The presented examples of modeling the pandemic development depending on two parameters: the time of possible dissemination of infection by one virus carrier and the probability of contamination of a healthy population member in a contact with an infected one per unit time, e.g., a day, is in qualitative agreement with the dynamics of COVID-19 pandemic. The proposed model is compared with the SIR model.

We analyze the origin of the unexpected deep subbarrier heavy-ion fusion hindrance in Ni-64 + Mo-100, Ni-64 + Ni-64, and Si-28 + Ni-64 reactions. Our analysis is based on the improved coupled-channels description, implemented by means of the finite element method. With the aid of the Woods-Saxon potential the experimental cross sections and the S factors of these reactions are remarkably well reproduced within the sudden approximation approach. We found that accounting for the nondiagonal matrix elements of the coupling matrix, traditionally neglected in the conventional coupled-channels approaches in setting the left boundary conditions inside the potential pocket, and its minimal value are crucially important for the interpretation of experimental data. We found as well a good agreement with the general trend of the experimental data for the S factor of the fusion reaction C-12 + C-12, which has no pronounced maximum for this system.

This review outlines the main results which show the dual nature of the chemical bond in diatomic beryllium molecule in the ground $X^1\Sigma_g^+$ state. It has been shown that the beryllium atoms are covalently bound at low-lying vibrational energy levels ({\nu}=0-4), while at higher ones ({\nu}=5-11) they are bound by van der Waals forces near the right turning points. High precision ab initio quantum calculations of Be$_2$ resulted in the development of the modified expanded Morse oscillator potential function which contains all twelve vibrational energy levels [A.V. Mitin, Chem. Phys. Lett. 682, 30 (2017)]. The dual nature of chemical bond in Be$_2$ is evidenced as a sharp corner on the attractive branch of the ground state potential curve. Moreover, it has been found that the Douglas-Kroll-Hess relativistic corrections also show a sharp corner when presented in dependence on the internuclear separation. The difference in energy between the extrapolated and calculated multi-reference configuration interaction energies in dependence on the internuclear separation also exhibits singular point in the same region. The other problems of ab initio quantum calculations of the beryllium dimer are also discussed. Calculated spectrum of vibrational-rotational bound states and new metastable states of the beryllium dimer in the ground state important for laser spectroscopy are presented. The vibration problem was solved for the modified expanded Morse oscillator potential function and for the potential function obtained with Slater-type orbitals [M. Lesiuk et al, Chem. Theory Comput. 15, 2470 (2019)]. The theoretical upper and lower estimates of the spectrum of vibrational-rotational bound states and the spectrum of rotational-vibrational metastable states with complex-valued energy eigenvalues and the scattering length in the beryllium dimer are presented.

The Russian Federation is suffering from African swine fever since november 2007. The main problem with African swine fever on the territory of the Russian Federation is determined by the formed enzootic zones and partially external cases of the disease on the administrative territories of the non-enzootic zones. Despite the measures taken, the disease tends to spread annually. For the period 2007-2020, 1840 outbreaks of this disease were registered in Russia, including 1077 domestic pigs and 737 wild boars. From our opinion, Russia has a unique opportunity in natural conditions to study the stages of the formation of the enzootic zone in the North Caucasus region. This territory is compactly located: the Chechen Republic, the Republic of Ingushetia, the Kabardino-Balkar Republic, the Republic of Dagestan, the Republic of North Ossetia-Alania, the Karachay-Cherkess Republic, the Republic of Adygea, where population, with the exception of the Republic of North Ossetia-Alania, confess the Islamic religion, which prohibits pork consumption. In this article, based on the analysis of the epizootic situation from 2007-2020. an opinion was expressed on the mechanism of the formation of enzootic zones for African swine fever, as well as on the establishment of zones where the disease appears in order to introduce restrictive and prohibitive measures. The position on the role of wild boars in the formation of enzootic zones and the spread of infection was determined. The understanding of the mechanism of formation of enzootic zones is the basis for the development of effective measures to eradicate African swine fever in Russia.

The computational scheme and calculation results of bound, metastable and Rydberg states of atomic and molecular systems important for laser spectroscopy are presented. The solution to the problem is performed using the authors' software package (see program libraries of the Computer Physics Communications journal and of the Joint Institute for Nuclear Research) that implement the high-accuracy finite element method. The FORTRAN procedure of matching tabulated potential functions with van der Waals asymptotic potential using interpolation Hermite polynomials which provides continuity of both the function itself and its derivative is presented. The efficiency of the proposed approach is demonstrated by calculated for the first time sharp metastable states with complex eigenenergies in a diatomic beryllium molecule and weakly bound Rydberg states of antiprotonic helium atom.

Phương pháp tách biến đóng vai trò quan trọng trong các bài toán vật lí toán, đặc biệt trong bài toán tán xạ có chứa phương trình dạng hyperbolic khi miền khảo sát được giới hạn bởi các bề mặt tọa độ có hình dạng bất kì. Trong bài báo này, chúng ta sẽ sử dụng phương pháp tách biến trong hệ tọa độ cầu cho việc tính toán tán xạ dừng vô hướng trên một phỏng cầu dài với tỉ lệ bất kì giữa bước sóng và kích thước của phỏng cầu.

We apply a new calculation scheme of a finite element method and upgraded program KANTBP in the coupled-channels calculations for heavy-ion fusion reactions. We diagonalize the matrix of close coupled effective potentials to formulate ingoing wave boundary conditions in point within a pocket of the potential well with a true set of thresholds. Efficiency of the proposed approach is shown by successfully described experimental data for the sub-barrier and above-barrier fusion cross section of some reaction systems.

We analyse the origin of the unexpected deep sub-barrier heavy-ion fusion hindrance in 64Ni+100Mo and 28Si+64Ni recations. Our analysis is based on the improved coupled-channels approach, implemented by means of the finite element method. With the aid of the Woods-Saxon potential the experimental cross sections and the S-factors of these reactions are remarkably well reproduced. We found that the account on the non-diagonal matrix elements of the coupling matrix, traditionally neglected in the conventional coupled-channels approaches in setting the left boundary conditions inside the potential pocket, and its minimal value are crucially important for the interpretation experimental data. Within our approach we found a good agreement with the experimental data for the S-factor of the fusion reaction 12C+12C, which has no a pronounced maximum for this system.

The paper contains the description of a new method for decomposition of arbitrary Hamiltonians and other operators into the tensor form. The method is based on a special kind of projection operators acting in the space of appropriate operators. The properties of these projection operators and the decomposition method are presented.

As a result of a study of the Pravdinsk reservoir, it was found that the radiation situ ation can be estimated as safe. According to microbiological indicators, the water quality was assessed as polluted with , -mesosaprobic, 34 classes of water quality. During the bloom of water by Cyanobacteria, we observed the toxic effect on the test organisms Daphnia magna and Ceriodaphnia affinis, in this period the consumption of phytoplankton by zooplankton was low. According to benthos biomass, the reservoir is assessed as a water body with a high food value. The ichthyofauna included 14 species of fish that belonged to families of Cyprinidae, Percidae, Gadidae and Esocidae. 11 species of parasites related to Microsporidia, Myxosporidia, Trematoda, Cestoda, Hirudinea and parasitic Crustacea were found. Metatercaria of trematoda Apophallus mehlingi pathogenic for humans were found in roach and perch.

The 1A1′,3E′,1E′,3A1′,1A2′,3A2′ representing the ground and the five excited states, which have the common character of being symmetrical with respect to reflection on the plane of the equilateral triangular H3+ molecule, are determined by an original three center wave function constructed by the use of the irreducible representations of the D3h point group. In contrast to past large one center or linear combinations of atomic orbitals functions, our model has the advantage of being well adapted to all internuclear distances, with limited number of basis functions including the electron-electron term. Our functions satisfy, by their nature, the triangular geometry of the molecule and thus permit the study the asymptotic behavior of the potential energy curves of the fundamental and excited levels for which, new experimental and theoretical results are needed to confirm astronomical observations. The results of this work and the implementation of the computational techniques employed opens the way to further studies on complex three center systems.

The problem of a quantum-mechanical description of a near-barrier fusion of heavy nuclei, that occurs at strong coupling of their relative motion to surface vibrations, is analyzed. To this end, an efficient finite-element method is proposed for numerically solving coupled Schrodinger equations with boundary conditions corresponding to total absorption. The method allows us to eliminate the instabilities in the numerical solutions that appear at a large number of coupled channels in some reactions. To illustrate the validity of our approach, the results of fusion cross section of the Ni-64 + Mo-100 and S-36 + Ca-48 reactions have been re-examined. The obtained results demonstrate a remarkable agreement with the available experimental data. It is found that experimental data can be reproduced with the use of the Woods-Saxon potential, without introducing the repulsive cores. It appears that the fusion cross sections at deep sub-barrier energies are sensitive to the potential pocket profile.