Pub Date : 2021-09-30DOI: 10.22363/2658-4670-2021-29-3-271-284
A. Baddour, M. Malykh
An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem Ρ{{Rho}} is considered, the solution of which results in a real number uu. To solve this problem, a numerical method is used, that is, the set Hℝ{Hsubset mathbb{R}} and the mapping uh:Hℝ{u_h:Htomathbb{R}} are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set HH and uh{u_h} can be expanded in a convergent series in powers of h:uh=u+c1hk+...{h:u_h=u+c_1h^k+...}. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for uu and cc from two values of uh{u_h}. The question of using a larger number of uh{u_h} values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.
{"title":"Richardson-Kalitkin method in abstract description","authors":"A. Baddour, M. Malykh","doi":"10.22363/2658-4670-2021-29-3-271-284","DOIUrl":"https://doi.org/10.22363/2658-4670-2021-29-3-271-284","url":null,"abstract":"An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem Ρ{{Rho}} is considered, the solution of which results in a real number uu. To solve this problem, a numerical method is used, that is, the set Hℝ{Hsubset mathbb{R}} and the mapping uh:Hℝ{u_h:Htomathbb{R}} are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set HH and uh{u_h} can be expanded in a convergent series in powers of h:uh=u+c1hk+...{h:u_h=u+c_1h^k+...}. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for uu and cc from two values of uh{u_h}. The question of using a larger number of uh{u_h} values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42407101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-30DOI: 10.22363/2658-4670-2021-29-3-205-220
V. Andreev, D. Chuprov
The purpose of this paper is to present the design and implementation of a reconfigurable remote control for performing plasma experiments with Hard-Real-Time (HRT) synchronization under jitter less than 1 microsecond. An additional requirement for a multichannel synchronization system is the use of high-speed optical converters to provide galvanic isolation between powerful modules of the setup and remote control in order to exclude any possibility of disruption of the physical experiment control system. Modeling and development of the software part of the maser remote control panel was performed in the LabVIEW application development environment with Real Time and FPGA modules. The hardware part of the control panel is implemented on a real-time controller working in conjunction with the Xilinx FPGA module. To ensure the optical isolation of synchronization signals, boards of electron-optical converters based on LED lasers with fiber-optic terminals were developed and manufactured. The control program is implemented in a two-module architecture with a HOST application and an FPGA application that exchange data over a 1000BASE-T Ethernet network.
{"title":"Modeling and design of an re-configurable isolated remote for plasma experiments with hard-real-time synchronization","authors":"V. Andreev, D. Chuprov","doi":"10.22363/2658-4670-2021-29-3-205-220","DOIUrl":"https://doi.org/10.22363/2658-4670-2021-29-3-205-220","url":null,"abstract":"The purpose of this paper is to present the design and implementation of a reconfigurable remote control for performing plasma experiments with Hard-Real-Time (HRT) synchronization under jitter less than 1 microsecond. An additional requirement for a multichannel synchronization system is the use of high-speed optical converters to provide galvanic isolation between powerful modules of the setup and remote control in order to exclude any possibility of disruption of the physical experiment control system. Modeling and development of the software part of the maser remote control panel was performed in the LabVIEW application development environment with Real Time and FPGA modules. The hardware part of the control panel is implemented on a real-time controller working in conjunction with the Xilinx FPGA module. To ensure the optical isolation of synchronization signals, boards of electron-optical converters based on LED lasers with fiber-optic terminals were developed and manufactured. The control program is implemented in a two-module architecture with a HOST application and an FPGA application that exchange data over a 1000BASE-T Ethernet network.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46917558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-27DOI: 10.22363/2658-4670-2021-29-4-361-386
A. Khvedelidze, D. Mladenov, A. Torosyan
Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_N/SU(N)}. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.
{"title":"Parameterizing qudit states","authors":"A. Khvedelidze, D. Mladenov, A. Torosyan","doi":"10.22363/2658-4670-2021-29-4-361-386","DOIUrl":"https://doi.org/10.22363/2658-4670-2021-29-4-361-386","url":null,"abstract":"Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_N/SU(N)}. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42203796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-2-141-153
O. K. Kroytor, M. Malykh, S. P. Karnilovich
The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.
{"title":"Kinematic support modeling in Sage","authors":"O. K. Kroytor, M. Malykh, S. P. Karnilovich","doi":"10.22363/2658-4670-2020-28-2-141-153","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-2-141-153","url":null,"abstract":"The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46492821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-3-274-288
A. Zorin
The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurements was earlier developed in detail for the quantum Kepler problem. For more complex quantum objects, such a construction is unknown. At the same time, the standard (non-constructive) model of Holevo-Helstrom quantum measurements is suitable for any quantum object. In this work, the constructive model of quantum measurements is generalized to a wider class of quantum objects, i.e., the optical spectrum of atoms and ions with one valence electron. The analysis is based on experimental data on the energy ordering of electrons in an atom according to the Klechkovsky-Madelung rule and on the substantiation of a single-particle potential model for describing the energy spectrum of optical electrons in alkali metal atoms. A representation of the perturbation of a single-particle potential in the form of a convolution of the potential of an electron in a hydrogen atom with the Wigner function of a certain effective state of the core in an alkali metal atom representation allows reducing all calculation algorithms for alkali metals to the corresponding algorithms for the hydrogen atom.
{"title":"Kuryshkin-Wodkiewicz quantum measurement model for alkaline metal atoms","authors":"A. Zorin","doi":"10.22363/2658-4670-2020-28-3-274-288","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-3-274-288","url":null,"abstract":"The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurements was earlier developed in detail for the quantum Kepler problem. For more complex quantum objects, such a construction is unknown. At the same time, the standard (non-constructive) model of Holevo-Helstrom quantum measurements is suitable for any quantum object. In this work, the constructive model of quantum measurements is generalized to a wider class of quantum objects, i.e., the optical spectrum of atoms and ions with one valence electron. The analysis is based on experimental data on the energy ordering of electrons in an atom according to the Klechkovsky-Madelung rule and on the substantiation of a single-particle potential model for describing the energy spectrum of optical electrons in alkali metal atoms. A representation of the perturbation of a single-particle potential in the form of a convolution of the potential of an electron in a hydrogen atom with the Wigner function of a certain effective state of the core in an alkali metal atom representation allows reducing all calculation algorithms for alkali metals to the corresponding algorithms for the hydrogen atom.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46324082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-3-216-229
Yu. L. Kalinovsky, A. Friesen, E. D. Rogozhina, L. I. Golyatkina
The aim of this work is to develop a set of programs for calculation the scattering amplitudes of the elementary particles, as well as automating the calculation of amplitudes using the appropriate computer algebra systems (Mathematica, Form, Cadabra). The paper considers the process of pion-pion scattering in the framework of the effective Nambu-Iona-Lasinio model with two quark flavours. The Package-X for Mathematica is used to calculate the scattering amplitude (starting with the calculation of Feynman diagrams and ending with the calculation of Feynman integrals in the one-loop approximation). The loop integrals are calculated in general kinematics in Package-X using the Feynman parametrization technique. A simple check of the program is made: for the case with zero temperature, the scattering lengths (a_0 = 0.147) and (a_2 = -0.0475) are calculated and the total cross section is constructed. The results are compared with other models as well as with experimental data.
本工作的目的是开发一套计算基本粒子散射振幅的程序，以及使用适当的计算机代数系统(Mathematica, Form, Cadabra)自动计算振幅。本文在具有两种夸克味的有效Nambu-Iona-Lasinio模型框架下考虑了介子-介子散射过程。使用Package-X for Mathematica计算散射振幅(从计算费曼图开始，以计算单环近似下的费曼积分结束)。利用费曼参数化技术对Package-X的一般运动学进行了环积分计算。对程序进行了简单的校核:在零温度情况下，计算了散射长度(a_0 = 0.147)和(a_2 = -0.0475)，并构造了总截面。结果与其他模型和实验数据进行了比较。
{"title":"Application of a computer algebra systems to the calculation of the (pipi)-scattering amplitude","authors":"Yu. L. Kalinovsky, A. Friesen, E. D. Rogozhina, L. I. Golyatkina","doi":"10.22363/2658-4670-2020-28-3-216-229","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-3-216-229","url":null,"abstract":"The aim of this work is to develop a set of programs for calculation the scattering amplitudes of the elementary particles, as well as automating the calculation of amplitudes using the appropriate computer algebra systems (Mathematica, Form, Cadabra). The paper considers the process of pion-pion scattering in the framework of the effective Nambu-Iona-Lasinio model with two quark flavours. The Package-X for Mathematica is used to calculate the scattering amplitude (starting with the calculation of Feynman diagrams and ending with the calculation of Feynman integrals in the one-loop approximation). The loop integrals are calculated in general kinematics in Package-X using the Feynman parametrization technique. A simple check of the program is made: for the case with zero temperature, the scattering lengths (a_0 = 0.147) and (a_2 = -0.0475) are calculated and the total cross section is constructed. The results are compared with other models as well as with experimental data.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48008956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-3-252-273
A. Sevastianov
The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwells equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwells equations.
{"title":"Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides","authors":"A. Sevastianov","doi":"10.22363/2658-4670-2020-28-3-252-273","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-3-252-273","url":null,"abstract":"The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwells equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwells equations.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48227884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-4-378-397
Konstantin P. Lovetski, A. Zhukov, M. Paukshto, L. Sevastianov, A. Tiutiunnik
The paper describes a methodology for determining the optical and physical properties of anisotropic thin film materials. This approach allows in the future designing multilayer thin-film coatings with specified properties. An inverse problem of determining the permittivity tensor and the thickness of a thin film deposited on a glass substrate is formulated. Preliminary information on the belonging of a thin-film coating to a certain class can significantly reduce the computing time and increase the accuracy of determining the permittivity tensor over the entire investigated range of wavelengths and film thickness at the point of reflection and transmission measurement Depending on the goals, it is possible to formulate and, therefore, solve various inverse problems: o determination of the permittivity tensor and specification of the thickness of a thick (up to 1 cm) substrate, often isotropic; o determination of the permittivity tensor of a thin isotropic or anisotropic film deposited on a substrate with known optical properties. The complexity of solving each of the problems is very different and each problem requires its own specific set of measured input data. The ultimate results of solving the inverse problem are verified by comparing the calculated transmission and reflection with those measured for arbitrary angles of incidence and reflection.
{"title":"Solving the inverse problem for determining the optical characteristics of materials","authors":"Konstantin P. Lovetski, A. Zhukov, M. Paukshto, L. Sevastianov, A. Tiutiunnik","doi":"10.22363/2658-4670-2020-28-4-378-397","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-4-378-397","url":null,"abstract":"The paper describes a methodology for determining the optical and physical properties of anisotropic thin film materials. This approach allows in the future designing multilayer thin-film coatings with specified properties. An inverse problem of determining the permittivity tensor and the thickness of a thin film deposited on a glass substrate is formulated. Preliminary information on the belonging of a thin-film coating to a certain class can significantly reduce the computing time and increase the accuracy of determining the permittivity tensor over the entire investigated range of wavelengths and film thickness at the point of reflection and transmission measurement Depending on the goals, it is possible to formulate and, therefore, solve various inverse problems: o determination of the permittivity tensor and specification of the thickness of a thick (up to 1 cm) substrate, often isotropic; o determination of the permittivity tensor of a thin isotropic or anisotropic film deposited on a substrate with known optical properties. The complexity of solving each of the problems is very different and each problem requires its own specific set of measured input data. The ultimate results of solving the inverse problem are verified by comparing the calculated transmission and reflection with those measured for arbitrary angles of incidence and reflection.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49143021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-3-205-215
Anastasia Kryukova
There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.
{"title":"On the rate of convergence for a class of Markovian queues with group services","authors":"Anastasia Kryukova","doi":"10.22363/2658-4670-2020-28-3-205-215","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-3-205-215","url":null,"abstract":"There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47810065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-3-230-251
I. Amirkhanov, Irina S. Kolosova, S. Vasilyev
The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.
{"title":"Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation","authors":"I. Amirkhanov, Irina S. Kolosova, S. Vasilyev","doi":"10.22363/2658-4670-2020-28-3-230-251","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-3-230-251","url":null,"abstract":"The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues \u0000are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44022219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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